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Organic Chemistry
5.11 Prochirality
Organic Chemistry5.11 Prochirality
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5.11 • Prochirality
Closely related to the concept of chirality, and particularly important in biological chemistry, is the notion of prochirality. A molecule is said to be prochiral if it can be converted from achiral to chiral in a single chemical step. For instance, an unsymmetrical ketone like 2-butanone is prochiral because it can be converted to the chiral alcohol 2-butanol by the addition of hydrogen, as we’ll see in Section 17.4.
Which enantiomer of 2-butanol is produced depends on which face of the planar carbonyl group undergoes reaction. To distinguish between the possibilities, we use the stereochemical descriptors Re and Si. Rank the three groups attached to the trigonal, sp2-hybridized carbon, and imagine curved arrows from the highest to second-highest to third-highest ranked substituents. The face on which the arrows curve clockwise is designated the Re face (similar to R), and the face on which the arrows curve counterclockwise is designated the Si face (similar to S). In this example, addition of hydrogen from the Re face gives (S)-2-butane, and addition from the Si face gives (R)-2-butane.
In addition to compounds with planar, sp2-hybridized atoms, compounds with tetrahedral, sp3-hybridized atoms can also be prochiral. An sp3-hybridized atom is said to be a prochirality center if, by changing one of its attached groups, it becomes a chirality center. The −CH2OH carbon atom of ethanol, for instance, is a prochirality center because changing one of its attached −H atoms converts it into a chirality center.
To distinguish between the two identical atoms (or groups of atoms) on a prochirality center, we imagine a change that will raise the ranking of one atom over the other without affecting its rank with respect to other attached groups. On the −CH2OH carbon of ethanol, for instance, we might imagine replacing one of the 1H atoms (protium) by 2H (deuterium). The newly introduced 2H atom ranks higher than the remaining 1H atom, but it remains lower than other groups attached to the carbon. Of the two identical atoms in the original compound, the atom whose replacement leads to an R chirality center is said to be pro-R and the atom whose replacement leads to an S chirality center is pro-S.
A large number of biological reactions involve prochiral compounds. One of the steps in the citric acid cycle by which food is metabolized, for instance, is the addition of H2O to fumarate to give malate. Addition of −OH occurs on the Si face of a fumarate carbon and gives (S)-malate as product.
As another example, studies with deuterium-labeled substrates have shown that the reaction of ethanol with the coenzyme nicotinamide adenine dinucleotide (NAD+), catalyzed by yeast alcohol dehydrogenase, occurs with exclusive removal of the pro-R hydrogen from ethanol and with addition only to the Re face of NAD+.
Determining the stereochemistry of reactions at prochirality centers is a powerful method for studying detailed mechanisms in biochemical reactions. As just one example, the conversion of citrate to cis-aconitate in the citric acid cycle has been shown to occur with loss of a pro-R hydrogen, implying that the OH and H groups leave from opposite sides of the molecule.
Problem 5-22
Identify the indicated hydrogens in the following molecules as pro-R or pro-S:
(a)
(b)
Problem 5-23
Identify the indicated faces of carbon atoms in the following molecules as Re or Si:
(a)
(b)
Problem 5-24
The lactic acid that builds up in tired muscles is formed from pyruvate. If the reaction occurs with addition of hydrogen to the Re face of pyruvate, what is the stereochemistry of the product?
Problem 5-25
The aconitase-catalyzed addition of water to cis-aconitate in the citric acid cycle occurs with the following stereochemistry. Does the addition of the OH group occur on the Re or Si face of the substrate? What about the addition of the H? Do the H and OH groups add from the same side of the double bond or from opposite sides?
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3401 | https://www.3blue1brown.com/lessons/inverse-matrices | 3Blue1BrownAnimated math
Linear Algebra
Chapter 7Inverse matrices, column space, and null space
Published Aug 15, 2016
Updated Sep 18, 2025
Lesson by Grant Sanderson
Text Adaption by River Way
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"To ask the right question is harder than to answer it."
— Georg Cantor
As you can probably tell by now, the bulk of this series is on understanding matrix and vector operations through the more visual lens of linear transformations. This chapter will be no exception, describing the concepts of inverse matrices, column space, rank, and null space through that lens.
A forewarning though, I’m not going to talk at all about the methods for computing these things, which some would argue is rather important. There are a lot of good resources for learning those methods outside this series, keywords “Gaussian elimination” and “row echelon form”. I think most of the actual value I have to offer here is on the intuition half. Plus, in practice, we usually get software to compute these operations for us anyway.
Linear Systems of Equations
You have a hint by now of how linear algebra is useful for describing the manipulation of space, which is useful for things like computer graphics and robotics. But one of the main reasons linear algebra is more broadly applicable, and required for just about any technical discipline, is that it lets us solve certain systems of equations. When I say “system of equations”, I mean you have a list of variables you don’t know and a list of equations relating them.
Unknown variablesxyz6x−3y+2z=7x+2y+5z=0Equations2x−8y−z=−2
These variables could be voltages across various elements in a circuit, prices of certain stocks, parameters in a machine learning network, or any situation where you might be dealing with multiple unknown numbers that somehow depend on one another.
How exactly they depend on each other is determined by the substance of your science, whether that means modeling the physics of circuits, the dynamics of the economy, or interactions in a neural network, but often you end up with a list of equations that relate these variables to one another. In many situations, these equations can get very complicated.
1−e2x−3y+4z1sin(xy)+z2x2+y2=1=y=e−z
But if you’re lucky, they might take a certain special form. Within each equation, the only thing happening to each variable is that it will be scaled by some constant, and the only thing happening to each of those scaled variables is that they are added to each other. There are no exponents, fancy functions, or multiplying two variables together.
2x+5y+3z4x+0y+8z1x+3y+0z=−3=0=2
The typical way to organize this sort of special system of equations is to throw all the variables on the left and to put any lingering constants on the right. It’s also nice to vertically line up all common variables, throwing in zero coefficients whenever one of the variables doesn’t show up in one of the equations. This is called a “linear system of equations”.
You might notice that this looks a lot like matrix-vector multiplication. In fact, you can package all the equations together into one vector equation where you have the matrix containing all the constant coefficients, and a vector containing all the variables, and their matrix-vector product equals some different, constant vector.
2x+5y+3z4x+0y+8z1x+3y+0z=−3=0=2→A^⎣⎡241503380⎦⎤Coefficientsx⎣⎡xyz⎦⎤Variables=v⎣⎡−302⎦⎤Constants
Which matrix-vector equation corresponds to this system: 3x+0y=52x−5y=2
Let’s name this coefficient matrix A, denote the vector holding all our variables with a boldfaced x, and call the constant vector on the right-hand side v. This is more than just a notational trick to get our system of equations written on one line, it sheds light on a wonderful geometric interpretation of the problem. The matrix A corresponds to some linear transformation, so solving Ax=v means we’re looking for a vector x which, after applying that transformation, lands on v.
Just think about what’s happening here for a moment. You can hold in your head this complex idea of multiple variables all intermingling with each other just by thinking about squishing and morphing space and trying to find which vector lands on another. Cool, right?
To start simply, let’s say you have a system with two equations and two unknowns. This means the matrix A is a 2×2 matrix, and x and v are both two dimensional vectors.
2x+2y1x+3y=−4=−1→A^[2123]x[xy]=v[−4−1]
The way we think about solutions to this equation depends on whether the transformation associated with A squishes all of space into a lower dimension, like a line or a point, or if it leaves everything spanning the full two dimensions where it started.
In the language of the last chapter, we subdivide into the case where A has zero determinant, and the case where A has nonzero determinant.
Inverses
Let’s start with the most likely case, where the determinant is nonzero and space does not get squished onto a zero-area line. In this case, there will always be one and only one vector x that lands on v, which you can find by playing the transformation in reverse.
Following where v goes as we rewind the tape like this, you will find the vector x such that Ax=v.
Inverse Matrix
When you play the transformation in reverse, it actually corresponds with a separate linear transformation, commonly called the inverse of A, denoted as A−1. For example, if A was a counterclockwise rotation by 90∘, the inverse of A would be a clockwise rotation by 90∘.
If A was a rightward shear that pushed j^ one unit to the right, the inverse of A would be leftward a shear that pushed j^ one unit to the left.
In general, A−1 is the unique transformation with the property that if you apply the transformation A, and follow it with the transformation A inverse, you end up back where you started.
What is the inverse matrix for a 180∘ rotation? [−100−1]
Since applying one transformation after another is captured algebraically with matrix multiplication, the core property of this reverse transform A−1 is that A−1 times A equals the matrix that corresponds to doing nothing. The transformation which does nothing is called the “identity” transformation. It leaves i^ and j^ where they are, unmoved, so its columns are [10] and [01].
A−1A=[1001]
There are computational methods to compute this inverse matrix. In the case of two dimensions there’s a commonly taught formula, which I have to admit, I can never remember. Once you find this inverse, which in practice you’d do with a computer, you can solve your equation by multiplying this inverse matrix by v. And again, what that means geometrically is that you’re playing the transformation in reverse and following v to end up at x.
A−1Axx=A−1v=A−1v
This nonzero determinant case corresponds with the idea that if you have two unknowns and two linear equations, it’s almost certainly the case that there is a single unique solution.
ax+cy=ebx+dy=fOne unique solution... probably
Higher Dimensions
This idea also makes sense in higher dimensions when the number of equations equals the number of unknowns. Again, the system of equations can be translated to the geometric interpretation where you have some transformation A and some vector v, and you’re looking for the vector x that lands on v.
Ax=v2x+5y+3z4x+0y+8z1x+3y+0z=−3=0=2→A^⎣⎡241503380⎦⎤x⎣⎡xyz⎦⎤=v⎣⎡−302⎦⎤
Just like in 2D, we can play a 3D transformation in reverse to find where the vector x came from when it landed on v.
As long as the transformation for A doesn’t squish all of space into a lower dimension, meaning its determinant is not zero, there will be an inverse transformation A−1. The inverse transformation has the property that if you first do A, then you do A−1, it’s the same as doing nothing. Multiplying that reverse transformation matrix by v, you get the answer x.
det(A)=0→A−1 exists
Irreversibility
Up to this point, we've only mentioned the case when the determinant isn't zero. But when the determinant is zero, and the transformation associated with the system of equations squishes space into a smaller dimension, there is no inverse. You cannot unsquish a line to turn it into a plane. At least, that’s not something a function can do.
That would require transforming each vector into a whole line of vectors, but functions can only take a single input to a single output.
Similarly, for 3 equations and 3 unknowns, there will be no inverse if the corresponding transformation squishes 3D space onto a plane, onto a line, or onto a point. Those all correspond to a determinant of zero, since any region is squished onto a zero-volume region.
Does A−1 exist when A=[3−9−13]?
Column Space
It’s still possible that a solution exists even when there is no inverse, it’s just that when your transformation squishes space onto, say, a line, you have to be lucky enough to have the vector v live somewhere on that line.
You might notice that some of these zero-determinant cases feel much more restrictive than others. Given a 3×3 matrix, it seems much harder for a solution to exist when it squishes space onto a line compared to when it squishes things onto a plane.
We have some language that’s a bit more specific than just saying zero-determinant. When the output of a transformation is a line, meaning it is one-dimensional, we say the transformation has a rank of 1.
If all the vectors land on some two-dimensional plane, we say the transformation has a rank of 2. So the word “rank” means the number of dimensions in the output of a transformation.
So for instance, in the case of 2×2 matrices, rank 2 is the best it can be, it means the basis vectors continue to span the full 2D space and the determinant is nonzero. But for 3×3 matrices, rank 2 means things have collapsed, but not as much as they would have for a rank 1 transformation.
If a 3D transform has a non-zero determinant and its output fills all of 3D space, it has a rank of 3. This set of all possible outputs for your matrix, whether it’s a line, a plane, or 3D space; is called the “column space” of your matrix. You can probably guess where the name comes from; the columns of your matrix tell you where the basis vectors land and the span of those transformed basis vectors gives you all possible outputs. In other words, the column space is the span of the columns of your matrix.
So a more precise definition of rank is that it’s the number of dimensions in the column space. When this rank is as high as it can be, equaling the number of columns in the matrix, the matrix is called “full rank”.
What is the rank of this matrix? ⎣⎡1−25−24−104−820⎦⎤
Null Space
Notice the zero vector will always be included in the column space since linear transformations must keep the origin fixed in place.
For a full-rank transformation, the only vector that lands at the origin is the zero vector itself. But for matrices that aren’t full rank, which squish to a smaller dimension, you can have a whole bunch of vectors land on zero. If a 2D transformation squishes space onto a line, there is a separate line in a different direction full of vectors that get squished on the origin.
If a 3D transformation squishes space onto a plane, there is a line full of vectors that land on the origin.
If a 3D transformation squishes all of space onto a line, there is a whole plane full of vectors that land on the origin.
This set of vectors that land on the origin is called the “null space” or the “kernel” of your matrix. It’s the space of vectors that becomes null, in the sense that they land on the zero vector. In terms of the linear system of equations, if v happens to be the zero vector, the null space gives you all possible solutions to the equation.
What does the null space of this matrix look like? ⎣⎡1−25−24−104−820⎦⎤
Conclusion
So that’s a very high-level overview of how to think about linear systems of equations geometrically. Each system has a linear transformation associated with it. When that transformation has an inverse, you can use that inverse to solve your system. Otherwise, the ideas of column space and null space let us know when there is a solution, and what the set of all possible solutions can look like.
Again, there’s a lot I haven’t covered, most notably how to compute these things. Also, I limited my scope of examples here to equations where the number of unknowns equals the number of equations. But my goal here is that you come away with a strong intuition for inverse matrices, column space, and null space that can make any future learning you do more fruitful.
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3402 | https://math.stackexchange.com/questions/3565733/find-conditions-on-the-parameters-such-that-a-polynomial-is-globally-nonnegative | maxima minima - Find conditions on the parameters such that a polynomial is globally nonnegative - Mathematics Stack Exchange
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Find conditions on the parameters such that a polynomial is globally nonnegative
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Let
f(x,y,z)=z 2+(β+2 c 3−4)y z+((β+1)c 3+c 6−4)y 2+z(1 2(10−c 3)x 2+c 6 x)+y((1 2(β+1)(10−c 3)+3 c 5+2 c 6−3)x 2+(β c 6+2 c 8)x)+c 5 x 4+(β c 5+3 c 8)x 3+c 8(β−1)x 2.
be a polynomial with c 3,c 5,c 6,c 8∈R and β>0. Could we find c 3,c 5,c 6,c 8 and β such that f(x,y,z)≥0 for all x,y,z∈R?
Any reference, suggestion, idea, or comment is welcome. Thank you!
polynomials
maxima-minima
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edited Mar 1, 2020 at 22:58
LCHLCH
asked Mar 1, 2020 at 19:59
LCHLCH
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My idea is to write f(x,y,z) as a sum of squares. However, it seems there will be lots of undetermined parameters...LCH –LCH 2020-03-01 20:58:16 +00:00 Commented Mar 1, 2020 at 20:58
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~~I am unable~~ It is impossible to find any c 3,c 5,c 6,c 8 and positive β to make f(x,y,z) globally non-negative (see update below). ~~However, there~~ There is a set of 14 inequalities for the coefficients. If the coefficients satisfy all of them, then f(x,y,z) will be globally non-negative.
Let A be the 4×4 matrix: [2 β+2 c 3−4 10−c 3 2 c 6 β+2 c 3−4 2(c 3(β+1)+c 6−4)(10−c 3)(β+1)+4 c 6+6 c 5−6 2 c 6 β+2 c 8 10−c 3 2(10−c 3)(β+1)+4 c 6+6 c 5−6 2 2 c 5 c 5 β+3 c 8 c 6 c 6 β+2 c 8 c 5 β+3 c 8 2 c 8(β−1)] For p=(x,y,z)∈R 3, let U p be the 4×1 column vector [z,y,x 2,x]T. In terms of A and U p, the function at hand equals to
f(x,y,z)=1 2 U T p A U p
When A is positive semi-definite, it is easy to see f(x,y,z) is globally non-negative.
If A isn't positive semi-definite, then one can find a non-zero U=[u 1,u 2,u 3,u 4]T such that U T A U is negative. If one perturb u 3,u 4 for sufficient small amount, U T A U remains to be negative. WOLOG, we can assume u 3,u 4≠0. For such a U, we find
f(u 3 u 4,u 2 u 3 u 2 4,u 1 u 3 u 2 4)=u 2 3 2 u 4 4 U T A U<0
This means f(x,y,z) is globally non-negative when and only when A is positive semi-definite.
By Sylvester's criterion for positive semidefinite matrices, f(x,y,z) will be globally non-negative when all principal minors of A have non-negative determinant.
Since A is a 4×4 matrix, it has 15=2 4−1 principal minors. 4 of them comes form diagonal elements of A. The inequality come from the first diagonal element is trivial. The other 3 inequalities are
c 3(β+1)+c 6−4≥0,
c 5≥0,
c 8(β−1)≥0
There are 11=6+4+1 more inequalities coming from principal minors obtained by removing 2, 1 or no rows/columns (the last one is simply det(A)≥0). I'm not giving to list all of them here. Please compute them yourselves using a CAS.
Update
We are going to show f(x,y,z) cannot be globally non-negative.
First, let use consider the case c 5>0.
Change variables to (β,t,u,v,w) such that (c 3,c 5,c 6,c 8)=(10−2 u t,t 2,v t,w t 2). We will assume t>0.
Let A¯i j be the determinant of principal minor of A with only i t h/j t h rows/columns are kept.
With help of a CAS, one can verify
A¯12=4 t v−(4 t u−15)2−(1−β)(7−β)A¯34=t 2 P⏞(−4 t v−12(β+1)t u−8 t 2+40 β+36)−(2 t v+(β+1)t u+t 2−3)2
When A is positive semidefinite, A¯12,A¯34≥0. Since the factor P in A¯34 can be expressed as a sum of A¯34 as square followed by division of t 2, we have P≥0. Adding a few more squares and simplify using an CAS, we obtain
5(β 2+6 β−11)=4(A¯12+P)+(8 t u+3 β−18)2+32 t 2≥0 This forces |β+3|≥√20⟹β≥√20−3
With help of CAS again, one find
A¯34=−t 4(9 w 2+(2 β+4)w+β 2) Treat this as a quadratic polynomial in w. Notice the coefficient for w 2 and w 0 are negative. In order for it to have a chance to be non-negative, its discriminant need to be non-negative. This leads to
(2 β+4)2−36 β 2=16(1−β)(1+2 β)≥0⟹β≤1 This contradicts with above result condition β≥√20−3>1.
What this means is when c 5>0, it is impossible for A¯12,A¯23,A¯34 to be non-negative at the same time and hence A cannot be positive semidefinite.
For the remaining case c 5=0, A has a zero at the 3 r d diagonal element. In order for A to be positive semidefinite, all entries in 3 t h row/column need to vansih. This implies c 3=10, c 6=3 2 and c 8=0. When c 8=0, A has a zero at the 4 t h diagonal element. However the 4 t h row/column has a non-zero entry and hennce A cannot be positive semidefinite.
Combine these, we can conclude there are no c 3,c 5,c 6,c 8 and positive β to make A positive semidefinite and hence f(x,y,z) is never globally non-negative.
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edited Mar 9, 2020 at 13:23
answered Mar 3, 2020 at 4:59
achille huiachille hui
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Thank! According to your calculations, I'm thinking about the problem: Is it possible to find those x,y,z for which f(x,y,z)<0 for given c i(i=3,5,6,8) and β. Also, we want to find c i(i=3,5,6,8) and β which minimize the region of x,y,z for which f(x,y,z)<0.LCH –LCH 2020-03-10 18:37:10 +00:00 Commented Mar 10, 2020 at 18:37
@LCH In principle, given c i and β, one can diagonalize A and use the eigenvectors of A to discover the set of U=[u 1,u 2,u 3,u 4]T which satisfy U T A U<0 and hence the set of (x,y,z) with f(x,y,z)<0. However, minimize the region where f(x,y,z)<0 seems really tough. There are too many variables which doesn't seem to have any structure among them.achille hui –achille hui 2020-03-10 18:58:02 +00:00 Commented Mar 10, 2020 at 18:58
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Write it out as a homogeneos polynomial, i.e. in x 2, x y, and so on. You are asking for that to be positive definite...
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answered Mar 2, 2020 at 0:27
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3404 | https://blogs.glowscotland.org.uk/nl/public/airdrieacadmaths/uploads/sites/28739/2016/02/Speed-Distance-Time-Worksheet.pdf | Speed, Distance, Time Worksheet. 1. A girl cycles for 3hrs at a speed of 40 km/h. What distance did she travel? 2. A train travels at a speed of 30mph and travel a distance of 240 miles. How long did it take the train to complete it’s journey? 3. A car travels a distance of 540km in 6 hours. What speed did it travel at? 4. John is a runner. He runs the 100m sprint in 10x6s. What speed did he travel at? (in m/s) 5. A cyclist travels 20km in 4hrs. What speed did the cyclist cycle at? 6. The distance between two cities is 144km, it takes me 3hours to travel between these cities. What speed did I travel at? 7. A coach travels from the station to the beach, a distance of 576km away in 6hrs. The coach is only allowed to travel at a maximum speed of 90km/h. Did the coach break the speed limit? 8. At the equator, the earth spins a distance of 25,992miles every day. What speed does the Earth spin at in mph? 9. Lauren walks 100m in half a minute. What must her speed have been to travel this distance? 10. A mouse runs a distance of 2metres in 15 seconds. What is it’s speed? 11. Jim travelled at a speed of 18km/h for 2 hours. What was the distance covered? 12. Marc was told his dinner would be ready at 18:00. He left his house at 12:00 and travelled in his car at an average speed of 45mph to his mum’s house 300 miles away. Did Marc make it home in time for dinner? 13. A whale swims at a constant speed of 8m/s for 17s. What distance did it travel? M Doran March ‘08 1 14. Callum writes down his jog times for each day. Mon – 15min Tue – 10min Wed – 12min Thu - 5min Fri – No jog. He jogs at a constant speed of 9km/h. Work out the distance he jogs each day. On which day did he jog the furthest? 15. How long does it take to drive a distance of 260 miles at a speed of 65mph? 16. How long does it take to travel a distance of 672km at a speed of 96km/h? 17. Carlisle is a distance of 135miles away from Airdrie. If I travelled at a constant speed of 45mph. How long would it take me to get there? 18. A beetle travels at a speed of 9cm/s., it travels a distance of 108cm before it is caught in a jar. How long did the beetle run for? 19. Neil travelled 36km at a speed of 8km/h. Grant travelled 48km at a speed of 10km/h a) Whose journey was quickest? b) By how many mins? 20. Susie estimated that she can run for hours at a steady rate of 8mph. She enters a marathon, a distance of 26miles. How long should it take her to complete the race? Give answer in hours/minutes. 21. Mr Dunn drives 64.8km from work at a speed of 48km/h. Mrs Dunn drives 81x2km from work at a speed of 58km/h. They both leave work at the same time. a) Who arrives home first? b) How many minutes later is it before the second person gets home? 22. The earth takes one year to go round the sun. The distance travelled is 584 million miles if there are 365 days in a year, what speed does the earth travel at in miles per day? Can you work out the speed of the earth in miles per hour? M Doran March ‘08 2 M Doran March ‘08 3 Speed, Distance, Time Answers. 1) 120km 2) 8 hours 3) 90km/h 4) 9.4m/s 5) 5km/h 6) 48km/h 7) Yes, it travelled at 96km/h 8) 1083mph 9) 3.33m/s 10) 0.13m/s 11) 36km 12) No, he arrived at 18:40 13) 136m 14) Mon – 2.25km Tue – 1.5km Wed – 1.8km Thu – 0.75km. He travelled furthest on Monday 15) 4 hours 16) 7 hours 17) 3 hours 18) 12s 19) a) Neil was quickest at 4.5 hours. Grant was 4.8 hours. b) 18 mins 20) 3 hours 15 minutes 21) a) Mr Dunn. He takes 1.35 hours. Mrs Dunn takes 1.4 hours b) 3 minutes 22) 1,600,000 miles per day. Which is 66,666.67 mph |
3405 | https://blog.prepscholar.com/completing-the-square-formula | How (and When) to Complete the Square: 5 Simple Steps · PrepScholar
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How (and When) to Complete the Square: 5 Simple Steps
Posted by Ashley Robinson
General Education
It’s pretty much a guarantee that you’ll see quadratic equations on the SAT and ACT. But they can be tricky to tackle, especially since there are multiple methods you can use to solve them.
In this article, we’re going to walk through using one specific method—completing the square—to solve a quadratic equation. In fact, we’ll give you step-by-step instructions for how to complete the square using the completing the square formula. By the end, you should have a better understanding of how and when to use this mathematical strategy!
Ready to learn more? Then let’s jump in!
Engineers use quadratic equations to design roller coasters!
What Is a Quadratic Equation?
In order to understand how to complete the square, you first have to know how to identify a quadratic equation. That’s because completing the square only applies to quadratic equations!
In math, a quadratic equation is any equation that has the following formula:
$ax^2 + bx + c = 0$
In this equation, $x$ represents an unknown number and $a$ cannot be 0. (If $a$ is 0, then the equation is linear, not quadratic!)
Quadratic equations have all sorts of real-world applications because they're used to calculate parabolas, or arcs. Construction projects like bridges use the quadratic equation to calculate the arc of the structure, and even roller coasters use quadratics to design adrenaline-pumping tracks. Quadratics even fuel popular video games like Angry Birds, where the arc of each bird is calculated using the quadratic formula!
So now that you know why quadratic equations are important, let’s look at one of the most common methods of solving them: completing the square.
What Is Completing The Square and When Do You Use It?
There are actually four ways to solve a quadratic equation: taking the square root, factoring, completing the square, and the quadratic formula. Unfortunately, taking the square root and factoring only work in certain situations.
For example, let’s look at the following quadratic equation:
$x^2 + 6x = -2$
Solving a quadratic equation by taking the square root involves taking the square root of each side of the equation. Because this equation contains a non-squared $\bi x$ (in $\bo6\bi x$), that technique won’t work.
Factoring, on the other hand, involves breaking the quadratic equation into two linear equations that are both equal to zero. Unfortunately, trying to factor this equation doesn’t result in two linear equations!
Both the quadratic formula and completing the square will let you solve any quadratic equation. (In this post, we’re specifically focusing on completing the square.) When you complete the square, you change the equation so that the left side of the equation is a perfect square trinomial. That’s just a fancy way of saying that completing the square is a technique that transforms your quadratic equation from an equation that can’t be factored into one that can.
Completing the square applies to even the trickiest quadratic equations, which you’ll see as we work through the example below.
Your Step-By-Step Guide for How to Complete the Square
Now that we’ve determined that our formula can only be solved by completing the square, let’s look at our example formula again:
$x^2 + 6x = -2$
Step 1: Figure Out What’s Missing
When you look at the equation above, you can see that it doesn’t quite fit the quadratic equation format ($ax^2 + bx + c = 0$). The number that should go in the $c$ spot, which is also known as the constant, is missing. So from a logical perspective, the equation actually looks like this:
$x^2 + 6x +$ __?__ $= -2$
In order to solve this equation, we first need to figure out what number goes into the blank to make the left side of the equation a perfect square. (This missing number is called the constant.) By doing that, we’ll be able to factor the equation like normal.
Step 2: Use the Completing the Square Formula
But at this point, we have no idea what number needs to go in that blank. In order to figure that out, we need to apply the completing the square formula, which is:
$x^2 + 2ax + a^2$
In this case, the $a$ in this equation is the constant, or the number that needs to go in the blank in our quadratic formula above.
Step 3: Apply the Completing the Square Formula to Find the Constant
As long as the coefficient, or number, in front of the $\bi x^\bo2$ is 1, you can quickly and easily use the completing the square formula to solve for $\bi a$.
To do this, you take the middle number, also known as the linear coefficient, and set it equal to $2ax$. Here’s what that would look like for our sample formula:
$6x = 2ax$
This equation is basically asking what number (this is $\bi a$) multiplied by 2 will give us 6.
Now that you know your equation, solving for $a$ is simple: divide each side of the equation by $2x$! So let’s see what that looks like:
$$6x = 2ax$$
Divide each side by $\bo2x$:
$${6x}/{2x} = {2ax}/{2x}$$
Result: $3 = a$
Look at that! We now know that $\bi a =\bo3$!
But we’re not quite done with the completing the square formula yet. In order to determine what the missing constant is, we need to plug our solution for $a$ back into the completing the square formula ($x^2 + 2ax + a^2$). Whatever the result is for $\bi a^\bo2$ is the constant that we’ll plug back into our first equation ($x^2+ 6x +$ __?__ $= -2$). So let’s take a look:
$x^2+ 2ax + a^2$ where $a = 3$
Add $\bi a$ into the equation: $x^2 + 2(3)x + 3^2$
Put in simplest terms: $x^2 + 6x + 9$
So now we know that our constant is 9.
Now it's time to plug in some numbers!
Step 4: Plug the Constant Into the Original Formula
Now that you know the constant, it’s time to put it into the blank in our original formula. Once you do that, the equation will look like this:
Original formula: $x^2 + 6x +$ __?__ $= -2$
Formula with constant: $x^2 + 6x + 9 = -2 + 9$
Put in simplest terms: $x^2+ 6x + 9 = 7$
You might be wondering why we’re adding 9 to the right side of the equation. Well, remember: in math, you can never do something to one side of an equation without doing it to the other side, too. So because we’re adding 9 to our equation to make it a perfect square, we also have to add 9 to the right side of the equation to keep things balanced.
If you forget to add the new constant to the right side of the equation, you won’t get the right answer!
Step 5: Factor the Equation
We’ve already done a lot of work, and there’s still a little more to go. Now it’s time for us to solve the quadratic equation by figuring out what x could be. But now that we’ve turned the left side of our equation into a perfect square, all we have to do is factor like normal.
Completed quadratic formula: $x^2 + 6x + 9 = 7$
Factor left side of the equation: $(x + 3)^2 = 7$
Take the square root: $√{(x + 3)^2}= √7$
Subtract 3: $x = ±√7 - 3$
Final solutions: $x =√{7} - 3$ and $x =√{-7} - 3$
What If There’s a Coefficient in Front of $x^2$?
The step-by-step guide we gave you above only works if there’s no coefficient, or number, in front of $x^2$. If there is a coefficient, you have to eliminate it. Once you do that, you can solve the quadratic equation through the method we outlined above.
So how do you remove the coefficient? Actually, it’s not as hard as it sounds.
To show you how, let’s look at a new quadratic equation:
$2x^2- 12x = -8$
How to Factor Out the 2
n order to remove the 2, you’ll need to divide both sides of the equation by 2. It’s really that simple! So let’s take a look at how that works:
Original formula: $2x^2- 12x = -8$Divide everything by 2: $x^2- 6x = -4$
By doing this, you’ve made the coefficient in front of the $x^2$ into 1, so now you can solve the equation by completing the square like we did above.
Additional Completing the Square Resources
We know that completing the square can be tricky, which is why we’ve compiled a list of resources to help you if you’re still having trouble with how to complete the square.
More Sample Problems
As you already know, practice makes perfect. That’s why it’s important to work as many quadratic equations as you need to in order to feel comfortable solving these types of problems. Luckily for you, completing the square can be used to solve any quadratic equation, so as long as the practice questions are quadratics, you can use them!
One great resource for this is Lamar University’s quadratic equation page, which has a variety of sample problems as well as answers. Another good resource for quadratic equation practice is Math Is Fun’s webpage. If you scroll to the bottom, they have quadratic equation practice questions broken up into categories by difficulty.
Completing the Square Tutorial Videos
If you’re a visual learner, you might find it easier to watch someone solve quadratic equations instead. Khan Academy has an excellent video series on solving quadratic equations, including one video dedicated to showing you how to complete the square. YouTube also has some great resources, including this video on completing the square and this video that shows you how to tackle more advanced quadratic equations.
Completing the Square Calculator
If you want to check your work, there are some completing the square calculators available online. It can be a good way to make sure you’re working problems correctly if you don’t have an answer guide. But be forewarned: relying on a tool like this won’t help you retain the information! Make sure you’re putting in the hard work to learn how to complete the square so you aren’t blindsided by these types of questions on test day.
Now What?
Working with quadratic equations is just one element of algebra you’ll need to master before taking the SAT and ACT. A good place to start is mastering systems of equations, which will help you brush up on your fundamental algebra skills, too.
One of the most helpful math study tools is a chart of useful mathematical equations. Luckily for you, we have a master list of the 31 formulas you must know to conquer the ACT.
If you think you need a more comprehensive study tool, test prep books are one way to go. Here’s a list of our favorite SAT Math prep books that will help set you on the path to success.
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3406 | https://www.quora.com/What-effect-does-k-have-on-the-graph-of-the-function-y-ab-x-k | Something went wrong. Wait a moment and try again.
Vertical Translations
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5
What effect does "k" have on the graph of the function y=ab^ {x} +k?
PhD in Mathematics, Iowa State University (Graduated 1988)
·
Author has 1.2K answers and 250.4K answer views
·
2y
Questions like this are PERFECT for the website
Desmos
. Go there and type in
y=ab^x+k
(or copy & paste). Accept the offered sliders for
a
,
b
, and
k
. Pick you favorite nonzero values for
a
&
b
, and slide to that value on the a-slider. Now slowly slide the
k
value on its slider, and see for youself what it does to the graph.
Nowadays we can do Natural History and observe these curves in their natural habitats!
Related questions
Why can you graph "Y = - X", but not "-Y = X"?
How do I draw the graph of the function Y=X as Y versus -X?
Do all exponential functions in the form of y=ab^(x-h) + k have y-intercepts?
Does the graph of k=xy represent a function?
What is the effect of changing k in y= (x^2+k) (x+1) (x-2)?
Bob Collier
Former EE Designed Specialized Computers for 33 Years.
·
Author has 3.1K answers and 1.6M answer views
·
2y
It’s already in your equation. It just happens that k=0 so you don’t see it.
So draw it again, but this time make every point that you have/know x & y values for plot at x=x but y at y+1 instead of just y+0./
You just changed that invisible “+0” to “+1”.
Your response is private
Was this worth your time?
This helps us sort answers on the page.
Absolutely not
Definitely yes
Alon Amit
PhD in Mathematics; Mathcircler.
·
Upvoted by
Fabio García
, MSc Mathematics, CIMAT (2018) and
Michael Jørgensen
, PhD in mathematics · Author has 8.8K answers and 173.8M answer views
·
Updated 2y
Related
Is
sin
x
=
x
∏
∞
k
=
1
(
x
−
k
π
)
(
x
+
k
π
)
valid?
No, but I can see why you’d want to know. It’s useful to understand why anyone would think this might be true, why it’s not, and how to fix it.
The function on the left, sin(x), is zero precisely when x=πn for an integer n. If x=πn then sin(x)=0, while if x≠πn then sin(x)≠0. This should be familiar from basic trigonometry.
Now, suppose you have a polynomial p(x), and you know that its roots are a1,a2,…,an, meaning p(ai)=0 for every i but p(x)≠0 whenever x≠ai for any i. You form the polynomial ∏nk=1(x−ai). This polynomial clearly also has the exact
No, but I can see why you’d want to know. It’s useful to understand why anyone would think this might be true, why it’s not, and how to fix it.
The function on the left, sin(x), is zero precisely when x=πn for an integer n. If x=πn then sin(x)=0, while if x≠πn then sin(x)≠0. This should be familiar from basic trigonometry.
Now, suppose you have a polynomial p(x), and you know that its roots are a1,a2,…,an, meaning p(ai)=0 for every i but p(x)≠0 whenever x≠ai for any i. You form the polynomial ∏nk=1(x−ai). This polynomial clearly also has the exact same roots. And two polynomials having the exact same roots are… the same? No? Kinda?
Well, almost. If two polynomials have the exact same roots (complex ones, and with the same multiplicities) then they are, indeed, very similar: one of them is some constant times the other. For example, x2−x−1 and 7x2−7x−7 have the same roots, because one of them is simply 7 times the other.
So, if I have a mystery polynomial f(x) and I tell you that its roots are 0,π and −π, you can form the polynomial g(x)=x(x+π)(x−π), and you know that f(x)=cg(x), for some appropriate constant c≠0. If you also know f(17), you can determine c by comparing it with g(17).
It’s a good exercise to understand why that is; for the purposes of this question I’ll assume you already know this, which is why you’re asking.
The philosophy underlying the question is that sin(x) and other “nice” functions behave like “polynomials of infinite degree”. For example, we could naively hope that once we understand the roots of sin(x), we can identify it with a product just like we did with the mystery polynomial. We now have a function sin(x) whose roots are 0,±π,±2π and so on, so we hope that it’s the same as g(x)=x(x+π)(x−π)⋯, an infinite product.
But it’s not that simple.
First of all, that infinite product. It seems to behave the way you want: if you plug x=πn into it, one of the terms is going to have k=n and then the product becomes zero. On the other hand, if x≠πn for any n then… well, none of the terms vanishes, so the product is… nonzero… right?
No. This is your first major problem. The infinite product x∏(x−kπ)(x+kπ) doesn’t converge for any value of x except the integer multiples of π. We know that sin(π/2)=1, for example, but plugging in x=π/2 into that infinite product doesn’t give 1. In fact it doesn’t give anything: it doesn’t converge.
That’s fixable, actually. Let’s try this:
g(x)=x∞∏k=1(1−xkπ)(1+xkπ)
See what we did here? We are still multiplying simple linear terms which vanish whenever x is a whole multiple of π, but now the terms we’re multiplying get closer and closer to 1. No matter how large x is, eventually we get to
(1+xMπ)
where M is huge, and so Mπ drowns x, and the whole thing is just 1 plus a tiny bit. This modified infinite product does converge. Everywhere.
So are we done? Can we say that sin(x)=g(x)? First we have to check that we don’t need some global multiplicative constant c, like before. Good news, we don’t. The easiest way to see this is to divide through by x:
sin(x)x?=∞∏k=1(1−xkπ)(1+xkπ)
When x=0, on the right we clearly have 1, and on the left we have a singularity but we know that sinc(x)=sin(x)/x tends to 1 as x→0, so we can patch this function up by defining sinc(0)=1 and all is good. The functions have the same roots and they agree at one non-root point, so they’re supposed to be equal.
But are they? They’re only supposed to be equal because of that wishful philosophy that nice functions are like infinite-degree polynomials.
In fact, they are, but obviously that philosophy isn’t proof, and it’s not even generally true. For example exg(x) also has the same roots as g(x), but it’s no longer equal to sin(x).
The idea of identifying functions with converging infinite products having the same roots goes back to Euler. He used the product formula we just developed to solve the Basel problem, for example, but he knew it’s a bit fishy, and later on he offered alternative solutions which bypassed the need for that product. I’ve no doubt that Euler was sure the product formula for sin(x) is correct, but he also knew he doesn’t have a complete justification for it (he didn’t care about “proofs” the way we do today).
There are many, many proofs of the infinite product formula for sin(x), which is a cornerstone of complex analysis and many results in the theory of elliptic functions and elsewhere. A fairly elementary proof was published by Eberlein
in 1977, and it may be the one requiring the least amount of preparation. Only basic aspects of complex functions and infinite products are needed.
The product formula is usually written
sin(x)=x∞∏k=1(1−x2π2k2)
which is clearly equivalent to what we have, just replacing (1−x/t)(1+x/t) with (1−x2/t2). Another way to write this is
sin(πx)=πx∞∏k=1(1−x2k2)
This is nice because the roots are now simply the integers 0,±1,±2 and so on. It’s really neat to see this infinite product in action. Here’s an interactive Desmos graph showing how it works.
Here’s sin(πx), nicely criss-crossing the x axis at the integers:
Here’s the first term in the infinite product, πx(1−x2):
The product vanishes at 0,1 and −1, matching the first few roots of sin(πx). Then it shoots off to infinity, as it must: polynomials always shoot off to infinity.
When we add another term, we get two more roots matching:
You can see that since we’re forcing the polynomial to double back so quickly, it has trouble doing so with grace. It overshoots the trig function quite badly at second crest and trough. But the roots are perfect.
Another term:
The proximity at those second crest and trough is a bit better, and the way our polynomial hugs the transcendental sin function around 0 is really amazing. Let’s push much further and take the first 12 terms:
You may be now get a feel for how this works. As we add more terms, we force the product polynomial to hit more and more roots. It smoothly settles down on the trig function in increasingly wide intervals around 0, and then it loses it and swings wildly up and down as you get farther out. It’s a polynomial, it kind of has to. When you force it into this tame sine wave form in the center, it has to compensate by going crazy far out.
When we push out to 100 terms, the fit is amazing…
…until we zoom out to see what’s going on beyond these 10 peaks!
Right. Total mayhem. But as we add more and more terms, a wider and wider region in the middle sees a better and better fit, and the explosive outskirts, though they get increasingly more explosive, are also increasingly pushed outwards, their region of control steadily diminishing out to infinity.
When you take the limit of that infinite product, it is exactly a sine wave, and “only” because it gets the roots right!
An “infinite polynomial”, indeed.
Footnotes
ScienceDirect
Gordon M. Brown
Math Tutor at San Diego City College (2018-Present)
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Author has 6.2K answers and 4.3M answer views
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3y
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For what values of k does the graph of y=-2x^2 +5x +k not cut the x-axis?
If you knew anything at all about the discriminant of a quadratic function, you would have no occasion to broach this question on Quora! I urge you to look up this concept, and study it carefully.
The discriminant of a quadratic function f(x) = ax^2 + bx + c is given by
b^2 - 4ac
If the discriminant is less than zero, then the quadratic has no real solutions, and so will never cut the x-axis. So just set up an inequality according to which the discriminant is less than 0:
b^2 - 4ac < 0
5^2 - 4 (-2) (k) < 0
25 + 8k < 0
8k < -25
k < -25/8
Observe that, in the graph below, when k is precisely equal to
If you knew anything at all about the discriminant of a quadratic function, you would have no occasion to broach this question on Quora! I urge you to look up this concept, and study it carefully.
The discriminant of a quadratic function f(x) = ax^2 + bx + c is given by
b^2 - 4ac
If the discriminant is less than zero, then the quadratic has no real solutions, and so will never cut the x-axis. So just set up an inequality according to which the discriminant is less than 0:
b^2 - 4ac < 0
5^2 - 4 (-2) (k) < 0
25 + 8k < 0
8k < -25
k < -25/8
Observe that, in the graph below, when k is precisely equal to -25/8, the graph of the parabola touches the x-axis at a single point. But if we make k ever-so-slightly less than -25/8, then the parabola will not cut the x-axis at all.
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Related questions
What is the graph of the function y=(1/3) x?
Which equation represents the graphed function? Y = 4x – 2 y = –4x – 2 y = x – 2 y = – x – 2?
What is the effect of the variables h and k on the graph y=(x-h) ^2 +k as compared to the graph y=x^2?
Is the sequence X^k-x^k divided by x-y?
What is the graph of y= -x?
Vishal Pal
basic knowledge about functions.
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Does the graph of k=xy represent a function?
Yes it is indeed
Given, xy= k,
If you try putting either x=0 or y=0 then you will find out that k=0 which is not given in question. (I assume k is a non-zero constant, if it is 0 then the curve xy=0 will actually represent either x =0 I.e y-z plane or y=0 I.e x-z plane).
So, the domain of the given function is R-{0} and its range is also R-{0}
The function y= k/x represents a rectangular hyberbola indeed!!
For example, the Pressure v/s Volume curve for an ideal gas at constant temperature.
I am hereby giving the graph of the function-
Hope that helps . Don't forget to upvote the answer if you underst
Yes it is indeed
Given, xy= k,
If you try putting either x=0 or y=0 then you will find out that k=0 which is not given in question. (I assume k is a non-zero constant, if it is 0 then the curve xy=0 will actually represent either x =0 I.e y-z plane or y=0 I.e x-z plane).
So, the domain of the given function is R-{0} and its range is also R-{0}
The function y= k/x represents a rectangular hyberbola indeed!!
For example, the Pressure v/s Volume curve for an ideal gas at constant temperature.
I am hereby giving the graph of the function-
Hope that helps . Don't forget to upvote the answer if you understand it clearly.
Alon Amit
PhD in Mathematics; Mathcircler.
·
Upvoted by
Aditya Garg
, M.Sc. Mathematics, Indian Institute of Technology, Delhi (2013) and
Rhys Thomas
, MSc Mathematics & Physics, De Montfort University (2002) · Author has 8.8K answers and 173.8M answer views
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5y
Related
Why can you graph "Y = - X", but not "-Y = X"?
Sure you can graph −y=x.
You can also graph cos(sin(x2+y2))≤cos(x+y).
Whoever told you that you can only graph things that look like y=f(x) was misinformed, confused or just meant something different. You can plot any sort of relation between x and y in the plane (or any relation between x,y and z in space). The points of the plane which satisfy the relation are colored, the ones that don’t are not. That’s all.
There’s an important sort of relation called a function, where every value of x has a single value of y satisfying the relation. But even functions can be described implicitly, ra
Sure you can graph −y=x.
You can also graph cos(sin(x2+y2))≤cos(x+y).
Whoever told you that you can only graph things that look like y=f(x) was misinformed, confused or just meant something different. You can plot any sort of relation between x and y in the plane (or any relation between x,y and z in space). The points of the plane which satisfy the relation are colored, the ones that don’t are not. That’s all.
There’s an important sort of relation called a function, where every value of x has a single value of y satisfying the relation. But even functions can be described implicitly, rather than explicitly, and −y=x is an expression which defines a perfectly legitimate function.
(Both images done with Desmos; try the second one here.)
Note: many old-style graphing calculators only support graphing relations of the form y=f(x). This is a limitation of those calculators, not any sort of mathematical constraint.
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Terry Moore
M.Sc. in Mathematics, University of Southampton (Graduated 1968)
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Author has 16.6K answers and 29.3M answer views
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3y
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Do all exponential functions in the form of y=ab^(x-h) + k have y-intercepts?
Do all exponential functions in the form of y=ab^(x-h) + k have y-intercepts?
Let’s assume that b>0, otherwise fractional powers of b are problematic.
The y intercept occurs when x=0. As y is well defined for all x, it is defined when x=0, so there is a y intercept for all a and k (even for a=0; then y=k).
However, if b<0 you have the problem of imaginary numbers and many valued functions.
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
8y
Related
How do you find the critical point on the graph y=x^(1/x)?
This is a far more interesting question than just differentiating the equation.
I became interested in what the graph looks like.
It was OK for positive x values but what about negative x values?
Well I calculated quite a few and found a lot of complex y values.
This meant that I needed an ordinary x axis but a complex y plane so I added an extra axis for the imaginary y values.
I plotted the points but I have not yet found a way of producing the strange curve for negative x values.
I hope you appreciate this attempt. If anyone can produce the equation for the left hand side curve, I would appreciat
This is a far more interesting question than just differentiating the equation.
I became interested in what the graph looks like.
It was OK for positive x values but what about negative x values?
Well I calculated quite a few and found a lot of complex y values.
This meant that I needed an ordinary x axis but a complex y plane so I added an extra axis for the imaginary y values.
I plotted the points but I have not yet found a way of producing the strange curve for negative x values.
I hope you appreciate this attempt. If anyone can produce the equation for the left hand side curve, I would appreciate it!
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Sarvaswa Tandon
Solving equations is my hobby
·
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Stewart Smith
, MMath Mathematics (2020)
·
8y
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Why does the graph y = x^x turn at 1/e?
A graph turns at its critical points i.e. the point of local extremum (minima or maxima). At this point the graph changes its slope from positive to negative or negative to positive.
To look for these critical points, slope of a curve y=f(x) is given by,
slope=dydx
Here the function y is given as,
y=xx
Take natural log on both sides,
ln(y)=x⋅ln(x)
Differentiate both sides w.r.t. x,
1y⋅dydx=1+ln(x)
dydx=xx⋅(1+ln(x))
At critical points, dydx=0
So, this implies,
xx=0, which is non-zero.
1+ln(x)=0
x=
A graph turns at its critical points i.e. the point of local extremum (minima or maxima). At this point the graph changes its slope from positive to negative or negative to positive.
To look for these critical points, slope of a curve y=f(x) is given by,
slope=dydx
Here the function y is given as,
y=xx
Take natural log on both sides,
ln(y)=x⋅ln(x)
Differentiate both sides w.r.t. x,
1y⋅dydx=1+ln(x)
dydx=xx⋅(1+ln(x))
At critical points, dydx=0
So, this implies,
xx=0, which is non-zero.
1+ln(x)=0
x=1e
Here is the curve of y=xx in the domain xϵ(0,1]
Gary Ward
MaEd in Education & Mathematics, Austin Peay State University (Graduated 1997)
·
Author has 4.9K answers and 7.6M answer views
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4y
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What is the difference between the graph of y=k/x and y=k/(x^2)?
What is the difference between the graph of y=k/x and y=k/(x^2)?
For k > 0, y = k/x has a vertical asymptote at x = 0, is positive for x > 0 and negative for x < 0. y = k/x² is positive on both sides of the same vertical asymptote. The curves cross at x = 1.
For k < 0, y = k/x has a vertical asymptote at x = 0, is negative for x > 0 and positive for x < 0. y = k/x² is negative on both sides of the same vertical asymptote. The curves cross at x = 1.
For k = 0 both curves are y = 0
The above is k = 1
What is the difference between the graph of y=k/x and y=k/(x^2)?
For k > 0, y = k/x has a vertical asymptote at x = 0, is positive for x > 0 and negative for x < 0. y = k/x² is positive on both sides of the same vertical asymptote. The curves cross at x = 1.
For k < 0, y = k/x has a vertical asymptote at x = 0, is negative for x > 0 and positive for x < 0. y = k/x² is negative on both sides of the same vertical asymptote. The curves cross at x = 1.
For k = 0 both curves are y = 0
The above is k = 1
Peter Shea
B. Sc in Mathematics & Computer Science, Monash University (Graduated 1972)
·
Author has 5.2K answers and 1.2M answer views
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2y
Related
How do you draw the graph of the function y=abs(x) +abs (x-2) +sgn(x)?
Here is the graph of abs(x):
i.e. y=
x, x≥0,
-x, x<0
Here is the graph of sign(x):
i.e. y =1, x≥0,
-1, x<0
Since they both alter their behaviour at x=0, it is easy to simply add then, giving this:
i.e. y=x+1, x≥0,
-x-1, x<0
Here is the graph of abs(x-2):
i.e. y=
x-2, x≥2,
2-x, x<2
Put them all together and you get:
What are the equivalent formulas at:
x < 0,
0 ≤ x ≤ 2,
2 ≤ x?
Here is the graph of abs(x):
i.e. y=
x, x≥0,
-x, x<0
Here is the graph of sign(x):
i.e. y =1, x≥0,
-1, x<0
Since they both alter their behaviour at x=0, it is easy to simply add then, giving this:
i.e. y=x+1, x≥0,
-x-1, x<0
Here is the graph of abs(x-2):
i.e. y=
x-2, x≥2,
2-x, x<2
Put them all together and you get:
What are the equivalent formulas at:
x < 0,
0 ≤ x ≤ 2,
2 ≤ x?
Uniquely Me
from Nanyang Polytechnic (NYP)
·
Author has 51 answers and 24.6K answer views
·
4y
Related
What is the effect of changing the value of k in the equation y= (x^2+k) (x-3)?
Take a look at this graph;
The red line shows y = (x² + 1)(x - 3)
The black line shows y = (x² + 2)(x - 3)
So as k increases the y intercept decreases and as k decreases the y intercept increases.
As k increases the curve becomes less steep and as k decreases the curve becomes more steep.
Take a look at this graph;
The red line shows y = (x² + 1)(x - 3)
The black line shows y = (x² + 2)(x - 3)
So as k increases the y intercept decreases and as k decreases the y intercept increases.
As k increases the curve becomes less steep and as k decreases the curve becomes more steep.
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist.
·
Author has 6.8K answers and 52.8M answer views
·
7y
Related
What is the minimum positive value of k if k=x^x+y^y?
I decided to draw the 3D graph z = x^x + y^y which produced the blue surface.
Then I constructed a horizontal plane z = a where “a” is a variable.
I increased “a” until it just touched the blue surface at the point marked A.
The 3D coordinates of A are (0.3679, 0.3679, 1.3844)
Then I thought, since the variables x and y were not actually combined by multiplication or division I decided that the minimu
I decided to draw the 3D graph z = x^x + y^y which produced the blue surface.
Then I constructed a horizontal plane z = a where “a” is a variable.
I increased “a” until it just touched the blue surface at the point marked A.
The 3D coordinates of A are (0.3679, 0.3679, 1.3844)
Then I thought, since the variables x and y were not actually combined by multiplication or division I decided that the minimum point must be when x^x is a minimum which also means y^y is a minimum too.
Related questions
Why can you graph "Y = - X", but not "-Y = X"?
How do I draw the graph of the function Y=X as Y versus -X?
Do all exponential functions in the form of y=ab^(x-h) + k have y-intercepts?
Does the graph of k=xy represent a function?
What is the effect of changing k in y= (x^2+k) (x+1) (x-2)?
What is the graph of the function y=(1/3) x?
Which equation represents the graphed function? Y = 4x – 2 y = –4x – 2 y = x – 2 y = – x – 2?
What is the effect of the variables h and k on the graph y=(x-h) ^2 +k as compared to the graph y=x^2?
Is the sequence X^k-x^k divided by x-y?
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What is the graph of function y=x/x+1?
Compare the graphs of y = x² and y = 2 (x - 2)² + 1. How will the values of a, h, and k affect the graph of y = x²?
For the function f(x) = x + 4, what is the ordered pair for the point on the graph when x = 3p?
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What is the minimum positive value of k if k=x^x+y^y?
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3408 | https://jnccn.org/abstract/journals/jnccn/22/2/article-e247002.xml | Update on the Sentinel Node Procedure in Vulvar Cancer
Willemijn L. van der Kolk, BSc 1 ; Joost Bart, MD, PhD 2 ; Ate J.G. van der Zee, MD, PhD 1 ;and Maaike H.M. Oonk, MD, PhD 1
ABSTRACT
Early-stage vulvar cancer is managed by a local excision of the pri-mary tumor and, if indicated, a sentinel node (SN) biopsy to assess the need for further groin treatment. With the SN procedure, many patients can be treated less radically and will experience less compli-cations and morbidity compared with an inguinofemoral lymphade-nectomy (IFL). Still, the SN procedure can be further optimized. Different tracers for detecting the SN are being investigated, aiming to optimize detection rates and decrease the burden of the proce-dure and short-term complications. Until now, no standardized pro-tocols exist for the pathologic workup of the SN, possibly leading to discrepancies in detection of metastases between institutes using dif-ferent methods. New techniques, such as one-step nucleic ampli fi ca-tion, seem to have potential in accurately detecting metastases in other cancers, but have not yet been investigated in vulvar squamous cell carcinoma (VSCC). Furthermore, several studies have investi-gated the possibility to broaden the indications for the SN proce-dure, such as its use in recurrent disease, larger tumors, or multifocal tumors. Although these studies show encouraging results, cohorts are small and further studies are needed. Prospective studies are cur-rently investigating these subgroups. Lastly, several studies investi-gated optimization of groin treatment of patients with a metastatic SN. Inguinofemoral radiotherapy is a good alternative to IFL in pa-tients with micrometastases in the SN, with comparable ef fi cacy and less treatment-related morbidity. Reduction of the radicality of groin treatment is also possible in other ways, such as omitting contralat-eral IFL in patients with lateralized tumors and a unilateral metastatic SN. In conclusion, the SN procedure is an established procedure in early-stage VSCC, although optimization of the technique, patho-logic workup, indications, and treatment in the setting of metastatic disease are the subject of ongoing research.
J Natl Compr Canc Netw 2024;22(2):e247002 doi:10.6004/jnccn.2024.7002
Vulvar Cancer: An Introduction
Vulvar cancer represents 3% of all gynecologic cancers, affecting 2.4 per 100,000 women on average in Western Europe in 2020. 1 The most common type, vulvar squa-mous cell carcinoma (VSCC), accounts for 90% of vulvar cancer cases. 2 VSCC can develop via 2 different pathways, although these are not yet completely understood. The pathways are based on the presence of HPV: an HPV-associated pathway, accounting for 17.4% to 25% of VSCC, and an HPV-independent pathway, mainly related to the presence of lichen sclerosus. 3–5 Multiple studies found better a prognosis for HPV-associated vulvar cancers. 6–9However, the possibility for less radical treatment ap-proaches has not yet been investigated. Early-stage VSCC is usually managed by a radical local excision of the primary tumor. For years it was recom-mended that one should aim for a margin of at least 8 mm, but evidence showing that this represents the optimal margin is insuf fi cient. Although the optimal margins of the excision remain unde fi ned, it is acceptable and often desirable to limit radicality to preserve midline structures, such as the clitoris, anus, and urethra. 10 ,11 If indicated, patients will undergo a sentinel node (SN) procedure for diagnostic purposes and determining further treatment. With this procedure, the fi rst tumor-draining lymph nodes are identi fi ed, removed, and exam-ined for metastases. Before the introduction of the SN proce-dure, inguinofemoral lymphadenectomy (IFL) was standard of care. With IFL, all lymph nodes in the groin were surgically removed. This procedure was ef fi cient and the recurrence rate was low. 12 However, removing a large number of lymph nodes in the groin causes major treatment-related morbidity, such as lymphoedema and recurrent infections. 13 The SN procedure replaced IFL as standard of care, after large studies showed its safety and ef fi cacy. 14 , 15 Moreover, treatment-associated morbidity decreased tremen-dously with this less-radical approach. 13 Groin treatment is in fl uenced by lymph drainage patterns, which are dependent on the location of the tu-mor. The location of the tumor on the vulva can be classi-
fi ed as lateralized ( .1 cm from the midline), near midline (#1 cm from the midline but not crossing the midline), or true midline (crossing the midline) (Figure 1). Lateralized
1Department of Gynaecologic Oncology, University Medical Center Groningen, Groningen, the Netherlands; and 2Department of Pathology, University Medical Center Groningen, Groningen, the Netherlands.
JNCCN.org
| Volume 22 Issue 2 | March 2024 1REVIEW tumors show bilateral drainage less often than midline tumors (25% vs 88%, respectively). Tumors located in a near-midline position show bilateral drainage in 66% of the cases. 16 If bilateral drainage is present, a bilateral SN procedure is indicated, and with unilateral drainage, a unilateral SN procedure is suf fi cient. In a true midline tumor, however, unilateral drainage cannot be accepted, and therefore an ipsilateral SN procedure and contralat-eral IFL should be performed. 16 ,17 Survival is good in patients with early-stage VSCC; patients with negative SNs have a 5- and 10-year disease-speci fi c survival of 93.5% and 90.8%, respectively. How-ever, if the patient has a metastasis in the SN, the 5- and 10-year disease-speci fi c survival decreases to 75.5% and 64.5%, respectively. 18
The SN Procedure
Today, patients are eligible for the SN procedure if they have a primary tumor ,4 cm, with a depth of invasion (DoI) .1 mm and no suspicious lymph nodes at pal-pation and imaging. 11 If there is a suspicious lymph nodes at imaging, fi ne-needle aspiration cytology is in-dicated to rule out lymph node metastases. If the DoI is ,1 mm, lymph node staging is not indicated, because the risk of lymph node metastases in these tumors is negligible. 11 The SN procedure consists of 3 steps: (1) mapping of the SN via tracer injection and performing lympho-scintigraphy (LSG), (2) surgical removal of the SN, and (3) workup by the pathologist.
SN Mapping
International guidelines recommend performing the SN procedure with at least the use of a radioactive tracer (usually a technetium-99m [Tc-99] –labeled nanocolloid). 11 Because the radioactive tracer will be taken up by the lymphatic system of the vulva and transported to the inguinofemoral region, the SNs can be visualized on an LSG prior to surgery. This facilitates the surgical proce-dure by providing information on the presence, localiza-tion, and number of SNs. 11 ,19 Early (30 minutes after injection) and late (2 –4 hours after injection) imaging is usually performed. In 2023, Thissen et al 20 showed that late imaging showed additional SNs in 35% of the pa-tients. Consequently, they stated that it is not safe to omit late scintigraphy. However, one can question whether these additional lymph nodes were clinically relevant, and the researchers did not investigate the rate of metas-tases in these additional SNs. In addition to the radioactive tracer, a blue dye can be used as an optical tracer to visualize the SN during surgery. In the operating room, the blue dye is injected at the same sites as the radioactive tracer, approximately 10 minutes prior to the surgical removal of the SN. Al-though the use of blue dye is not mandatory, the gold standard for SN mapping is a combination of blue dye and a radioactive tracer, because this has been shown to have the highest detection rate. 11 ,21 Furthermore, the tech-nique is easier to learn when the nodes are visible, and therefore will also have a faster learning curve for less ex-perienced gynecologic oncologists. In order to supplement or even replace the previ-ously described methods, multiple techniques are being developed and investigated for clinical implementation. For example, indocyanine green (ICG) is an optical tracer, which is fl uorescent in light at wave lengths in the near infra-red range. Use of ICG has been investigated and validated in a variety of cancers, such as early-stage breast cancer. Its use is easily reproducible, and has a steep learning curve. 22 ICG is invisible to the naked eye, and therefore has the advantage of not coloring the surgical fi eld. Furthermore, it has a pene-tration depth of 5 to 8 mm, which makes lymph nodes visi-ble through overlying tissue, allowing for transcutaneous detection. On the other hand, this means the technique is only optimal in lean patients, with a maximal distance be-tween femoral artery and skin of 24 mm. For obese patients, a skin incision may be needed to localize the SN, and the ad-vantage of transcutaneous detection is lost. 23 Several studies investigated the accuracy of ICG in VSCC, either as a hybrid tracer, bound to human serum albumin (HSA), or alone. Theoretically, HSA improves the fl uorescence of ICG, although differences have not yet been observed in practice. 24 Table 1 shows studies investigating ICG and its detection rates in VSCC. All clinical studies (n 510) un-til 2023 with full-text availability were included. 23 ,25 –33 The
Lateralized >1 cm Near midline ≤1 cm Midline
Figure 1. Classi fi cation of potential tumor locations. Lateralized tumors are located .1 cm from the midline; near midline tumors are located #1 cm from the midline but not crossing the midline; true midline tumors cross the midline.
REVIEW van der Kolk et al 2 © JNCCN —Journal of the National Comprehensive Cancer Network | Volume 22 Issue 2 | March 2024 detection rate varies, ranging from 76.8% to 100%. Until now, the use of ICG alone has not been considered for vul-var cancer guidelines, but only as an alternative for blue dye in combination with a radioactive tracer. 11 An optimal technique for use of ICG in the SN procedure has not yet been identi fi ed. Another novel tracer is superparamagnetic iron oxide (SPIO), which can be visualized on MRI prior to surgery, but can also be detected by a magnetometer probe during the procedure. Although SPIO is widely used in breast cancer, only one study has been published on this tracer in vulvar cancer. 34 In this study, 20 patients received SPIO and Tc-99, and a magnetometer probe was used to detect the SPIO. Sensitivity was 98.5% for the superparamagnetic technique and 93.8% for the radioactive tracer. Although the superpar-amagnetic technique seems to be feasible, studies on its safety in vulvar cancer are lacking. A review on the use of SPIO with ultra-small particles of 20 to 50 mm (USPIO) with MRI visualization in head and neck cancer has shown promising results for its use in detecting SNs. 35 Use of a single tracer without the need for a radioactive tracer has many advantages. First, the procedure is no longer dependent on the availability of medical isotopes. Addition-ally, planning of the procedure is more fl exible. Moreover, the whole procedure can be performed under general anesthesia because early injection of a tracer is no longer necessary, making the procedure more patient-friendly. On the other hand, new techniques also require new equip-ment or machines, which are expensive because they are not frequently used. For these reasons, organizations might not be tempted to invest in these new techniques.
Excision of the SN
The SN procedure is performed preferably prior to the re-moval of the primary tumor. However, several studies have shown that scar injection (when the primary tumor has al-ready been removed in a previous procedure) of the tracer might also be feasible to accurately visualize the SN. 36 –38 The SNs are removed according to the fi ndings on LSG. For the LSG to be acceptable, at least a unilateral SN should be found in lateralized and near-midline tumors. For real midline tumors, SNs should be found bilaterally. If this is not the case, this should be noted as a false-negative result and IFL should be performed on that side. 16 ,17 Once all identi fi ed SNs are removed, they are separately labeled and sent to the pathology department for routine workup.
Histopathology
Lymph nodes are fi xated in formalin and embedded in paraf fi n. Then, the SN will be investigated by routine hematoxylin-eosin (HE) staining on one slide per 2 mm of lymph node tissue. If these slides do not contain tumor cells, ultrastaging is performed. With ultrastaging, the en-tire lymph node will be cut into serial sections and stained with HE and cytokeratin AE1/3. 11 Both the European Society of Gynaecological Oncology (ESGO) and the Groningen Iternational Study on Sentinel Nodes in Vulvar Cancer (GROINSS-V) study protocol advise to include at least 3 slides per millimeter. 11 ,39 Despite the fact that stan-dardized protocols have been described, many variations between institutes exist. Alternatively, the procedure can be started with up-front ultrastaging to reduce turnaround times. It may be worthwhile to investigate which procedure renders optimal results. Frozen sectioning is normally not performed, pre-sumably due to the high rate of false-negatives and to prevent loss of diagnostic tissue. Literature describing frozen section analysis, such as a study by Swift et al, 40 has shown a sensitivity of 89.7% and speci fi city of 99.5%. Brunner et al 41 showed similar results, with a sensitivity
Table 1. Studies on ICG for SN Detection in Vulvar Cancer Tracer Author Retrospective/Prospective Patients, n Detection Rate
ICG Benmoulay-Rigollot et al 25 Retrospective 30 76.8% Crane et al 23 Prospective 10 89.7% Prader et al 26 Retrospective 64 93.6% Rundle et al 27 Prospective 50 78% Soergel et al 28 Prospective 27 100% Verbeek et al 29 Prospective 12 100%
ICG-HSA Hutteman et al 30 Prospective 9100%
ICG 1Tc-99 Deken et al 31 Prospective 48 92.5% Math eron et al 32 Prospective 15 96% Broach et al 33 Retrospective 114 100% Rundle et al 27 Prospective 50 84% Abbreviations: HSA, human serum albumin; ICG, indocyanine green; SN, sentinel node; Tc-99, technetium-99m.
Sentinel Node Procedure in Vulvar Cancer REVIEW JNCCN.org | Volume 22 Issue 2 | March 2024 3of 88.5% and speci fi city of 100%. However, an additional analysis of GROINSS-V I by Oonk et al 42 showed a much lower sensitivity of 48% in 315 patients who underwent frozen sectioning. Furthermore, low-volume metastases were even less accurately detected. Theoretically, low-volume metastases can be missed with frozen sectioning because these are smaller than 2 mm, and lymph nodes are usually not cut every 2 mm. Smaller nodes are usually cut in half. Although frozen sectioning of the SN offers the possibility that the surgical procedure could be per-formed in one session, and therefore is regarded as a patient-friendly option, it also has logistic drawbacks, such as prolonged duration of the surgical procedure. For this reason, a 2-step approach is considered equivalent. The OSNA technique, in which the whole lymph node can be assessed by detecting mRNA cytokeratin 19, has not yet been described for vulvar cancer. 43 In breast cancer, multiple studies have investigated the accuracy of OSNA, and sensitivity is high. 44 –46 Because OSNA accu-rately detects macrometastases ( .2 mm), it appears to be a promising technique. 46 However, Tiernan et al 47 argue it is not reliable for detecting micrometastases (#2 mm) because on histopathologic examination, 21% of patients needed to be reclassi fi ed from macrometa-stasis to micrometastasis. For breast cancer, this will result in overtreatment when using OSNA alone and is therefore not recommended outside of a clinical re-search setting. Moreover, a criterion for the use of OSNA is that the tumor expresses mRNA cytokeratin 19. In head and neck cancer, sensitivity of OSNA is high, ranging from 82.4% to 90%. 48 –50 However, expression of cytokera-tin 19 in head and neck squamous cell carcinoma is ap-proximately 60% to 80%, and even lower in early-stage head and neck tumors. 51 ,52 The rate of expression of mRNA cytokeratin 19 in VSCC has yet to be determined.
SN Metastases
If a metastasis is found, it can either be a macrometasta-sis, a micrometastasis, or isolated tumor cells (ITCs). A previous study from GROINSS-V showed that a patient with a micrometastasis (de fi ned as a metastasis #2 mm) has a better prognosis than a patient with a macrometa-stasis. 42 Nevertheless, the 2021 FIGO (International Fed-eration of Gynecology and Obstetrics) classi fi cation uses a cutoff value of 5 mm for lymph node metastases be-tween stage IIIA and IIIB. 53 GROINSS-V previously found comparable prognosis among patients with metastases between 2 and 5 mm and those with a metastases .5 mm, questioning the clinical relevance of a 5-mm cutoff value. Furthermore, the 2021 FIGO staging states that ITCs are not categorized as positive lymph nodes. Indeed, in breast cancer, detection of micrometastases and ITCs is less im-portant because the vast majority of the patients will un-dergo adjuvant locoregional radiotherapy and/or systemic therapy. In vulvar cancer, however, adjuvant therapy is not standard after SN biopsy. An analysis of GROINSS-V I data showed that 4.2% of patients with only ITCs in the SN had additional metastases found at lymphadenectomy. 42 In GROINSS-V II, patients with ITCs or metastases #2 mm in their SN (n 556) had no groin recurrences when they underwent inguinofemoral radiotherapy (n 545), and 1 groin recurrence was observed in the 11 patients who did not undergo any additional treatment. 39 Therefore, all patients with SN metastases, including those with ITCs, should undergo additional groin treatment.
Broadening Indications
To this day, the SN procedure is performed in a well-selected group of patients with vulvar cancer. Several studies aim to investigate the possibility of broadening the indication for the SN procedure, for example in tu-mors $4 cm, multifocal tumors, or patients with locally recurrent disease. Currently, local recurrences are managed by per-forming a radical local excision of the recurrent tumor and a unilateral or bilateral IFL in patients who did not undergo lymphadenectomy at primary treatment. 11 In a retrospective evaluation of a series of 27 patients who had undergone repeat SN procedure, van Doorn et al 54 reported that the procedure was successful in 78% of the patients, although it appeared to be more challenging for the gynecologic oncologist. The ongoing V2SLN study by van Doorn et al 55 is prospectively evaluating the repeat SN procedure. Considering larger tumors, Levenback et al 15 evaluated outcomes of the SN procedure in patients with tumors 4 to 6 cm. They found that the false-negative predictive value was 7.4%, compared with 2.0% for patients with tumors
,4 cm. In 2019, Nica et al 56 investigated whether the SN procedure could be used in patients with tumors .4 cm. Finding that 9% of the patients (1/11) had a groin re-currence after negative SNs by the SN procedure, they concluded that performing the SN procedure in patients with larger tumors might not be safe. However, the co-hort was very small. Garganese et al 57 investigated the safety of the SN procedure in patients with clinically negative nodes but who, according to guidelines, were not fi t for the SN pro-cedure. All patients underwent a preoperative PET/CT to con fi rm negative nodes. The included patients had larger tumors ( .4 cm), multifocal tumors, complete tumor exci-sion, unilateral nodal involvement, or a local recurrence. In all groups (n 547) the SN biopsy false-negative rate was 0%. Larger prospective studies are needed to con fi rm these results. Furthermore, Zach et al 58 are currently conducting a prospective multicenter study on this subject, investigating
REVIEW van der Kolk et al 4 © JNCCN —Journal of the National Comprehensive Cancer Network | Volume 22 Issue 2 | March 2024 the feasibility of the SN procedure in patients with larger tu-mors, multifocal tumors, and local recurrences.
Treatment of SN Metastases
Further treatment of nodes in the groin is determined based on the outcomes of the SN workup. In the case of a negative SN, no further treatment is required. On the other hand, if the SN is found to be metastatic, further groin treatment should be performed, independent of the size of the SN metastases. 42 Additional groin treatment can be limited to the groin where the metastatic SN(s) was found, because the risk on contralateral metastases ap-pears to be low in patients with unilateral SN involvement. 16 IFL is the standard of care when a macrometastasis is present. In 2021, the results of GROINSS-V II showed that radiotherapy is a safe alternative in the case of a mi-crometastatic SN, with a groin recurrence rate of only 1.6%. Additionally, treatment-related morbidity was less frequent compared with IFL. 39 In GROINSS-V II, the groin recurrence rate for patients with a macrometastasis in the SN was high, therefore standard of care (IFL) was continued. Currently, GROINSS-V III is investigating whether pa-tients with a macrometastatic SN can be treated with che-moradiation instead of radiotherapy to prevent groin recurrences, but also to reduce treatment-related morbid-ity for these patients. 59 NCCN already recommends use of concurrent chemotherapy with radiotherapy for the treat-ment of SN metastases. 10 However, this recommendation is based on studies mainly investigating patients with advanced vulvar cancer. To date, chemotherapy and chemoradiation have insuf fi ciently been studied as treat-ments for SN metastases in early-stage vulvar cancer.
Future Perspectives
The SN procedure in early-stage vulvar cancer is a safe and ef fi cient option for assessing SN status. Even though it is a widely used technique, optimization of the procedure is still possible and is the subject of current research in this
fi eld. Several prospective studies are currently being con-ducted, investigating whether eligibility for the SN proce-dure can be extended to more categories of patients. This would reduce treatment-related morbidity in these patient groups as well. Although large studies recently showed progress in the treatment of patients with a micrometa-static SN, ample room remains for further improvement. Furthermore, in the search for a more patient-friendly SN procedure, larger prospective studies are needed with new tracers for SN detection to provide data on safety in terms of false-negative rates of the SN procedure. Lastly, there is a need to de fi ne the optimal pathologic workup for SNs, with consid-eration toward minimizing the workload for the pathologist without decreasing the detection rate of metastasis.
Conclusions
The SN procedure is an established procedure in treating early-stage vulvar cancer, although optimization of the technique, pathologic workup, indications, and treat-ment in the case of metastatic disease are the subject of ongoing research.
Submitted October 2, 2023; fi nal revision received December 22, 2023; accepted for publication January 4, 2024.
Disclosures: The authors have disclosed that they have no fi nancial interests, arrangements, af fi liations, or commercial interests with the manufacturers of any products discussed in this article or their competitors.
Correspondence: Willemijn L. van der Kolk, BSc, UMC Groningen, University of Groningen, Hanzeplein 1, 9713 GZ Groningen, the Netherlands. Email: w.l.van.der.kolk@umcg.nl
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A novel automated as-say for the rapid identi fi cation of metastatic breast carcinoma in sentinel lymph nodes. Cancer 2011;117:2599 –2607. 46. Shi F, Liang Z, Zhang Q, et al. The performance of one-step nucleic acid ampli fi cation assay for intraoperative detection of sentinel lymph node macrometastasis in breast cancer: an updated meta-analysis. Breast 2018; 39:39 –45. 47. Tiernan JP, Verghese ET, Nair A, et al. Systematic review and meta-analy-sis of cytokeratin 19-based one-step nucleic acid ampli fi cation versus his-topathology for sentinel lymph node assessment in breast cancer. Br J Surg 2014;101:298 –306. 48. Goda H, Nakashiro KI, Oka R, et al. One-step nucleic acid ampli fi cation for detecting lymph node metastasis of head and neck squamous cell carcinoma. Oral Oncol 2012;48:958 –963. 49. Matsuzuka T, Takahashi K, Kawakita D, et al. 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REVIEW van der Kolk et al 6 © JNCCN —Journal of the National Comprehensive Cancer Network | Volume 22 Issue 2 | March 2024 |
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一个数学概念
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子集：数学界的“套娃”艺术
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【集合重点】“子集”的概念与考法总结 【高中数学大合集】
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高中数学必修一：子集与真子集详解
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【高中数学 】高中数学必修一同步全集 2026新版 人教A版 新教材 高中数学必修一数学 新课标新教材数学
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集合间的基本关系：子集与包含关系的图解
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真包含与包含的区别
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本词条由“科普中国”科学百科词条编写与应用工作项目审核 。
子集（subset）是数理科学中描述集合间包含关系的数学概念，指集合A的任意元素都属于集合B时，则称A为B的子集，记作A⊆B。定义可表述为：若∀a∈A均有a∈B，则A⊆B 。
当A是B的子集且存在B中元素不属于A时，A称为B的真子集，记作A⊂B 。子集具有自反性（任何集合是其自身子集）、包含空集（空集是任何集合的子集）及传递性（若A⊆B且B⊆C，则A⊆C）等基本性质。
中文名 子集
外文名 subset
应用领域 数理科学
应用类别 集合
表 示∀a∈A，均有a∈B，则A⊆B
目录
1定义
2性质
定义
播报
02:27
数学初高衔接视频之子集
如果集合 A 的任意一个元素都是集合 B 的元素（任意 a∈A 则 a∈B），那么集合 A 称为集合 B 的 子集，记为A⊆B或 B⊇A，读作“集合 A包含于集合 B”或集合 B 包含集合 A”。
即：∀a∈A 有 a∈B，则 A⊆B。
真子集
如果集合 A 是 B 的子集，且 A≠B，即 B 中至少有一个元素不属于 A，那么 A 就是 B 的 真子集，可记作：A⊂ B。
符号语言：若∀a∈A，均 有 a∈B，且
x∈B使x∉A，则 A⊊B。
图1
如图1所示，集合A就是集合B的真子集。
性质
播报
一、根据子集的定义，我们知道 A⊆A。也就是说，任何一个集合是它本身的子集。
二、对于空集∅，我们规定∅⊆A，即空集是任何集合的子集。
说明：若A=∅，则∅⊆A仍成立。
证明：给定任意集合A，要证明∅是A的子集。这要求给出所有∅的元素是A的元素；但是，∅没有元素。对有经验的数学家们来说，推论“∅没有元素，所以∅的所有元素是A 的元素"是显然的；但对初学者来说，有些麻烦。 因为∅没有任何元素，如何使"这些元素"成为别的集合的元素？ 换一种思维将有所帮助。
为了证明∅不是A的子集，必须找到一个元素，属于∅，但不属于A。 因为∅没有元素，所以这是不可能的。因此∅一定是A的子集。
三、若A、B、C是集合，则：
自反性：A=A
反对称性：当且仅当
且
时，
传递性：若
且
，则
这个命题说明：包含是一种偏序关系。
四、
，
这个命题说明：对任意集合S，S的幂集按包含排序是一个有界格，与上述命题相结合，则它是一个布尔代数。
五、： 对任意两个集合 A 和 B，下列所有表述等价：
A ⊆ B
A ∩ B =A
A ∪ B = B
A−B=A (当A∩B=∅) ；A−B=C𝖠(A∩B)（当A∩B≠∅）
B′ ⊆ A′
这个命题说明：表述 "A ⊆ B " 和其他使用并集，交集和补集的表述是等价的，即包含关系在公理体系中是多余的。
六、假设非空集合A 中含有 n 个元素，则有：
A的子集个数为2 n。
A的真子集的个数为2 n-1。
A的非空子集的个数为2 n-1
A的非空真子集的个数为2 n-2。
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参考资料图册(1张)
1/1
参考资料
1 苏州教育出版社．数学 必修1．苏州．苏州教育出版社．2012
2 人民教育出版社．数学 必修1．北京．人民教育出版社．2012
3 曲一线．五年高考 三年模拟 理数．2014．北京．首都师范大学出版社
4 张三元主编；陈锦辉，金义明副主编. 离散数学[M]. 杭州：浙江科学技术出版社, 2002.01.P1.
学术论文
内容来自
．年
温邦彦.自然数和偶数的个数一样多吗?——无穷理论的新方案(1)．《WanFang》，2008
艾士薇.通往真理的事件——论阿兰·巴迪欧的"事件哲学"的理论基础．《CNKI》，2013
徐坤林.和不重数集合及其应用．《CNKI》，1983
刘明，高月，肖瑞，张伯礼.中药组方原则"君臣佐使"的模糊数学量化描述．《药学学报》，2009
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Derivative of {f(x)}^2
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h(x)=f 2(x)h(x)=f 2(x)
Find h′(x)h′(x).
I initially thought that h′(x)=2 f(x)h′(x)=2 f(x) but it is wrong. When I googled it, I found it to be h′(x)=0 h′(x)=0 but I'm not sure if that's true.
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edited Feb 11, 2021 at 6:35
Indian135
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asked Feb 11, 2021 at 6:10
Susanne MendezSusanne Mendez
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1 Do you want to find derivative of h(x)=(f(x))2 h(x)=(f(x))2? if so, you need to use chain rule. (see this: en.wikipedia.org/wiki/Chain_rule)Soheil –Soheil 2021-02-11 06:13:40 +00:00 Commented Feb 11, 2021 at 6:13
There is a way to post typeset mathematical notation here. See this brief introduction and its links to more comprehensive information.hardmath –hardmath 2021-02-11 06:22:47 +00:00 Commented Feb 11, 2021 at 6:22
DOn't know how you googled it but both the goggled result and you initial though are both wrong. Use the chain rule. [f(x)2]′=2 f(x)⋅f′(x)[f(x)2]′=2 f(x)⋅f′(x).fleablood –fleablood 2021-02-11 07:11:52 +00:00 Commented Feb 11, 2021 at 7:11
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Well, if
h(x)=(f(x))2 h(x)=(f(x))2
then using the chain rule we get
h′(x)=2 f(x)f′(x)h′(x)=2 f(x)f′(x)
So, I'm not sure how you're getting h′(x)h′(x) to be 0 0, the derivative is 0 0 only when the function is a constant so h′(x)h′(x) being 0 0 means that h(x)=c h(x)=c where c is some constant.
Now if that's the case f(x)f(x) would be the square root of c c so f(x)=c√f(x)=c and this would make h′(x)=0 h′(x)=0
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answered Feb 11, 2021 at 6:17
AsvAsv
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If u(x)=x 2 u(x)=x 2, then h(x)=u(f(x))h(x)=u(f(x)), and by the chain rule h′(x)=u′(f(x))f′(x)=2 f(x)f′(x)h′(x)=u′(f(x))f′(x)=2 f(x)f′(x).
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answered Feb 11, 2021 at 6:13
Hamish BHamish B
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3411 | https://web.auburn.edu/holmerr/1617/Textbook/relatedrates-screen.pdf | Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 1 of 15 Back Print Version Home Page 27. Related rates 27.1. Method When one quantity depends on a second quantity, any change in the second quantity effects a change in the first and the rates at which the two quantities change are related. The study of this situation is the focus of this section.
A rate of change is given by a derivative: If y = f(t), then dy dt (meaning the derivative of y) gives the (instantaneous) rate at which y is changing with respect to t (see 14).
27.1.1 Example The radius of a circle is increasing at a constant rate of 2 cm/s. Find the rate at which the area of the circle is changing when the radius is 5 cm.
Solution Let r denote the radius of the circle and let A denote the circle’s area.
The given information and the quantity to be found, expressed using symbols, are as follows: Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 2 of 15 Back Print Version Home Page Given: dr dt = 2 Want: dA dt r=5 (As long as the units are consistent, there is no need to continually write them. Our policy is to convert to consistent units at the outset of the solution, if necessary, and then suppress the units until the end.) The relationship between the area A and the radius r is as follows: Relationship: A = πr2 Implicit differentiation of this equation with respect to time t gives d dt [A] = d dt πr2 dA dt = 2πrdr dt = 2πr(2) = 4πr, where we have used the given information.
Evaluating at r = 5 gives the answer: dA dt r=5 = 4π(5) = 20π cm2/s (or approximately 62.8 cm2/s).
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 3 of 15 Back Print Version Home Page The example illustrates the steps one typically takes in solving a related rates problem.
Solving a related rates problem.
(i) Sketch a diagram showing the ongoing situation and label relevant quantities.
(ii) Express the given information and the quantity to be found using symbols.
(iii) Write an equation expressing the relationship between the quanti-ties.
(iv) Use implicit differentiation to find the desired derivative.
(v) If required, evaluate the derivative at the specified value(s).
27.2. Examples 27.2.1 Example An airplane flying horizontally at an altitude of 3000 m and a speed of 480 k/hr passes directly above an observer on the ground. How fast is the distance from the observer to the airplane increasing 30 s later?
Solution We begin by sketching a diagram: Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 4 of 15 Back Print Version Home Page The units in the statement of the problem are mixed. In the diagram, we have converted the 3000 m altitude to 3 k. Converting 30 s to 1/120 hr below leaves us with consistent units (kilometers and hours), and we can safely forget about units until the end.
Given: dx dt = 480 Want: ds dt t=1/120 The relationship between the variables comes from the Pythagorean theorem: Relationship: s2 = 32 + x2 Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 5 of 15 Back Print Version Home Page Differentiating implicitly with respect to t, we get d dt s2 = d dt 32 + x2 2sds dt = 2xdx dt ds dt = 480x s , where we have used the given information. The final step is to evaluate ds/dt at t = 1/120.
The formula we obtained requires that we find x and s corresponding to this particular time. The plane, moving at a constant rate of 480 k/hr, travels 4 k in 1/120 hr, so x = 4.
The corresponding s is 5 as can be seen from the relationship. Therefore, ds dt t=1/120 = ds dt x=4 s=5 = (480)(4) 5 = 384 k/hr.
27.2.2 Example A circular oil slick of uniform thickness is caused by a spill of 1 m3 of oil. The thickness of the oil slick is decreasing at a rate of 0.1 cm/hr. At what rate is the radius of the slick increasing when it is 8 m?
Solution The oil slick has the shape of a cylinder: Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 6 of 15 Back Print Version Home Page After converting 0.1 cm/hr to 0.001 m/hr, we have Given: V = 1, dh dt = −0.001 Want: dr dt r=8 (The negative sign is required since the thickness of the oil slick is decreasing with time.) The volume of a cylinder is the area of its base times its height: Relationship: 1 = V = πr2h Using implicit differentiation, we have d dt = d dt πr2h 0 = d dt πr2 h + πr2 d dt [h] 0 = 2πrdr dt h + πr2 dh dt , so dr dt = −r 2h · dh dt = 0.0005r h , the last step using the given information.
Before we evaluate dr/dt at r = 8, we use the relationship to find that the corresponding thickness of the oil slick is h = 1/(64π).
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 7 of 15 Back Print Version Home Page Therefore, dr dt r=8 = dr dt r=8 h=1/(64π) = (0.0005)(8) 1/(64π) = 0.256π m/hr (or approximately 0.804 m/hr).
27.2.3 Example Soybeans, pouring down from a chute at a constant rate of 2 m3/min, form a conical hill. Assuming that the height of the hill is always twice the radius of its base, find the rate at which the height is increasing at the moment the height is 1 m, and also when the height is 4 m.
Solution The hill of beans looks like this: The units are already consistent, so we suppress them until the end.
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 8 of 15 Back Print Version Home Page Given: dV dt = 2, h = 2r Want: dh dt h=1 and dh dt h=4 The relationship is from the formula for the volume of a cone: Relationship: V = 1 3πr2h = 1 12πh3 (The given information h = 2r has been used to get the final expression.) Using implicit differentiation, we get d dt [V ] = d dt 1 12πh3 dV dt = 1 4πh2 dh dt dh dt = 8 πh2 , the last step using the given information. The two desired rates are dh dt h=1 = 8/π m/min ≈2.5 m/min and dh dt h=4 = 1/2π m/min ≈0.2 m/min.
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 9 of 15 Back Print Version Home Page 27.2.4 Example A particle is moving along the graph of y = x2 in such a way that its x-coordinate is increasing at a constant rate of 10 units per second. Let θ be the angle between the positive x-axis and the line joining the particle to the origin. Find the rate at which θ is changing when the x-coordinate of the point is 3.
Solution The following graph shows the ongoing situation: The units are already consistent, so we suppress them until the end.
Given: dx dt = 10 Want: dθ dt x=3 Using the indicated triangle in the diagram, we get the following relationship between θ and x: Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 10 of 15 Back Print Version Home Page Relationship: tan θ = x2 x = x Differentiating this equation implicitly with respect to t, we get d dt [tan θ] = d dt [x] sec2 θ · dθ dt = dx dt , so dθ dt = 10 cos2 θ, where we have used the given information. In order to evaluate this last equation when x = 3 we need to find the corresponding angle θ. Actually, it is sufficient just to find the cosine of this corresponding angle. The relationship gives tan θ = 3, so, in a right triangle with one angle equal to θ, we have o/a = 3/1, for instance The Pythagorean theorem says that the length of the hypotenuse is √ 10, so cos θ = a/h = 1/ √ 10.
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 11 of 15 Back Print Version Home Page Therefore, dθ dt x=3 = 10 1 √ 10 2 = 1 rad/s.
27.2.5 Example A water trough of length 10 m has cross section an equilateral tri-angle of side 1 m. The trough is being filled at a constant rate of 2 m3/min. Find the rate at which the level of the water in the trough is rising when the water is 50 cm deep.
Solution The following diagram shows the ongoing situation: For later use, we have recorded the height of the trough, which can be determined either from the memorized 30-60-90 triangle or from the Pythagorean theorem. We get consistent units by converting 50 cm to 0.5 m, and so we suppress the units until the end.
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 12 of 15 Back Print Version Home Page Given: dV dt = 2 Want: dh dt h=0.5 The relationship comes from the formula for the volume of a triangular prism: Relationship: V = (area of end) · (length) = 1 2bh (10) = 5bh.
The similar triangles at the end of the trough relate b to h: b h = 1 √ 3/2, so that b = 2h √ 3.
Substitution of this expression for b into the relationship gives V = 10 √ 3h2.
We differentiate this last equation implicitly with respect to t: d dt [V ] = d dt 10 √ 3h2 dV dt = 20 √ 3hdh dt , Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 13 of 15 Back Print Version Home Page so dh dt = √ 3 20h dV dt = √ 3 10h, where we have used the given information. Evaluation at h = 0.5 gives the answer: dh dt h=0.5 = √ 3 10(0.5) = √ 3 5 m/min ≈35 cm/min.
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 14 of 15 Back Print Version Home Page 27 – Exercises 27 – 1 A meteor (spherical in shape) enters earth’s atmosphere and starts burning up in such a way that its surface area decreases at a constant rate of 100 cm2/s. Find the rate at which the diameter is changing when the radius is 5 m.
Hint: The surface area of a sphere of radius r is 4πr2.
27 – 2 The width of a rectangle is increasing at a rate of 3 cm/s while its height is decreasing at a rate of 4 cm/s. Find the rate at which the area of the rectangle is changing when the width is 12 cm and the height is 20 cm.
27 – 3 A particle moves on the line y = x + 2 in such a way that its x-coordinate changes at the constant rate of 5 units per second. A right triangle is formed by the line joining the particle to the origin, the vertical line from the particle to the x-axis, and the x-axis. Find the rate at which the area of this triangle is changing when the particle is at the point (3, 5).
27 – 4 Zoe is flying a kite. The kite is initially directly over her head at a height of 30 m. Then the wind starts carrying the kite in one direction at a rate of 40 m/min while Zoe starts running in the opposite direction at a rate of 200 m/min. Assuming that the height of the kite remains constant and that the string forms a straight line, find the rate at which the string is paying out after 10 s.
Related rates Method Examples Table of Contents ◀◀ ▶▶ ◀ ▶ Page 15 of 15 Back Print Version Home Page 27 – 5 The ice cream in a sugar cone (conically shaped) has all melted and is dripping out a hole in the bottom of the cone at a constant rate of 2 cm3/min. Assuming that the cone is 12 cm tall and has a radius at the top of 4 cm, find the rate at which the level of the melted ice cream is dropping when the level is 3 cm.
Hint: The volume of a cone of radius r and height h is 1 3πr2h. |
3412 | https://leprosyreview.org/admin/public/api/lepra/website/getDownload/5f7aee8b2bea3041ef678ba5 | Does clofazimine prevent Erythema Nodosum Leprosum (ENL) in leprosy? A retrospective study, comparing the experience of multibacillary patients receiving either 12 or 24 months WHO-MDT
MARIVIC BALAGON, PAUL R. SAUNDERSON & ROBERT H. GELBER
Cebu Skin Clinic, Leonard Wood Memorial Center for Leprosy Research, Cebu, Philippines
Accepted for publication 04 March 2011
Summary Objective: To compare the occurrence, duration and severity of ENL in leprosy patients treated with either 12 or 24 months of standard multi-drug therapy (MDT).
Materials and Methods: Study population: 296 patients treated with MDT for 2 years, between 1985 and 1992 and followed up as part of a relapse study; and 293 patients, treated between 1998 and 2004, with MDT for 1 year and also followed up as part of a relapse study. The Chi squared test and multiple logistic regression analysis were used to test for statistical significance.
Results: ENL was not significantly more common, but it was longer-lasting and more severe in patients receiving only 12 months of MDT, as compared with those receiving 24 months treatment. A high BI at the start of treatment significantly increased the risk of severe ENL by a factor of between 6 and 12, while treatment with 12 instead of 24 months of MDT significantly increased the risk by a factor of between 3 and 10.
Conclusions: This study provides further evidence that a high initial BI is the key risk factor for ENL. It also suggests that the difference between these two cohorts in their experience of ENL as demonstrated in this study, may be related to the different amounts of clofazimine which the two cohorts were given in the early years of their treatment. Further studies are needed to determine whether clofazimine could be used more specifically to reduce the severity of ENL in the small group of patients at high risk for the condition.
Introduction
Erythema Nodosum Leprosum (ENL), also known as Type II reaction, is a well recognised complication of leprosy, occurring exclusively in BL and LL patients. 1 ENL has been Correspondence to: Marivic Balagon, Cebu Skin Clinic, Leonard Wood Memorial Center for Leprosy Research, Cebu, Philippines (e-mail: csc_epi@yahoo.com)
Lepr Rev (2011) 82, 213–221 0305-7518/11/064053+09 $1.00 q Lepra 213 regarded as one of the most severe complications of the disease and although leprosy is not generally associated with a high mortality, those deaths that did occur in earlier times were often related to chronic ENL, secondary amyloidosis and renal failure. 2 Steroids were discovered at the start of the antibiotic era and in leprosy during the 1950s were used exclusively to treat ENL; 3 only later were they found to be useful in suppressing the neuropathy associated with reversal reactions. Treatment is difficult because of the potentially severe side-effects of the drugs currently available and the lack of good evidence on which to base an effective therapeutic strategy. 4 There is great interest in the search for safer and more effective medication for this condition. 5The pathology of ENL is poorly understood, involving immune complex deposition, 6as well as a localised cell-mediated immune response with vasculitis. Levels of certain cytokines, including TNF alpha and IL-6, have been shown to be consistently raised, while others, such as IL-4 are low. 7 While the clinical features are most obvious in the skin, ENL is a systemic condition affecting many organs, including the eyes, joints and kidneys. An important characteristic of ENL is its chronic nature. Even today, when the infecting organism, M. leprae , is rapidly killed at the start of effective chemotherapy with bactericidal drugs, the condition waxes and wanes for up to 5 years or more, in severe cases, eventually resolving completely. The management of ENL has changed very little in recent years. The suppressive effect of clofazimine has been known for some time, 8 while the two additional drugs used most widely are prednisolone and thalidomide. 6 Prednisolone is effective in controlling ENL, but side-effects are a major problem in the more severe cases, which may need treatment at high doses for several years. Thalidomide was initially thought to act through its anti-TNF alpha action, but recent evidence suggests this is not the case, 9 and the search continues for new drugs that can mimic its potent anti-ENL effects while avoiding its well-known teratogenic side-effects, which currently severely restrict its use. Clofazimine is an orange-coloured imino-phenazine dye with a weakly bactericidal action against M. leprae . Its mechanism of action is unknown and very few cases of drug resistance have been reported. It is unevenly distributed throughout the body, but very persistent in the tissues and, at high doses, may even precipitate out to form crystals, especially in the intestinal mucosa, lymph nodes and fatty tissue. The most common side effect is discolouration of the skin, with increased pigmentation, although this gradually fades over a period of 1 to 2 years once the drug is discontinued. 10 Different societies and social groups have differing degrees of tolerance of this problem. 11 The formation of crystals in the gastrointestinal tract, may lead to abdominal pain, nausea and diarrhoea, 12 which may be fatal in severe cases. 13 Clofazimine has long been regarded as having a mild immuno-suppressive effect, beneficial in ENL reactions. 8,14,15 This effect is not seen in Type I (or reversal) reactions. In severe, chronic cases of ENL, an increased dose is recommended, although care must be taken to avoid the dose-related side-effects in the gastrointestinal tract. While clofazimine is not adequate on its own in severe ENL, it may significantly reduce the requirement for steroids. 16 ENL is less frequent and less troublesome when patients are treated with a drug regimen which includes clofazimine. 8,16 The mechanism for this suppressive effect is unknown. Clofazimine is given orally as a daily dose of 50 mg; because of its long half-life in the body, a monthly supervised dose of 300 mg was included in the WHO-recommended regimen for multibacillary patients. The purpose of this was to try to provide some protection against
M. Balagon et al. 214
rifampicin monotherapy in patients who, for whatever reason, did not adhere correctly to the regimen and missed some or all of the unsupervised daily doses of dapsone and clofazimine. In severe ENL, clofazimine may be given in a dose of 300 mg daily for one month, followed by 200 mg daily for 5 months, followed by 100 mg daily indefinitely, while the condition persists. Surprisingly, little attention has been paid to the prevention of ENL. It is generally regarded as an inevitable complication in a proportion of patients with a high bacillary load. At the Cebu Skin Clinic (CSC) in the Philippines, it was noted that ENL appeared to be more of a problem when the length of treatment with MDT was reduced from 24 months to 12 months for multibacillary cases, following the meeting of the WHO Seventh Expert Committee on Leprosy in 1998. 17 We therefore decided to compare the experience of ENL in two cohorts of patients, given either 12 or 24 months of MDT, who had been recruited and rigorously followed up for relapse studies being conducted at the CSC.
Materials and Methods
A cohort of multibacillary patients had been recruited for a relapse study of 2 year MDT during the late 1980s and early 1990s, as described by Cellona et al. 18 Although 316 subjects were originally recruited at the Cebu Skin Clinic, for the current study we excluded those lost to follow-up and those treated with additional MDT, usually because of relapse; thus 296 from that cohort are analysed here. A similar cohort of 293 patients was recruited between 1998 and 2003, to determine the relapse rate after 1 year’s MDT. Both cohorts were recruited at the Cebu Skin Clinic, using the same procedures, which included recruiting each consecutive eligible case who completed treatment correctly and who consented to participate in the long follow-up. Subjects were examined and actively followed up annually by the same clinicians under the leadership of one of the authors (MB), and were managed in the same way, apart from the length of MDT. Annual slit-skim smears were examined. The records of both cohorts were examined to extract details of ENL experienced by patients. The two MDT regimens were identical apart from the duration of treatment, and contained the following drugs (only the adult doses are given): rifampicin 600 mg monthly, clofazimine 50 mg daily and 300 mg monthly, dapsone 100 mg daily; the monthly doses were supervised and the daily doses were unsupervised. When ENL was noted, it was graded as either mild or severe, according to clinical criteria which remained the same for both cohorts. This grading was done by clinicians at the time ENL was being diagnosed and managed and is recorded in the medical records; it is not a retrospective classification. Mild reactions were those with less than 20 papulonodules and no systemic signs; severe reactions were those with more than 20 papulonodules, or any of the following signs and symptoms: joint pains, constitutional symptoms, nerve involvement, oedema, or ulceration. The duration of each episode of ENL was also noted in the medical record. For regression analysis, we used a duration of more than 20 weeks as another indicator of severity. Thalidomide is not available for use in the Philippines, so patients were managed with prednisolone and in some cases, additional clofazimine. In general, the same management policies were applied to both cohorts. The use of additional clofazimine was determined by the severity of ENL and its response to steroid treatment.
Does clofazimine prevent ENL in leprosy? 215
Regarding treatment of ENL, mild cases received non-steroidal anti-inflammatory drugs. Steroids were only given in severe cases of ENL, particularly if reactions were associated with neuritis or if ENL lesions were located on the face. Prednisolone was given at 20– 40 mg daily. A higher dose of50 mg daily was given if ENL was associated with neuritis. Steroid doses were tapered every 2 weeks and adjusted according to weight, severity and clinical response to treatment. Clofazimine was only given if steroids could not be tapered below 20 mg daily after 12 weeks. Clofazimine was given at 200– 300 mg daily for one month; 100– 200 mg daily for the second month;100 mg daily for the third month, and 50 mg daily until no ENL lesions were noted for 4 successive weeks. As an additional measure of severity, the total dose of prednisolone prescribed to each patient with ENL was recorded in the medical records and was available for retrospective analysis. For the regression analysis, we used a total dose of .2 gm as a marker of severity (a typical 12 –week course of steroids which is widely used as a standard course, gives a total dose of 1·68 gm). Four outcome indicators were examined: the occurrence of ENL and three measures of the severity of ENL, namely, the clinical assessment, the duration of symptoms and the total dose of prednisolone given. The data were analysed using Statcalc for the x2 test for comparing proportions and Epi Info 3.4.3 for the multiple logistic regression analysis of risk factors.
Results
The two cohorts were recruited at different times, because the primary purpose of the studies was to examine the relapse rate following the MDT regimens currently recommended by WHO. The recommended duration was reduced from 24 months to 12 months in 1998. Table 1 shows the demographic and disease characteristics of the two cohorts. Of the characteristics described in Table 1, the only significant difference between the two cohorts was the initial BI, the average of the BIs at all six sites taken at one time for the slit-skin smear test. The difference in the proportions with a BI of 4 or more, was 0·17 (95% CI: 0·09– 0·25), indicating that the 1-year MDT cohort had a significantly higher bacillary load and thus an increased risk of ENL. For this reason, any comparison between the two groups must include a multivariate analysis. Figure 1 shows the occurrence of the first episode of ENL during each year of follow-up. During the first year, while both cohorts received the same treatment, the results are not
Table 1. Baseline characteristics of the two cohorts One year MDT Two years MDT
N¼293 N¼296 Mean age in years (range) 30 (6– 73) 29·7 (9–68) Gender M:F 3·5: 1 4·3: 1 Initial average BI (mean) 3·9 3·6 Initial average BI (range) 1·0–5·2 0·2–5·2 Proportion with BI of 4 or more 60% 43% Leprosy type LL 137 (47%) 140 (47%) BL 156 (53%) 156 (53%)
M. Balagon et al. 216
significantly different. During the second year, when one group has stopped MDT, that group has a significantly greater number of subjects developing ENL ( p , 0·001) and also a significantly greater proportion getting severe, rather than mild ENL ( p , 0·005). In years 3 and 4, the same trend appears to be present, but as the numbers are much smaller, the differences are not statistically significant. Table 2 shows three indicators of severity of ENL. Firstly, the clinical examination showing signs and symptoms meeting the criteria for severe disease; secondly, the duration in weeks of each episode was noted and summed to produce an overall total, so that for each person affected, the number of weeks with ENL was recorded; typically this would be the total for a number of distinct episodes added together, as the condition waxes and wanes over time. Thirdly, the total dose of prednisolone was recorded for each patient. In these cohorts, no cases in the 2 year MDT group received additional clofazimine whereas about 30% of the ENL cases in the 1 year MDT group received additional clofazimine (with steroids) to suppress ENL. Likewise, the total amount of steroid intake in each case was determined by both the severity and the total duration of ENL – that is, total steroid intake was significantly
Cases with ENL by year after MDT ( n=96) 0510 15 20 25 30 35 12-mths: year 1 12-mths: year 2 12-mths: year 3 12-mths: year 4 24-mths: year 1 24-mths: year 2 24-mths: year 3 24-mths: year 4 Regimen and year after MDT Number of cases Mild ENL Severe ENL
Figure 1. The occurrence and severity of the first episode of ENL, over four years of follow-up. In this Figure, the terms mild and severe refer only to the clinical assessment.
Table 2. Characteristics of ENL reactions in each group One year MDT Two years MDT Total number of cases with ENL N¼60 N¼36 Clinical diagnosis of severe ENL: n (%) 55 (92%) 14 (39%) Total duration of ENL in weeks: Mean 49·7 12·4 Range 8 –125 3– 44 Duration .20 weeks: n (%) 48 (80%) 8 (22%) Average number of episodes per patient 2·9 2·4 Average duration of each episode 17 weeks 5·3 weeks Total dose of prednisolone (gm) given: Mean 10·9 0·9 Range 0·0–34·7 0·1–4·2 Cases given .2 gm prednisolone: n (%) 51 (85%) 5 (14%)
Does clofazimine prevent ENL in leprosy? 217 higher in patients with more severe, prolonged or recurrent ENL, compared to those who had only mild, short term ENL, as was the case of most ENL patients in the two year MDT group. Table 3 examines risk factors for all four outcomes (occurrence of ENL and three measures of severity), using multiple logistic regression analysis. Leprosy type (either BL or LL disease) is closely correlated with the initial BI, so we chose to use the latter for the analyses, which were therefore carried out using four independent variables – age, sex, the initial BI and the MDT regimen. With the appearance of any episode of ENL as the outcome, the multivariate analysis showed that this difference is best explained by the difference in BI between the two groups. The different treatment regimens do not have a significant effect on the occurrence of ENL.
Table 3. Risk factors for occurrence and severity of ENL Multivariate analysis Factor Level Cases (%) Adjusted odds ratio 95% CI Outcome: occurrence of ENL at any time (96 cases) Age 0– 19 yrs 25/124 (20%) 1·3 0·8–2·3 20 þ yrs 71/465 (15%) 1Sex Male 80/469 (17%) 1·3 0·7–2·4 Female 16/120 (13%) 1Initial BI ,4·0 13/285 (5%) 14·0 þ 83/304 (27%) 7·3 3·9–13·5 MDT regimen 24 months 36/296 (12%) 112 months 60/293 (20%) 1·5 0·9–2·4 Outcome: clinically severe ENL (69 cases) Age 0– 19 yrs 18/124 (15%) 1·3 0·7–2·4 20 þ yrs 51/465 (11%) 1Sex Male 58/469 (12%) 1·4 0·7–2·9 Female 11/120 (9%) 1Initial BI ,4·0 10/285 (4%) 14·0 þ 59/304 (19%) 5·5 2·7–11·1 MDT regimen 24 months 14/296 (5%) 112 months 55/293 (19%) 3·9 2·1–7·3 Outcome: duration of ENL .20 weeks (56 cases) Age 0– 19 yrs 14/124 (11%) 1·2 0·6–2·4 20 þ yrs 42/465 (9%) 1Sex Male 48/469 (10%) 1·6 0·7–3·6 Female 8/120 (7%) 1Initial BI ,4·0 4/285 (1%) 14·0 þ 52/304 (17%) 11·8 4·2–33·5 MDT regimen 24 months 8/296 (3%) 112 months 48/293 (16%) 5·8 2·7–12·7 Outcome: total dose of prednisolone .2 gm (56 cases) Age 0– 19 yrs 13/124 (10%) 1·1 0·5–2·2 20 þ yrs 43/465 (9%) 1Sex Male 48/469 (10%) 1·6 0·7–3·6 Female 8/120 (7%) 1Initial BI ,4·0 4/285 (1%) 14·0 þ 52/304 (17%) 11·7 4·1–33·2 MDT regimen 24 months 5/296 (2%) 112 months 51/293 (17%) 10·3 4·0–26·4
M. Balagon et al. 218 On the other hand, all three indicators of severity were significantly associated with both the initial BI and the treatment regimen given. For a person with a BI of 4 or more, the adjusted odds ratios were, for clinical severity, 5·5; for duration greater than 20 weeks, 11·8; and for total prednisolone dosage greater than 2 gm, 11·7. For a person treated with only 12 months MDT, the adjusted odds ratios were, for clinical severity, 3·9; for duration, 5·8; and for prednisolone dosage, 10·3. We also looked at an aggregate measure of severity, combining all three individual measures, which produced adjusted odds ratios of 2·6 (95% CI: 1·5– 4·5) for the 1 year MDT group and 6·1 (95% CI: 3·1– 11·9) for the high BI subjects. In general therefore, a high BI at the start of treatment increased the risk of severe ENL by a factor of between 6 and 12, while treatment with 12 instead of 24 months of MDT increased the risk by a factor of between 3 and 10. Regarding reversal reactions, the results were very similar in the two cohorts, year by year, although during the first year of treatment, the severity was significantly greater in the more recent (1 year) cohort, when the actual treatment being given (MDT) was exactly the same. This suggests that awareness of reactions on the part of both patients and health staff may have increased during the period between the recruitment of each cohort.
Discussion
Two cohorts of multibacillary patients were reviewed for their experience of ENL reactions. The study confirmed previous reports that a high initial BI is the most important risk factor for ENL. The two groups took different MDT regimens, and, although there was some difference in the initial BI of the two groups, the difference in the anti-leprosy treatment is associated with a significant difference in the severity of ENL experienced, although not in the proportion of subjects who experienced episodes of ENL. We have demonstrated that ENL was more commonly severe following 1 year rather than 2 year MDT. Whether this observation is a function of more prolonged MDT or clofazimine, the only component of MDT known to ameliorate established ENL, is unclear. In the treatment of ENL clofazimine at 300 mg daily is known to augment prednisone treatment and reduce the required prednisone dosage. In the WHO MDT regimen for MB leprosy, clofazimine is administered once monthly (supervised) in a dose of 300 mg and daily in a dose of 50 mg (unsupervised). Thus, if the clofazimine component of WHO MDT were the critical determinant resulting in less severe ENL as noted here following 2 year MDT, its prophylactic dose must be considerable lower than the dosage required to affect established ENL. Clofazimine is unlikely to be included in new regimens because of its ability to kill
M. leprae , as it is now surpassed by many newer drugs in this respect. We suggest, however, that clofazimine may play a role in suppressing ENL reactions in those patients with an initially high BI (an average BI of 4 or more). ENL remains amongst the most troublesome of the complications of leprosy and it would be unfortunate if it becomes more of a problem in future because clofazimine is no longer a component of the ‘best’ bactericidal regimens. It should be noted that, although clofazimine is a very safe drug when used correctly, it is not easily available in some countries, because it is of limited use outside the field of leprosy. Thus, for example, it is difficult to prescribe clofazimine in the United States and some multibacillary patients there are being treated with alternative antibiotics (D. Scollard, personal communication). In other countries, including the Philippines,
Does clofazimine prevent ENL in leprosy? 219
clofazimine is available in the MDT blister-packs supplied by WHO, but is difficult to get as a single drug. The study reported here has several deficiencies. It was retrospective and compared two different cohorts of patients, one of which had a significantly higher initial BI. The grading of severity took place before the development of severity scales for leprosy reactions and could therefore be regarded as somewhat subjective. The use of additional indicators of severity, however, including duration of ENL and the total dose of prednisolone prescribed, each of which gave essentially the same result, and the use of multiple logistic regression analysis, lend some confidence to the overall finding. However, it is clear that the general awareness of reactions has improved: patients may be more willing to complain of symptoms and staff may be more willing to take complaints seriously during the period in which the 1 year cohort was being treated. In summary, this study suggests that an extended period of coverage with clofazimine may reduce the severity of ENL in the relatively small number of high risk patients, namely those with the most multibacillary form of leprosy (LL, or average initial BI of 4 or more). Further research is required to confirm this tentative finding and, if confirmed, to identify the best way of using clofazimine to minimize the effects of ENL.
Acknowledgements
We thank the staff of the Cebu Skin Clinic, including the former Chief of the Clinic, Dr R Cellona, for their hard work in managing the two relapse studies from which the current data were taken. The relapse studies were sponsored by WHO, the Sasakawa Memorial Health Foundation, the Pacific Leprosy Trust and American Leprosy Missions.
References
1
Saunderson P, Gebre S, Byass P. ENL reactions in the multibacillary cases of the AMFES cohort in central Ethiopia: incidence and risk factors. Lepr Rev , 2000; 71 : 318–324. 2 McAdam K, Anders R, Smith S et al. Association of amyloidosis with erythema nodosum leprosum reactions and recurrent neutrophil leucocytosis. Lancet , 1975; ii : 572–575. 3 Cochrane R. Leprosy in theory and practice . J. Wright & Sons Ltd, Bristol, 1959. 4 Van Veen NHJ, Lockwood DNJ, van Brakel WH et al. Interventions for erythema nodosum leprosum. A Cochrane review. Lepr Rev , 2009; 80 : 355–372. 5 Walker SL, Waters MFR, Lockwood DNJ. The role of thalidomide in the management of erythema nodosum leprosum. Lepr Rev , 2007; 78 : 197–215. 6 Lockwood D. The management of erythema nodosum leprosum: current and future options. Lepr Rev , 1996; 67 :253–259. 7 Kahawita IP, Lockwood DNJ. Towards understanding the pathology of erythema nodosum leprosum. Trans R Soc Trop Med Hyg , 2008; 102 : 329–337. 8 Cellona RV, Fajardo TTJ, Kim DI et al. Joint chemotherapy trials in lepromatous leprosy conducted in Thailand, the Philippines, and Korea. Int J Lepr Other Mycobact Dis , 1990; 58 : 1 –11. 9 Shannon E, Noveck R, Sandoval F, Kamath B. Thalidomide suppressed IL-1beta while enhancing TNF-alpha and IL-10, when cells in whole blood were stimulated with lipopolysaccharide. Immunopharmacol Immunotoxicol ,2008; 30 : 447–457. 10 Kroger A, Pannikar V, Htoon MT et al. International open trial of uniform multi-drug therapy regimen for 6 months for all types of leprosy patients: rationale, design and preliminary results. Trop Med Int Health , 2008; 13 :594– 602. 11 Deps PD, Nasser S, Guerra P et al. Adverse effects from multi-drug therapy in leprosy: a Brazilian study. Lepr Rev , 2007; 78 : 216–222.
M. Balagon et al. 220 12 Matthew BS, Pulimood AB, Prasanna CG et al. Clofazimine induced enteropathy – a case highlighting the importance of drug induced disease in differential diagnosis. Trop Gastroenterol , 2006; 27 : 87 –88. 13 Jadhav MV, Sathe AG, Deore SS et al. Tissue concentration, systemic distribution and toxicity of clofazimine – an autopsy study. Indian J Pathol Microbiol , 2004; 47 : 281–283. 14 Browne SG, Harman DJ, Waudby H, McDougall AC. Clofazimine (Lamprene, B663) in the treatment of lepromatous leprosy in the United Kingdom. Int J Lepr Other Mycobact Dis , 1981; 49 : 167–176. 15 Burte NP, Chandorkar AG, Muley MP et al. Clofazimine in lepra (ENL) reaction, one year clinical trial. Lepr India , 1983; 55 : 265–277. 16 Schreuder PAM, Naafs B. Chronic recurrent ENL, steroid dependent: long-term treatment with high dose clofazimine. Lepr Rev , 2003; 74 : 386–389. 17 WHO Expert Committee on Leprosy. Seventh Report. Technical Report Series 1998; 874 .18 Cellona RV, Balagon MFV, dela Cruz EC et al. Long-term efficacy of 2 year WHO multiple drug therapy (MDT) in multibacillary (MB) leprosy patients. Int J Lepr Other Mycobact Dis , 2003; 71 : 308–319.
Does clofazimine prevent ENL in leprosy? 221 |
3413 | https://www.khanacademy.org/math/cc-fifth-grade-math/multi-digit-multiplication-and-division | Khan Academy
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3414 | https://www.ck12.org/flexi/precalculus/sum-and-difference-identities/prove-that-sin(ab)-sina.cosb--cosa.sinb./ | Prove that sin(a+b)= sina.cosb + cosa.sinb.
@$$\begin{align}\sin(a + b) & = \cos \left [\frac{\pi}{2} - (a+b) \right ] && \text{Set}\ \theta = a + b\ & = \cos \left [\left (\frac{\pi}{2} - a \right ) - b \right ]&& \text{Distribute the negative} \ & = \cos \left (\frac{\pi}{2} - a \right ) \cos b + \sin \left (\frac{\pi}{2} - a \right ) \sin b && \text{Difference Formula for cosines} \ & = \sin a \cos b + \cos a \sin b && \text{Co-function Identities}\end{align}@$$ In conclusion, @$\begin{align}\sin(a + b) = \sin a \cos b + \cos a \sin b\end{align}@$, which is the sum formula for sine.
@$$\begin{align}\sin(a + b) & = \cos \left [\frac{\pi}{2} - (a+b) \right ] && \text{Set}\ \theta = a + b\
& = \cos \left [\left (\frac{\pi}{2} - a \right ) - b \right ]&& \text{Distribute the negative} \
& = \cos \left (\frac{\pi}{2} - a \right ) \cos b + \sin \left (\frac{\pi}{2} - a \right ) \sin b && \text{Difference Formula for cosines} \
& = \sin a \cos b + \cos a \sin b && \text{Co-function Identities}\end{align}@$$
In conclusion, @$\begin{align}\sin(a + b) = \sin a \cos b + \cos a \sin b\end{align}@$, which is the sum formula for sine.
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3415 | https://www.reddit.com/r/askmath/comments/1g5zn6f/do_sequences_start_with_the_0th_or_1st_term/ | Do sequences start with the 0th or 1st term? : r/askmath
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Do sequences start with the 0th or 1st term?
Discrete Math
I already know the answer is “It doesn’t matter”, but I was wondering if one is more accepted than the other. In english, you start with 1st and in computer science you start with 0th. I’m inclined to think it’s more traditional to start with 0 since 0 is the first (or 0th) number in set theory, but wanted some opinions.
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3416 | https://sites.math.rutgers.edu/~tb822/Math361-HW10-Sols.pdf | MATH 361 Homework 10-Solution (due Dec 9) Dec 2, 2022 Problem 1. Suppose that ⟨𝐴, <𝐴⟩is a well-ordered set. Prove that if 𝑓: 𝐴→𝐴is order-preserving then 𝑓= 𝑖𝑑𝐴.
Solution. Otherwise, let 𝐷= {𝑥∈𝐴| 𝑓(𝑥) ≠𝑥} ≠∅. Let 𝑥∗= min(𝐷).
Then for every 𝑥<𝐴𝑥∗, 𝑓(𝑥) = 𝑥by minimality of 𝑥∗. Hence 𝑓(𝑥∗) >𝐴𝑥∗ (otherwise, 𝑓is not one-to-one). Since 𝑓is an isomorphism, there is 𝑦∈𝐴 such that 𝑓(𝑦) = 𝑥∗. Since 𝑓(𝑦) <𝐴𝑓(𝑥∗), then 𝑦<𝐴𝑥∗, but then 𝑥∗= 𝑓(𝑦) = 𝑦, contradiciton.
Problem 2. Prove that if 𝐴is countable the 𝐴can be well-ordered.
Instruction: Split into two cases- first prove that every linear strong order on a finite set is a well order. If 𝐴is infinitely countable, then by taking any bijection 𝑓: N →𝐴, we can define <𝐴on 𝐴by 𝑎<𝐴𝑏if and only if 𝑓−1(𝑎) < 𝑓−1(𝑏). Prove that ⟨𝐴, <𝐴⟩≃⟨N, <⟩and deduce that ⟨𝐴, <𝐴⟩is a well ordered set.
Solution. Take any injection 𝑓: 𝐴→N, we can define <𝐴on 𝐴by 𝑎<𝐴𝑏 if and only if 𝑓(𝑎) < 𝑓(𝑏). Then by definition, 𝑓is order-preserving and injective. Note that 𝑎<𝐴𝑏<𝐴𝑐then 𝑓(𝑎) < 𝑓(𝑏) < 𝑓(𝑐) and therefore 𝑓(𝑎) < 𝑓(𝑐) so 𝑎<𝐴𝑐(namely <𝐴is transitive). If 𝑎<𝐴𝑏then 𝑓(𝑎) < 𝑓(𝑏) so 𝑓(𝑏) ≮𝑓(𝑎) and therefore 𝑏≮𝐴𝑎. It is linear since for every 𝑎, 𝑏∈𝐴, 𝑓(𝑎), 𝑓(𝑏) are comparable hence 𝑎, 𝑏are <𝐴-comparable. So far we proved that <𝐴is a strong linear order on 𝐴. Let us prove that it is a well order.
Let 𝑋⊆𝐴, be a non-empty set. Then 𝑓′′𝑋⊆N is non-empty and therefore there is 𝑛∗= min(𝑓′′𝑋).
Let 𝑥∈𝑋be such that 𝑓(𝑥) = 𝑛∗, then for every 𝑦∈𝑋, 𝑓(𝑦) ≥𝑛∗= 𝑓(𝑥) and therefore 𝑦≥𝐴𝑥.
It follows that 𝑥= min<𝐴(𝑋).
1 MATH 361 Homework 10-Solution (due Dec 9) Dec 2, 2022 Problem 3. Prove that if ⟨𝐴, <𝐴⟩is a well-ordered set and 𝑋⊆𝐴is an initial segment (i.e. ∀𝑥∈𝑋∀𝑎∈𝐴, 𝑎<𝐴𝑥⇒𝑎∈𝑋) then either 𝑋= 𝐴or ∃𝑎∈𝐴such that 𝑋= 𝐴<𝐴[𝑎].
Hint: If 𝑋≠𝐴let 𝑎= min<𝐴(𝐴\ 𝑋) (why does it exists?), prove that 𝑋= 𝐴<𝐴[𝑎].
Solution. If 𝑋= 𝐴we are done. Otherwise, 𝑋⊊𝐴, so 𝐴\ 𝑋≠∅. Let 𝑎= min<𝐴(𝐴\ 𝑋) and let us prove that 𝐴<𝐴[𝑎] = 𝑋by double inclusion.
If 𝑏<𝐴𝑎, then 𝑏∈𝐴and 𝑏∉𝐴\ 𝑋(otherwise this would contradict the minimality of 𝑎) and therefore 𝑏∈𝑋. If 𝑥∈𝑋, then 𝑥, 𝑎are <𝐴-comperable.
If 𝑎= 𝑥, then 𝑎∈𝑋, contradiction. If 𝑎<𝐴𝑥, then 𝑎∈𝑋since 𝑋is an initial segment. Hence 𝑥<𝐴𝑎, namely 𝑥∈𝐴<𝐴[𝑎].
Problem 4. Prove that the axiom of foundation implies that there is no 𝑥 such that 𝑥∈𝑥.
Solution. Suppose otherwise, and let 𝑥∈𝑥. Define {𝑥}. By the axiom of foundation there is 𝑦∈{𝑥} such that 𝑦∩{𝑥} = ∅. But then 𝑦= 𝑥and 𝑥∈𝑥∩{𝑥}, contradiction.
Problem 5. Prove that if 𝐴is a set of ordinals then Ð 𝐴is an ordinal Solution. First let us note that Ð 𝐴is a transitive set. If 𝑥∈𝑦∈Ð 𝐴, then there is 𝛼∈𝐴such that 𝑦∈𝛼. Since 𝛼is transitive it follows that 𝑥∈𝛼 and therefore 𝑥∈Ð 𝐴. To see that ∈well-orders Ð 𝐴, let 𝑥∈𝑦∈𝑧all in Ð 𝐴, there there are 𝛼, 𝛽, 𝛾∈𝐴such that 𝑎∈𝛼, 𝑦∈𝛽, 𝑧∈𝛾. Since every two ordinals are comparable, WLOG 𝛼, 𝛽⊆𝛾and therefore 𝑥, 𝑦, 𝑧∈𝛾.
Since ∈well orders 𝛾, 𝑥∈𝑧. So ∈is transitive on Ð 𝐴. It is then strongly 2 MATH 361 Homework 10-Solution (due Dec 9) Dec 2, 2022 anti-symmetric, since if 𝑥∈𝑦and 𝑦∈𝑥, there are 𝛼, 𝛽∈𝐴such that 𝑎∈𝛼 and 𝑦∈𝛽. WLOG 𝛼≤𝛽, 𝑥, 𝑦∈𝛽but ∈well orders 𝛽, contradiction. A similar argument shows that ∈is linear on Ð 𝐴. Let 𝑋⊆Ð 𝐴such that 𝑋≠∅. Take any 𝛼∈𝐴such that 𝑋∩𝛼≠∅(there exists such 𝛼since 𝑋⊆Ð 𝐴is nonempty) The 𝑋∩𝛼is a non-empty subset of 𝛼and since ∈ well-orders 𝛼, there is 𝑥= min∈(𝑋∩𝛼). We claim that 𝑥= min∈(𝑋). Let 𝑦∈𝑋, then 𝑦∈∪𝐴, then there is 𝛽such that 𝑦∈𝛽. If 𝛽≤𝛼, then 𝑦∈𝛼 and therefore 𝑦∈𝑋∩𝛼in which case 𝑥≤𝑦. Otherwise, 𝛼< 𝛽and then 𝑥, 𝛼, 𝑦∈𝛽so 𝑥, 𝑦are ∈-comparable if 𝑦∈𝑥then 𝑦∈𝛼which then imply that 𝑦∈𝑋∩𝛼contradicting the minimality of 𝑥. Otherwise, 𝑥≤𝑦as wanted.
and moreover Ð 𝐴= sup(𝐴) i.e.: 1. Ð 𝐴is an upper bound for 𝐴, namely, for every 𝛼∈𝐴, 𝛼≤Ð(𝐴).
Solution. If 𝛼∈𝐴, then 𝛼⊆Ð 𝐴and therefore by the lemma we saw in class 𝛼≤Ð 𝐴.
2. If 𝛽∈𝑂𝑛is an upper bound for 𝐴then 𝛽≥Ð 𝐴.
Solution. If 𝛽is an upper bound for 𝐴, then for every 𝛼∈𝐴, 𝛼≤𝛽, namely 𝛼⊆𝛽. It follows that Ð 𝐴is a union of subseteq pf 𝛽and therefore itself a subset of 𝛽. we conclude that Ð 𝐴≤𝛽.
Additional problems Problem 6. Suppose that ⟨𝐴, <𝐴⟩, ⟨𝐵, <𝐵⟩are well ordered sets such that 𝐴∩𝐵= ∅. Define <+ on 𝐴⊎𝐵by 𝑥<+ 𝑦if: 3 MATH 361 Homework 10-Solution (due Dec 9) Dec 2, 2022 • 𝑥, 𝑦∈𝐴and 𝑥<𝐴𝑦. or • 𝑥, 𝑦∈𝐵and 𝑥<𝐵𝑦. or • 𝑥∈𝐴and 𝑦∈𝐵.
Prove that <+ is a well ordering of 𝐴⊎𝐵.
Problem 7. Suppose that ⟨𝐴, <𝐴⟩, ⟨𝐵, <𝐵⟩are well orders. Define the lexi-cographic order on 𝐴× 𝐵as follows: ⟨𝑎, 𝑏⟩<𝐿𝑒𝑥⟨𝑎′, 𝑏′⟩iff 𝑎<𝐴𝑎′ ∨(𝑎= 𝑎′ ∧𝑏<𝐵𝑏′) Prove that ⟨𝐴× 𝐵, <𝐿𝑒𝑥⟩is a well ordering.
Problem 8. Prove that if 𝛼is an ordinal then 𝛼∪{𝛼} is an prdinal.
Problem 9. Prove that if 𝐶≠∅is a set of ordinals then Ñ 𝐶is an ordinal and Ñ 𝐶= min∈(𝐶).
Problem 10. Prove that if 𝑋is transitive than 𝑃(𝑋) is transitive.
4 |
3417 | https://mathequalslove.net/combining-like-terms-cut-and-paste-activity/ | Combining Like Terms Cut and Paste Activity | Math = Love
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Home » Algebra » Expressions » Combining Like Terms » Combining Like Terms Cut and Paste Activity
Combining Like Terms | Graphic Organizers | INBs | Miscellaneous Activities | Most Popular Posts
Combining Like Terms Cut and Paste Activity
September 10, 2016 July 29, 2025
Combining Like Terms Activity (Version 3.0)
I modified my combining like terms strip activity that I have used for the past two years. This year, I added a box for students to write their simplified answer in. I think it really helped them organize their work better!
Close-ups of each problem:
Looking for a fun combining like terms puzzle? I suggest checking out my combining like terms maze!
Combining Like Terms Activity (Version 2.0)
Our third skill this year in Algebra 1 is to be able to rewrite expressions, equations, and inequalities by applying the distributive property and combining like terms. I was able to take how I taught these concepts last year and improve them (hopefully) for the better.
I have a few students taking Algebra 1 with me for a second time due to failing last year, and they commented that I taught this way better this year than last year.
I was multi-tasking (aka eating lunch) while I was doing this, and I ended up making a few arithmetic mistakes. I think I fixed them all, though.
I gave my students the three expressions on a small piece of paper. They cut them into individual strips of expressions.
Then, one at at a time, we cut the strips into their individual terms. After making a pile of terms, I asked students to apply what we had just learned about the definition of like terms to put their terms into piles. I would ask students how many groups of terms they ended up with.
Different students grouped them differently, and this led to great discussions amongst my students. They would back up their points of view with the definition which was awesome!
Here’s a student’s work on putting the terms for the first expression into groups:
Once we were all happy with how the terms were grouped, we glued them in.
I overheard a student say “This is fun!” as she was deciding how to put the terms into groups. It was great formative assessment for me as a teacher to see how the lesson was going. As a class, we combined the coefficients or constants to form our final, simplified answer.
On the next day, we continued combining like terms in the context of applying the distributive property. I wanted to do this in a way that tied directly into the “grouping” of like terms we had been doing on the day before.
I’m pretty proud of the way I came up with to do this! We followed up this combining like terms activity with a distributive property foldable.
Combining Like Terms Activity (Version 1.0)
This year, I decided to really emphasize combining like terms with my Algebra 1 students. In retrospect, I should have done the same thing in Algebra 2 because they were still struggling with what they can and cannot combine.
I thought this would be a one day lesson, but it ended up taking my students two days to work through it. There were lots of great conversations happening, so I think it was definitely worth it!
I gave students a quarter sheet of paper that had a note box and three polynomial expressions.
We began by taking some notes over what like terms are. I really wanted to emphasize to my students that xy and yx are like terms, so I really pushed the “order doesn’t matter” this year.
I had them copy down the first polynomial strip in their interactive notebooks.
Then, the students had to cut the strip into terms. This led to a great discussion of what a term is. Students had to make sure they cut the strip so that each term contained the sign in front of it.
Students were super careful to make sure they were cutting the strips correctly which is exactly what I was hoping for.
Next, I instructed students to group the terms into groups that were like terms. This is where the best conversations happened. After students sorted their terms, I asked them how many groups they had.
When students realized they had sorted into a different number of groups, they started justifying their groupings to their classmates. It was just awesome to see them pointing each other back to the definition of like terms.
Finally, we decided on how the terms should be grouped.
Next, I instructed students to glue in their groupings. I intentionally did not tell them how to group them in. Luckily, the students glued them in different orders which let us discuss the fact the order of the terms doesn’t matter.
Finally, we circled the groups and combined the coefficients. Since the students glued the groups in in different orders, their terms ended up in different orders. I emphasized that this was okay as long as the sign in front of 21x was negative, the sign in front of 2x^2 was negative, and the sign in front of 4 was positive.
Next, they proceeded to do the next two problems in their groups.
The zero coefficients and invisible one coefficients freaked some of my students out, but they persevered.
Last problem:
We finished the class period off with two additional practice problems. The kids were quite miffed that I did not give them strips to cut because how else would they figure out what the terms were. To remedy this, many students drew “cut lines” between the terms to separate them.
I like this activity got students actually separating terms, grouping them, and combining them. I hope I made an abstract concept a little more concrete and understandable for my students.
Files for Combining Like Terms Activity
Version 3.0
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Version 2.0
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Version 1.0
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Post Tags: #high#middle
Sarah Carter
Sarah Carter teaches high school math in her hometown of Coweta, Oklahoma. She currently teaches AP Precalculus, AP Calculus AB, and Statistics. She is passionate about sharing creative and hands-on teaching ideas with math teachers around the world through her blog, Math = Love.
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16 Comments
Jen Wsays: September 11, 2016 at 4:10 am Thank you so much for blogging about this! I saw your Twitter post and had my fingers (and toes!) crossed that you'd share your files. As always, LOVE the awesome things you come up with and LOVE that you share!
Meg Craigsays: September 11, 2016 at 3:49 pm Idea…what if you had a student pick a value for x and plug it into the first expression and the last? and HOLY COW they're equal! then have a student pick another value and HOLY COW they're equal! I'm not sure that I ever really focused on the fact that when we simplify, these things are still the equivalent for any value of x. Like I would focus on the "how" but not the "what we're actually accomplishing"
Then when you get to equations, start with an equation that has a weird answer like 147. Again ask students to pick values of x (and hope they don't pick 147). Oh, wait, these sides aren't equal! It ONLY works for this one special value of x (147). Or maybe have the answer be a smaller number that they might try (3?) so it would work if they chose that, but not for others?
I'm kind of thinking out loud here but if I had to teach Alg I again I would really want to focus on the fact that expressions are true for any x; equations are true for a limited number of x, and functions are true for a limited-but-infinite (whoa…mind blown!) set of x and y values that we can display as a graph.
Also one of teachers at school has the Ss draw a vertical "river" below the equation sign and you only change if you cross the river. I think the vertical line visual is very useful when they're just starting out, otherwise I get students try to simplify 2 + 3x + 5 by subtracting the 2 from the 2 and the 5!
3. Unknownsays: September 13, 2016 at 3:37 pm Love this! I'm using the distribute worksheet today. Small mistake on the last row? Should it be -2×2 + 7x – 2?
4. Unknownsays: September 15, 2016 at 2:42 am Hello Sarah! I'm a college student studying to be a Middle School or High School Math Teacher. One of my Math Education teachers encouraged us to start following Math Blogs to gain knowledge and insight for our future classrooms. I stumbled upon your blog for an assignment last semester, so I've decided to return to it to better prepare myself as a future educator.
I really like how the first day of the lesson really gave students the power to group the terms however they would like to do so. Not only that, but you have examples for one, two, and three-variable expressions. Having students construct their own expression makes it seem fun and easier when it comes time to distribute and combine like terms.
The foldable is a nice way to tie things together, the grouping and the distributive property. Drawing the arrows from the number that gets distributed is a nice visual representation, as well as having each column of the table be one step in the simplification process. It's very easy to follow and gives students structure to build off. My one comment/suggestion would be to maybe incorporate a variable somewhere on the outside of the foldable. A student who is used to A,B, and C being numbers may get confused when they see the variable and not distribute it otherwise.
Overall, I think this is a fun way to combine the distributive property with like terms while still adding valuable content to the interactive math notebook.
5. Anonymoussays: October 17, 2016 at 4:05 pm Thank you so much for this. I think it will be a great way for them to organize it and understand it as well!
6. Unknownsays: November 23, 2016 at 5:05 am Thank you! I really appreciate you sharing your work with us!
1. Sarah Carter (@mathequalslove)says: November 25, 2016 at 10:25 pm You're welcome, Jennifer!
Anonymoussays: December 7, 2016 at 9:59 pm Thank you for sharing! I'll be using this for an extra intervention period/activity with my 8th graders 🙂
Unknownsays: October 9, 2017 at 2:46 am Do you have the files available to download? Not sure if I just missed where they were or is this just pictures of what you did with explanation.
Amy Hurleysays: October 30, 2017 at 1:54 am Hi, Sarah, I wanted to make sure you were aware that the last problem should read -2x^2 + 7x – 2 just in case you didn't catch that little mistake. Thank you for this foldable; I'm using it tomorrow! 🙂
Kate Rsays: September 5, 2018 at 7:46 am Hi, Sarah. Do you still have the documents available to share? Thanks!
P. Michele D.says: September 17, 2018 at 10:02 pm I've done something similar with my students, but this is exactly what I need to help the kids understand. Thanks so much for making this available!!
Unknownsays: September 28, 2018 at 7:37 pm This is great! I used it for my Sped kiddo and he understood it perfectly. Thanks for sharing!
Unknownsays: March 13, 2019 at 9:20 pm How do you teach students to figure out what the coefficient will be once you have sorted into like terms? For example -2x, -4x, -15x. Do your kids recognize these as negative numbers once they are alone? Do you have them add all the numbers as negatives (i.e., -2+-4+-15) or just use the signs that are there (i.e., -2-4-15)?
Unknownsays: November 15, 2019 at 1:51 am I tried the link but it doesn't work. Did it get removed?
Unknownsays: September 27, 2020 at 10:05 pm I made a digital version based on this post for my 8th grade CCSM class. I did the interactive notebook for like terms and a practice activity as google slides.
The link will create a copy in your google drive.
Comments are closed.
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3418 | https://www.cureus.com/articles/240784-a-distinct-instance-of-palindromic-rheumatism-disguised-as-polymyalgia-rheumatica.pdf | Received 03/31/2024 Review began 04/06/2024 Review ended 06/01/2024 Published 06/04/2024 © Copyright 2024 Farooq et al. This is an open access article distributed under the terms of the Creative Commons Attribution License CC-BY 4.0., which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
DOI: 10.7759/cureus.61644 A Distinct Instance of Palindromic Rheumatism Disguised as Polymyalgia Rheumatica Omer Farooq , Mashal Awais , Tijin Mathew , Nabeel A. Siddiqui , Justin G. Hovey 1. Internal Medicine, Southeast Health Medical Center, Dothan, USA 2. Internal Medicine: Pediatrics, Alabama College of Osteopathic Medicine (ACOM), Dothan, USA Corresponding author: Omer Farooq, dromerayaz@gmail.com Abstract In this case report, we highlight a rare case of palindromic rheumatism (PR) presenting as polymyalgia rheumatica (PMR). Many challenges and complexities are associated with diagnosing and treating PR.
Literature reviews showed only a few case reports of this unique presentation. PR has a distinct presentation that often goes unnoticed and is misinterpreted by medical professionals. A more thorough clinical approach is required to identify and treat this condition. We hope sharing such uncommon cases will help the medical community better understand PR and develop improved diagnostic and therapeutic options. This case also demonstrates the need for further research to better understand the pathogenesis of this uncommon condition.
Categories: Internal Medicine, Rheumatology, Orthopedics Keywords: joint pain and stiffness, oral corticosteroids, seronegative rheumatoid arthritis, palindromic rheumatism, polymyalgia rheumatica Introduction In 1928, Philip S Hench, MD of the Mayo Clinic in Rochester, USA, identified the first case of palindromic rheumatism (PR). He encountered a 21-year-old female with recurring episodes of pain and edema in multiple joints, lasting 12 to 36 hours, with only a single joint affected at a time. Hench and Edward Rosenberg saw numerous comparable clinical appearances and published them in 1944. They noted that these attacks were unexpected, with periarticular/para-articular inflammation present. Hench and Rosenberg distinguished PR from rheumatoid arthritis (RA) based on clinical/remitting trends . The clinical and imaging characteristics of PR indicate significant variations from RA and underlying molecular differences between the two disorders. PR's actual nature is uncertain; it may be regarded as a separate illness, an early stage of RA, or merely pre-RA syndrome. The genetic predisposition of PR is the same as RA . PR has the potential to develop into a chronic rheumatic disease.
The observation that most patients with PR have RA-related autoantibodies and that many eventually develop RA has led to PR often being viewed as a relapsing-remitting variant of RA . PMR is an inflammatory rheumatic condition characterized clinically by aches and morning stiffness.
Symptoms mainly involve the shoulders, hip girdle, and neck. It may be associated with giant cell arteritis . In contrast, PR presents with flares of joint pain and periarticular stiffness. Clinical symptoms of PMR typically include fatigue, weakness, mild fever, shoulder and hip girdle pain, and stiffness. It's crucial to differentiate between various illnesses like PMR and seronegative rheumatism, which can present with intermittent patterns of arthritis. This case report delves into a rare instance of palindromic rheumatism (PR) and outlines its diagnostic workup. Steroids, although infrequently employed in PR management, play a pivotal role in this particular case. Furthermore, this report elucidates the judicious use of steroids alongside nonsteroidal anti-inflammatory drugs (NSAIDs) and disease-modifying agents as effective measures in controlling PR flares.
Case Presentation A 69-year-old male patient presented at Southeast Health Clinic, Dothan, USA, in December 2022 with complaints of bilateral hip pain, shoulder pain, and fatigue. Physical exam findings were positive for tender spots on the shoulder girdle. He was initially diagnosed with PMR, and the patient commenced treatment with prednisolone at a dosage of 1 mg/kg administered in divided doses twice daily for one month, which was subsequently tapered off. The patient underwent close monitoring every three months, and within 13 months, there was an evolution of his symptoms, prompting concerns of RA. These symptoms included bilateral joint pain involving his wrists, knees, proximal interphalangeal joints, and simultaneous periarticular muscle stiffness. The physical exam findings were positive for motion restriction of bilateral knees and wrists due to pain, redness, and swelling.
His clinical presentation and symptoms strongly suggested seronegative RA. Consequently, he commenced treatment with hydroxychloroquine at a dosage of 200 mg orally twice daily for four months. Subsequently, he was initiated on methotrexate at a dosage of 15 mg orally once weekly for three months. He could not 1 1 1 1 2 Open Access Case Report Published via Alabama College of Osteopathic Medicine Research How to cite this article Farooq O, Awais M, Mathew T, et al. (June 04, 2024) A Distinct Instance of Palindromic Rheumatism Disguised as Polymyalgia Rheumatica.
Cureus 16(6): e61644. DOI 10.7759/cureus.61644 tolerate either medication due to side effects, including generalized rash and hair loss. He experienced recurrent symptoms of arthritis and muscular stiffness, with periods of no symptoms in between. Joint X-rays were done to exclude seronegative RA, and they were negative for findings like narrowing of joint space or erosion of joints, as shown in Figure 1.
FIGURE 1: Posteroanterior view X-ray of the right knee Arrow indicates no concerning findings for seronegative rheumatoid arthritis (RA) such as narrowing of joint space and erosion of joints.
PR was suspected based on clinical symptoms in the form of flares and studies such as joint X-rays that ruled out seronegative RA. He was referred to a rheumatologist for further investigations. Despite an extensive autoimmune workup, including testing for rheumatic factor, antinuclear antibody, anti-double-stranded DNA antibody, anticitrullinated peptide antibody, anti-Sjögren’s syndrome-related antigen A, anti-Sjögren’s syndrome-related antigen B, and antineutrophil cytoplasmic antibody, all results were negative.
Additionally, skeletal imaging of the affected joints showed no evidence of destructive changes. Notably, the patient exhibited a positive response to corticosteroids during acute episodes. Based on these findings, the diagnosis of PR was confirmed. Currently, the patient encounters approximately 10 flares annually, yet his symptoms are effectively managed with prednisolone at a dosage of 1 mg/kg administered in divided doses twice daily for five days, as required. The patient maintains regular follow-up appointments with the rheumatology clinic, scheduling visits every eight to 10 weeks. Despite his condition, he continues to lead an active lifestyle.
Discussion PR is an articular and periarticular inflammation lasting from a few hours to several days and resolving spontaneously. It can lead to an increase in inflammatory markers during relapsing episodes. The most commonly involved joints are the fingers, wrists, and knees, but other joints can also be involved .
Diagnosing PR poses a formidable clinical challenge due to the transient nature of inflammatory signs, often Published via Alabama College of Osteopathic Medicine Research 2024 Farooq et al. Cureus 16(6): e61644. DOI 10.7759/cureus.61644 2 of 4 subsiding during the patient's clinic visit. Diagnostic criteria require the following: 1) Recurrent attacks of sudden-onset mono or polyarthritis or of peri-articular tissue inflammation, lasting from a few hours to weeks; 2) Verification by a physician of at least one attack; 3) Subsequent attacks in at least three different joints; 4) Exclusion of other forms of arthritis .
It is noteworthy that approximately one-third of individuals initially presenting with PR ultimately progress to RA . Therefore, close surveillance of patients with PR is warranted for progression to RA . Moreover, no laboratory or clinical characteristics enable early distinction between PMR and RA with PMR-like onset.
Patients with PR typically lack the radiological findings of RA, like joint erosion and joint narrowing.
The treatment of this condition presents a unique challenge, primarily relying on clinical judgment and expertise. Non-steroidal anti-inflammatory drugs (NSAIDs) can serve as a treatment option for managing acute episodes of this condition. Disease-modifying anti-rheumatic drugs can be used to reduce symptoms and flare-ups. The response of PR patients to hydroxychloroquine provides evidence for the potential link between PR and its role as an initial manifestation of RA . In a study of 113 PR patients, hydroxychloroquine lowered the risk of chronic rheumatic diseases by 20%, indicating its potential as a preventive treatment for severe rheumatic conditions .
As shown in our case, PMR symptoms are distinct from PR symptoms due to the disease's flare-like nature and various joint pains, which set them apart from the latter's typical presentation of fatigue and pain in the shoulder and hip girdles. PMR usually does not impact the knee joint. PR can be distinguished from seronegative RA by obtaining a thorough history, performing baseline testing, such as joint X-rays, and looking for abnormalities such as narrowing of the joint space and joint erosion. NSAIDs and corticosteroids can be used to treat PR symptoms, much as in our case study. It is not required for the patient to take steroids continuously; instead, steroids can be administered to reduce their symptoms during flare-ups.
Table 1 shows the differences between PR, PMR, and seronegative rheumatic disease.
PR PMR Seronegative Rheumatic Disease Recurrent, self-limiting joint inflammation, often affecting one or a few joints.
Pain and stiffness, typically in shoulders, neck, and hips.
Heterogeneous group; absence of specific autoantibodies (RF, anti-CCP).
Episodes last hours to days and are reversible with no permanent joint damage between attacks.
Persistent symptoms lasting weeks to months.
Variable, depending on the specific disease within the group.
It can occur at any age.
It primarily affects older adults.
It can occur across various age groups.
Not always consistently elevated.
Elevated inflammatory markers (ESR, CRP).
Variable: some conditions may have elevated markers.
Monoarticular with Hand predominance, especially MCP joints Shoulders, neck, hips; may involve other areas.
Variable; may involve peripheral joints, spine, or entheses.
One or a few joints during episodes.
Shoulders, neck, hips; may involve other areas.
Variable; may involve peripheral joints, spine, or entheses.
Generally seronegative (lack of RF or anti-CCP).
Seronegative for RF and anti-CCP.
The absence of specific autoantibodies is a defining feature.
TABLE 1: Differences between PR, PMR, and seronegative rheumatic disease PR = Palindromic rheumatism; PMR = Polymyalgia rheumatica; RF = Rheumatoid factor; Anti-CCP = Anti-cyclic citrullinated peptide; ESR = Erythrocyte sedimentation rate; CRP = C-reactive protein; MCP = Metacarpophalangeal Conclusions This case report emphasizes the significance of examining PR as a possible diagnosis in individuals with symptoms similar to PMR. It demonstrates the difficulties in distinguishing between these conditions due to overlapping clinical features and emphasizes the importance of a thorough clinical approach and differential diagnosis. PR patients usually do not exhibit radiological signs of seronegative RA or RA such as joint space narrowing and erosion of the joints. Strong indicators of palindromic rheumatism include relapsing-remitting joint pain, positive response to steroids, and lack of radiological findings for RA. The unique presentation of PR frequently eludes detection and is subject to misinterpretation by healthcare practitioners, necessitating a more diligent clinical approach to recognize and appropriately manage this condition. In addition to disease-modifying agents, steroids can also serve as a treatment option for managing flare-ups of PR, similar to other NSAIDs.
We hope this case improves the medical community's knowledge of PR and helps with the early detection of afflicted persons. More studies are necessary to clarify the underlying pathophysiology of PR and improve diagnostic and treatment strategies. In summary, this case highlights the significance of ongoing research endeavors, interdisciplinary cooperation, and clinical vigilance in augmenting our comprehension and handling of PR. Published via Alabama College of Osteopathic Medicine Research 2024 Farooq et al. Cureus 16(6): e61644. DOI 10.7759/cureus.61644 3 of 4 Additional Information Author Contributions All authors have reviewed the final version to be published and agreed to be accountable for all aspects of the work.
Concept and design: Nabeel A. Siddiqui, Omer Farooq, Justin G. Hovey, Tijin Mathew, Mashal Awais Acquisition, analysis, or interpretation of data: Nabeel A. Siddiqui, Omer Farooq, Justin G. Hovey, Tijin Mathew, Mashal Awais Drafting of the manuscript: Nabeel A. Siddiqui, Omer Farooq, Justin G. Hovey, Tijin Mathew, Mashal Awais Critical review of the manuscript for important intellectual content: Nabeel A. Siddiqui, Omer Farooq, Justin G. Hovey, Tijin Mathew, Mashal Awais Supervision: Nabeel A. Siddiqui, Omer Farooq, Justin G. Hovey, Tijin Mathew, Mashal Awais Disclosures Human subjects: Consent was obtained or waived by all participants in this study. Conflicts of interest: In compliance with the ICMJE uniform disclosure form, all authors declare the following: Payment/services info: All authors have declared that no financial support was received from any organization for the submitted work. Financial relationships: All authors have declared that they have no financial relationships at present or within the previous three years with any organizations that might have an interest in the submitted work. Other relationships: All authors have declared that there are no other relationships or activities that could appear to have influenced the submitted work.
References 1. Hench P, Rosenberg E: Palindromic rheumatism. “New,” oft recurring disease of joints (arthritis, periarthritis, para-arthritis) apparently producing no articular residues— report of thirty-four cases; its relation to “angioneural arthrosis,” “allergic rheumatism” and rheumatoid arthritis. JAMA Intern Med.
1944, 72:293-321. 10.1001/archinte.1944.00210160025004 2. Sanmartí R, Haro I, Cañete JD: Palindromic rheumatism: a unique and enigmatic entity with a complex relationship with rheumatoid arthritis. Expert Rev Clin Immunol. 2021, 17:375-84.
10.1080/1744666X.2021.1899811 3. Acharya S, Musa R: Polymyalgia Rheumatica. StatPearls [Internet], Treasure Island (FL); 2024.
4. Myong-hak R, Po-hum R, Song-phil P, Yong-jin R, Paek-hwa K, Ok-i J: Early and advanced stages in palindromic rheumatism patients: test characteristics of three classification criteria and discrimination potential. Egypt Rheumatol. 2022, 44:63-7. 10.1016/j.ejr.2021.08.007 5. Mankia K, Emery P: Palindromic rheumatism as part of the rheumatoid arthritis continuum . Nat Rev Rheumatol. 2019, 15:687-95. 10.1038/s41584-019-0308-5 6. Aletaha D, Neogi T, Silman AJ, et al.: 2010 rheumatoid arthritis classification criteria: an American College of Rheumatology/European League Against Rheumatism collaborative initiative. Arthritis Rheum. 2010, 62:2569-81. 10.1002/art.27584 7. Gonzalez-Lopez L, Gamez-Nava JI, Jhangri G, Russell AS, Suarez-Almazor ME: Decreased progression to rheumatoid arthritis or other connective tissue diseases in patients with palindromic rheumatism treated with antimalarials. J Rheumatol. 2000, 27:41-6. Published via Alabama College of Osteopathic Medicine Research 2024 Farooq et al. Cureus 16(6): e61644. DOI 10.7759/cureus.61644 4 of 4 |
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Herpangina is an illness caused by a virus, characterized by small blister-like bumps or ulcers that appear in the mouth, usually in the back of throat or the roof of the mouth. If your child has herpangina, she will probably have a high fever. Herpangina is very contagious and is usually seen in children between the ages of 1 and 4. It's seen most often in the summer and fall.
You can help prevent your child from getting herpangina by ensuing that her hands are kept clean. If your child has herpangina, good handwashing is also necessary to help prevent the spread of the disease.
Herpangina | Symptoms & Causes
What are the symptoms of herpangina?
The following are the most common symptoms of herpangina. However, each child may experience symptoms differently.
Blister-like bumps in the mouth, usually in the back of the throat and on the roof of the mouth
Headache
Quick onset of fever
High fever, sometimes up to 106F
Pain in the mouth or throat
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Decrease in appetite
What causes herpangina?
Herpangina is caused by a virus. The most common viruses that cause herpangina are:
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Echovirus
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Herpangina is usually diagnosed based on a complete history and physical examination of your child. The lesions of herpangina are unique and usually allow for a diagnosis simply on physical examination.
How do we treat herpangina?
The goal of treatment for herpangina is to help decrease the severity of the symptoms. Since it's a viral infection, antibiotics are ineffective. Treatment for your child may include:
Increased fluid intake
Acetaminophen for any fever
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Decimal To Binary Program In C
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Table of contents
Decimal Numbers in C
Binary Numbers in C
Different Methods to convert Decimal to Binary programs in C
Algorithm to convert Decimal to Binary Number in C
Conclusion
Frequently Asked Questions
Converting decimals to binary is perhaps one of those exercises that embrace computer science fundamental studies upon which the computer's way of thinking can be mastered. In day-to-day life, we deal with decimals or base 10 (0 to 9), but computers use binary numbers base 2 (0 or 1) numbers. This decimal to binary program in C conversion process is very important, especially in programming, networking and data processing.
Number System
4 Major Types of Number Systems
1. Decimal Number System (Base-10)
The decimal system is the number system we use every day. It has ten digits from 0 to 9. Each digit's place in a number is important because it tells us how many tens, hundreds, or thousands it represents. For example, in 345, the 3 means 300, the 4 means 40, and the 5 means 5. All these add up to 345. This system works well for humans and day-to-day counting.
2. Binary Number System (Base-2)
The binary system is the language of computers. It only has two digits: 0 and 1. These represent off-and-on states in electronics. In binary, each place value is a power of 2. For example, the binary number 101 means 1x 4 (or )+ 0 x 2 (or + 1 x 1 (or When you add that up, 101 in binary equals 5 in decimal. Computers use binary because it’s simple and matches their internal circuits.
3. Octal Number System (Base-8)
The octal system uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each place value in octal is a power of 8. For example, the octal number 123 means 1X64 (or 8^2) + 2 x 8 (or 8^1) +3 x 1 (or8^0). Adding these gives 83 in decimal. Octal is helpful in computing because it makes binary numbers shorter. Three binary digits can fit into one octal digit, making long binary numbers easier to read and write.
4. Hexadecimal Number System (Base-16)
The hexadecimal system uses sixteen symbols: 0 to 9 and A to F. A stands for 10, B for 11, and so on up to F, which is 15. Each place value in hexadecimal is a power of 16. For example, the hexadecimal number 1A means 1 x 16 (or 16^1) + 10 x 1 (or 16^0) . When you add these, 1A in hexadecimal equals 26 in decimal. Hexadecimal is often used in computing because it makes binary numbers shorter and easier to work with. One hexadecimal digit equals four binary digits.
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Decimal Numbers in C
Decimal numbers are base-10 numbers, and these are the type we use in everyday life. In C programming, they're represented using the float or double data types. A float uses 4 bytes of memory and provides a precision of about 6-7 digits. If you need higher precision, double uses 8 bytes and offers around 15-16 digits of precision.
Binary Numbers in C
Binary numbers are base-2 numbers, made up of just 0s and 1s. In C, they're usually represented by the int or long int data types, which use 32 or 64 bits, respectively. To write a binary number in C, you use the prefix 0b followed by a sequence of 0s and 1s, like 0b1010.
Different Methods to Convert Decimal to Binary Programs in C
1. Division Method
The division method involves repeatedly dividing the decimal number by 2 and storing the remainder. These remainders represent the binary digits. The process continues until the quotient becomes 0. The binary number is then read by reversing the order of the remainders.
2. Stacks
Using stacks helps store binary digits (remainders) as they are calculated which allows for easy reversal. The decimal number is divided by 2 in a loop, and the remainders are pushed onto the stack. After the loop ends, the stack is popped to display the binary number in the correct order.
3. While Loop
A while loop is used to repeatedly divide the decimal number by 2, and store each remainder. The loop runs until the decimal number becomes 0. Binary digits are calculated in each iteration and either printed directly or stored for later use.
4. For Loop
A for loop performs the same task as a while loop but with defined initialization, condition, and increment/decrement in one statement. The loop divides the number by 2 and calculates remainders to derive the binary number step by step.
5. Bitwise Operators
Bitwise operators like & and >> can convert decimal to binary. The & 1 operation extracts the least significant bit, while >> shifts the bits to the right. This approach is efficient and avoids arithmetic operations like division.
6. Recursion
Recursion uses a function that repeatedly calls itself with the quotient of the number divided by 2. Each remainder is calculated and printed after the recursive call returns, and the binary number is displayed in the correct order.
7. User-Defined Function
A user-defined function encapsulates the logic of binary conversion. For example, a function can take the decimal number as input, it then handles the conversion process internally, and return or print the binary result.
8. Math Library
The math library can assist in binary conversion using logarithms and powers. For instance, base-2 logarithms can determine the highest bit position. It helps in manual binary representation without traditional loops or recursion.
9. Negative Numbers
Converting negative numbers to binary involves using two’s complement representation. First, calculate the binary form of the absolute value of the number. Then, flip all the bits (1 becomes 0 and 0 becomes 1), and add 1 to the result. This method is used in computer systems to represent signed integers.
10. Floating Decimals
For floating-point numbers, the integer part is converted using standard methods, and the fractional part is multiplied by 2 repeatedly. The integer parts of the results are recorded to derive the binary equivalent of the fractional part.
Algorithm to Convert Decimal to Binary Number in C
Here’s a step-by-step algorithm to convert a decimal number to binary in C:
Declare integer variables to store the decimal number, quotient, remainder, and binary number.
Read the decimal number from the user.
Initialise the quotient with the decimal number.
Initialise the binary number to 0.
Loop while the quotient is not 0:
Calculate the remainder by taking the modulus of the quotient with 2 (quotient % 2).
Multiply the binary number by 10 and add the remainder to it.
Update the quotient by dividing it by 2.
Print the binary number.
Now let’s write some code for a C program to convert decimal to binary.
C Program to Convert Decimal to Binary Number Using FOR Loop
Algorithm:
1. Initialize Variables: Declare variables for the decimal number, remainder, and a binary array to store the binary digits.
2. Input Decimal Number: Prompt the user to enter a decimal number.
3. Convert to Binary:
Use a for loop to divide the decimal number by 2 repeatedly.
Store the remainder (0 or 1) in the binary array during each iteration.
Update the decimal number by dividing it by 2 in each loop.
4. Output the Binary Number:
Iterate through the binary array in reverse order.
Print the binary digits to display the result.
Here’s the binary to decimal C code using For loop:
```
include int main() { int binary; // Array to store binary digits (supports up to 32-bit numbers) // Prompt the user for input printf("Enter a decimal number: "); scanf("%d", &decimal); i = 0; // Convert decimal to binary using a for loop for (; quotient > 0; i++) { remainder = quotient % 2; quotient = quotient / 2; // Print the binary number in reverse order printf("Binary equivalent: "); for (int j = i - 1; j >= 0; j--) { printf("%d", binary[j]); printf("\n"); return 0;
```
Output:
Enter a decimal number: 3 Binary equivalent: 11
How the code works
Input: The user enters a decimal number.
Initialize: The quotient is set to the entered number.
Conversion Loop:
In each iteration, the number is divided by 2.
The remainder (0 or 1) is stored in an array.
The quotient is updated to match the division's result.
Repeat until the quotient becomes 0.
Reverse Output:
The binary digits are stored in reverse order.
Print the array elements backward to display the correct binary number.
C Program to Convert Decimal to Binary Number Using While Loop
Algorithm:
1. Initialize Variables: Declare variables for the decimal number, remainder, and binary number (initialized to 0).
2. Input Decimal Number: Ask the user to enter a decimal number.
3. Conversion Process:
Start a while loop that runs as long as the decimal number is greater than 0.
Inside the loop:
Compute the remainder by dividing the decimal number by 2.
Multiply the binary number by 10 and add the remainder to it to build the binary representation.
Update the decimal number by dividing it by 2.
4.Output Binary Number: Print the binary result after the loop ends.
Here’s a binary to decimal C using the While loop:
```
include int main() { int binary; // Array to store binary digits (supports up to 32-bit numbers) int i = 0; // Prompt the user for input printf("Enter a decimal number: "); scanf("%d", &decimal); // Convert decimal to binary using a while loop while (quotient > 0) { remainder = quotient % 2; quotient = quotient / 2; // Print the binary number in reverse order printf("Binary equivalent: "); for (int j = i - 1; j >= 0; j--) { printf("%d", binary[j]); printf("\n"); return 0;
```
Output:
Enter a decimal number: 6 Binary equivalent: 110
How the code works:
Input: The user enters a decimal number.
Initialize: The quotient is set to the input number, and the index i is initialised to 0.
Conversion Loop: While the quotient is greater than 0:
Calculate the remainder by taking quotient % 2.
Store the remainder in the binary array.
Update the quotient by dividing it by 2.
Increment i to track the array index.
Reverse Output:
The binary digits are stored in reverse.
Print the array elements in reverse order to get the correct binary number.
C Program to Convert Decimal to Binary Number Using Function
Algorithm:
1. Initialize Variables:
Declare variables to store the decimal number, remainder, and an array to store the binary digits.
2. Input Decimal Number:
Prompt the user to enter a decimal number.
3. Calculate Binary Digits:
Start a for loop to iterate through the decimal number:
Divide the decimal number by 2 and store the remainder.
Store the remainder in the array for binary digits.
Update the decimal number by dividing it by 2.
4. Reverse Binary Digits:
After the loop finishes, print the array from the last element to the first (to reverse the binary digits).
5. Output the Binary Number:
Print the binary number after all digits have been printed in the correct order.
Here’s the decimal to binary C using the function:
```
include // Function to convert decimal to binary void decimalToBinary(int decimal) { int binary; // Array to store binary digits (supports up to 32-bit numbers) int i = 0; // Convert decimal to binary while (decimal > 0) { binary[i] = decimal % 2; decimal = decimal / 2; // Print the binary number in reverse order printf("Binary equivalent: "); for (int j = i - 1; j >= 0; j--) { printf("%d", binary[j]); printf("\n"); int main() { // Prompt the user for input printf("Enter a decimal number: "); scanf("%d", &decimal); // Call the function to convert decimal to binary return 0;
```
Output:
Enter a decimal number: 8 Binary equivalent: 1000
How the code works:
Input: The user enters a decimal number.
Function Call: The decimalToBinary function is called with the input decimal number as an argument.
Inside Function:
The function initialises an array to store binary digits.
A while loop converts the decimal number to binary by dividing it by 2 and storing the remainder in the array.
The quotient is updated after each division.
Reverse Output:
Once the conversion is complete, the binary digits are printed in reverse order.
This ensures the correct binary number is displayed.
C Program to Convert Decimal to Binary Number Using For Stack
Algorithm:
Initialize Variables:
Declare an integer array to represent the stack and a variable for the stack's top index.
Input Decimal Number:
Prompt the user to input a decimal number.
Push Remainders onto Stack:
Start a while loop that continues as long as the decimal number is greater than 0.
Inside the loop:
Divide the decimal number by 2 and compute the remainder.
Push the remainder onto the stack (increment the stack's top index).
Update the decimal number by dividing it by 2.
Pop and Print Binary Digits:
After the loop finishes, start popping elements from the stack one by one.
Print each popped value to form the binary number.
Output the Binary Number:
The binary number is printed as the stack elements are popped, which displays the binary representation of the decimal number.
In this, we’ll look into decimal to binary C code using for stack:
```
include // Function to convert decimal to binary using stack (array) void decimalToBinary(int decimal) { int binary; // Array to store binary digits (supports up to 32-bit numbers) int top = - 1; // Stack top pointer // Convert decimal to binary using a for loop and stack for (int quotient = decimal; quotient > 0; quotient /= 2) { top++; // Increment stack pointer binary[top] = quotient % 2; // Push remainder onto stack // Print the binary number by popping it from the stack printf("Binary equivalent: "); for (int i = top; i >= 0; i--) { printf("%d", binary[i]); // Pop and print each binary digit printf("\n"); int main() { // Prompt the user for input printf("Enter a decimal number: "); scanf("%d", &decimal); // Call the function to convert decimal to binary return 0;
```
Output:
Enter a decimal number: 10 Binary equivalent: 1010
How the code works:
Input: The user enters a decimal number.
Function Call: The decimalToBinary function is invoked with the decimal number as an argument.
Inside Function:
The function initialises a stack (array) to store binary digits, with the top variable tracking the stack's current position.
A for loop is used to convert the decimal number to binary. Each division by 2 gives a remainder, which is pushed onto the stack.
Reverse Output:
Once all binary digits are stored in the stack, the program "pops" and prints each digit starting from the top of the stack.
This ensures that the binary number is displayed in the correct order.
C Program to Convert Decimal to Binary Number Using Bitwise Operator
Algorithm:
Initialize Variables:
Declare an integer variable to store the decimal number and a variable to store the binary digits.
Input Decimal Number:
Prompt the user to input a decimal number.
Bitwise Operation Loop:
Start a while loop that runs as long as the decimal number is greater than 0.
Inside the loop:
Use the bitwise AND operator (&) to get the least significant bit (binary digit) by performing decimal & 1.
Shift the decimal number one bit to the right using the bitwise right shift operator (>>) to prepare for the next iteration.
Store the result in the binary output (can be printed directly or stored in a variable).
Output the Binary Number:
The binary digits are printed in reverse order, as the least significant bit is calculated first. You can reverse the output or store it for correct printing.
End the Program:
The program ends once the binary representation is printed.
Now let’s look at decimal to binary C code using the bitwise operator:
```
include // Function to convert decimal to binary using bitwise operator void decimalToBinary(int decimal) { int isLeadingZero = 1; // Flag to skip leading zeros // Check if the decimal number is 0 if (decimal == 0) { printf("Binary equivalent: 0\n"); return; printf("Binary equivalent: "); // Iterate through each bit using the bitwise right shift for (i = 31; i >= 0; i--) { int bit = (decimal >> i) & 1; // Right shift and mask to get the bit at position i // Skip leading zeros if (bit == 1) { isLeadingZero = 0; // Once we find the first 1, stop skipping zeros if (!isLeadingZero) { printf("%d", bit); // Print the bit if it's not a leading zero printf("\n"); int main() { // Prompt the user for input printf("Enter a decimal number: "); scanf("%d", &decimal); // Call the function to convert decimal to binary return 0;
```
Output:
Enter a decimal number: 15 Binary equivalent: 1111
How the code works:
Input: The user enters a decimal number.
Function Call: The decimalToBinary function is invoked with the decimal number as an argument.
Bitwise Conversion:
A for loop iterates through each of the 32 bits of the decimal number, starting from the most significant bit.
The >> (right shift) operator shifts the bits, and the & 1 operation checks if the bit is 1 or 0.
Leading zeros are skipped using the isLeadingZero flag.
Output the Binary Number:
The binary digits are printed, starting from the most significant bit, without leading zeros, to give the correct binary representation of the decimal number.
C Program to Convert Decimal to Binary Number Without Using Array
Algorithm:
1. Initialize Variables:
Declare an integer variable to store the decimal number.
Declare a variable to store the remainder (binary digit).
2. Input Decimal Number:
Prompt the user to input a decimal number.
3. Conversion Process:
Start a while loop that continues as long as the decimal number is greater than 0.
Inside the loop:
Calculate the remainder by dividing the decimal number by 2 (using modulus %).
Print the remainder (binary digit).
Update the decimal number by dividing it by 2 (using integer division).
4. Output the Binary Number:
The binary digits are printed in reverse order since the least significant bit is printed first.
To correct the order, you can print the digits from last to first (in reverse) or use a different method to store and print.
5. End the Program:
The program ends after printing the binary digits.
Now let’s convert binary to decimal in C without using an array:
```
include // Function to convert decimal to binary without using an array void decimalToBinary(int decimal) { // Handle the case for zero if (decimal == 0) { printf("Binary equivalent: 0\n"); return; printf("Binary equivalent: "); int isLeadingZero = 1; // Flag to skip leading zeros // Iterate through each bit using the bitwise right shift for (int i = 31; i >= 0; i--) { int bit = (decimal >> i) & 1; // Right shift and mask to get the bit at position i // Skip leading zeros if (bit == 1 || !isLeadingZero) { printf("%d", bit); // Print bit and stop skipping leading zeros isLeadingZero = 0; // After printing the first 1, stop skipping printf("\n"); int main() { // Prompt the user for input printf("Enter a decimal number: "); scanf("%d", &decimal); // Call the function to convert decimal to binary return 0;
```
Output:
Enter a decimal number: 20 Binary equivalent: 10100
How the code works:
Input: The user enters a decimal number.
Bitwise Conversion:
The program uses a for loop to iterate through each bit, starting from the most significant bit (leftmost).
The >> (right shift) operator shifts the bits of the number to the right, and the & 1 operation checks if the current bit is 1 or 0.
Skip Leading Zeros:
The variable isLeadingZero helps skip leading zeros in the binary output. The first 1 encountered stops the skipping, and the rest of the bits are printed.
Output:
The binary digits are printed, ensuring no leading zeros and giving the correct binary representation of the decimal number.
Conclusion
This exercise of converting decimals to binary in different ways is crucial for students to grasp the fundamentals of coding. By implementing various approaches, such as using loops, functions, bitwise operators, and arrays, students learn to think in different ways and choose the most efficient solution. It strengthens their understanding of data manipulation, logical operations, and problem-solving. Mastering these concepts lays a strong foundation for more advanced topics in programming, enhances their coding skills, and prepares them for real-world programming challenges. If you wish to learn more and build a strong foundation that prepares you for a job, enroll on the CCBP Academy 4.0 program.
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Frequently Asked Questions
1. What is the purpose of converting decimal to binary?
Decimal to binary conversion is needed for computer science as these machines use binary (base-2) to process and store data. Understanding this conversion helps you to learn how data is represented and manipulated at the machine level.
2. What is the division 2 conversion method?
The division by 2 methods for conversion means calculating the division of a decimal number with 2, and each digit is noted in terms of the remainder. Taking the division of each digit in a definite number from a unit digital number gives a remainder, which is 0 or 1, which forms a binary digit. The binary number is then obtained by reading the remainder from the last towards the first that was obtained during the division exercise. This process is done until the quotient is zero, and all the remainder offered in binary form signify the Decimal value of a number.
3. What about negative decimal numbers in conversion?
For negative numbers, you can use what is known as a two-s complement to represent them in binary position. This pertains to converting the bits of the positive equivalent and making its complement by adding one more.
4. How does using a function improve decimal to binary program in C?
Using functions makes the code modular and easier to manage. It helps break down the problem into smaller, manageable parts, improving code readability and maintainability.
5. Why are bitwise operators used in decimal to binary programs?
Bitwise operators allow direct manipulation of individual bits in a number. It is ideal for tasks like converting decimals to binary ones.
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Cholinesterase Inhibitors for Alzheimer’s Disease
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NATHAN HITZEMAN, M.D., Sutter Health Family Medicine Residency Program, Sacramento, California
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Am Fam Physician. 2006;74(5):747-749
Clinical Scenario
A 72-year-old woman is brought into the office by her daughter, who complains that her mother has become increasingly forgetful over the past two years. On a Mini-Mental State Examination (MMSE) she scores 20 out of 30 points, and after an appropriate evaluation you diagnose Alzheimer’s disease. Her daughter asks if there are medicines that could help.
Clinical Question
Do cholinesterase inhibitors improve function in persons with mild, moderate, or severe dementia caused by Alzheimer’s disease, and is one cholinesterase inhibitor better tolerated or more effective than the others?
Evidence-Based Answer
Cholinesterase inhibitors produce a small benefit on several cognitive and noncognitive function scales. Although data for patients with severe dementia are sparse, there is no evidence to suggest a difference in effectiveness among patients with mild, moderate, or severe dementia. Compared with placebo, adverse reactions are significantly more common in treatment groups. Limited evidence suggests donepezil (Aricept) is better tolerated than rivastigmine (Exelon). There is no consistent evidence to suggest treatment reduces health care costs or prolongs time until institutionalization.1
Cochrane Abstract
Background. Since the introduction of the first cholinesterase inhibitor in 1997, most clinicians, and probably most patients, would consider the cholinergic drugs donepezil (Aricept), galantamine (Razadyne [previously Reminyl]), and rivastigmine (Exelon) to be first-line pharmacotherapy for mild to moderate Alzheimer’s disease.
The drugs have slightly different pharmacologic properties, but they all work by inhibiting the breakdown of acetylcholine, an important neurotransmitter associated with memory, by blocking the enzyme acetylcholinesterase. The most that these drugs could achieve would be to modify the manifestations of Alzheimer’s disease. Cochrane reviews of each cholinesterase inhibitor for Alzheimer’s disease have been completed.
Objectives. To assess the effects of donepezil, galantamine, and rivastigmine in patients with mild, moderate, or severe dementia caused by Alzheimer’s disease.
Search Strategy. The Cochrane Dementia and Cognitive Improvement Group’s Specialized Register was searched using the terms donepezil, E2020, Aricept, galanthamin, galantamin, reminyl, rivastigmine, Exelon, ENA 713, and ENA-713 on June 12, 2005. This Register contains up-to-date records of all major health care databases and many ongoing trial databases.
Selection Criteria. All unconfounded, blinded, randomized trials in which treatment with a cholinesterase inhibitor at the usual recommended dose was compared with placebo or another cholinesterase inhibitor for patients with mild, moderate, or severe dementia caused by Alzheimer’s disease.
Data Collection and Analysis. Data were extracted by one reviewer and pooled where appropriate and possible, and the pooled treatment effects, or the risks and benefits of treatment, were estimated.
Primary Results. The results of 13 randomized, double-blind, placebo-controlled trials demonstrate that treatment for six months with donepezil, galantamine, or rivastigmine at the recommended dose for persons with mild, moderate, or severe dementia caused by Alzheimer’s disease produced improvements in cognitive function, on average −2.7 points (95% confidence interval,−3.0 to −2.3,P< .00001), in the midrange of the 70-point Alzheimer’s Disease Assessment Scale for Cognition (ADAS-Cog). Study clinicians rated global clinical state more positively in treated patients. Benefits of treatment also were seen on measures of activities of daily living and behavior. None of these treatment effects were large. The effects were similar for patients with severe dementia, although there is very little evidence, from only two trials. More patients leave cholinesterase inhibitor treatment groups (29 percent) on account of adverse events than leave the placebo groups (18 percent).
There is evidence of more adverse events in total in the patients treated with a cholinesterase inhibitor than with placebo. Although many types of adverse event were reported, nausea, vomiting, and diarrhea were significantly more frequent in the cholinesterase inhibitor groups than in the placebo groups.
There is no evidence of a difference between the effects of donepezil and those of rivastigmine on cognitive function, activities of daily living, or behavioral disturbance at two years. There were fewer reports of adverse events among patients taking donepezil than among those taking rivastigmine.
Reviewers’ Conclusions. The three cholinesterase inhibitors studied are effective for mild to moderate Alzheimer’s disease. Despite the slight variations in the mode of action of the three drugs, there is no evidence of any differences among them with respect to effectiveness. The evidence from one large trial shows fewer adverse effects associated with donepezil compared with rivastigmine.
These summaries have been derived from Cochrane reviews published in the Cochrane Database of SystematicReviews in the Cochrane Library. Their content has, as far as possible, been checked with the authors of the originalreviews, but the summaries should not be regarded as an official product of the Cochrane Collaboration; minorediting changes have been made to the text (
Practice Pointers
Alzheimer’s disease affects 4.5 million persons in the United States, including one half of all nursing home residents.2 Family physicians sometimes can feel pressured by family members to attempt to slow the course of the disease.
This Cochrane review identified 13 randomized controlled trials of cholinesterase inhibitors.1 The average age of participants in each trial typically was between 72 and 75 years, and treatment durations were between six and 12 months. Multiple measures were used to assess cognition and well-being: the Alzheimer’s Disease Assessment Scale for Cognition (ADAS-Cog), the MMSE, the Severe Impairment Battery scale, global clinical state assessments, activities of daily living scales, and a behavioral disturbance scale. After six to 12 months, pooled data showed modest benefits favoring the treatment groups. The reviewers found a 2.7-point improvement on the 70-point ADAS-Cog and a 1.4-point improvement on the 30-point MMSE in patients receiving treatment compared with those receiving placebo. The U.S. Preventive Services Task Force equates a difference of two to three points on the ADAS-Cog after one year to a delay in disease progress of about two to seven months.3
Patients taking a cholinesterase inhibitor are more likely to discontinue treatment as a result of adverse reactions than are those taking placebo (29 versus 18 percent, respectively; number needed to harm = 9). Common reactions include nausea, vomiting, and diarrhea. In a study sponsored by the manufacturer of rivastigmine, rivastigmine was less well tolerated than donepezil.1
Although their benefit is modest, cholinesterase inhibitors remain first-line treatment for Alzheimer’s disease and should be titrated as tolerated to target dosage.4 Donepezil is started at a dosage of 5 mg daily and increased to 10 mg daily after one to four weeks. Galantamine (Razadyne [previously Reminyl]) is started at a dosage of 4 mg twice daily and gradually titrated to 12 mg twice daily with stepups at one- to four-week intervals. Rivastigmine is started at a dosage of 1.5 mg twice daily and titrated over one to three months to 6 mg twice daily. The costs of one month’s supply of the target dosages are $166, $175, and $188, respectively.5
When Alzheimer’s disease becomes moderate to severe, memantine (Namenda), an N-methyl-D-aspartate antagonist, sometimes is added to therapy. In limited studies of short duration (six to seven months), memantine was found to produce modest improvement in cognition, function, and behavior when used alone or in conjunction with donepezil.6 In these studies, patients taking memantine did not experience more adverse reactions than those taking placebo. Target dosing is 10 mg twice daily, and the cost is comparable to that of a cholinesterase inhibitor.5
Many patients with Alzheimer’s disease live with and are cared for by family members; therefore, family physicians can help by referring families to the Alzheimer’s Association Web site,
expand_more Author Information
NATHAN HITZEMAN, M.D., is a faculty physician at Sutter Health Family Medicine Residency Program in Sacramento, Calif., where he also completed his residency. He received his medical degree from the University of California at Los Angeles School of Medicine.
Address correspondence to Nathan Hitzeman, M.D., Sutter Health Family Medicine Residency Program, 1201 Alhambra Blvd., Suite #300, Sacramento, CA 95816 (e-mail:hitzemn@sutterhealth.org). Reprints are not available from the author.
expand_more Reference(s)
Birks J. Cholinesterase inhibitors for Alzheimer’s disease. Cochrane Database Syst Rev. 2006;1:CD005593.
Herbert LE, Scherr PA, Bienias JL, Bennett DA, Evans DA. Alzheimer disease in the U.S. population: prevalence estimates using the 2000 census. Arch Neurol. 2003;60:1119-22.
U.S. Preventive Services Task Force. Screening for dementia: recommendations and rationale. Rockville, Md.: Agency for Healthcare Research and Quality, 2003. Accessed May 18, 2006, at:
Doody RS, Stevens JC, Beck C, Dubinsky RM, Kaye JA, Gwyther L, et al. Practice parameter: management of dementia (an evidence-based review). Reportof the Quality Standards Subcommittee of the American Academy of Neurology. Neurology. 2001;56:1154-66.
Red Book Montvale, N.J.: Medical Economics Data, 2006.
Areosa SA, Sherriff F, McShane R. Memantine for dementia. Cochrane Database Syst Rev. 2005;3:CD003154.
These are summaries of reviews from the Cochrane Library.
This series is coordinated by Corey D. Fogleman, MD, assistant medical editor.
A collection of Cochrane for Clinicians published in AFP is available at
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3422 | https://byjus.com/chemistry/ph-to-poh/ | What is pH?
pH or potential of hydrogen ion is a scale used to determine the hydrogen ion (H+) concentration in a solution. It is a quantitative measure of the acidity or alkalinity of a solution. It equals the negative log of hydrogen ion (H+) concentration.
pH = – log [H+]
What is pOH?
pOH or potential of hydroxide ion is a scale used to determine the hydroxide ion (OH–) concentration in a solution. It is a quantitative measure of the acidity or alkalinity of a solution. It equals the negative log of hydroxide ion (OH–) concentration.
pH = – log [OH–]
Table of Content
Understanding pH and pOH in detail
pH to pOH
pH and pOH value of a few compounds
Difference between pH and pOH
Frequently Asked Questions – FAQs
Understanding pH and pOH in detail
pH or potential of hydrogen ion is a scale used to determine the hydrogen ion (H+) concentration in a solution. It is a quantitative measure of the acidity or alkalinity of a solution. It equals the negative log of hydrogen ion (H+) concentration.
pH = – log [H+]
At pH 7 solution is found to be neutral. In contrast, if the pH value is less than 7, the solution will be acidic, and if the pH value is more than 7, the solution will be basic.
In contrast, pOH or potential of hydroxide ion is a scale used to determine the hydroxide ion (OH–) concentration in a solution. It is a quantitative measure of the acidity or alkalinity of a solution. It equals the negative log of hydrogen ion (OH–) concentration.
pH = – log [OH–]
At pOH 7 solution is found to be neutral. In contrast, if the pOH value is less than 7, the solution will be basic, and if the pOH value is more than 7, the solution will be acidic.
Both pH and pOH are related to each other. pH is inversely proportional to pOH, i.e. pH increases with decreasing pOH.
pH ∝ 1 / pOH
pH to pOH
Consider a reaction,
H2O ⇆ H+ + OH–
For the above reaction dissociation constant, Kw would be,
Kw = [H+][OH–]
Here, Kw refers to dissociation constant, and [H+] and [OH–] refer to hydrogen and hydroxide ion concentrations.
We are taking the negative logarithm of both sides.
-log Kw = – log [H+] [OH–]
-log Kw = -(log [H+] + [OH–] )
-log Kw = -log [H+] – log [OH–] . . . 1
We know that,
Kw = 1 X 10 -14 at 298 K
-log [H+] = pH
-log [OH–] = pOH
Putting value in equation 1
-log 1 X 10 -14 = pH + pOH
14 = pH + pOH
pOH = 14 – pH
pH = 14 – pOH
Hence if the pH is known, we can quickly determine the pOH value.
pH and pOH value of a few molecules
| S No. | Name of the Molecule | pH at 1 mM concentration | pOH at 1 mM concentration |
--- --- |
| 1. | Sodium Hydroxide | 1 | 13 |
| 2. | Potassium Hydroxide | 1.48 | 12.52 |
| 3. | Sulfuric acid | 2.75 | 11.25 |
| 4. | Hydrobromic Acid | 3.01 | 10.99 |
| 5. | Nitric Acid | 3.01 | 10.99 |
| 6. | Hydrochloric Acid | 3.01 | 10.99 |
| 7. | Hydroiodic Acid | 3.01 | 10.99 |
| 8. | Orthophosphoric Acid | 3.06 | 10.94 |
| 9. | Hydrofluoric Acid | 3.27 | 10.73 |
Difference between pOH and pH
| S No. | pH | pOH |
---
| 1. | pH is the potential of hydrogen ions. | pOH is the potential of hydroxide ions. |
| 2. | It is a scale used to determine the solution’s hydrogen ion (H+) concentration. | It is a scale used to determine the solution’s hydroxide ion (OH–) concentration. |
| 3. | It equals the negative log of hydrogen ion (H+) concentration. pH = – log [H+] | It equals the negative log of hydroxide ion (OH– ) concentration. pH = – log [OH–] |
| 4. | If the pH value is less than 7, the solution will be acidic. | If the pOH value is less than 7, the solution will be basic. |
| 5. | If the pH value is more than 7, the solution will be basic. | If the pOH value is more than 7, the solution will be acidic. |
Recommended Videos
Acids and Bases
2,72,279
Frequently Asked Questions on pH to pOH
Q1
What is pH?
pH or potential of hydrogen ion is a scale used to determine the hydrogen ion (H+) concentration in a solution. It is a quantitative measure of the acidity or alkalinity of a solution. It equals the negative log of hydrogen ion (H+) concentration.
At pH 7 solution is found to be neutral. In contrast, if the pH value is less than 7, the solution will be acidic, and if the pH value is more than 7, the solution will be basic.
Q2
What is pOH?
pOH or potential of hydroxide ion is a scale used to determine the hydroxide ion (OH–) concentration in a solution. It is a quantitative measure of the acidity or alkalinity of a solution. It equals the negative log of hydrogen ion (OH–) concentration.
At pOH 7 solution is found to be neutral. In contrast, if the pOH value is less than 7, the solution will be basic, and if the pOH value is more than 7, the solution will be acidic.
Q3
Are pH and pOH the same?
No, pH and pOH are not the same. However, they were found to be associated. pH is inversely proportional to pOH, i.e. pH increases with decreasing pOH.
pH ∝ 1 / pOH
Q4
What is the difference between pH and pOH?
| S No. | pH | pOH |
---
| 1. | pH or potential of hydrogen ion is a scale used to determine the hydrogen ion (H+) concentration in a solution. | pH or potential of hydrogen ion is a scale used to determine the hydroxide ion (OH–) concentration in a solution. |
| 2. | It equals the negative log of hydrogen ion (H+) concentration. pH = – log [H+] | It equals the negative log of hydroxide ion (OH– ) concentration. pH = – log [OH–] |
Q5
Calculate the pOH of a solution having a pH value of 1?
Given,
pH = 1,
We know that,
pH + pOH = 14
pOH = 14 – pH
pOH = 14 – 1
pH = 13
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Volume 42, Issue 1 pp. 261-276
Major Revision from Eurographics Conference
Open Access
Improved Evaluation and Generation Of Grid Layouts Using Distance Preservation Quality and Linear Assignment Sorting
K. U. Barthel,
Corresponding Author
K. U. Barthel
barthel@htw-berlin.de
orcid.org/0000-0001-6309-572X
Visual Computing Group, HTW Berlin, Berlin, Germany
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N. Hezel,
N. Hezel
hezel@htw-berlin.de
orcid.org/0000-0001-7607-3124
Visual Computing Group, HTW Berlin, Berlin, Germany
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K. Jung,
K. Jung
Klaus.Jung@htw-berlin.de
orcid.org/0000-0001-9797-5362
Visual Computing Group, HTW Berlin, Berlin, Germany
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K. Schall,
K. Schall
Konstantin.Schall@htw-berlin.de
orcid.org/0000-0002-9465-3533
Visual Computing Group, HTW Berlin, Berlin, Germany
Search for more papers by this author
K. U. Barthel,
Corresponding Author
K. U. Barthel
barthel@htw-berlin.de
orcid.org/0000-0001-6309-572X
Visual Computing Group, HTW Berlin, Berlin, Germany
barthel@htw-berlin.de
Search for more papers by this author
N. Hezel,
N. Hezel
hezel@htw-berlin.de
orcid.org/0000-0001-7607-3124
Visual Computing Group, HTW Berlin, Berlin, Germany
Search for more papers by this author
K. Jung,
K. Jung
Klaus.Jung@htw-berlin.de
orcid.org/0000-0001-9797-5362
Visual Computing Group, HTW Berlin, Berlin, Germany
Search for more papers by this author
K. Schall,
K. Schall
Konstantin.Schall@htw-berlin.de
orcid.org/0000-0002-9465-3533
Visual Computing Group, HTW Berlin, Berlin, Germany
Search for more papers by this author
First published: 16 November 2022
Citations: 15
Abstract
Images sorted by similarity enables more images to be viewed simultaneously, and can be very useful for stock photo agencies or e-commerce applications. Visually sorted grid layouts attempt to arrange images so that their proximity on the grid corresponds as closely as possible to their similarity. Various metrics exist for evaluating such arrangements, but there is low experimental evidence on correlation between human perceived quality and metric value. We propose distance preservation quality (DPQ) as a new metric to evaluate the quality of an arrangement. Extensive user testing revealed stronger correlation of DPQ with user-perceived quality and performance in image retrieval tasks compared to other metrics. In addition, we introduce Fast linear assignment sorting (FLAS) as a new algorithm for creating visually sorted grid layouts. FLAS achieves very good sorting qualities while improving run time and computational resources.
Graphical Abstract
We propose Distance Preservation Quality, a new metric for evaluating the quality of sorted grid layouts with strong correlation to user-perceived quality. We also present Fast Linear Assignment Sorting, a new algorithm for creating visually sorted grid layouts that achieves very good sorting qualities while improving runtime and computational resources.
1 Introduction
It is difficult for humans to view large sets of images simultaneously while maintaining a cognitive overview of its content. As set sizes increase, the viewer quickly loses their perception of specific content contained in the set (Figure 1 left). For this reason, most applications and websites typically display no more than 20 images at a time, which in many cases is only a tiny fraction of the images available. However, if the images are sorted according to their similarity, up to several hundred can be perceived simultaneously. It has been shown that a sorted arrangement helps users to identify regions of interest more easily and thus find the images they are looking for more quickly [SA11, QKTB10, RRS13, HZL15]. The simultaneous display of larger image sets is particularly interesting for e-commerce applications and stock photo agencies.
In order to be able to sort images according to their similarity, a suitable measure of this similarity must be specified. Image analysis methods can generate visual feature vectors and image similarity is then expressed by the similarity of their feature vectors. While low-level feature vectors generated by classical image analysis techniques represent the general visual appearance of images (such as colours, shapes and textures), vectors generated with deep neural networks can also describe the content of images [BSCL14, ZIE18, RTC19, CAS20]. The dimensions of these vectors are on the order of a few tens for low-level features, while deep learning vectors generated with neural networks typically have up to thousands of dimensions.
If the images are represented as high-dimensional (HD) vectors, their similarities can be expressed by appropriate visualization techniques. A variety of dimensionality reduction techniques have been proposed to visualize HD data relationships in two dimensions. Often a distinction is made between methods that use vectors or pairwise distances. However, these methods can be converted from one another; pairwise distances can be calculated from the vectors, and the rows of a distance matrix can be used as vectors.
Numerous techniques (principal component analysis (PCA) [Pea01], multi-dimensional scaling (MDS) [Sam69], locally linear embedding (LLE) [RS00], Isomap [TdSL00] and others) are described in Sarveniazi [Sar14]. Other methods that work very well are t-distributed stochastic neighbourhood embedding (t-SNE) [vdMH08], uniform manifold approximation and projection (UMAP) [MH18] and subset embedding networks [XTL21].
Visualization is achieved by projecting the HD data onto a two-dimensional plane. However, all of the above techniques are of limited use if the images themselves are to be displayed. The centre of Figure 1 shows a t-SNE projection of the relative similarity of 1024 flower images. Due to the dense positioning of the projected images, some overlap and are partially obscured. Furthermore, only a fraction of the display area is used. Using techniques such as DGrid [HMJE21] would solve the overlap problem, but would still not make best use of the available space.
To arrange or sort a set of images by similarity while maximizing the display area used, three requirements must be satisfied:
The images should not overlap.
The image arrangement should cover the entire display area.
The HD similarity relationships of the image feature vectors should be preserved by the 2D image positions.
Requirements 1 and 2 can only be met if the images are positioned on a rectangular grid. For the 3rd requirement, the images have to be positioned such that their spatial distance corresponds as closely as possible to the HD distance of their feature vectors, despite the given grid structure.
The self-organizing map is one of the oldest methods for organizing HD vectors on a grid [Koh82, Koh13]. Self-sorting maps [SG11, SG14] are a more recent technique that orders images using a hierarchical swapping method. Other approaches first project the HD vectors to two dimensions, which are then mapped to the grid positions. Various metrics exist for assessing the quality of such arrangements, but there is little experimental evidence of correlation between human-perceived quality and these metrics.
In our paper, we first describe other existing quality metrics for evaluating sorted grid layouts, then we give an overview of existing algorithms for generating sorted 2D grid layouts.
The key contributions of our work are: 1. Inspired by the k-neighbourhood preservation index [FFDP15], we propose distance preservation quality (DPQ) as a new metric for evaluating grid-based layouts. 2. We then propose linear assignment sorting (LAS), an algorithm that very efficiently produces high-quality 2D grid layouts. 3. We conducted an extensive user study examining different metrics and show that distance preservation quality better reflects the quality perceived by humans. We furthermore performed qualitative and quantitative comparisons with other sorting algorithms. 4. In the last section, we show how to generate arrangements with special layout constraints with our proposed sorting method.
A preprint of this paper has been published in Barthel et al. [BHJS22]. This paper is based on our previous work [BH19], in which we proposed a predecessor of the arrangement quality metric and a combination of SOM and SSM. The differences between this paper and these previous approaches are described in Sections 3.2 and 4.1.
2 Related Work
2.1 Quality evaluation of distance preserving grid layouts
A high-quality image arrangement is one that provides a good overview, places similar images close to each other and images being searched can be found quickly. An evaluation metric expresses the quality of a sorted arrangement with a single number. This value should highly correlate with the quality perceived by humans. We review commonly used evaluation metrics and examine their properties and problems.
Grid-based arrangement of HD data X consists of finding a mapping (a sorting function) or , where is the ith HD vector, whereas is the ith position vector on the grid in . The distance between HD vectors is denoted by whereas denotes the corresponding spatial distance of positions of the 2D grid.
Mean average precision. The mean average precision (mAP) is the commonly used metric to evaluate image retrieval systems.
(1)
AP is the average precision, N the number of total images, the number of positive images per class. represents the precision at rank k for the query q, is a binary indicator function (1 if q and the image at rank k have the same class and 0 otherwise).
The mAP metric defines a sorting as ‘good’ if the nearest neighbours belong to the same class. In most cases, the mAP cannot be used because typically images do not have class information. Another problem is that mAP only considers images of the same class and ignores the order of the other images (see Figure 2).
k-Neighbourhood preservation index. The k-neighbourhood preservation index is similar to the mAP in that it evaluates the extent to which the neighbourhood of the HD data set X is preserved in the projected grid Y. It is defined as
(2)
where k is the number of considered neighbours, is the set of the k nearest neighbours of in the HD space, whereas is the set of the k nearest neighbours to on the 2D grid.
The k-neighbourhood preservation index has several problems: The quality of an arrangement is not described by a single value, but by individual values for each neighbour size k. Because of the discrete 2D grid, many spatial distances λ are equal, which means that there is no unique ranking of the grid elements. However, the biggest problem is a high sensitivity to noisy or similar distances.
Cross-correlation. The cross-correlation is used to determine how well the distances of the projected grid positions correlate with the distances of the original vectors:
(3)
The main problem of cross-correlation is that differences of large distances have a higher impact than differences of small distances. It may be problematic to assess the quality of an image arrangement with cross-correlation as it is equally important to maintain both small and large distances to keep similar images together and prevent dissimilar images from being arranged next to each other.
Normalized energy function. The normalized energy function measures how well the distances between the data instances are preserved by the corresponding spatial distances on the grid.
(4)
The normalized energy function has essentially the same properties and problems as cross-correlation. The parameter p can be used to tune the balance between small and large distances. Usually, p values of 1 or 2 are used. Throughout this paper, we use with a range of [0,1] with larger values representing better results.
2.2 Algorithms for Sorted Grid Layouts
2.2.1 Grid arrangements
Since our new sorting method is based on both self-organizing map and self-sorting map, we present them here in more detail.
A self-organized map (SOM) uses unsupervised learning to produce a lower dimensional, discrete representation of the input space. A SOM consists of a rectangular grid of map vectors M having the same dimensionality as the input vectors X. To adapt a SOM for image sorting, the input vectors must all be assigned to different map positions, since multiple assignments would result in overlapping images. Algorithm 1 describes the SOM sorting process.
Algorithm 1. SOM
A self-sorting map (SSM) arranges images by initially filling cells (grid positions) with the input vectors. Then for sets of four cells, a hierarchical swapping procedure is used by selecting the best permutation from 4! = 24 swap possibilities. Algorithm 2 describes the sorting process with a SSM. In [SJGE13], an alternative to SSMs is described that uses more sophisticated swapping strategies to achieve better global correlation, but at a much higher computational cost.
Algorithm 2. SSM
Level-of-detail grid (LDG) is a recent method for creating hierarchical grid layouts [Fre22]. A progressive optimization method based on local search generates hierarchical grids. The method is based solely on pairwise distances and jointly optimizes homogeneity within interior nodes and between grid neighbours.
2.2.2 Graph matching
Kernelized sorting (KS) [QSS08] and Convex KS [DGV12] generate distance-preserving grids and find a locally optimal solution to a quadratic assignment problem [BK57]. KS creates a matrix of pairwise distances between HD data instances and a matrix of pairwise distances between grid positions. A permutation procedure on the second matrix modifies it to approximate the first matrix as well as possible, resulting in a one-to-one mapping between instances and grid cells.
IsoMatch also uses an assignment strategy to construct distance preserving grids [FDH15]. First, it projects the data into the 2D plane using the Isomap technique [TdSL00] and creates a complete bipartite graph between the projection and the grid positions. Then, the Hungarian algorithm [Kuh55] is used to find the optimal assignment for the projected 2D vectors to the grid positions. IsoMatch uses the normalized energy function E1 trying to maximize the overall distance preservation.
Similarly, DS++ presents a convex quadratic programming relaxation to solve this matching problem [DML17]. KS, IsoMatch and DS++ are not limited to rectangular grids. They can create layouts of any shape. As with IsoMatch, any other dimensionality reduction methods (such as t-SNE or UMAP) can be used to first project the HD input vectors onto the 2D plane, and then re-arrange them on the 2D grid. A fast placing approach can be found in Hilasaca et al. [HMJE21]. Any linear assignment scheme like the Jonker–Volgenant Algorithm [JV87] can be used to map the projected 2D positions to the best grid positions. Many non-linear dimensionality reduction methods have been recently proposed, but the question of their assessment and comparison remains open. Methods comparing HD and 2D ranks are reviewed in Refs. [LV08, LMBH11].
3 A New Quality Metric for Grid Layouts
Our goal is to develop a metric that better reflects perceived quality. The quality is to be expressed with a single value, where 0 stands for a random and 1 for a perfect arrangement. There are two approaches when designing a suitable quality function for grid layouts. The first option would be to refer to the best possible 2D sorting that can theoretically be achieved. However, this approach is not applicable because the best possible sorting is usually not known. The only viable way is to refer to the distribution of the HD data. A perfect sorting here means that all 2D grid distances are proportional to the HD distances. However, depending on the specific HD distribution, it is usually not possible to achieve this perfect order in a 2D arrangement (see Figure 3).
3.1 Neighbourhood preservation quality
Our initial approach towards a new evaluation metric was to combine the k-neighbourhood preservation index values to a single quality value. The values for a perfect arrangement Sopt and the expected value for random arrangements Srand are
(5)
where k is the evaluated neighbourhood size, K is the maximum number of neighbours which is the number of HD data elements−1. The expected NPk value for a random arrangement is , since more and more correct nearest neighbours are found as k increases.
For a given 2D arrangement S, we define the neighbourhood preservation gain as the difference between the actual value and the expected value for random arrangements.
(6)
The maximum is taken because theoretically an arrangement can be worse than a random arrangement. This happens very rarely, but if it does, the negative values are very small. Since an optimal arrangement preserves all HD neighbourhoods perfectly, we define
(7)
Figure 4 shows an example of four different primary colours, each used 64 times. All colours were slightly changed by some noise, resulting in 256 different colours. On the left side, two arrangements and the colour histogram are shown. The ΔNP curves are shown on the right. Here the optimal HD order cannot be preserved in 2D.
We determine the vectors of neighbourhood preservation gains of the actual 2D arrangement and of the perfect HD arrangement. We define the neighbourhood preservation quality as the ratio of the norms of these vectors. is close to 0 for a random and 1 for a perfect sorting.
(8)
Each 2D grid position has several other positions with the same spatial distance λ (e.g. the four nearest neighbours with a distance of 1). To determine for all k-values, the mean k-neighbourhood preservation index values must be determined for equal spatial distances. This ensures that isometries (rotated or mirrored arrangements) result in equal NPQ values.
To determine the neighbourhood preservation quality , the value for the p-norm must be chosen. Higher values give more weight to NPk values with smaller k, so the preservation of nearer neighbours becomes more important. In the case of very large p values, only the four adjacent positions are taken into account.
One problem with the proposed neighbourhood preservation quality is its sensitivity to noisy distances in the HD data. This often occurs when using visual feature vectors. As image analysis is not perfect, a feature vector can be considered as a ‘perfect’ vector that is disturbed by some noise. This effect can be seen in Figure 4. Although sorting S is rather good, the values are very low, especially for near neighbours. The top row of Figure 5 shows the resulting order when ranking different arrangements of this data set according to their NPQ values for . It can be seen that NPQ does not reflect the perceived sorting quality well.
3.2 Distance preservation quality
The problem of the proposed neighbourhood preservation quality consists of the fact that only the correct ranking of the neighbours is taken into account. The actual similarity of wrongly ranked neighbours is not considered. To address the noise-induced degradation of the neighbourhood preservation quality, we propose not to compare the correspondence of the closest neighbours, but to compare the averaged distances of the corresponding neighbourhoods and . For this, the average neighbour distances for the k closest neighbours are determined in HD and 2D:
(9)
It should be noted that the distances δ of the HD vectors are used for both the HD and the 2D neighbourhoods. The only difference is that the sets of the actual k nearest neighbours in HD and 2D are not the same if the 2D arrangement is not optimal.
Similar to the neighbourhood preservation quality, we compare the average neighbourhood distance with the expectation value of the average neighbourhood distance of random arrangements, which is equal to the global average distance of all HD vectors .
(10)
Analogous to , we define the distance preservation gain as the difference between the average neighbourhood distance of a random arrangement and the sorted arrangement.
(11)
Compared to ΔNP, the order of subtraction is reversed for ΔD, since a higher distance is considered instead of a lower neighbour preservation. Taking the difference between and the average neighbour distance ensures is approximately 0 for random arrangements. In theory, the division by is not necessary, but limiting the values to a range from 0 to 1 improves the numerical stability when calculating the norm of the distance preservation gain for larger p values. Figure 6 shows the ΔD curves of the previous example. It shows that for the sorted arrangement S, the values are much higher for small neighbourhoods k, indicating that close neighbours on the grid are similar. For neighbours with equal 2D distances, the mean of the corresponding HD distances was used.
The distance preservation quality is defined as the ratio of the p-norms of the distance preservation gains of the actual arrangement to a perfect arrangement:
(12)
For a random arrangement, will be approximately 0, for a perfect arrangement, will be 1. The influence of p is evaluated in the user study in Section 5.
In Barthel and Hezel [BH19], we have proposed a predecessor of this approach. For all vectors, the differences between the global average distance and the weighted distances to their neighbourhood vectors was determined in HD and in 2D. A projection quality was obtained as the ratio of the means of these differences in HD and 2D. A Gaussian based on the normalized neighbour ranks was used as the weighting function. The weighting of distances for higher ranks could be controlled by σ. Both, the choice of the weighting function and its parametrization were in some sense arbitrary.
Again, the problem of equal spatial distances must be considered when determining the average distances for the k nearest neighbours on the grid. There are two ways to approach this: One is to use the mean HD distance for neighbours with equal 2D distance. The other possibility is to sort these neighbours by their HD distance. The former would be a pessimistic estimate of , whereas the latter would be an optimistic estimate (see Figure 7). The use of mean HD distances for equal 2D distances is denoted as . Whereas DPQp denotes the use of sorted HD distances. Figure 5 shows a better ranking of arrangements when evaluated with DPQ quality than with NPQ (for ).
4 Our new Sorting Algorithm: Linear Assignment Sorting
First, we show how SOM and SSM can be optimized for speed and quality, which in combination leads to our new sorting scheme.
4.1 Speed and quality optimizations of SOM and SSM
The SOM described in Section 2.2 assigns each input vector to the best map vector and updates its neighbourhood. The map update can be thought of as a blending of the map vectors with the spatially low-pass filtered assigned input vectors, where the filter radius corresponds to the current neighbourhood radius. We propose to replace this time-consuming updating process: First, all input vectors are copied to the most similar unassigned map vector. Then, all map vectors are spatially filtered using a box filter. It is possible to achieve constant complexity independent of the kernel size by using uniform or integral filters [Lew94, VJ01]. Due to the sequential process of the SOM, the last input vectors can only be assigned to the few remaining unassigned map positions. This results in isolated, poorly positioned vectors.
The SSM avoids the problem of isolated, bad assignments by swapping the assigned positions of four input vectors at a time. To find the best swap, the SSM uses a brute force approach that compares the four input vectors with the four mean vectors of the blocks to which each swap candidate belongs. Due to the factorial number of permutations, adding more candidates would be computationally too complex. In order to still be able to use more swap candidates, we propose optimizing the search for the best permutation by linear programming. Another problem of the SSM is the use of a single mean vector per block, which incorrectly implies that all positions in the block are equivalent when they are swapped. The usage of a single mean vector per block can be considered as a sub-sampled version of the continuously filtered map vectors. Therefore, in Barthel and Hezel [BH19], we proposed using map filtering without sub-sampling, as this allows a better representation of the neighbourhoods of the map vectors. The block sizes of the SSM remain the same for multiple iterations, this can be seen as repeated use of the same filter radius. We propose continuously reducing the filter radius.
4.2 Linear assignment sorting
Our proposed new (image) sorting scheme called linear assignment sorting (LAS) combines ideas from the SOM (using a continuously filtered map) with the SSM (swapping of cells) and extends this to optimally swapping all vectors simultaneously. The principle of the LAS algorithm can be described as follows: Initially all map vectors are randomly filled with the input vectors. Then, the map vectors are spatially low-pass filtered to obtain a smoothed version of the map representing the neighbourhoods. In the next step, all input vectors are assigned to their best matching map positions. This is done by finding the optimal solution by minimizing the cost C:
(13)
is a binary assignment value, whereas is the distance between the input vectors and the map vectors . The power q allows the distances to be transformed in order to balance the importance of large versus small distances. Since the number of possible mappings is factorial, we use the Jonker–Volgenant linear assignment solver [JV87] to find the best swaps with reduced run time complexity and memory complexity .
The actual sorting is achieved by repeatedly assigning the input vectors and filtering the map vectors with a successively reduced filter radius. The principle of the LAS sorting scheme for a grid of size is summarized in Algorithm 3.
Algorithm 3. LAS
The only parameters of the LAS algorithm are the initial filter radius and the radius reduction factor, which controls the exponential decay of the filter radius and thus the quality and/or the speed of the sorting. Examining different q values for transforming the distances between the input and map vectors did not reveal much difference; in the interest of faster computations, we use .
LAS is a simple algorithm with very good sorting quality (see next section for results). However, for larger sets in the range of thousands of images, the computational complexity of the LAS algorithm becomes too high. However, with a slight modification of the LAS algorithm, very large image sets can still be sorted. Fast linear assignments sorting (FLAS) is able to handle larger quantities of images by replacing the global assignment with multiple local swaps, as described in Algorithm 4. This approach allows much faster sorting while having little impact on the quality of the arrangement. Comparisons between LAS and FLAS are given in the next section.
Algorithm 4. FLAS
The selection of FLAS parameters allows the control of the quality and speed of the sorting process. In this way, we generated many sorted arrangements of different quality, which were then used in the user study in Section 5. For an example implementation of the LAS and FLAS algorithms, the distance preserving quality DPQp together with the images, and the feature vectors used in this paper, see
5 User Study
5.1 Experiment design
To evaluate the proposed DPQp metric and the new sorting schemes (LAS and FLAS), an extensive user study was conducted. In a first experiment, we determined the correlation between user preferences and the quality metrics described in Sections 2.1 and 3.2. In a second experiment, we examined the relationship between the time required to find images in arrangements and the metrics' quality scores and the users' ratings, respectively.
5.1.1 Image sets
Figure 8 shows the four image sets used in the experiments. The first set consists of 1024 random RGB colours. The random selection implies that there is no specific low-dimensional embedding that can be exploited to project the data to 2D. While the RGB colour set is a somewhat artificial set, we also used image sets.
In advance, we conducted tests with various image sets of different sizes and colour distributions. It became apparent that some image sets, regardless of the arrangement, were too difficult for users to search. With smaller image sets, on the other hand, the arrangement of the images had little effect on the speed of the search, as the searched images could often be spotted immediately. The three chosen image sets covered different scenarios, where a significant difference in search performance between sorted and random arrangements can be observed. The first set consists of 169 images of traffic signs taken from Pixabay [BS10] in 2017, excluding photos of real traffic signs and nearly identical images. This set contains several groups of visually similar images as might be found when searching for signs or logos. The second set consists of 256 images of kitchen items crawled from the IKEA website in 2016. From a total of 10,262 images, all images with kitchen items were selected. Images with multiple or small objects, duplicates and many very colourful images that are potentially easy to find were removed. This set is an example of what one might find on e-commerce websites. Some of these images are very similar, which makes it difficult to find them. The last set consists of 400 images for 70 unrelated concepts crawled from the Internet. This set was chosen because the low-level feature vectors used in the experiment are capable of describing images of the same class with similar feature vectors. This ensures that a supposedly bad arrangement is not due to a poor description by the feature vectors used. This set has an uncharacteristically high proportion of coloured images, but there are many images that belong to different object categories despite similar appearance, potentially complicating visual retrieval.
5.1.2 Feature vectors
For the RGB colour set, the R, G and B values were taken directly as vectors. Theoretically, the Lab colour space would be better suited for human colour perception, but even the Lab colour space is not perceptually uniform for larger colour differences. To ensure easy reproducibility of the results, we kept the RGB values.
For images, one might expect that feature vectors from neural networks would be best suited to describe them, which is definitely true for retrieval tasks. However, when neural feature vectors are used to visually sort larger sets of images, the arrangements often look somewhat confusing because images can have very different appearances even though they represent a similar concept (see Figure 9). Since people pay strong attention to colours and visually group similar-looking images when viewing larger sets of images, feature vectors describing visual appearance are usually more suitable for arrangements that are perceived as ‘well-organized’. For this reason, in our experiment, we used 50 dimensional low-level feature vectors describing the colour layout, the fuzzy YCoCg colour histogram and the MPEG-7 edge histogram of the images. However, the choice of feature vectors has limited impact on the experiments performed, since all sorting methods use the same feature vectors and the metrics indicate how well their similarities are preserved.
5.1.3 Implementation
We organized the experiment as online user tests, where participants could take part in a raffle after completing the experiments. It was possible to perform the experiment more than once, but it was ensured that participants would not see the same arrangements twice. In total, more than 2000 people participated in the experiment. About half of them were employees and students of our university, coming from different fields of study, and only a small proportion was from the field of computer science. Little is known about the other participants, as the experiment was advertised as a raffle on various websites.
5.1.4 Investigated sorting methods and metrics
In our experiments, we used sorted arrangements generated with the following methods: SOM, SSM, IsoMatch, LAS, FLAS and the t-SNE 2D projection that was mapped to the best 2D grid positions (indicated as t-SNEtoGrid). Several of these generated arrangements were then selected based on the range of variation in sorting results per method. The UMAP method was not investigated because in many cases, its KNN graph broke into multiple components, which made an arrangement onto the 2D grid impossible. In order to also have examples of low quality for comparison, some sorted arrangements were generated with FLAS using poor parameter settings (indicated as Low Qual.).
The evaluated quality metrics were the Energy function and (Equation 4) and the distance preservation quality DPQp (Equation 12) with different p values. As the normalized energy function and cross-correlation provide an almost identical quality ranking for different arrangements, we did not evaluate the cross-correlation metric.
5.2 Evaluation of user preferences
In the first experiment, pairs of sorted image arrangements were shown. Users were asked to decide which of the two arrangements they preferred in the sense that ‘the images are arranged more clearly, provide a better overview and make it easier to find images they are looking for’.
Figure 10 shows a screenshot of this experiment. All users had to evaluate 16 pairs and decide whether they preferred the left or the right arrangement. They could also state that they considered both to be equivalent. To detect misuse, the experiment contained one pair of a very good and a very bad sorting. The decisions of users who preferred the bad sorting here were discarded. The number of different arrangements were 32 for the colour set and 23 each for the three image sets, (giving 496 pairs for the colour set and 253 pairs for each image set). Each pair was evaluated by at least 35 users.
For each comparison of with , the preferred arrangement gets one point. In case of a tie, both get half a point each. Let be the points received by in the rth out of R comparisons between and . Let
(14)
be the probability that receives a higher quality assessment in comparison to , (). The final user score for is defined by
(15)
Because the number of comparisons per pair was quite high and nearly constant, sophisticated methods for unbalanced pairwise comparison such as the Bradley–Terry model [BT52, Hun04] were not necessary since they provided an equal ranking.
The overall result of the user evaluation of the arrangements is shown in Figure 11. Figures 12 and 13 show the relationship between user ratings and the values of the and metrics for the colour set and the three image sets. It can be seen that the Pearson correlation is significantly higher for compared to . In the case of RGB colours, users liked the LAS arrangements the best. For the image sets, there is no clear winning method. The t-SNEtoGrid method obtained rather low scores for the RGB colours, but much higher ones for the image sets.
Figure 14 shows the degree of correlation between user scores and quality metrics for different p values of the and the DPQp metrics. For all four sets, the correlation of DPQ with the user scores is higher than that of and for all p values. For predicting user scores, values (using mean HD distances for equal 2D distances) with higher p values give the best results (left of Figure 14). The high correlation for larger p values with the user scores could indicate that users essentially pay attention to how well the immediate nearest neighbours have been preserved.
5.3 Evaluation of user search time
In the second part of the user study, the users were shown different arrangements in which they were asked to find four images in each case. The four images to be searched were randomly chosen and shown one after the other. As soon as one image was found, the next one was displayed. Participants were asked to pause only when they had found a group of four images, but not during a search. At the beginning, users were given a trial run to familiarize themselves with the task. Here, the time was not recorded. Figure 15 shows a screenshot of the search experiment. The overall set of arrangements was identical to those from the first part of the experiment, in which the pairs had to be evaluated.
Obviously, the task of finding specific images varies in difficulty depending on the image to be found. In addition, the participants are characterized by their varying search abilities. However, a total of more than 28,000 search tasks were performed, each with four images to be found. This means that for each arrangement, more than 400 search tasks were performed for four images each. This compensated for differences in both the difficulty of the search and the abilities of the participants.
The search times required for each of the 23 arrangements per image set were recorded. It was found that the time distribution of the searches is approximately log-normal. Search times that fell outside the upper three standard deviations were discarded to filter out experiments that were likely to have been interrupted. Figure 16 shows the search time distribution of different arrangements for the three image sets. The median values of the search times of the different arrangements are shown as coloured markers. Again it can be seen that the correlation of the median search times is higher with the DPQ16 metric than with the metric.
Figure 17 shows the median values of the search times for all arrangements, sorted from the arrangement where the images were found the fastest to the one where the image search took the longest. The standard error of the median search times was determined by bootstrapping with 10,000 runs. While the ranking order of the algorithms is similar to the order of the user preferences, it occasionally differs, suggesting that an apparently well-sorted arrangement is only conditionally indicative of finding images quickly.
To evaluate the degree of correlation between search time and the quality metrics, Figure 18 compares the normalized energy function ( and ) with the distance preservation quality (DPQq and ). Again it can be seen that DPQp outperforms . Contrary to the user preference evaluation, for image retrieval tasks, DPQq performs slightly better than . Both show maximum correlation for high p values. This in turn indicates that it seems to be most important for people to locate similar images very close to each other in order to find them quickly, as hypothesized in Figure 7.
We also investigated the use of the squared L2 distance instead of the L2 distance to calculate the DPQ, the correlations remained similar, but were slightly lower. Other distance transformations could be investigated, but this is beyond the scope of this paper.
6 Qualitative and Quantitative Comparisons
6.1 Quality and run time comparison
To get a better understanding about the behaviour of FLAS and other 2D grid-arranging algorithms using different hyperparameter settings, we conducted a series of experiments. Since the run time strongly depends on the hardware and implementation quality, the numbers given in this section only serve as comparative values. In the previous section, DPQ16 has shown high correlation with user preferences and performance, we, therefore, use it when comparing algorithms in terms of their achieved ‘quality’ and the run time required to generate the sorted arrangement.
At this point, it is important to emphasize that LAS and FLAS, and likewise SOM and SSM, do not optimize an objective quality function or metric, unlike many other dimensionality reduction methods. The only used methods that perform an optimization in terms of a quality function or a metric are IsoMatch and t-SNE. IsoMatch attempts to maximize the normalized energy function , while the t-SNE objective is to make HD and 2D similarity distributions as similar as possible by minimizing their Kullback–Leibler divergence.
Our test machine is a Ryzen 2700x CPU with a fixed core clock of 4.0 GHz and 64GB of DDR4 RAM running at 2133 MHz. The tested algorithms were all implemented in Java and executed with the JRE on Windows 10. Only the single-threaded sorting time was measured. As much code as possible was re-used (e.g. the solver of LAS, FLAS, IsoMatch and t-SNEtoGrid) to make the comparison as consistent as possible.
The Isomap and t-SNE projection implementation is from the popular library SMILE [Li14] (version 2.6). The SSM code is an implementation adapted from Strong and Gong [SG14] to match the characteristics of our implementation of SOM, LAS and FLAS. At startup, all data is loaded into memory. Then the averaged run time and DPQ16 value of 100 runs were recorded. We ensured the algorithms received the same initial order of images for all runs.
There are different hyperparameters that can be tuned. Some of them affect the run time and/or the quality of the arrangement, while others result in only minor changes. Figure 19 shows the relationship between speed and quality when varying the hyperparameters. For t-SNE, SSM and SOM, the number of iterations were changed, the t-SNE learning rate (eta) was set to 200. For LAS and FLAS, the radius reduction factor was gradually reduced from 0.99 to 0, while the initial radius factor was 0.35 and 0.5, respectively. FLAS used nine swap candidates per iteration. IsoMatch has only the k-neighbour setting which does not influence the quality nor the run time and therefore produces only a single data point in the plots. For small data sets like the 256 kitchenware images, FLAS offers the best trade-off between DPQ and computation time. LAS and t-SNE can produce higher DPQ16 values but are 10–100 times slower. There is no reason to use a SSM or SOM, since both are either slower or generate inferior arrangements. For the 1024 random RGB colours, LAS and FLAS yielded the highest DPQ.
In order to compare the scalability of the analysed algorithms, three data sets of different sizes were analysed, containing 256, 1024 and 4096 random RGB colours. The hyperparameters of the points marked with a ⦿ in Figure 19 were used for all the tests shown in Figure 20. It can be seen that FLAS and SSM have the same scaling properties, while FLAS exhibits better qualities. Even higher DPQ values can be achieved by LAS at the expense of run time. As the number of colours increases, arrangements with smoother gradients become possible, resulting in better quality. Most approaches can exploit this property, except for IsoMatch and t-SNEtoGrid. Both initially project to 2D and rely on a solver to map the overlapping and cluttered data points to the grid layout. Since the number of grid cells is equal to the number of data points, it is difficult for the solver to find a good mapping. This often results in hard edges, as can be seen in Figure 21.
To summarize, LAS can be used for high quality arrangements, while FLAS should be used if the number of images is very high (several thousand) or if fast execution is important.
6.2 Visual comparison
While the quantitative qualities of the various algorithms are quite similar in some cases, visual inspection reveals specific differences.
For the 1024 RGB colour data set, the run closest to the DPQ16 mean was selected from the 100 test runs used for Figure 20. The corresponding arrangements are shown in the order of their distance preservation quality DPQ16 in the top of Figure 21. In addition, the normalized energy function value () is given. LAS has the smoothest overall arrangement, followed by FLAS and SSM. The SOM arrangement is disturbed by isolated, poorly positioned colours, while the t-SNEtoGrid approach shows boundaries between regions. This is due to previously separate groups of projected vectors being re-distributed over the grid, resulting in visible boundaries where these regions touch. The noisy looking arrangement of IsoMatch is caused by the normalized energy function trying to equally preserve all distances. This leads to a kind of dithering of the vectors respectively the colours.
Most of these effects are less visible when real images are used instead of colours. The lower row of Figure 21 shows the arrangements from our user study 5.3, which required searching images in the web images set. The DPQ16 values and the values are given. The arrangements are ordered by the median time it took users to find the images they were looking for (fastest on the left to slowest on the right). t-SNEtoGrid again shows some boundaries between regions, but this time the boundaries apparently help to better identify the individual groups of images, reducing the time needed to find them. The LAS, FLAS, SSM and SOM arrangements have a similar appearance for this data set. The dithered appearance of the IsoMatch arrangement apparently makes it difficult for users to quickly find the images they are looking for.
7 Applications
In this section, we present a variety of applications that use our algorithms introduced in Section 4.2 to efficiently manage or search larger sets of images.
7.1 Image management systems
For browsing local images on a computer, a visually sorted display of the images can help to view more images at once. Given user-scrolling of a view tends to be in a vertical direction, it is important that the images on one horizontal line are similar to each other, and one perceives obvious changes in the vertical direction. This can be achieved by using a larger filter radius for horizontal filtering. See Figure 22 showing the PicArrange app [Jun21] as an example.
7.2 Image exploration
For very large sorted sets with millions of images, it may be useful to use a torus-shaped map which gives the impression of an endless plane. If one can navigate this plane in all directions, it is possible to bring regions of interest into view. Such an arrangement can be achieved by using a wrapped filter operation. This means part of the low-pass filter kernel uses vectors from the opposite edge of the map. Figure 23 shows a (small) example of such a torus-shaped arrangement. If zooming is possible, images of interest can be found and inspected very easily. This idea can be combined with a hierarchical pyramid of sorted maps that allows visual exploration for huge image sets. The user can explore the image pyramid with an interface similar to the mapping service (e.g. Google Maps). By dragging or zooming the map, other parts of the pyramid can be explored. The online tool wikiview.net [Bar19] allows the exploration of millions of Wikimedia images using this approach.
7.3 Layouts with special constraints
Although our algorithms work with rectangular grids, other shapes can also be sorted. The map has the size of the rectangular bounding box of the desired shape, however, only the map positions within the shape can be assigned. Figure 24 shows an example where the colours to be sorted were only allowed within the shape of a heart. The corresponding algorithm remains the same, the only difference is that the vectors of the assigned map positions are used to fill the rest of the the map positions. Each unassigned map position is filled with the nearest assigned vector of the map's constrained positions.
Sometimes it is desirable to keep some images fixed at certain positions (see Figure 25). This is possible with two minor changes to the algorithm: In a first step, the images or the corresponding vectors are assigned to the desired positions. These positions are then never changed again. Also, an additional weighting factor is introduced for filtering, where the fixed positions are weighted more. This results in neighbouring map vectors becoming similar to these fixed vectors, which in turn results in similar images also being placed nearby. The rest of the algorithm remains the same.
8 Conclusions
We presented a new evaluation metric to assess the quality of grid-based image arrangements. The basic idea is not to evaluate the preservation of the HD neighbour ranks of an arrangement, but the preservation of the average distances of the neighbourhood. Furthermore, we do not weight all distances equally either, because for humans, the preservation of small distances seems to be more important than that of larger distances. User experiments have shown that DPQ better represents human-perceived qualities of sorted arrangements than other existing metrics. If the overall impression of an arrangement is to be evaluated, the metric had a small advantage. For predicting how fast images can be found for an arrangement, DPQp was better. In general, however, these differences are small and we recommend always using DPQp.
Large p values lead to the highest correlations with user perception. This implies that for a ‘good’ arrangement, it is essentially important that the sum of the distances to the immediate neighbours on the 2D grid is as small as possible. The same is true for one-dimensional sorting, which is ‘optimal’ only if the sum of the differences to the direct neighbours is minimal. It remains to be investigated whether DPQ is a useful metric for evaluating the quality of other, non-grid-based dimensionality reduction methods.
Furthermore, we have presented LAS which is a simple but at the same time very effective sorting method. It achieves very good arrangements according to the new metric as well as for other metrics. The FLAS variant can achieve better arrangements than existing methods with reduced complexity. In single-threaded execution, a CPU can sort tens of thousands of data vectors in a fraction of a second and over a million in less than 30 s. The FLAS algorithm is fully parallel because different swappings do not interfere with each other. Therefore, the algorithm can run efficiently on parallel hardware.
The ideas presented in this paper can be developed in numerous directions. Since the new DPQ metric allows a better prediction of the quality of an arrangement, it also better predicts the expected search time. If a sorting scheme were optimized in terms of DPQ, searched images would also be found faster. In this context, it remains to be investigated how a sorting algorithm can be optimized directly in the sense of a high DPQ value.
The filtering approach used determines mean HD distances for equal 2D distances. We will investigate whether further improvement is possible by exploiting the fact that for equal 2D distances on the grid, the sorted HD distances better represent the perceived quality.
Currently, our sorting method only supports regular grids. We want to investigate how this approach can be extended to densely packed rectangles of various sizes.
Acknowledgement
Open access funding enabled and organized by Projekt DEAL.
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sequences and series both arithmetic and Geometric
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benjibooSenior | Next Rank: 100 Posts
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sequences and series both arithmetic and Geometric
bybenjiboo » Wed Oct 28, 2009 6:40 pm
I want to make this more clear for people who stumble on this post in the future. The following is meant to help one understand the entire topic that this falls under.
Please note that the formatting is a little weird on this forum, so feel free to view it ... 85969.html here. Its the SAME THING but one the exponents are without the ^ sign.
PART I
Topic: Sequences and Series
There are two types of sequences and two types of series. They are geometric sequences and arithmetic sequences, and geometric series and arithmetic series.
Geometric sequence vs arithmetic sequence
An arithmetic sequence is a sequence of numbers where each new term after the first is formed by adding a fixed amount called the common difference to the previous term in the sequence.
Set A={1,2,3,4,5,6,7,8,9,10}
Set B={2,4,6,8,10,12,14}
Set C={3,8,13,18,23,28}
In 'set A', the common difference is the fixed amount of one. In 'set B' the common difference is the fixed amount of two, and in 'set C' the common difference is the fixed amount of five. As you most likely noticed already, the common difference is found by finding the difference between two consecutive terms within the sequence. For example, in 'set C', to find the common difference compute (8-3=5).
A geometric sequence on the other hand, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio.
Set D={2,4,8,16,32}
Set E={3,9,27,81}
Set F={5,10,20,40,80}
You might notice that the difference between consecutive numbers in the above three sets are not a fixed amount. For instance, in 'set F', the first two terms (5 and 10) have a smaller difference than the last two terms (40 and 80). Therefore, the above sets are geometric sequences. The difference between two consecutive numbers is therefor the common ratio. To find the common ratioyou simply take the ratio one consecutive number to the one before it. In 'set F' this would be (10/5=2). Therefore, n 'set F' the common ratio is two. In 'set E' the common ratio is (27/9=3). In 'set' D' the common ratio is two (32/16=2).
Summary
The difference between the two types of sequences is that in arithmetic sequences the consecutive numbers in a set differ by a fixed amount known as the common difference whereas in a geometric sequence the consecutive numbers in a set differ by a fixed number known as the common ratio.
Sequence vs Series
This is quite simple. A sequence is a list of numbers. A series is created by adding terms in the sequence. There you go, now you know the difference. So if you take 'set A' and add the terms then you have an arithmetic series. If you take 'set D' and add the terms, then you have a geometric series.
Sequence: {1, 3, 5, 7, 9, ...}
Series: {1+3+5+7+9+...}
What the GMAT could ask us to do with sequences and series and how to do it!
There is no limit to what the GMAT can ask you to find when dealing with series and sequences. Here are some examples of things you may be asked to find/do with them.
(1) The sum of numbers in a series (which can be asked in many tricky ways such as the sum of all the numbers, sum of just the even numbers, sum of just the odd numbers, sum of only the numbers which are multiples of 7, sum of the first 10 numbers, and many more tricky ways!)
(2) The nth term in a sequence
(3) How many integers are there in a sequence
Anyway, now that you get the point... lets give you the formulas that will allow you to answer any question regarding series and sequences. I will then show you how to use the formulas to answer some questions that might not be intuitive of non math geniuses.
Formula for geometric sequence (when there is a common ratio)
dark green means subscript
Recursive (to find just the next term):
a n = a n-1 r
Explicit (to find any nth term):
a n = a 1 [m]r^n-^1[/m]
a n = nth term
a 1 = the first term
r = common ratio
In reality, you only need to know the explicit formula, because you can find any term with it. I only put the recursive formula for understanding.
Formula for arithmetic sequence (when there is a common difference)
dark green means subscript
recursive:
a n = a n-1 + d
Explicit:
a n = a 1 + (n-1)d
a n = nth term
a 1 = first term
d = common difference
Again, you only need to know the explicit formula, because you can find any term with it. I only put the recursive formula for understanding.
Formula for geometric series (when there is a common ratio)
dark green means subscript
S n = a 1[m]fraction/(1-r)[/fraction][/m]
S n = Sum of first nth terms
a 1 = first term
r = common ratio
n = nth term
Formula for arithmetic series (when there is a common ratio)
dark green means subscript
S n = [m][fraction]n/2[/fraction]/m
or
S n = [m][fraction]n/2[/fraction]/m
The above two equations are the same (I put them in both ways because some prep programs teach "first + last" but it is important to see that in the first of the two, the last term is identified as a n. Well what if you do not know the last term? Then you have to calculate it using the equation for the nth term (solving for a n) of an arithmetic sequence which is listed above... or you can substitute the formula for a n into the first one of these two by replacing an with what is equals and simplifying. You get the following:
S n = [m][fraction]n/2[/fraction][/m][2 a + (n-1)d]
Sn = sum of the series
a 1 = the first term
a n = the nth term
n = the number of terms
d = the common difference
PART 2
Now that we know all this information, there are some important things that are understood as well to ensure that the formulas are used correctly.
How to find the number of integers in a set
(Last term - First term) + 1
A mistake is that people will forget to add the 1. The number of terms between 3 and 10 is not 7, it is 8. A common mistake is that people will calculate (10-3=7)... but this is wrong. Remember, as Manhattan GMAT says, "Add one before you are done".
Notice how I used the word "term" and not number. This is important because sometimes you don't always just put the first and last number you are given. For example, If you are asked to find the number of even integers between 1 and 30, you don't use the "first number" in the set. The first number is "1", which is odd, and we are only speaking about even numbers. Therefore, the first term is "2", not "1", even though the set or question might have stated "from 1-30". Same goes with the last term. There is another step needed to answer this question though.
Find number of odd integers (or even) in a set
([m]fraction/2[/fraction][/m] + 1
If the question is to find the number of odd integers between 2 and 30, then your first term is 3, and your last term is 29. They must be odd to fit in the set you are asked to analyze.
If the question is find the number of even integers between 3 and 29, then your first term is 4, and your last term is 28.
Find number of integers that are a multiple of a certain number in a set
GMAT questions can get tricky, but luckily not too tricky. For example... What if you are asked to "find the number of multiples of 7 between 2 and 120"?
([m]fraction/increment[/fraction][/m]) + 1
[m][fraction/4[/fraction]][/m]+ 1
All you have to do is instead of dividing our old formula by 2, you divide it by the increment. Also, notice how my first and last terms are the first term that is a multiple of 7 and the last term that is a multiple of seven within the set!
Sum of odd numbers in a series
This seems to be a popular topic on GMAT forums. Its quite simple. You already know everything you need to after reading this post. It is a two step problem. Here are the two steps:
(1) Find the number of odd terms. This is you "n" value now.
(2) Plug in the "n" value into the formula for an arithmetic series.
PART III
There are some short cuts and concepts that you should know about this topic.
(1) The mean and the medium of any arithmetic sequence is equal to the average of the first and last terms.
(2) The sum of an arthritic sequence is equal to the mean (average) times the number of terms.
(3) The product of n consecutive integers is always divisible by n! So, 4x5x6 (456=120) is divisible by 3!
(4) If you have an odd number of terms in consecutive set, the sum of those numbers is divisible by the number of terms.
(5) number four (above) does not hold true for consecutive sets with an even amount of terms.
Thank you
Benjiboo
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3425 | https://pmc.ncbi.nlm.nih.gov/articles/PMC9156507/ | Hormonal treatments for endometriosis: The endocrine background - PMC
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Hormonal treatments for endometriosis: The endocrine background
Silvia Vannuccini
Silvia Vannuccini
1 Obstetrics and Gynecology, Department of Experimental, Clinical and Biomedical Sciences, University of Florence, Careggi University Hospital, Florence, Italy
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1, Sara Clemenza
Sara Clemenza
1 Obstetrics and Gynecology, Department of Experimental, Clinical and Biomedical Sciences, University of Florence, Careggi University Hospital, Florence, Italy
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Margherita Rossi
1 Obstetrics and Gynecology, Department of Experimental, Clinical and Biomedical Sciences, University of Florence, Careggi University Hospital, Florence, Italy
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Felice Petraglia
1 Obstetrics and Gynecology, Department of Experimental, Clinical and Biomedical Sciences, University of Florence, Careggi University Hospital, Florence, Italy
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1 Obstetrics and Gynecology, Department of Experimental, Clinical and Biomedical Sciences, University of Florence, Careggi University Hospital, Florence, Italy
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Accepted 2021 Jun 15; Issue date 2022.
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PMCID: PMC9156507 PMID: 34405378
Abstract
Endometriosis is a benign uterine disorder characterized by menstrual pain and infertility, deeply affecting women’s health. It is a chronic disease and requires a long term management. Hormonal drugs are currently the most used for the medical treatment and are based on the endocrine pathogenetic aspects. Estrogen-dependency and progesterone-resistance are the key events which cause the ectopic implantation of endometrial cells, decreasing apoptosis and increasing oxidative stress, inflammation and neuroangiogenesis. Endometriotic cells express AMH, TGF-related growth factors (inhibin, activin, follistatin) CRH and stress related peptides. Endocrine and inflammatory changes explain pain and infertility, and the systemic comorbidities described in these patients, such as autoimmune (thyroiditis, arthritis, allergies), inflammatory (gastrointestinal/urinary diseases) and mental health disorders.
The hormonal treatment of endometriosis aims to block of menstruation through an inhibition of hypothalamus-pituitary-ovary axis or by causing a pseudodecidualization with consequent amenorrhea, impairing the progression of endometriotic implants. GnRH agonists and antagonists are effective on endometriosis by acting on pituitary-ovarian function. Progestins are mostly used for long term treatments (dienogest, NETA, MPA) and act on multiple sites of action. Combined oral contraceptives are also used for reducing endometriosis symptoms by inhibiting ovarian function. Clinical trials are currently going on selective progesterone receptor modulators, selective estrogen receptor modulators and aromatase inhibitors. Nowadays, all these hormonal drugs are considered the first-line treatment for women with endometriosis to improve their symptoms, to postpone surgery or to prevent post-surgical disease recurrence. This review aims to provide a comprehensive state-of-the-art on the current and future hormonal treatments for endometriosis, exploring the endocrine background of the disease.
Keywords: Activin, AMH, Aromatase inhibitors, CRH, Dienogest endometriosis, Estrogens, Progesterone-resistance, GnRH agonist, GnRH antagonist, Hormones, Inflammation, Inhibin, Progestin, SERMs, SPRMs, Stress
Introduction
Endometriosis is a chronic disease characterized by the presence of endometrium-like tissue outside the uterine cavity, affecting women of reproductive age with pelvic pain and infertility . The prevalence ranges between 2 and 10% of women in reproductive age, 30–50% among infertile women, and 5 to 21% among women with severe pelvic pain . However, the true prevalence is uncertain, because estimates vary widely among population samples and diagnostic approaches .
The pathophysiology of endometriosis is still a matter of investigation, but endocrine and inflammatory backgrounds are well characterized, recognizing an estrogen-dependency and a progesterone-resistance . The main mechanisms involved in the ectopic location of endometrial cells include retrograde menstruation, vascular and lymphatic spread and/or metaplasia/stem cells. The most accepted theory is the retrograde menstruation, according to which menstrual endometrial fragments migrate through the fallopian tubes to the peritoneal cavity, where they implant, proliferate and invade pelvic peritoneum. The back flow of endometrial cells into the pelvis is physiologic, resulting apoptosis/autophagy and cell-mediated immunity the scavenger system for eliminating these cells, while in endometriotic patients hormonal influences and genetic/epigenetic factors determine an impairment of these mechanisms, promoting cell survival, proliferation and peritoneal invasion. . Increased estrogen receptors activity, estrogen production in endometriotic lesions and progesterone-resistance are the determinants of impaired apoptosis, reduced immune function and increased inflammation [7–9]. Thus, endometriotic cells attach, penetrate and invade the peritoneum, determining growth of lesions which undergo cyclic bleeding with repeated tissue injury and repair , neoangiogenesis and neurogenesis . Fibroblast–myofibroblast transdifferentiation contributes to collagen production and fibrogenesis , with entrapment of nerve fibers which, associated with chronic inflammation, explain pain symptoms.
According to the location of the lesions three phenotypes of endometriosis are recognized: ovarian endometriomas (OMA) (the most common, characterized by typical chocolate cysts) superficial peritoneal endometriosis (SUP) and deep infiltrating endometriosis (DIE) (the most severe forms developing deeper than 5 mm under the peritoneal surface also infiltrating the muscularis propria of bladder or bowel) . In addition, extraperitoneal locations are described, i.e. pleura, diaphragm or umbilicus and 30% of cases endometriosis are associated to adenomyosis (infiltration by endometrial stroma and glands into the myometrium) [15, 16].
The most common symptoms of endometriosis are menstruation-related pain, i.e. dysmenorrhea, dyspareunia, dysuria and dyschezia, and noncyclic pelvic pain may also occurs in these patients. Since these symptoms are not specific to endometriosis and may be signs of other gynecological or non-gynecological conditions, misdiagnosis or a significant delay in endometriosis identification is frequently reported .
Painful symptoms and infertility are also associated with psychological stress, low self-esteem, and depression impairing physical, mental, and social well-being and reducing quality of life (QoL) . Therefore, these patients, other than hypothalamus-pituitary-ovary axis (HPO) changes, show also an impairment of hypothalamus–pituitary–adrenal axis (HPA) and thyroid function, and comorbidities associated with inflammation and immune dysfunction.
In the last two decades an increased diagnosis/incidence of endometriosis has been observed and its chronic and progressive nature determines a relevant impact in a lifelong perspective among these patients. In the past, surgery was considered the definitive treatment, but recent evidences showed that it does not solve the pathogenetic mechanisms and patients need a long-term management. Treatment goals are pain control and fertility improvement maximizing the use of medical treatment, but also postsurgical prevention of symptoms and lesions recurrence, in order to avoid repeated surgical procedures [20, 21]. In fact, surgery in women with endometriosis is associated with the risk of urological, intestinal, vascular and neurological complications and pain may recur or persist in case of incomplete excision of endometriosis lesions [22, 23]. Presently, the medical therapy is considered the first-line treatment for most of women with endometriosis to improve their symptoms, but also to plan the most adequate timing of surgery or assisted reproductive technologies (ART) treatment, or to prevent post-surgical disease recurrence [1, 24, 25]. The choice of the most appropriate therapy is based on the intensity of pain, age, desire to conceive, but also on the impact of the disease on QoL on each patient .
Currently, hormonal treatments are the most effective drugs for the treatment of endometriosis and are based on the pathogenic mechanisms involved in the disease. The goal is to stop cyclic menstruation: by blocking ovarian estrogen secretion or by causing a pseudopregnancy state . The endocrine background provides the rationale for the current and future hormonal drugs for treating women with endometriosis.
Endocrine changes in endometriosis
HPO axis hormones
FSH and LH
No significant difference in terms of serum follicle-stimulating hormone (FSH) and luteinizing hormone (LH) levels were found between women with endometriosis and controls. However, some FSHR and LHR single nucleotide polymorphisms (SNPs) have been observed in endometriosis patients . FSHR 680Ser-Ser/GG genotype and ‘‘GG/307Ala680Ser’’ haplotype were more frequently found in fertile women with endometriosis, while the presence of "GA/307Ala680Asn" haplotype lowers the likelihood of disease onset and progression [28, 29]. Besides, the SS (680 Ser/Ser) or AA (307 Ala/Ala) genotype are associated with a reduced risk to develop stage 3–4 endometriosis compared to the stage 1–2 endometriosis . FSHR 680Asn/Asn induces aromatase activity resulting in higher estrogens levels and proliferation of endometriotic lesions . Among LHR SNPs, a polymorphic insertion in exon 1 of LH receptor (LHR) gene (insLQ) is common in women with endometriosis and infertility, and it is thought to boost LHR activity, by decreasing the half maximal effective concentration and increasing the cell surface expression .
Estrogens and ERs
Although serum estrogen levels in endometriosis patients are not significantly different from those in healthy women, it is clear that estrogen-mediated alterations play a role in the etiology of endometriosis: an estrogen dominance is caused by to a local estrogens synthesis and to an increased ERs activity in endometriotic cells (Fig.1). Estrogens are a significant biologic driver of chronic inflammation, promoting endometriotic cell survival and lesion progression. Clear data show that endometriotic tissue expresses the entire set of steroidogenic genes, including aromatase, allowing to local de-novo estradiol (E2) production. . Local E2 levels are increased in endometriosis due to upregulation of the aromatase gene CYP19A1 and reduction of 17-hydroxysteroid dehydrogenase type 2 (17HSD2), which normally (induced by P4) converts E2 to the less potent estrone [35, 36]. The endometriotic stromal cells are epigenetically dysregulated and express steroidogenic proteins and enzymes such as steroidogenic acute regulatory protein (STAR) and convert the precursor molecule cholesterol to E2. The fundamental event in E2 synthesis is the recruitment of the nuclear receptor steroidogenic factor-1 (SF-1) to the promoters of these steroidogenic genes is the key event for E2 synthesis .A feed-forward loop connects hyperestrogenic stimulation with inflammation: the overexpression of cyclooxygenase 2 (COX2) and CYP19A1 increases local production of prostaglandins and estrogen, causing a vicious circle . The overproduction of estradiol in endometriosis drives ERβ signaling to support endometriotic tissue survival and inflammation.
Fig. 1.
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Estrogen receptors activity and local estrogens production in endometriosis. 17β-HSD, 17β-hydroxysteroid dehydrogenase. E2, estradiol. E3, estrone
In terms of alterations in ERs activity, an overexpression of ERβ and a downregulation of ERα [39, 40] have been observed in endometriosis. Changes in promoter methylation may be a cause for the increased ERβ/ERα ratio in endometriotic cells, since regions of the ERα promoter become hypermethylated leading to a decreased expression, whereas a CpG island in the ERβ promoter becomes hypomethylated causing an increased expression [41, 42] (Fig.1). When compared to controls, ERα levels are higher in the eutopic endometrium of women with endometriosis, resulting in enhanced estrogenic activity and proliferation, which impacts endometrial function. ERβ expression is unchanged in eutopic endometrium of women with endometriosis, although an increased ERβ/ERα ratio has been observed . An important role is played by steroid receptor coactivators (SRCs) , and expression profiling of SRCs in endometriotic lesions identified SRC-1 as the predominant SRC . Despite a drop in overall SRC-1, levels of a truncated form are increased in animal and human models. This new isoform of SRC-1 in vitro decreases tumor necrosis factor alpha (TNFα)-mediated apoptosis in endometriotic cells, promoting increased cell survival and invasion and reflecting the in vivo disease pathophysiology . In addition, SRC-1 isoform and ERβ may mediate a synergistic role in promoting cell survival in endometriosis .
Estrogens have a major role for endometriotic tissue attachment to peritoneum, lesion survival, production of inflammatory substances (metalloproteinases, cytokines, or prostaglandins and growth factors) and angiogenesis. ERβ triggers pathways that enhance lesion survival, remodel pelvic peritoneal tissue, and produce inflammatory substances, which stimulate nociceptors in pelvic tissues, leading to pain . The pathologic levels of local estradiol biosynthesis seems to induce also decrease of apoptosis in endometriotic stromal and epithelial cells compared with eutopic endometrial tissues [48–50]. Estrogens also mediate immune system dysregulation in endometriotic lesion. Peritoneal fluid macrophages from women with endometriosis upregulate the expression of ERβ and in a mouse model of endometriosis E2 treatment increases the macrophages present in lesions as well as the expression of macrophage migration factor [51, 52].
Progesterone and PRs
Circulating progesterone (P4) levels are similar to those found in healthy women. In endometriosis a typical dysregulation of progesterone signaling and an endometrial tissue inability to appropriately respond to progesterone exposure identifies the condition of progesterone resistance. It manifests in endometriosis as failed induction of PRs activation, or P4 target gene transcription in presence of bioavailable P4 . Progesterone resistance has been well-established in both the endometriotic lesions and eutopic endometrium of women with endometriosis . Since P4 signaling is required to counteract E2-induced proliferation and to promote decidualization , the loss of P4-responsiveness leads to both an increased growth of endometriotic lesions and to a non-receptive endometrium [33, 56].
Changes in the expression of the nuclear PR isoforms PR-A and PR-B, of steroid receptor coactivators, and of multiple downstream effectors in endometriotic lesions and eutopic endometrium from women with endometriosis represent the molecular cause of progesterone resistance. (Fig.2). The concept of progesterone resistance was suggested by the finding that in endometriotic lesions of PR-B was undetectable and PR-A was markedly lower compared with normal endometrium . Promoter hypermethylation and microRNA dysregulation were suggested to be potential mechanisms for PR-B loss in endometriosis. In fact, aberrations in genetic and epigenetic regulation of PRs and their targets have been demonstrated . Progesterone receptor (PR) gene polymorphism may also promote the susceptibility to endometriosis . Among polymorphisms described in the PR gene of patients with endometriosis, the PROGINS polymorphism affects ligand-binding and downstream signaling in the cellular context of endometriosis, and is involved in progesterone resistance [59, 60].
Fig. 2.
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The mechanisms of progesterone-resistance in endometriosis
Furthermore, in endometriotic tissue, P4 does not induce epithelial 17β-HSD-2 expression , an enzyme which in normal endometrium induces the expression of the enzyme 17β-hydroxysteroid dehydrogenase type 2 (17β-HSD-2), which metabolizes the biologically active estrogen E 2 to estrone. This additional deficiency, when combined to excessive estradiol production due to aberrant aromatase activity, contributes to the abnormally high estradiol activity in endometriosis. P4 also influences inflammatory pathways, suppressing the signaling of members of the nuclear factor kappa light-chain-enhancer of activated B cells (NF-kB) family of proteins in endometrial cells. This signaling network has been implicated in endometriosis as a factor leading to the establishment and maintenance of endometriosis implants .
Inhibin, activin and follistatin
Inhibins and activins belong to the transforming growth factors (TGF)-ß superfamily and are involved in the regulation of cellular proliferation, differentiation and apoptosis of endometrial cells. Inhibin A, inhibin B and activin A are detectable in the peritoneal fluid of women with pelvic endometriosis and cultured endometriotic cells expressed the mRNA of inhibin, ßA-, ßB-subunits, and activin receptors types II and IIB . Both α and ß A subunits are expressed in glands and stroma of OMA, and their dimers inhibin A and activin A are more concentrated in the cystic fluid than in peritoneal fluid , suggesting that they may be contributing to both implantation defects in eutopic endometrium and to the development of ectopic locations of endometriosis . Indeed, in cultured human endometrial stromal cells from women with endometriosis, activin A increases IL-6 and IL-8 secretion [65, 66]. An impaired expression of OMA and endometrial cripto (activin receptor antagonist), and follistatin (activin-binding protein) indicates an impaired activin pathway in endometriosis . Furthermore, nodal, a growth factor highly expressed in high turnover tissues and acting through SMAD proteins, has shown only subtle changes in endometriosis, differentiating the high proliferation of endometriosis cells from malignancies . Serum activin A and follistatin are not significantly altered in SUP or DIE phenotypes and have limited diagnostic accuracy in the diagnosis of OMA .
Anti-mullerian hormone (AMH)
AMH is a dimeric glycoprotein belonging to the transforming growth factor-β superfamily and besides its functional role in the ovary, it reflects the number of preantral follicles that comprise the oocyte pool, thus serum AMH level serves as a marker of ovarian reserve .
A significantly decrease of serum AMH levels was reported in women with OMA compared with age-matched fertile controls . Conversely, the adverse effect of surgical removal of OMA on ovarian reserve parameters including AMH levels is well recognized [72–74]. Thus, the impact of endometriosis and OMA per se on the ovarian reserve is still subject to controversy . Moreover, infertility patients with endometriosis showed lower AMH level compared to women with a primary diagnosis of male factor infertility . This conclusion is confirmed by recent data indicating that serum AMH level in infertile patients with OMA is significantly lower than in the control group and patients with bilateral OMAs have lower AMH levels than those with unilateral OMA. Moreover, patients with previous cystectomy had a considerably lower mean serum AMH level than individuals with OMA who had never had surgery. These findings suggest that OMA per se is associated with reduced ovarian reserve, and laparoscopic cystectomy can further exert significant damage on ovarian reserve. Anyhow, patients with OMA experience a progressive decline in serum AMH levels, which is faster than that in healthy women . The contrasting observation that AMH levels are not diminished in women with endometriosis, including those with presence of uni- or bilateral OMAs unless they had had previous OMA surgery, was based on data from women undergoing surgery without information on infertility, thus biasing the results .
The mechanism of OMA inducing ovarian reserve damage is still elusive. The inflammatory response to the endometriosis implants may cause microscopic alterations of the follicular and vascular patterns. Moreover, the compression of surrounding ovarian cortex by the cyst could hamper circulation and cause follicle loss . However, further studies are needed to elucidate the mechanisms of ovarian reserve damage induced by OMA. There are limited data regarding the effect of SUP or DIE, without OMA, on ovarian reserve , showing that the effect of extraovarian endometriosis on ovarian reserve is less pronounced than that of OMA.
AMH is also produced by eutopic and ectopic endometriotic cells and it secreted in peritoneal fluid . The treatments with AMH in vitro decreases the proliferative activity and increases the intracellular signal of apoptosis, suggesting a role of AMH in the pathogenesis of the disease [82–84].
Other endocrine aspects
HPA axis and stress hormones
Endometriosis-related pain and infertility elicit a stress response: from one side, infertility provokes family issues and fear of frustrating social expectations [85, 86], on the other pelvic pain causes sexual dysfunction and work absenteeism , all of which contribute to increase anxiety and chronic stress. Since endometriosis is also surrounded by apprehension about the disease progression, the long-term health risks and the prospect of having children may be additional sources of stress [87–89]. Furthermore, women with endometriosis experience a delay of 4 to 7 years from first presentation of symptoms to the diagnosis [90, 91], which may further enhance the levels of stress perceived by the patient. Women with endometriosis and severe endometriosis-related pain (dysmenorrhea, pelvic pain, dyspareunia) usually present with very high scores of perceived stress . However, if on one hand surgical treatment of symptomatic women reduces perceived stress, women undergoing to multiple surgeries reported high stress scores impairing the QoL . In fact, endometriosis has a significantly negative impact on health-related QoL scores and the factors involved are mainly linked to pain symptoms . A recent study by Marki et al. reported that both physical pain symptoms and emotional regulation difficulties, the latter being mediated by psychological stress, reduced health-related QoL of women with endometriosis . Furthermore, other aspects, such as self-confidence, body esteem, and emotional self-efficacy, play a role in the psychological health and stress perception, being impaired among women with endometriosis.
A dysfunction of HPA axis is found in patients with endometriosis (Fig.3) and it is related to an attenuated cortisol response, a condition known as burnout. A paradoxical hypocortisolism like an adrenal fatigue may exacerbate painful symptoms by reducing the endogenous analgesia associated with stress (stress-induced analgesia) . Supporting this assumption, a blunted early morning cortisol response to CRH test was associated with greater menstrual and non-menstrual pain in endometriosis . Low salivary cortisol levels and a high degree of perceived stress were found to be associated with poor QoL in patients with endometriosis and chronic pelvic pain , as well as salivary hypocortisolism, which was linked to infertility and dyspareunia but not dysmenorrhea . On the other hand, higher hair cortisol levels were found in patients with endometriosis compared to healthy women of similar age, parity, education level and BMI . Moreover, increased serum cortisol levels were detected in infertile women with endometriosis, especially in those with advanced stage of disease . Interestingly, physical and psychological interventions have been shown to normalize salivary cortisol levels of women with endometriosis-related chronic pain .
Fig. 3.
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Hypothalamus–pituitary–adrenal (HPA) axis and stress hormones in endometriosis
CRH and urocortin (Ucn) are also produced by endometrium and locally act modulating tissue differentiation (decidualization of endometrial stroma, embryo implantation and maintenance of pregnancy) and inflammation (Fig.3). Eutopic endometrium highly express CRH, CRHR type 1 and 2, as well as urocortin mRNA and protein , thus suggesting that a deranged CRH and Ucn mRNA expression associated with an impaired CRH-R1 activity may affect the process of decidualization and contribute to infertility in these patients. In fact, cultured endometrial cells from endometriotic patients have a reduced decidualization capacity, reducing the secretion of prolactin, CRH and Ucn . The most intense immunostaining for CRH and Ucn is observed in DIE lesions, with an increased expression of CRH-R1 and R2 and inflammatory enzymes PLA2G2A and COX2 . Since CRH and Ucn significantly increase COX2 expression (effect was reversed by the CRH-R2 antagonist astressin) and endometriotic tissue expresses both Ucn 2 and Ucn 3 (which modulate TNF-α and IL-4 secretion) an involvement of this stress pathway in inflammation is suggested .
High levels of CRH-binding protein in peritoneal fluid from women with endometriosis than in controls has been observed, suggesting possible changes also in circulating levels . Plasma urocortin levels are twice as high in women with OMA, and levels are significantly higher in the cystic fluid of OMA than in the peritoneal fluid and plasma . Besides, the preoperative blood testing of Ucn among symptomatic women undergoing surgery for suspect of endometriosis showed that confirmed cases had higher plasma Ucn levels compared to patients with no lesions and elevated plasma Ucn1 levels are found among all endometriosis phenotypes. However, no cutoff could accurately distinguish endometriosis from other pathological conditions, thus it is not useful .
Thyroid hormones
Autoimmune thyroid disorders are frequently found in endometriosis patients suggesting a pathogenic association between these two conditions [113–115]. A relationship between endometriosis and the presence of thyroid autoantibodies is found, leading either to hypothyroidism or hyperthyroidism. The relative risk of endometriosis is significantly increased in women tested positive for thyroperoxidase (TPO) antibodies , similarly a high prevalence of anti-TSHR antibodies, pathognomonic of Grave’s disease, is shown in patients with endometriosis . It is not clear whether these antibodies or thyroid hormones play a role in the pathogenesis of endometriosis. A microarray analysis of mild versus severe endometriosis confirmed a potential involvement of thyroid hormone homeostasis and metabolism in the pathophysiology of endometriosis .
A recent ex vivo study on thyroid transcripts in patients with endometriosis described an overexpression of TSHR and a decreased biosynthesis of T3 and an accumulation of T4 in ectopic endometrium. The direct stimulation of estrogen receptors on endometrial cells by thyroid hormones was suggested to cause cell proliferation. In fact, in vitro studies demonstrated that TSH activates the proliferation of all endometriotic and control cells, T4 has a specific proliferative effect on epithelial and stromal ectopic endometrial cells, whereas T3 only acts on epithelial cells. In addition, thyroid hormones cause ROS production by ectopic endometrial cells, that may favor, in turn, endometriotic cell proliferation .
Thyroid hormones may also contribute to the pathogenesis of endometriosis by modulating the immune response, as they can active neutrophils and macrophages to locally promote a proinflammatory environment . Therefore, an increase of serum TSH or of T4 could be hypothesized as a participating factor for endometriosis development and progression. A more sever chronic pelvic pain and disease score in endometriotic patients with a thyroid disorder confirm that endometriosis should be carefully monitored in patients with comorbid thyroid disease .
Clinical implications: pain, infertility and systemic comorbidities in endometriosis
Endometriosis is a heterogeneous disease also in terms of clinical presentation. Common symptoms include dysmenorrhea and non-menstrual pelvic pain, which may develop into chronic pelvic pain . with a relevant impact on daily life . Other endometriosis-related pain are dyspareunia, dyschezia, and dysuria, usually associated with DIE lesions [121, 122]. According to the anatomical involvement of bowel, patients may alternate constipation and diarrhea, dischezia or blood in the stool (in particular perimenstrually) [122, 123] or when urinary tract is affected, recurrent dysuria, cyclic macrohaematuria or interstitial cystitis are observed . Chest and shoulder pain should be considered suspecting diaphragmatic endometriosis , whereas endometriosis in the ileo-caecal or peri-appendiceal region has been significantly associated to abdominal pain, nausea, vomiting and diarrhea [126, 127].
Regarding the physiopathology of endometriosis-related pain, nociceptive (including inflammatory), neuropathic and a combination of these mechanisms are involved , under the influence of hormonal aberrations, stress, inflammation, and the interplay between the peripheral and central nervous systems [129–131]. Neurogenic factors, such as brain-derived neurotrophic factor (BDNF) and nerve growth factor (NGF) are reported to be overexpressed in the peritoneal fluid and in endometriotic lesions of affected women . Neurotrophic factors are also responsive to estrogens, prostaglandin and cytokine and stimulate the growth and sensitization of sensory nerve fiber terminals [133, 134], particularly in DIE, presenting high nerve fibers density . The development of a vicious cycle characterized by nociceptor sensitization and local neo-neurogenesis, triggered by inflammatory and immune mediators, is observed in endometriosis . Endometriotic lesions themselves send noxious signals to dorsal root spinal cord neurons and activate spinal microglia to maintain pain stimuli, resulting in a central sensitization . In fact, a number of central changes are observed: alterations in the behavioral and central response to noxious stimulation, changes in brain structure, altered activity of both the HPA and the autonomic nervous system and psychological distress , with larger volume in regions involved in pain modulation and endocrine function regulation [137–139]. Indeed, chronic pain and stress experienced by patients with endometriosis might cause multiple psychiatric diseases (Fig.3) and the somatoform disorder is the most common . Anxiety and depression traits, and a higher tendency of pain catastrophizing are commonly present in endometriosis patients and can amplify the perception of pain [141, 142]. Another frequently present, but often neglected, symptom in women with endometriosis is chronic fatigue, although the exact mechanism remains not fully understood .
Women with endometriosis show a higher prevalence of systemic comorbidities has been shown, even though it is still unclear whether a common endocrine, immune and inflammatory background predispose to the development of those conditions or a high levels of perceived stress [144, 145]. An increased risk of inflammatory bowel diseases (Chron’s disease, ulcerative colitis) allergic manifestations [sinus allergic rhinitis, and food allergy) , autoimmune diseases (systemic lupus erythematosus, rheumatoid arthritis, Sjogren’s syndrome, multiple sclerosis, fibromyalgia) are more likely to be diagnosed in women with endometriosis, also underlying a neuroendocrine–immune imbalance [148–153].
Infertility is the other major symptom of endometriosis, even though a diagnosis of endometriosis does not always imply infertility. Endometriosis is identified in approximately 30% of women in infertile couples . The disease adversely affects fertility by different mechanisms acting at the level of pelvic cavity, ovary and uterus . Pelvic cavity is an hostile environment because the chronic inflammatory changes in the peritoneal fluid and the distortion of normal anatomy of the fallopian tubes hindering tubo-ovarian contact and affecting sperm-oocyte interaction; ovary produces low quality oocytes, impaired folliculogenesis, and luteal function, with ovarian reserve reduced by OMA and/or by surgery. Besides, in endometriosis uterus itself has an altered endometrial receptivity mainly due to local growth factors changes (integrin, LIF, activin, CRH), to hormonal aberrations (ER and PR) and to dysperistalsis of myometrium, due to the association to adenomyosis [156, 157]. However, the evidences supporting the impairment of endometrial receptivity in endometriosis are still controversial. The endometrial chronic inflammation, together with progesterone resistance, estrogen dominance, aberrant cell signaling pathways and reduced expression of key homeostatic proteins in women with endometriosis, are disruptive to endometrial receptivity . On the contrary, data from in vitro fertilization (IVF) and egg donations, other than basic data regarding the transcriptomic signature of the endometrium, seem to indicate that endometrial receptivity gene signature during the window of implantation is similar between infertile women with and without endometriosis, suggesting a major effected played by embryo and oocyte quality more than to the endometrial factor itself .
Endocrine background of hormonal treatments for endometriosis
Hormonal therapies are the most common used for treating women with endometriosis. The goal is to block menstruations by causing a state of iatrogenic menopause or pseudopregnancy. Current hormonal medical therapy does not cure definitively the disease, but it is able to control pain symptoms in order to prevent or postpone surgery and to long term manage the disease [21, 160]. First-line hormonal therapies include progestins, while second-line therapy are represented by GnRH agonists (GnRH-a) and antagonists. The off-label use of combined oral contraceptives (COCs) is common. New hormonal drugs (aromatase inhibitors, selective estrogen receptor modulators (SERM), selective progesterone receptor modulators (SPRM)) are under investigation for the treatment of endometriosis (Fig.4).
Fig. 4.
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Hormonal targets of currently used drugs for endometriosis
Gonadotropin releasing hormone agonists (GnRH-a)
GnRH-a (goserelin, leuprolide, nafarelin, buserelin, and triptorelin) are labelled drugs used since the ‘90 s to treat endometriosis. They bind to the GnRH receptors and, during the first 10 days of treatment, stimulate the pituitary to produce LH and FSH . Subsequently, the prolonged and continuous exposure to these agents cause downregulation of the GnRH receptors, thus decreasing LH and FSH levels and suppressing estrogen ovarian production (Fig.4). The induced hypoestrogenism with subsequent amenorrhoic state leads to the regression of endometriotic lesions . Several trials have shown that GnRH-a improved endometriosis-associated pain [163–166] and a meta-analysis of 41 trials comparing the use of GnRH-agonists at different doses, regimens and routes of administration, reports that GnRH-a are more effective than placebo and as effective as other progestins for relieving pain . In particular, the administration of GnRH-a for a period of 3 to 6 months prior to ART in women with endometriosis may increase the odds of clinical pregnancy by fourfold .
However, treatment with GnRH-a is associated with significant hypoestrogenic side effects, including amenorrhea, vasomotor symptoms, sleep disturbance, urogenital atrophy, and accelerated bone loss. Therefore, GnRH-a should be used carefully in adolescents since these women may not have reached maximum bone density . The addition of add-back therapy (low-dose COCs, estrogen or progestins alone, bisphosphonates, tibolone or raloxifene), may reduce these adverse effects, without reducing the efficacy of pain relief. With the addition of add-back therapy, the administration of GnRH-a, which was initially limited to 6 months, is allowed for longer time . Some clinical trials and cohort studies have demonstrated that a GnRH-a plus steroid add-back therapy can be effective from 30 months to up to 10 years [171, 172].
GnRH antagonists
GnRH antagonists suppress gonadotropin hormone production, by competing with endogenous GnRH for its pituitary receptors (Fig.4). Contrary to GnRH-a, antagonists do not provoke the initial flare-up phase and cause a rapid onset of the therapeutic effect . They have also the advantage of being administered orally because of its non-peptide structure which avoids the gastrointestinal proteolysis.
Elagolix, a short-acting GnRH antagonist, has been recently approved in USA for the management of moderate to severe pain associated with endometriosis . Compared to the classic GnRHa, elagolix, by blocking endogenous GnRH signalling, causes a dose-related suppression of LH and FSH, and a consequent modulation of estradiol levels. Thus, it provides relief of endometriosis-related pain avoiding severe hypoestrogenism . The FDA approved elagolix for the treatment of endometriosis related pain following the results of two multicenter, double-blind, randomized, phase 3 trials which compared two distinct doses of elagolix (150 mg once daily or 200 mg twice daily) with placebo. In both trials, during the 6 months treatment, elagolix dramatically reduced dysmenorrhea and non-menstrual pelvic discomfort. Also in women who still menstruated, a lower proportion of menstrual period days with moderate or severe dysmenorrhea compared with placebo was shown, indicating pain reduction despite continued menses . Positive results were found in two phase 3 extension studies , which evaluated long-term efficacy and safety of elagolix for 12 months decreasing dysmenorrhea, nonmenstrual pelvic pain and dyspareunia. Moreover, treatment with elagolix improves QoL [175, 176], decreasing the use of analgesic agents and fatigue levels . Although it inhibits ovarian function in a dose-dependent manner, elagolix especially the higher dose, causes hypoestrogenic side effect, such as hot flash, decrease in BMD and increase in serum lipid levels. Based on those observations, two ongoing Phase III trials are currently examining the safety and efficacy of both elagolix alone and elagolix plus E2 and NETA for the treatment and management of moderate to severe pain in premenopausal women with endometriosis over a 24-months period (NCT03343067 and NCT03213457). Further studies are also needed to evaluate the drug effects on ovarian function, as a number of pregnancies have been reported while taking elagolix; as a result,, patients should use non-hormonal contraceptive systems during the treatment [178, 179].
Relugolix and linzagolix are the two new oral GnRH antagonists, in an advanced stage of clinical development for the management of pain associated with endometriosis [180, 181]. A Phase 2, multicenter, randomized, double-blind, placebo-controlled study on oral administration of relugolix for 12 weeks demonstrated efficacy in alleviating endometriosis-associated pain in a dose–response manner with some adverse events (hot flush, heavy menstrual bleeding, and irregular menstruation, and bone mineral density decrease). However, oral relugolix at the dose of 40 mg was generally well tolerated and showed similar efficacy and safety compared with those of leuprorelin . A Phase 3 extension trial is ongoing aiming to assess the tong-term efficacy and safety of relugolix 40 mg once daily co-administered with low-dose estradiol and norethindrone acetate on endometriosis-associated pain. A Phase 2b, double-blind, placebo-controlled, dose-ranging study with linzagolix, has been performed in women surgically confirmed endometriosis and moderate-to-severe endometriosis-associated pain . Doses ≥ 75 mg resulted in a significantly greater proportion of responders for overall pelvic pain, dysmenorrhea and non-menstrual pelvic pain after 12 and 24 weeks treatment. Serum estradiol was suppressed, QoL improved, and the rate of amenorrhea increased in a dose-dependent fashion. Also mean BMD loss (spine) increased in a dose-dependent manner and was < 1% at 24 weeks at doses of 50 and 75 mg and up to 2.6% for 200 mg. The most frequently reported adverse events related to the trial treatments were hot flushes and headaches .
Progestins
Progestins are compounds with multiple actions on PRs: decreased secretion of FSH and LH, anovulation, relatively hypoestrogenic state and amenorrhea that help suppressing endometriosis and preventing dysmenorrhea. Moreover, they have antiestrogenic effect causing endometrial pseudodecidualisation, inhibit inflammatory response, provoke apoptosis of endometriotic cells, reduce oxidative stress, inhibit angiogenesis and suppress expression of matrix metalloproteinases [5, 26] (Fig.4). All these mechanisms induced by progestins have a beneficial effect on the progression of endometriosis and associated-pain.
According to the ESHRE guidelines, progestins are considered as a first choice for the treatment of endometriosis , because they are as effective in reducing scores and pain as GnRH agonists, and have a lower cost and a lower incidence of adverse effects.
Progestins can be administered by an oral, intramuscular, subcutaneous, or intrauterine route . The progestins most commonly used for the treatment of endometriosis-related pain include dienogest (DNG) norethindrone acetate (NETA) and medroxyprogesterone acetate (MPA) [169, 184]. DNG is approved in Europe, Japan, Australia and Singapore, while NETA and MPA are currently approved by the USA Food and Drug Administration (FDA). Alternative progestin treatment options include gestrinone, desogestrel, danazol, the etonogestrel implant and the levonorgestrel intrauterine system (LNG-IUS). Side effects of progestins include irregular uterine bleeding/spotting, weight gain, mood changes (eg, depression), and bone loss (specific to long-term use of depot MPA). Although these side effects are frequent, they rarely cause therapy abandonment. Overall, progestins are safe and about two thirds of patients are satisfied with their use for symptomatic endometriosis .
Dienogest
DNG, a 19-nortestosterone derivative, is the most recent progestin available and labelled for endometriosis and according to a number of evidences it substantially improves endometriosis-related pain symptoms in long term treatment . Both surgically and clinically diagnosed patients described comparable pain reduction, as well as women with or without prior treatment . Compared to danazol, MPA and goserelin, DNG is the most efficient alternative to treat pelvic pain associated to endometriosis . Furthermore, no effects on bone mineral density were reported compared to treatment with leuprolide, maintaining a stable bone turnover . Regarding the effects of DNG according to different endometriosis phenotypes, it causes a significant reduction in both diameter and volume of OMA, whereas the ovarian reserve appears to be preserved . Among women with ultrasound identified OMAs followed up for 12 months, DNG reduced the volume of OMAs up to 76% from the initial size. Besides, a reduction of 74.05% for dysmenorrhea, 42.71% for dyspareunia and 48.91% for chronic pelvic pain were observed . Furthermore, DNG alone has been shown to be superior to COCs, containing DNG, to reduce the size of OMAs . A recent study showed that in women with OMA DNG reduces the size of ovarian cysts, effective in reducing endometriosis related symptoms both after 6 and 12 months of treatment and well tolerated .
DNG also appears to be effective in controlling pain caused by rectovaginal endometriosis , bladder endometriosis [195, 196] and DIE . In a prospective cohort study including 30 women with a sonographic diagnosis of DIE (intestinal and posterior fornix) treated with DNG for 12 months, the treatment was effective to control symptoms of pain related to DIE (dysmenorrhea, dyspareunia, dischezia), improving QoL, even without reducing the volume of DIE nodules .
Patients treated with DNG have also shown an improvement of sexual functioning and QoL [187, 199]. Among Asian women, DNG therapy decreased Endometriosis Health Profile-30 (EHP-30) scores in all assessed domains, especially the "pain" domain was improved in 78.4% of patients. Both surgically and clinically diagnosed patients described comparable pain reduction . In randomized controlled trial among Chinese women with endometriosis, DNG for 24 weeks provided significantly greater reduction in endometriosis-associated pelvic pain than placebo, and maintained or enhanced efficacy after 28 weeks of additional treatment .
Long-term (60-month) treatment effectively reduced endometriosis-associated pain and avoided pain recurrence post-surgery without sever adverse effects [202, 203], especially on bone mineral density (BMD). Therefore, its use as first-line therapy for long-term management of debilitating and chronic endometriosis-associated pain represents an interesting option.Regarding efficacy and tolerability, a large study conducted in Korea showed that satisfaction scores were mostly favorable. The most frequently reported side effects are abnormal uterine bleeding (4.1%), weight gain (2.5%) and headache (1.2%). The number of patients with favorable bleeding patterns was observed to increase as the duration of treatment increases, till amenorrhea .
DNG is an effective treatment also as postoperative treatment in order to reduce recurrences, to avoid re-interventions and to control pain symptoms. DNG is as effective and tolerable as GnRH agonist and add–back therapy using 17b-estradiol and NETA for 6 months for the prevention of pelvic pain recurrence after laparoscopic surgery for endometriosis . A prospective cohort study on women undergoing to surgery for OMA, receiving postoperative medical treatment with DNG for 24 months no cases of OMA recurrence were found . In case of recurrent OMA after surgery, DNG therapy early after recurrence appears to be viable for reducing the risk of repeated surgery, given that after 24 months of treatment with DNG, a reduction of size and complete resolution of recurrent OMA was achieved in 57.1% .
Norethindrone acetate (NETA)
NETA, another 19-nortestosterone derivative, is effective on pain relief in women with endometriosis. NETA has strong progestogenic effects and a androgenic activity, that may cause side effects due to its residual androgenic activity (weight gain, acne, and seborrhea) . The continuous administration of NETA (5 mg/day) for the treatment of endometriosis is approved by the US FDA. Low-dose NETA 2.5 mg/day orally is considered an effective, tolerable and inexpensive first choice for symptomatic rectovaginal endometriosis, significantly decreasing VAS scores for dysmenorrhea and dyspareunia . A pilot study on women with bowel endometriosis showed that low dose oral NETA determined a significant improvement in the intensity of chronic pelvic pain, deep dyspareunia, dyschezia and the disappearance of symptoms related to the menstrual cycle (dysmenorrhea, constipation during the menstrual cycle, diarrhea during the menstrual cycle and cyclical rectal bleeding) . Recently, a long term study 5-year therapy with NETA [2.5 mg/day up to 5 mg/day) is safe and well tolerated by women with rectovaginal endometriosis, who were satisfied or very satisfied in 68.8% of cases. Due to its low cost and good pharmacological profile, it may represent a good candidate for long-term treatment for endometriosis . Low dose of NETA was also found to have less side effects, such as unscheduled bleeding, compared to extended-cycle COCs, despite the same effectiveness in terms of pain control . The comparison between low dose NETA and DNG as first line drug used in new diagnosed endometriosis women showed that treatment was well tolerated by 58% of NETA users compared with 80% of DNG users . However, in a subpopulation of symptomatic women with rectovaginal endometriosis “NETA "resistant”, who had pain persistence, DNG was effective in treating pain and improving QoL .
Medroxyprogesterone acetate (MPA)
MPA is a 17-OH progesterone derivative, available as oral formulation or depot formulation, which can be administered intramuscularly and subcutaneously every 3 months. MPA appeared to be more effective than placebo and as effective as danazol and GnRH-agonists [216, 217] in reducing endometriosis-related pain. In particular, depot MPA (dMPA) reduces pain as effectively as leuprolide and improves quality of life and productivity. The major source of concern regarding continuous use of depot MPA is the loss of BMD with an increased risk of fracture, due to estrogens deficiency. Therefore, the FDA have suggested that it should be administrated only if other methods are unsuitable or unacceptable, and have limited its maximum use to 2 years . On the contrary, the American College of Obstetricians and Gynecologists support the use of dMPA as current longitudinal and cross-sectional evidence suggests the recovery of BMD after discontinuation of dMPA, and, considering the modest increase in the risk of fracture, benefits of dMPA use surpasses the risks.
Danazol
Danazol is a derivative of 17α-ethynyl testosterone and since 1971 is approved by FDA to treat endometriosis. Its mechanisms of action include inhibition of pituitary gonadotropin secretion, direct inhibition of ovarian enzymes responsible for estrogen production, modulation of immunological function, suppression of cell proliferation and inhibition of endometriotic implant growth [170, 219]. Danazol is effective in treating endometriosis-related pain and its efficacy seems to persist also after the discontinuation of therapy . However, its use is limited by the androgenic-type adverse effects such as seborrhea, hypertrichosis, weight gain, HDL levels decrease, and LDL levels increase . Danazol is typically given orally (400 to 800 mg/day). Good efficacy and better tolerability has been reported with danazol-loaded intrauterine device and with off-label vaginal administration (200 mg/day) [222, 223], particularly for women with DIE and rectovaginal endometriosis . A significant reduction of painful symptoms in patients with DIE was observed, with less recurrences and a decreased volume of endometriosis lesions . Furthermore, long-standing use of vaginal danazol suppositories resulted in favourable control of postoperative pelvic pain associated with pelvic endometriosis without significant adverse side effects . Low-dose vaginal danazol (200 mg per day for 6 months) is effective also for the treatment of pain in recurrent endometriosis after surgery for severe disease, with reduction of VAS pain intensity . With low doses and vaginal route of administration, side- effects are seldom observed, and lipid parameters and liver function are reported to be unaltered.
Other progestins
Desogestrel
Desogestrel (DSG) (75 mg/day) is an effective, safe and low cost therapy for endometriosis related pain [227, 228] with a good satisfaction rate and causing also a significant improvement in QoL. DSG treatment of women with symptomatic rectovaginal endometriosis induced volume size reduction and improvement of gastrointestinal symptoms, chronic pelvic pain, and deep dyspareunia. At 12‐month follow up, the rate of satisfied patients was higher in those treated with the desogestrel‐only pill compared to those on sequential estro-progestin pill . DSG resulted effective also in A significant improvement of both pelvic pain and dysmenorrhea after 6-months treatment in endometriosis recurrence. Breakthrough bleeding is the main adverse effect reported during DSG treatment .
Levonorgestrel intrauterine device (LNG-IUS)
The effect of the LNG-IUS on endometriosis has been assessed in several RCTs. LNG induces endometrial glandular atrophy and decidual transformation of the stroma, reduces endometrial cell proliferation and increases apoptotic activity. After the first year of use, a 70–90% reduction in menstrual blood loss is observed. The LNG-IUS has proven effective in relieving pelvic pain symptoms caused by peritoneal and rectovaginal endometriosis and in reducing the risk of recurrence of dysmenorrhea after conservative surgery . In fact, LNG-IUS use after surgery was associated with a significantly lower dysmenorrhea recurrence rate compared to with expectant management [231–233]. Dyspareunia and dysmenorrhea were clearly reduced after 12-months of treatment with few adverse events and very low discontinuation rate . A recent study evaluated the efficacy of LNG-IUS versus DNG treatment compared to no post-operative therapy after laparoscopic surgery for endometriosis. At 6 and 12 months, the median pain scores in treatment groups were significantly lower and both treatments had significantly lower recurrence rate than control group (3.8% and 9.7%, respectively, vs 32.5%). In addition, patients with LNG-IUS showed lower rate of discontinuation, suggesting that LNG-IUS is effective for postoperative pain control and for preventing recurrence . However, no or limited effect was observed in preventing OMA recurrence. In fact, in a randomized clinical trial including 80 patients with OMA undergoing laparoscopic cystectomy followed by six cycles of GnRH-a, and then allocated to LNG-IUS insertion or not for 30 months, LNG-IUS was able to control pain symptoms but it was not effective for preventing OMA recurrence . However, a recent meta-analysis on the efficacy of different hormonal regimens for the prevention of OMA recurrence in women who have undergone conservative surgery showed that among cohort studies LNG-IUS ranked highest .
Gestrinone
In a meta-analysis including two small studies, treatment with gestrinone resulted effective in reduction of pain . However, the use of gestrinone for endometriosis is limited due to its side effects. In fact, because of its androgenic, anti-estrogenic and anti-progestogenic properties, it may cause acne, seborrhea, hirsutism, weight gain, liver dysfunction and osteoporosis .
Etonogestrel-releasing subdermal implant (ENG- implant)
Few data are available also on the use of the etonogestrel-releasing subdermal implant (ENG- implant) for the treatment of women with endometriosis, resulting effective in decreasing dyspareunia, dysmenorrhea and nonmenstrual pelvic pain [239, 240]. A recent study evaluating the efficacy of ENG-implant versus the 52-mg LNG-IUS in the control of endometriosis-associated pelvic pain showed that both contraceptives improved significantly pelvic pain, dysmenorrhea, and health-related quality of life in endometriosis .
Combined oral contraceptives (COCs)
COCs are currently used off label for the treatment of endometriosis, however they are commonly used as empirical therapy for women with suspect of endometriosis, without a confirmed surgical diagnosis of the disease . ESHRE guidelines classifies as Grade B the COCs prescription to reduce dyspareunia, dysmenorrhea and non-menstrual pain. On the other hand, Grade C evidence has been provided to COCs continuous use in women suffering endometriosis-associated pain . The advantages of using COCs for the treatment of endometriosis include the good tolerability as well as the low costs, but they contain estrogens. COCs reduce menstrual flow, cause decidualisation of endometriotic implants and decrease cell proliferation . Ovarian function is inhibited as well as the metabolism of arachidonic acid to prostaglandins, resulting effective in reducing pelvic pain and menstrual cramps. Although COCs are widely used in clinical practice since decades, given their effectiveness for dysmenorrhea, high level evidence of their effectiveness for the treatment of endometriosis does not exist.
Only two trials [244, 245], both conducted in Japan, compared COCs with placebo in women with endometriosis. In these studies, COCs treatment was associated with an improvement in dysmenorrhea, cyclical non-menstrual pain, dyspareunia and dyschezia. However, the formulation of COCs used in these studies (ethinylestradiol 35 mcg + norethisterone 1 mg in cyclic regimen and ethinylestradiol 20 mcg + drospirenone 3 mg in flexible regimen) may not be readily available globally and it is unknown if different formulations may have different effects .
In a recent systematic review about patient response to medical therapies for endometriosis , the rate of patients experiencing pain symptoms at the end of treatment was higher with COC, vaginal ring and patch compared to GnRH-a or progestins. The observation that about 50% of patients have partial or no improvement in symptoms of endometriosis under COCs and about 70% of women had used multiple COCs for relief of pain and over 40% had been prescribed between 3 and 10 different COCs support the conclusion that this treatment is not completely effective . Despite the low dose COCs (20–30 μg is equivalent to 4 to 6 times the physiologic dose of estrogens) and, given ER and PR alterations in endometriosis, the administration of COCs may result in estrogen dominance in the presence of progesterone resistance . Studies also showed an increased risk of endometriosis in past users of COCs .
Some studies showed that COCs prevent and reduce frequency and severity of recurrent dysmenorrhea and relapse of endometriosis after surgery [252–256]. The continuous use of COCs after conservative surgery is more beneficial than the cyclic use [253, 256, 257]. However, COCs after previous surgery has similar or less efficacy in pain relief than GnRH-a . In conclusions, despite their wide use in clinical practice, further research is needed to fully evaluate the role of COCs in the management of endometriosis-related pain.
Drugs under development for endometriosis
Selective progesterone receptor modulators (SPRM)
SPRMs are progesterone receptor ligands that act as tissue-selective progesterone agonist, antagonist, or partial agonist/antagonist on various progesterone target tissues. Although SPRMs inhibit the ovulation, they are not associated with the systemic effects of estrogen deprivation as estradiol secretion is not affected and circulating levels of estradiol remain in the physiological range. Furthermore, SPRMs inhibit the endometrial proliferation, suppress endometrial bleeding through a direct effect on endometrial blood vessels, and reduce endometrial prostaglandin production in a tissue-specific manner (Fig.4). Therefore, a potential good efficacy of SPRMs on endometriosis was suggested [260–262], but no SPRMs are used in clinical practice.
Ulipristal acetate (UPA), telapristone acetate, vilaprisan and tanaproget are SPRMs which were proposed for the treatment of endometriosis [263, 264]. SPRMs are generally well tolerated. Common adverse effects are headache, abdominal pain, nausea, dizziness, and heavy menstrual bleeding. Mifepristone and asoprisnil were the most studied SPRMs. Mifepristone-induced regression of endometriotic lesions has been variable and appears to be dependent on the duration of treatment [265, 266]. A small prospective open-label trial suggested the possible efficacy of mifepristone for endometriosis-associated pain . Similar results were found in a phase II/III trial; however, 3,4% of patients reported a significant increase in hepatic transaminases . In a randomized placebo-controlled trial, asoprisnil caused a higher decrease of dysmenorrhea among women affected by endometriosis compared to placebo . The effect of UPA was assessed on endometriosis lesions and symptoms in women treated over a 27-month study period prior to surgery. In 58% of cases progesterone receptor modulator-associated endometrial changes (PAECs) were observed in both eutopic endometrium and ectopic lesions; those cases reported all pain reduction and amenorrhea . However, there are insufficient data to permit firm conclusions about their safety and effectiveness .
Selective estrogen receptor modulators (SERMs)
SERMs bind to estrogen receptors (ER-α and ER-β) in target cells acting as ER agonist in some tissues and ER antagonist in others (Fig.4), and therefore they have been proposed for the treatment of endometriosis and are under investigation. Raloxifene (RLX), a common drug approved for the prevention and treatment of osteoporosis, has estrogenic effects in bone and antiestrogenic effects in endometrium and breast tissue . Tested in animal studies RLX induces regression of endometriosis implant [270, 271]. In a double-blind prospective study , patients with endometriosis-related pelvic pain following surgical treatment were randomly assigned to RLX or placebo for 6 months. However, this study was halted prematurely because women in RLX group experienced an earlier relapse of pelvic pain and sooner surgery than the placebo group.
Bazedoxifene (BZA) is a novel SERM used for the treatment of osteoporosis and antagonizes estrogen-induced uterine endometrial stimulation . In a rat model, BZA reduces the size of endometriotic lesions and decreases proliferating cell nuclear antigen and estrogen receptor expression in the endometrium . A tissue-selective estrogen complex (TSEC) containing BZA and conjugated estrogens (CE) also decreased endometriotic lesion size in a mouse model. The addition of estrogens to BZA did not induce endometrial growth or endometrial hyperplasia and did not reduce the efficacy of the SERM . Therefore, TSEC is a potential novel therapy for endometriosis that could have a high level of efficacy without the side effects of currently available treatments.
SR-16234 is another experimental SERM with antagonistic activity on ERα and partial agonistic activity on ERβ. SR has a regressive effect on the development of murine endometriosis-like lesions, by acting on cell proliferation, angiogenesis, inflammation, and NF-κB phosphorylation A recent trial that investigated this drug in a small group of women with endometriosis and adenomyosis showed that SR-16234 was able to decrease the intensity of pelvic pain and dysmenorrhea .
Aromatase inhibitors
Aromatase is expressed by endometriotic lesions and in the eutopic endometrium of women with endometriosis causing a local secretion of estrogens, which promote the growth and invasion of endometriotic lesions and favour the onset of pain and prostaglandin-mediated inflammation (Fig.4) .
Aromatase inhibitors (AIs) block estrogen synthesis both in the periphery and in the ovaries . Some clinical studies have shown that third- generation nonsteroidal AIs, such as letrozole and anastrozole, effectively reduced the severity of endometriosis-related pain symptoms; however, their use is limited by several adverse events, such as bone and joint pain, muscle aches, and fatigue .
The ESHRE guidelines only recommend the use of AIs in association with COCs or progestins or GnRH-a in patients with drug-resistant pain and surgery-resistant recto-vaginal endometriosis . Currently, a randomized, double-blind, parallel-group, multicenter phase IIb trial is evaluating the efficacy and safety of BAY98-7196 (an intravaginal ring with different doses of anastrozole and LNG), in comparison with placebo and LEU (subcutaneous depot) for treating women with symptomatic endometriosis over a 12-week period (NCT02203331).
Conclusions
Endometriosis is a chronic disease requiring a lifelong management. Based on patient’s symptoms and the desire of pregnancy, an individualized approach aiming to reduce pain, stress, stress-related comorbidities and to improve QoL should be used for an adequate management [1, 21, 25]. Until a few years ago, the suspect of endometriosis represented an indication for surgery, also used to make the diagnosis through the visualization and histology confirmation of endometriotic lesions. Research development has shown a clear endocrine pathogenesis for endometriosis and thus hormonal therapies represent now a cornerstone of its management, as first choice, before surgery and after surgery in order to reduce the risk of recurrence. The goal is to limit non-indicated surgical procedures because of disease recurrence risk, surgical complications [22, 23, 278], and negative effects on ovarian reserve . The modern approach for endometriosis requires a life-long management plan with the aim of maximizing the use of medical treatment, that can be safely prescribed without histological confirmation of the disease [182, 280–282], and avoiding repeated surgical procedures . Medical hormonal treatment should be the first-line therapeutic option also for patients who have not an immediate desire to become pregnant. Currently, hormonal treatments are the most effective drugs for the treatment of endometriosis and are based on the pathogenic mechanisms involved in the disease. The block of menstruation through an inhibition of HPO axis and consequent amenorrhea or pseudodecidualisation impairs the development or the activity of endometriotic implants. A modern endometriosis management includes a patient-focused approach taking care of overall wellbeing, considering stress, QoL and systemic comorbidities.
Abbreviations
AIs
Aromatase inhibitors
AMH
Anti-Müllerian hormone
ART
Assisted reproductive technologies
BDNF
Brain-derived neurotrophic factor
BMD
Bone Mass Density
BZA
Bazedoxifene
CE
Conjugated estrogens
COCs
Combined oral contraceptives
COX2
Cyclooxygenase 2
CRH
Corticotropin-releasing hormone
CRHR
Corticotropin-releasing hormone receptor
DIE
Deep infiltrating endometriosis
DNG
Dienogest
DSG
Desogestrel
E2
Estradiol
ENG-
Implant Etonogestrel-releasing subdermal implant
ESHRE
European Society of Human Reproduction and Embryology
FDA
Food and Drug Administration
FSH
Follicle-stimulating hormone
Gn-RH
Gonadotropin-releasing hormone
HDL
High-density lipoprotein
HPA
Hypothalamus–pituitary–adrenal axis
HPO
Hypothalamus-pituitary-ovary axis
IL
Interleukin
IVF
In vitro fertilization
LDL
Low-density lipoprotein
LH
Luteinizing hormone
LNG
Levonorgestrel
LNG-IUS
Levonorgestrel intrauterine system
MPA
Medroxyprogesterone acetate
NETA
Norethisterone Acetate
NF-kB
Nuclear factor kappa light-chain-enhancer of activated B cells
NGF
Nerve growth factor
OMA
Ovarian endometriomas
P4
Progesterone
PAECs
Progesterone receptor modulator-associated endometrial changes
PR
Progesterone receptor
QoL
Quality of life
RLX
Raloxifene
ROS
Reactive oxygen species
SERMs
Selective estrogen receptor modulators
SF-1
Steroidogenic factor-1
SNPs
Single nucleotide polymorphisms
SPRMs
Selective progesterone receptor modulators
SRCs
Steroid receptor coactivators
STAR
Steroidogenic acute regulatory protein
SUP
Superficial peritoneal endometriosis
TGF
Transforming growth factors
TNFα
Tumor necrosis factor alpha
TPO
Thyroperoxidase
TSEC
Tissue-selective estrogen complex
TSH
Thyroid stimulating hormone
TSHR
Thyroid stimulating hormone receptor
UCN
Urocortin
UPA
Ulipristal acetate
17β-HSD-2
17β-Hydroxysteroid dehydrogenase type 2
Author contributions
FP had the idea for the article, SV, SC and MR performed the literature search, and SV drafted the manuscript. FP critically revised the work. All authors approved the final manuscript.
Funding
Open access funding provided by Università degli Studi di Firenze within the CRUI-CARE Agreement. No funding was received to assist with the preparation of this manuscript.
Declarations
Conflicts of interest
The authors report no conflict of interest.
Footnotes
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Abstract
Introduction
Endocrine changes in endometriosis
Other endocrine aspects
Clinical implications: pain, infertility and systemic comorbidities in endometriosis
Endocrine background of hormonal treatments for endometriosis
Conclusions
Abbreviations
Author contributions
Funding
Declarations
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References
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. 2003 Jan-Mar;7(1):15–18.
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Accuracy of Laparoscopic Diagnosis of Endometriosis
L Mettler
L Mettler, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1,✉, T Schollmeyer
T Schollmeyer, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1, E Lehmann-Willenbrock
E Lehmann-Willenbrock, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1, U Schüppler
U Schüppler, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1, A Schmutzler
A Schmutzler, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1, D Shukla
D Shukla, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1, A Zavala
A Zavala, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1, A Lewin
A Lewin, MD
1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
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1 Department of Obstetrics and Gynaecology; University of Kiel, Kiel, Germany
✉
Address reprint requests to: L. Mettler, Prof Dr. Med, Department of Obstetrics and Gynaecology, University of Kiel, Michaelisstr. 16, 24105 Kiel, Germany. Telephone: 0049 431 5972086, Fax: 0049 431 5972116, E-mail: endo-office@email.uni-kiel.de
✉
Corresponding author.
© 2003 by JSLS, Journal of the Society of Laparoendoscopic Surgeons.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial No Derivatives License ( which permits for noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited and is not altered in any way.
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PMCID: PMC3015470 PMID: 12722993
This article has been corrected. See JSLS. 2003 Jul-Sep;7(3):290.
Abstract
Background and Objectives:
Laparoscopy is the standard method to visually identify endometriotic lesions under magnification within and outside the minor pelvis. The aim of this study was to analyze the accuracy of laparoscopic visualization in diagnosing the various endometriotic sites as confirmed histologically.
Methods:
Presumed endometriotic sites were observed in 164 patients operated on under the clinical suspicion of endometriosis. Targeted biopsies were performed for histologic corroboration, comparing the laparoscopic findings and diagnosis to the histological results.
Results:
The histological reports of the biopsies confirmed the presence of endometriosis in 138 patients (84.1%), but in 26 patients (15.9%), no evidence of endometriosis was observed. 100% of “red” lesions, 92% of “black” lesions, and 31% of “white” lesions turned out to be endometriosis. Of the 264 various suspected endometriotic sites observed, 142 (53.8%) were confirmed histologically. The most accurate diagnosis was in lesions on the parietal peritoneum of the pelvis, confirmed in 9/9 cases (100%); the ovarian fossa, confirmed in 8/12 cases (66.7%); and the uterosacral ligaments and posterior surface of the broad ligament, confirmed in 83/138 cases (60.1%). As for the other sites, the histologic confirmation rates in the ovarian surface, bowel serosa, and vesicouterine fold of the peritoneum were 48%, 40%, and 13%, respectively.
Conclusions:
Endometriosis has a multiple appearance, and the lesions may be confused with nonendometriotic lesions. It is clear that a nonhistology-based diagnosis may lead to unnecessary prolonged medical treatment and operations and may delay the proper treatment measures from being applied. Therefore, a meticulous histological confirmation should still be the first step in the laparoscopic diagnosis and treatment of suspected endometriosis.
Keywords: Endometriosis, Laparoscopy, Histology
INTRODUCTION
Early and accurate diagnosis of endometriosis may improve the quality of life of patients and provide costeffective and long-lasting treatment. Various methods are available to diagnose endometriosis as a genetic, immunologic, and endocrine-based disease. Although suspicion of endometriosis may be diagnosed with the patient's history and complaints; physical examination, especially the rectovaginal palpation; imaging techniques, such as ultrasound, MRI and computerized tomography (CT); and large, nonspecific tumor markers, such as CA 125, dysmenorrhea, dyspareunia, and chronic abdominal pain, a certain diagnosis can be verified only by histological examination. Laparoscopy is the standard method for visually identifying the endometriotic lesions under magnification within and outside the minor pelvis, and for performing targeted biopsies for histologic corroboration.1,2 Various published reports have shown that the presence of endometriosis observed at laparoscopy or laparotomy could be confirmed histologically in the majority of cases.2–6 Yet, the drawbacks of performing a laparoscopic diagnosis derive from the diversity of endometriotic appearances according to the site of the endometriotic lesion. For example, in a frozen pelvis, adhesions may completely cover endometriotic lesions. It is the aim of this study to analyze the value of laparoscopy in diagnosing the various endometriotic sites as confirmed histologically.
MATERIALS AND METHODS
Patients
Laparoscopic data on 164 endometriosis patients recorded in the German Complications Register were analyzed, comparing the laparoscopic description to the histological data. The German Complications Register is a computerized database established by the Institute of Natural Intelligence in Bremen, which compiles data from 41 German endoscopic surgery centers. In our evaluation, however, only the data from the Department of Obstetrics and Gynecology at the University of Kiel were evaluated. The evaluation period was from January 1998 until September 2000.
Laparoscopic Approach
All 164 patients were operated on under the clinical suspicion of endometriosis, comparing the laparoscopic findings and diagnosis to the histological results. Laparoscopy was performed with the patient under general anesthesia. Magnification was used to obtain a better view of the abdominal wall and the organs of the minor pelvis. Under observation, any lesion was taken as suspicious for endometriosis. To verify the diagnosis, biopsies were taken by grasping the “red,” “black,” or “white” lesion and punching it out with punch biopsy forceps. The biopsy wounds were then coagulated either by endocoagulation7 or by bipolar coagulation. In cases of ovarian endometriomas, the cysts were enucleated in the typical manner in an attempt to extract the endometriotic lesion. The base of the ovarian wound was endocoagulated at 80° to 100°C, and in most cases, the wound edges were coapted with endosutures by utilizing an extracorporal knotting technique.8
Classification of Endometriosis
Laparoscopically, the endoscopic endometriosis classification9 was applied. This classification is comparable to the AFS Classification.10 In our cases, the aim was to excise all visible red, black, or white endometriotic lesions and to verify the diagnosis histologically. The histologic diagnosis of endometriosis was determined by the presence of endometrial glands, stroma, fibrosis, and hemosiderin-carrying macrophages.4
RESULTS
The majority of patients, 98 (59.8%), had stage I endometriosis, 14 (8.5%) had stage II, 28 (17%) stage III, and 24 (14.6%) stage IV endometriosis (Table 1). The majority of patients, 111 (67.7%), were found to have multiple lesions, and 53 (32.3%) had single lesions (Table 2).
Table 3, arranged according to the site of the endometriosis, reveals that of the 264 stated sites in 164 patients (multiple sites included), lesions in the uterosacral ligament and the posterior surface of the broad ligament, suspected laparoscopically in 138 cases, were confirmed histologically in 83/138 (60.1%). Lesions on the ovarian surface were confirmed in 37/77 cases (48%); they were all black lesions. Lesions on the vesicouterine fold of the peritoneum were confirmed in only 3/23 cases (13%). Lesions in the ovarian fossa were confirmed in 8/12 cases (66.6%). Lesions in the parietal peritoneum of the pelvis were confirmed in all 9 cases. Lesions of the bowel serosa were confirmed in 2/5 cases (40%). Altogether, of the 264 various suspected endometriotic sites observed, 142 (53.8%) were confirmed histologically. Yet, when the confirmation of endometriosis in the evaluated patients is considered, the histological reports of the biopsies demonstrated endometriosis with or without fibromuscular, fibrofatty, or fibrovascular tissue in 138/164 patients (84.1%). In 26/164 patients (15.9%), no evidence of endometriosis was observed, only fibrous tissue with fat and smooth muscle (Table 4), although one may argue that some of the fibrotic lesions may derive from endometriotic damage. All (82) of the red lesions biopsied were endometriosis. In 69% (22) of the white lesions and in 8% (4) of the black lesions, no endometriosis was histologically detectable.
DISCUSSION
The laparoscopic diagnosis of endometriosis as described in the literature varies widely because of the presence of a wide range of presumably characteristic lesions.1–3,11–13 The promptness and accuracy of diagnosis is an important contribution to the application of early treatment and the prevention of scarring and adhesion and compromise of fertility.
Usually the laparoscopic diagnosis derives from the identification of the typical black or dark bluish or deep red spots on the peritoneal surface. One can easily miss the presence of endometriosis when a less marked discoloration is present. These “faint” lesions described by Jansen and Russel11 include white opacification of the peritoneum, red flame-like lesions, yellowish patches, peritoneal defects, and adhesions. These lesions may be more common and possibly more active than the dark lesions.11,14,15 An exfoliative cytologic examination was also applied in an attempt to widen diagnostic accuracy. It was shown to be of no value in the diagnosis, because in 46.5% of cases with positive histology the peritoneal aspirates failed to reveal the characteristics of endometriosis.5 Furthermore, our study demonstrates that even in the face of presumably certain endometriosis, as judged by the operators, histology failed to confirm endometriosis in almost half of the sites, and we could not describe any appearance to be a symptom of endometriosis. Nevertheless, the overall diagnostic accuracy of the presence of endometriosis in the operated on patients was high, because in 138/164 patients (84.1%), histology corroborated the laparoscopic diagnosis of endometriosis in the patients.
A careful inspection of the peritoneum and laparoscopic magnification may help in the detection of minor lesions,16 but laparoscopic magnification may also contribute to the over diagnosis that we have observed in this study.
Obviously, some endometriotic lesions are more easily recognized than others, especially the scarred blue/black, red, and brown lesions resulting from the accumulation over time of blood pigments, but a diversity of peritoneal lesions exists that may be mistaken for endometriotic lesions. Among these are chronic inflammation, foreign body reaction (black punctations resulting from the reaction to previous sutures), electrocautery and laser carbonized burns, metastases of ovarian and breast cancer, epithelial inclusions, hemangiomas, and others.2,12,16 Another confounding factor for the laparoscopic diagnosis may be the frequent combination of endometriosis with smooth muscle or fibrofatty tissue observed in half of the patients, 82 (50%) (Table 3), confirming previous observations.1 In our study, as clearly demonstrated in Table 4, endometriosis was histologically determined mainly in red and black lesions, but seldom in white lesions. No histological verification of endometriosis was obtained in 26 patients (15.9%). Of these 26 patients, 4 had black lesions; and 22 had white fibrous lesions.
As for the sites of lesions, it was previously demonstrated that the most common site of endometriosis is on the uterosacral ligaments.12 This was corroborated by our study, because of the suspected endometriotic sites by laparoscopy on the uterosacral ligaments and the posterior surface of the broad ligament, suspected in 138/264 cases (52.3 %), 83/264 (31.4%) were confirmed histologically. The low incidence of confirmation of endometriotic lesions on the vesicouterine fold of the peritoneum could be attributed to too careful and superficial sampling. Obviously, our results do not refer to deep infiltrating endometriotic lesions and microscopic implants, not being appreciated visually. In this study, no effort was made to differentiate between active and passive endometriosis; however, more histologically detectable lesions were found in red, followed by black, and least frequently in white lesions.
CONCLUSION
Endometriosis has a multiple appearance, and the lesions may be confused with other nonendometriotic lesions, as well as endometriotic lesions that are nonendometriotic by appearance or deep infiltrating ones that may be missed on visual diagnosis. It is also clear that a nonhistology-based diagnosis may lead to unnecessary, prolonged medical treatment and operations and may delay the proper treatment measures from being applied. Therefore, a meticulous histological confirmation should still be the first step in the laparoscopic diagnosis and treatment of suspected endometriosis.
References:
Anaf V, Simon P, Fayt I, Noel J. Smooth muscles are frequent components of endometriotic lesions. Hum Reprod. 2000;15:767–771 [DOI] [PubMed] [Google Scholar]
Martin DC, Hubert GD, Vender Zwang R. Laparoscopic appearance of peritoneal endometriosis. Fertil Steril. 1989;51:63–67 [DOI] [PubMed] [Google Scholar]
Nisolle M, Paindaveine B, Bourden A, Berliere M, Casansas Roux, Donnez J. Study of peritoneal endometriosis in infertile women. Fertil Steril. 1990;53:984–988 [DOI] [PubMed] [Google Scholar]
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Berube S, Marcoux S, Maheux R. Characteristics related to the prevalence of minimal or mild endometriosis in infertile women. Canadian Collaborative Group on Endometriosis. Epidemiology. 1998;9:504–510 [DOI] [PubMed] [Google Scholar]
Semm K. Endocoagulation: a new field of endoscopic surgery. J Reprod Med. 1976;16:195–203 [PubMed] [Google Scholar]
Mettler L, Caesar G, Neunzling S, Semm K. Value of endoscopic ovarian surgery - critical analysis of 626 pelviscopically operated ovarian cyst at the Kiel University Gynecologic clinic 1990-1991. Geburtshilfe Frauenheilkd. 1993;53:253–257 [DOI] [PubMed] [Google Scholar]
Mettler L, Semm K. Three step medical and surgical treatment of endometriosis. Ir J Med Sci. 1983;152(suppl 2):26–28 [DOI] [PubMed] [Google Scholar]
10.The American Fertility Society Revised American Fertility Society Classification of Endometriosis 1985. Fertil Steril. 1985;43:351–352 [DOI] [PubMed] [Google Scholar]
Jansen RPS, Russel P. Non pigmented endometriosis clinical laparoscopic and pathologic definition. Am J Obst Gynecol. 1986;155:1154–1158 [DOI] [PubMed] [Google Scholar]
Redwine DB. The visual appearance of endometriosis and its impact on our concept of disease. Current concepts in endometriosis. New York, NY: Alan R Liss; 1990:393–412 [Google Scholar]
Stripling MC, Martin DC, Chatman DC, Vander Zwang R, Poston WM. Subtle appearance of pelvic endometriosis. Fertil Steril. 1988;49:427–431 [DOI] [PubMed] [Google Scholar]
Wild RA, Wilson EA. Clinical presentation and diagnosis. In Wilson EA. ed. Endometriosis. New York, NY: Alan R. Liss; 1987:53 [Google Scholar]
Vernon MW, Beard JS, Graves K, et al. Classification of endometriotic implants by morphologic appearance. Fertil Steril. 1986;46:801–804 [PubMed] [Google Scholar]
Redwine DB. The distribution of endometriosis in the pelvis by age groups and fertility. Fertil Steril. 1987;47:173–175 [DOI] [PubMed] [Google Scholar]
Articles from JSLS : Journal of the Society of Laparoendoscopic Surgeons are provided here courtesy of Society of Laparoscopic & Robotic Surgeons
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3427 | https://www.varsitytutors.com/practice/subjects/act-math/help/lines/coordinate-geometry/geometry/perpendicular-lines | Perpendicular Lines
Help Questions
ACT Math › Perpendicular Lines
Questions 1 - 10
1
Which of the following lines is perpendicular to the line ?
Explanation
Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.
The slope of this line is . The negative reciprocal will be , which will be the slope of the perpendicular line.
Now we need to find the answer choice with this slope by converting to slope-intercept form.
This equation has a slope of , and must be our answer.
2
What line is perpendicular to and passes through ?
Explanation
Convert the given equation to slope-intercept form.
The slope of this line is . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.
The perpendicular slope is .
Plug the new slope and the given point into the slope-intercept form to find the y-intercept.
So the equation of the perpendicular line is .
3
Which of the following is the equation of a line perpendicular to the line given by:
?
Explanation
For two lines to be perpendicular their slopes must have a product of .and so we see the correct answer is given by
4
Which of the following is the equation of a line perpendicular to the line given by:
?
Explanation
For two lines to be perpendicular their slopes must have a product of .and so we see the correct answer is given by
5
What line is perpendicular to and passes through ?
Explanation
Convert the given equation to slope-intercept form.
The slope of this line is . The slope of the line perpendicular to this one will have a slope equal to the negative reciprocal.
The perpendicular slope is .
Plug the new slope and the given point into the slope-intercept form to find the y-intercept.
So the equation of the perpendicular line is .
6
Which of the following lines is perpendicular to the line ?
Explanation
Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.
The slope of this line is . The negative reciprocal will be , which will be the slope of the perpendicular line.
Now we need to find the answer choice with this slope by converting to slope-intercept form.
This equation has a slope of , and must be our answer.
7
Which of the following lines is perpendicular to the line with the given equation:?
Explanation
First we must recognize that the equation is given in slope-intercept form, where is the slope of the line.
Two lines are perpendicular if and only if the product of their slopes is . In other words, the slope of a line that is perpendicular to a given line is the negative reciprocal of that slope.
Thus, for a line with a given slope of 3, the line perpendicular to that slope must be the negative reciprocal of 3, or .
To double check that that does indeed give a product of when multiplied by three simply compute the product:
8
Which of the following lines is perpendicular to the line with the given equation:?
Explanation
First we must recognize that the equation is given in slope-intercept form, where is the slope of the line.
Two lines are perpendicular if and only if the product of their slopes is . In other words, the slope of a line that is perpendicular to a given line is the negative reciprocal of that slope.
Thus, for a line with a given slope of 3, the line perpendicular to that slope must be the negative reciprocal of 3, or .
To double check that that does indeed give a product of when multiplied by three simply compute the product:
9
What is the slope of a line that is perpendicular to the equation given by:
Explanation
Perependicular lines have slopes whose product is .
and so the answer is
10
Calculate the slope of a line perpendicular to the line with the following equation:
None of these
Explanation
Perpendicular lines will have slopes that are negative reciprocals of one another. Our first step will be to find the slope of the given line by putting the equation into slope-intercept form.
The slope of this line is .
First let's find the negative of the current slope.
Now, we need to find the reciprocal of . In order to find the reciprocal of a number we divide one by that number; therefore, we can calculate the following:
The negative reciprocal will be or which will be the slope of the perpendicular line.
Page 1of 7
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3428 | https://en.wikipedia.org/wiki/Threshold_energy | Jump to content
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Threshold energy
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From Wikipedia, the free encyclopedia
Particle creation energy in physics
In particle physics, the threshold energy for production of a particle is the minimum kinetic energy that must be imparted to one of a pair of particles in order for their collision to produce a given result. If the desired result is to produce a third particle then the threshold energy is greater than or equal to the rest energy of the desired particle. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle.
The threshold energy should not be confused with the threshold displacement energy, which is the minimum energy needed to permanently displace an atom in a crystal to produce a crystal defect in radiation material science.
Example of pion creation
[edit]
Consider the collision of a mobile proton with a stationary proton so that a meson is produced:
We can calculate the minimum energy that the moving proton must have in order to create a pion.
Transforming into the ZMF (Zero Momentum Frame or Center of Mass Frame) and assuming the outgoing particles have no KE (kinetic energy) when viewed in the ZMF, the conservation of energy equation is:
Rearranged to
By assuming that the outgoing particles have no KE in the ZMF, we have effectively considered an inelastic collision in which the product particles move with a combined momentum equal to that of the incoming proton in the Lab Frame.
Our terms in our expression will cancel, leaving us with:
Using relativistic velocity additions:
We know that is equal to the speed of one proton as viewed in the ZMF, so we can re-write with :
So the energy of the proton must be MeV.
Therefore, the minimum kinetic energy for the proton must be MeV.
Example of antiproton creation
[edit]
At higher energy, the same collision can produce an antiproton:
If one of the two initial protons is stationary, we find that the impinging proton must be given at least of energy, that is, 5.63 GeV. On the other hand, if both protons are accelerated one towards the other (in a collider) with equal energies, then each needs to be given only of energy.
A more general example
[edit]
Consider the case where a particle 1 with lab energy (momentum )
and mass impinges on a
target particle 2 at rest in the lab, i.e. with lab energy and mass .
The threshold energy to produce three particles of masses , ,
, i.e.
is then found by assuming that these three particles are at rest in the center of mass frame (symbols with
hat indicate quantities in the center of mass frame):
Here is the total energy available in the center of mass frame.
Using , and
one derives that
References
[edit]
^ Jump up to: a b c Michael Fowler. "Transforming Energy into Mass: Particle Creation". Particle Creation. Archived from the original on Aug 15, 2022.
^ Jackson, John (14 August 1998). Classical Electrodynamics. Wiley. pp. 533–539. ISBN 978-0-471-30932-1.
Retrieved from "
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3429 | https://math.stackexchange.com/questions/2102552/how-do-i-find-arctan-left-tan-left-frac2-pi3-right-right | algebra precalculus - How do I find $\arctan\left(\tan\left(\frac{2\pi}{3}\right)\right)$ - Mathematics Stack Exchange
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How do I find arctan(tan(2 π 3))arctan(tan(2 π 3))
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How do I find arctan(tan(2 π 3))arctan(tan(2 π 3))?
My thought process
So I first drew a right triangle on the 2nd quadrant as 3√2 3 2 and −1 2−1 2 as the legs of the triangles but then I get the answer as −3–√−3 for tangent of the angle in the triangle that i drew and I dont know how to evaluate the arctan.
algebra-precalculus
trigonometry
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edited Jan 18, 2017 at 4:36
zipirovich
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asked Jan 18, 2017 at 4:08
John RawlsJohn Rawls
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You use principle value.snowfall512 –snowfall512 2017-01-18 04:10:31 +00:00 Commented Jan 18, 2017 at 4:10
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Points to remember:
(1)(1) The range of arctan arctan is from (−π 2,π 2)(−π 2,π 2).
(2)(2) We have tan 2 π 3=tan(π−π 3)=−tan π 3=tan(−π 3)tan2 π 3=tan(π−π 3)=−tanπ 3=tan(−π 3).
Hope you can take it from here.
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edited Jan 18, 2017 at 4:23
answered Jan 18, 2017 at 4:16
user371838 user371838
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Range, not domain. The domain of arctan arctan is all reals.zipirovich –zipirovich 2017-01-18 04:21:03 +00:00 Commented Jan 18, 2017 at 4:21
@zipirovich Thanks for correcting.user371838 –user371838 2017-01-18 04:24:22 +00:00 Commented Jan 18, 2017 at 4:24
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For a straightforward solution to this specific question, you have a good start: since indeed tan 2 π 3=−3–√tan2 π 3=−3, the expression at hand now takes the form arctan(tan 2 π 3)=arctan(−3–√)arctan(tan2 π 3)=arctan(−3). Now you need to recall the (standard) definition of the arctangent function:
arctan a=φ arctana=φ means that tan φ=a tanφ=a and φ∈(−π 2,π 2)φ∈(−π 2,π 2).
That's why 2 π 3 2 π 3 itself cannot be the answer: it's outside the range of the arctangent function. You need to find an angle within this range whose tangent is also −3–√−3. Since tan(−π 3)=−3–√tan(−π 3)=−3 and −π 3∈(−π 2,π 2)−π 3∈(−π 2,π 2), the final answer is
arctan(tan 2 π 3)=arctan(−3–√)=−π 3.arctan(tan2 π 3)=arctan(−3)=−π 3.
For an arbitrary angle θ θ, for the same reason arctan(tan θ)≠θ arctan(tanθ)≠θ in general; this is only true if θ∈(−π 2,π 2)θ∈(−π 2,π 2). To find the value of arctan(tan θ)arctan(tanθ) for any other θ θ you'll need to do something similar to find an appropriate angle within the range of arctangent. The concept of reference angles can help with that.
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answered Jan 18, 2017 at 4:30
zipirovichzipirovich
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Let θ=tan−1(tan(2 π 3))θ=tan−1(tan(2 π 3))
tan θ=tan(2 π 3)tanθ=tan(2 π 3)
tan θ=tan(π−π 3)tanθ=tan(π−π 3)
[as tan is in the 2nd quadrant]
tan θ=−3–√tanθ=−3 ......(1)
As tan π 3=3–√tanπ 3=3
So tan(−π 3)=−3–√tan(−π 3)=−3
From equation (1),
tan θ=tan(−π 3)tanθ=tan(−π 3)
θ=−π 3 θ=−π 3
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answered Jan 18, 2017 at 4:33
Kanwaljit SinghKanwaljit Singh
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3430 | https://pmc.ncbi.nlm.nih.gov/articles/PMC10670724/ | Clinical Characteristics and Immune Responses in Children with Primary Ciliary Dyskinesia during Pneumonia Episodes: A Case–Control Study - PMC
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Children (Basel)
. 2023 Oct 24;10(11):1727. doi: 10.3390/children10111727
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Clinical Characteristics and Immune Responses in Children with Primary Ciliary Dyskinesia during Pneumonia Episodes: A Case–Control Study
Danli Lu
Danli Lu
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
2 Key Laboratory of Birth Defects and Related Diseases of Women and Children, Sichuan University, Ministry of Education, Chengdu 610000, China
3 NHC Key Laboratory of Chronobiology, Sichuan University, Chengdu 610000, China
4 The Joint Laboratory for Lung Development and Related Diseases of West China Second University Hospital, School of Life Sciences of Fudan University, West China Institute of Women and Children’s Health, West China Second University Hospital, Sichuan University, Chengdu 610000, China
5 Sichuan Birth Defects Clinical Research Center, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Danli Lu
1,2,3,4,5, Wenhao Yang
Wenhao Yang
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
2 Key Laboratory of Birth Defects and Related Diseases of Women and Children, Sichuan University, Ministry of Education, Chengdu 610000, China
3 NHC Key Laboratory of Chronobiology, Sichuan University, Chengdu 610000, China
4 The Joint Laboratory for Lung Development and Related Diseases of West China Second University Hospital, School of Life Sciences of Fudan University, West China Institute of Women and Children’s Health, West China Second University Hospital, Sichuan University, Chengdu 610000, China
5 Sichuan Birth Defects Clinical Research Center, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Wenhao Yang
1,2,3,4,5, Rui Zhang
Rui Zhang
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Rui Zhang
1, Yan Li
Yan Li
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Yan Li
1, Tianyu Cheng
Tianyu Cheng
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
2 Key Laboratory of Birth Defects and Related Diseases of Women and Children, Sichuan University, Ministry of Education, Chengdu 610000, China
3 NHC Key Laboratory of Chronobiology, Sichuan University, Chengdu 610000, China
4 The Joint Laboratory for Lung Development and Related Diseases of West China Second University Hospital, School of Life Sciences of Fudan University, West China Institute of Women and Children’s Health, West China Second University Hospital, Sichuan University, Chengdu 610000, China
5 Sichuan Birth Defects Clinical Research Center, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Tianyu Cheng
1,2,3,4,5, Yue Liao
Yue Liao
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Yue Liao
1, Lina Chen
Lina Chen
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
2 Key Laboratory of Birth Defects and Related Diseases of Women and Children, Sichuan University, Ministry of Education, Chengdu 610000, China
3 NHC Key Laboratory of Chronobiology, Sichuan University, Chengdu 610000, China
4 The Joint Laboratory for Lung Development and Related Diseases of West China Second University Hospital, School of Life Sciences of Fudan University, West China Institute of Women and Children’s Health, West China Second University Hospital, Sichuan University, Chengdu 610000, China
5 Sichuan Birth Defects Clinical Research Center, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Lina Chen
1,2,3,4,5, Hanmin Liu
Hanmin Liu
1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
2 Key Laboratory of Birth Defects and Related Diseases of Women and Children, Sichuan University, Ministry of Education, Chengdu 610000, China
3 NHC Key Laboratory of Chronobiology, Sichuan University, Chengdu 610000, China
4 The Joint Laboratory for Lung Development and Related Diseases of West China Second University Hospital, School of Life Sciences of Fudan University, West China Institute of Women and Children’s Health, West China Second University Hospital, Sichuan University, Chengdu 610000, China
5 Sichuan Birth Defects Clinical Research Center, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Find articles by Hanmin Liu
1,2,3,4,5,
Editors: Davor Plavec, Mirjana Turkalj
Author information
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1 Department of Pediatric Pulmonology and Immunology, West China Second University Hospital, Sichuan University, Chengdu 610000, China
2 Key Laboratory of Birth Defects and Related Diseases of Women and Children, Sichuan University, Ministry of Education, Chengdu 610000, China
3 NHC Key Laboratory of Chronobiology, Sichuan University, Chengdu 610000, China
4 The Joint Laboratory for Lung Development and Related Diseases of West China Second University Hospital, School of Life Sciences of Fudan University, West China Institute of Women and Children’s Health, West China Second University Hospital, Sichuan University, Chengdu 610000, China
5 Sichuan Birth Defects Clinical Research Center, West China Second University Hospital, Sichuan University, Chengdu 610000, China
Correspondence: liuhm@scu.edu.cn
Roles
Davor Plavec: Academic Editor
Mirjana Turkalj: Academic Editor
Received 2023 Aug 28; Revised 2023 Oct 18; Accepted 2023 Oct 20; Collection date 2023 Nov.
© 2023 by the authors.
Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (
PMC Copyright notice
PMCID: PMC10670724 PMID: 38002818
Abstract
Objective: This study explored the clinical features and immune responses of children with primary ciliary dyskinesia (PCD) during pneumonia episodes. Methods: The 61 children with PCD who were admitted to hospital because of pneumonia were retrospectively enrolled into this study between April 2017 and August 2022. A total of 61 children with pneumonia but without chronic diseases were enrolled as the control group. The clinical characteristics, levels of inflammatory indicators, pathogens, and imaging features of the lungs were compared between the two groups. Results: The PCD group had higher levels of lymphocytes (42.80% versus 36.00%, p = 0.029) and eosinophils (2.40% versus 1.25%, p = 0.020), but lower neutrophil counts (3.99 versus 5.75 × 10 9/L, p = 0.011), percentages of neutrophils (46.39% versus 54.24%, p = 0.014), CRP (0.40 versus 4.20 mg/L, p< 0.001) and fibrinogen (257.50 versus 338.00 mg/dL, p = 0.010) levels. Children with PCD and children without chronic diseases were both most commonly infected with Mycoplasma pneumoniae (24.6% versus 51.9%). Children with PCD had significantly more common imaging features, including mucous plugging (p = 0.042), emphysema (p = 0.007), bronchiectasis (p< 0.001), mosaic attenuation (p = 0.012), interstitial inflammation (p = 0.015), and sinusitis (p< 0.001). Conclusion: PCD is linked to immune system impairment, which significantly contributes to our understanding of the pathophysiology of this entity.
Keywords: primary ciliary dyskinesia, pneumonia, inflammation, immunology, pathogen, imaging examination
1. Introduction
Primary ciliary dyskinesia (PCD) is a rare genetic disorder affecting upper and lower airway motile cilia, causing recurrent chronic respiratory infections, starting in infancy . So far, the incidence rate of PCD in children is between 1/20,000 and 1/10,000 . The prevalence of PCD in children with recurrent respiratory infections is up to 5%, and in children with bronchiectasis, it is estimated to be 26% [3,4]. Unexplained respiratory distress syndrome in newborns and frequent respiratory infections in children can present in PCD . Early embryonic development, and the brain and genital systems can also be involved, manifesting as situs inversus, hydrocephalus, and infertility . The quality of life of PCD patients is significantly lower than that of healthy individuals [6,7].
The need for early diagnosis is underscored by data showing that delayed diagnosis is linked to poorer lung function and quality of life . However, there is no independent gold-standard diagnostic test for PCD . Specific PCD-causative genes can cause different clinical manifestations and disease severity, which makes PCD a very distinct disease and increases the difficulty of diagnosis [9,10]. Nearly 37% of patients showed PCD-related symptoms but could not be diagnosed with PCD according to the recommended diagnostic guidelines . Nasal nitric oxide measurement (nNO), transmission electron microscopy (TEM), and gene testing are used to diagnose PCD, most of which are expensive or unavailable [12,13]. As a relatively cheap and non-invasive method, nNO has a sensitivity rate of 98% and a specificity rate of 99% in detecting PCD. However, this test requires a high degree of cooperation and is usually only suitable for children over 5 years of age . Moreover, approximately only 70% of individuals with clinical manifestations of PCD have known ultrastructure defects or mutations . In addition, the current treatment strategy for PCD mainly refers to cystic fibrosis, a similar chronic respiratory disease, mainly focusing on promoting the airway clearance of mucus and anti-infection treatments . According to a recent multicenter study, azithromycin is now the only evidence-based medication available that can reduce the aggravation of respiratory symptoms . These apparent issues in diagnosis and treatment reflect the fact that physicians lack an appropriate understanding of this disease.
Motile cilia are responsible for beating rhythmically and sweeping out fluid, mucus, and pathogenic particles, which are important for mucosal defense in the respiratory tracts . Recurrent respiratory infections in children with PCD are currently thought to be due to decreased ciliary clearance [16,17,18]. Recurrent respiratory infection symptoms in PCD patients include coughing, sputum production, nasal congestion, and rhinorrhea, which are non-specific compared with ordinary pneumonia . However, previous studies suggest that PCD patients have specific alterations in inflammatory indicators. According to prior research, PCD patients have higher neutrophil counts, neutrophil elastase levels, and CXCL8/IL-8 levels in their sputum than healthy individuals . The activity of neutrophils extracted from PCD patients is reduced, which may be crucial in the occurrence of recurrent respiratory infections [21,22,23]. The count and percentage of neutrophils and other immune cells in the blood could directly reflect the level of immune response. However, most previous studies have focused on the levels of immune cells or cytokines in the sputum, which only reflect the local immune response of the airway. Few clinical studies have focused on the composition of peripheral blood immune cells in PCD patients or explored the differences in clinical manifestations and systematic immune responses between healthy children and children with PCD during the early stages of acute infection.
Thus, we performed this retrospective study to explore the differences in clinical features and immune responses between children with and without PCD during pneumonia episodes. These findings should improve our comprehension of the clinical characteristics of PCD in the acute early stages of infection. In addition, the discovery of PCD-related immune responses may guide the development of therapeutic treatments for PCD.
2. Materials and Methods
2.1. Study Design
2.1.1. Patient Enrollment
In this single-center retrospective study, we reviewed the records of children with pneumonia and unexplained recurrent respiratory infections between April 2017 and August 2022 who met the following criteria: (1) presented with two or more of the four highly suspected PCD clinical manifestations, including unexplained neonatal respiratory distress during the neonatal period, year-round daily cough starting before the age of 6 months, year-round daily nasal congestion starting before the age of 6 months, or organ laterality defects, in accordance with the ATS PCD screening criteria ; or (2) their score was greater than 5 points on the PICADAR scoring scale .
To provide a reference for assessing immune responses in children with PCD, we matched the enrolled children with PCD to control group children who suffered pneumonia at our institution during the same period. Control patients were randomly selected and matched to children with PCD at a ratio of 1:1. Age, sex, and severity of pneumonia were matched accordingly, in order to reduce bias. Children with pneumonia treated in our hospital were included according to the following criteria: (1) no history of recurrent respiratory infections; and (2) no history of chronic disease.
Children with congenital immunodeficiency, cystic fibrosis, idiopathic pulmonary fibrosis, other airway and lung developmental malformations, foreign body aspiration, or chronic heart, liver, kidney, and blood system diseases were excluded. All children met the diagnostic criteria and hospitalization criteria for community-acquired pneumonia according to the British Thoracic Society guidelines . Additionally, a PCD exacerbation was deemed to exist if three or more of the subsequent seven conditions were present, in accordance with expert consensus : (1) increased coughing, (2) altered sputum volume and/or color, (3) increased shortness of breath perceived by the patient or parent, (4) decision to begin or alter antibiotic therapy due to perceived pulmonary symptoms, (5) malaise, tiredness, fatigue, or lethargy, (6) new or increased haemoptysis, or (7) temperature > 38 °C.
2.1.2. Criteria for Diagnosing PCD
PCD can be diagnosed in children with highly suspected PCD who meet one of the following three conditions: low nasal nitric oxide level after excluding CF, PCD-associated genes with biallelic pathogenic variations, or a recognized ciliary ultrastructural defect .
2.2. Data Extraction
The following data for each patient was obtained from electronic medical records: age, gender, body mass index, clinical diagnosis, symptoms, lung function, TEM, results of laboratory tests, chest and nasopharynx computed tomography, and respiratory secretion bacterial culture results. The primary outcomes were differences in immune-related indicators including white blood cells (WBC), neutrophils, monocytes, lymphocytes, eosinophils, C-reactive proteins (CRPs), and fibrinogen from peripheral blood. Clinical features, pathogens, and imaging features of the PCD and control groups constituted the secondary outcome.
2.3. Laboratory Evaluations
Our hospital has a standardized admission process and strict requirements for the timing of blood collection for children. All of the children were admitted to the hospital within 48 h of onset. The nurse collected the venous blood on the day of the child’s admission, and sent it to the Laboratory Department of our hospital within 2 h for analysis of the blood routine examination, CRP levels, and other indicators. Immune cell counts and percentages in the peripheral blood were used to assess the systemic immune status of the PCD. CRP and fibrinogen levels were used to assess the level of inflammation. Respiratory secretions were collected, including sputum and nasopharyngeal swabs, and sent for microbial testing within 30 min. Sputum cultures were used to identify bacterial and fungal infections. Seven patients in the control group had completed etiological examinations in other hospitals before admission, and their family members refused to repeat the tests after admission. In order to ensure consistency in the test results, we performed statistical analysis after excluding patients who had completed examinations from other hospitals. An alveolar lavage fluid examination was performed for all PCD patients but only for some control patients, so this part of the data was not selected. The virus was detected in the nucleic acid of the nasopharyngeal swabs from the patients’ respiratory tracts. Serological methods were relied upon to detect atypical pathogens, namely Mycoplasma pneumoniae (MP) and Chlamydia pneumoniae (CP). Serum IgM antibody titers ≥1:160 were used to detect MP infections, and serum IgM antibody titers ≥1:32 were used to detect CP infections. Computed tomography of the chest and nasopharynx were completed within 48 h of admission for all patients. All CT scans were assessed by a radiologist with eight years of clinical experience reading chest CT scans (SD). Children who did not meet the above conditions were excluded.
2.4. Ethics
The Ethical Review Board of the West China Second University Hospital, Sichuan University, granted consent for this study (approval number: 2022YFS076). All participants or their legal guardians gave their informed permission.
2.5. Statistical Analyses
Continuous variables were presented as means and standard deviations (SDs) for data with a normal distribution, while categorical variables were presented as frequencies (percentages). If data were normally distributed, inter-group differences in continuous variables were assessed for significance using the independent-samples t-test. If the data were skewed or had an uneven variance, the Mann–Whitney U test was applied. Differences in categorical variables were assessed using the chi-squared and Fisher’s exact tests. p< 0.05 was considered statistically significant. All statistical analyses were performed using SPSS 23.0 (IBM, Armonk, NY, USA).
3. Results
A total of 167 children met the PCD screening criteria of the ATS guidelines or had a PICADAR score > 5 points. From this, a further 106 children were excluded, as due to inadequate diagnostic facilities and clinician training, they had an insufficient diagnostic basis and failed to meet the diagnostic criteria. A total of 15 children were confirmed to have PCD through transmission electron microscopy (TEM), 18 children were confirmed to have PCD through TEM and genetic testing, and 28 children without recognized ciliary ultrastructural defect were confirmed to have PCD through genetic testing. In total, 61 children with PCD and 61 control children were enrolled in this study (Figure 1).
Figure 1.
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Flowchart of patient enrollment.
3.1. Patient Characteristics
Of all the patients with PCD, 32 (52.46%) were male and 29 (47.54%) were female. There were no statistical differences in gender, age, height, weight, or body mass index (BMI). Common respiratory symptoms, such as coughing, expectoration, gasping, wheezing, stuffy nose, and running nose, did not show distinctly different patterns between the PCD group and the control group. A total of 6.56% of children with PCD (4/61) had congenital heart defects (atrial septal defect, patent ductus arteriosus, and patent foramen ovale) and 11% (7/61) had asthma. Impaired lung function in PCD manifested as a decrease in FEV1% (ratio of forced expiratory volume in one second, actual measured to predicted values; mean value < 80%), FEV1/FVC% (ratio of FEV1 to forced vital capacity, actual measured to predicted values; mean value < 92%), FEV1/VC MAX% (ratio of FEV1 to max vital capacity, actual measured to predicted values; mean value < 92%), and MMEF (maximal midexpiratory flow curve, actual measured to predicted values; mean value < 65%), a combination of obstructive and restrictive ventilatory dysfunctions, and small airway dysfunction. All children with PCD underwent TEM when disease stable: eleven cases (18.03%) were found not to have abnormal cilia structures, thirty-three cases (54.10%) had typical PCD ultrastructure defects (thirty with outer dynein arm and inner dynein arm deficiency, three with the absence of a central pair), and seventeen cases (27.87%) had unidentifiable cilia structures or were difficult to classify (Table 1). To illustrate these findings, the TEM images of a child with a normal ultrastructure of the ciliary axonemes and two patients with typical PCD ultrastructure deficiency (outer dynein arm and inner dynein arm deficiency, and central pair complex abnormalities) are given in Figure 2.
Table 1.
Characteristics of children in the PCD and control groups.
| Characteristic | PCD Group (n = 61) | Control Group (n = 61) | p |
:---: :---: |
| Mean age, years (SD) | 5.60 ± 3.43 | 5.54 ± 3.39 | 0.932 |
| Female | 29 (47.54%) | 32 (52.46%) | 0.587 |
| Height, m (SD) | 1.18 ± 0.23 | 1.13 ± 0.24 | 0.292 |
| Z-scores for height (SD) | −0.07 ± 1.32 | −0.18 ± 1.27 | 0.648 |
| Weight, kg (SD) | 21.55 ± 10.76 | 21.09 ± 11.21 | 0.819 |
| Z-scores for weight (SD) | −0.003 ± 1.99 | −0.08 ± 1.36 | 0.808 |
| Body mass index, kg/m 2 (SD) | 16.27 ± 2.60 | 16.04 ± 3.29 | 0.712 |
| Symptoms | | | |
| Cough | 61 (100.00%) | 61 (100.00%) | NA |
| Expectoration | 50 (81.97%) | 55 (90.16%) | 0.191 |
| Gasp | 13 (21.31%) | 16 (26.23%) | 0.523 |
| Wheezing | 27 (44.26%) | 17 (27.87%) | 0.059 |
| Stuffing nose | 26 (42.62%) | 20 (32.79%) | 0.262 |
| Running nose | 28 (45.90%) | 23 (37.70%) | 0.359 |
| Congenital heart disease | 4 (6.56%) | 0 (0) | 0.042 |
| Asthma | 7 (11.00%) | 0 (0) | 0.006 |
| Lung function | | | |
| FEV1 percent predicted | 77.94 ± 14.14 | NA | NA |
| FEV1/FVC percent predicted | 89.67 ± 7.97 | NA | NA |
| FEV1/VC MAX percent predicted | 89.27 ± 7.85 | NA | NA |
| MMEF percent predicted | 50.31 ± 23.36 | NA | NA |
| TEM | | | |
| Normal ultrastructure | 11 (18.03%) | NA | NA |
| Fail to recognize | 17 (27.87%) | NA | NA |
| Abnormal ultrastructure | 33 (54.10%) | NA | NA |
| ODA and IDA deficiency | 30 (90.90%) | NA | NA |
| CP abnormalities | 3 (9.10%) | NA | NA |
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Values are means ± standard deviation or n (%), unless otherwise noted. p< 0.05. NA, not application. FEV1, forced expiratory volume in one second. FEV1/FVC: ratio of FEV1 to forced vital capacity. FEV1/VC: ratio of FEV1 to vital capacity. MMEF: maximal midexpiratory flow curve. TEM, transmission electron microscopy. ODA, outer dynein arm. IDA, inner dynein arm. CP, central pair complex.
Figure 2.
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TEM images of the ultrastructure of the ciliary axonemes from a normal child and patients with PCD: (a) Normal ciliary axonemes in a healthy individual, indicated by the green arrow. (b) The lack of ODA and IDA in the patient, indicated by the red arrows. (c) The lack of CP in the patient, indicated by the red arrows.
3.2. Inflammatory Factors
The neutrophil counts, percentages of neutrophils, and CRP and fibrinogen levels in the control group were significantly higher than those in the PCD group. Children with PCD had high percentages of lymphocytes as well as high percentages of eosinophils. The white blood cell counts, monocyte levels, and percentages of monocytes in the control group were higher than in the PCD group, while the lymphocyte and eosinophil levels were higher in the PCD group, although there were no statistical differences (Table 2).
Table 2.
Inflammatory parameters for the two children’s groups.
| Parameters | PCD Group (n = 61) | Control Group (n = 61) | p |
:---: :---: |
| White blood cell, ×10 9/L | 8.39 ± 3.57 | 9.89 ± 4.90 | 0.055 |
| Lymphocytes, ×10 9/L | 3.52 ± 2.16 | 3.21 ± 1.93 | 0.397 |
| Percentage of lymphocytes, % | 42.80 ± 17.03 | 36.00 ±17.09 | 0.029 |
| Monocytes, ×10 9/L | 0.52 (0.40, 0.76) | 0.60 (0.43, 0.78) | 0.303 |
| Percentage of monocytes, % | 7.05 ±2.20 | 7.46 ±3.14 | 0.408 |
| Neutrophils, ×10 9/L | 3.99 ± 2.72 | 5.75 ± 4.55 | 0.011 |
| Percentage of neutrophils, % | 46.39 ±17.08 | 54.24 ±17.76 | 0.014 |
| Eosinophils, ×10 9/L | 0.18 (0.08, 0.34) | 0.11 (0.02, 0.23) | 0.059 |
| Percentage of eosinophiles, % | 2.40 (1.25, 4.70) | 1.25 (0.13, 2.83) | 0.020 |
| C-reactive protein, mg/L | 0.40 (0.40, 0.95) | 4.20 (0.40, 16.50) | <0.001 |
| Fibrinogen | 257.50 (229.25, 289.75) | 338.00 (232.00, 437.00) | 0.010 |
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Values are means ± standard deviation or median (interquartile range), unless otherwise noted. p< 0.05.
3.3. Prevalence of Respiratory Pathogens
The prevalence of bacteria differed significantly between the children with PCD and the control group children. The children with PCD and the control group children were both more commonly infected with Haemophilus influenzae among all kinds of bacteria, without any significant differences. Although the prevalence of atypical pathogens was high in both groups, the prevalence was significantly higher in the control group children than in the children with PCD. Compared with the children with PCD, the Mycoplasma pneumoniae infection rate was higher in the control group. The infection rates of fungus and viruses were low in the two groups, and there were no statistical differences. However, the fungal infection rate in the PCD group tended to be higher than in the control group, and there were no fungal infections detected in the control group (Table 3). All culture-positive patients turned negative after receiving standard anti-infection treatments.
Table 3.
Respiratory pathogen results for the PCD and control groups.
| Species | PCD Group (n = 61) | Control Group (n = 54) | p Value |
:---: :---: |
| n | Frequency, % | n | Frequency, % |
| Bacteria | 21 | 34.4 | 9 | 16.7 | 0.030 |
| Haemophilus influenzae | 12 | 19.7 | 6 | 9.8 | 0.207 |
| Streptococcus pneumoniae | 2 | 3.3 | 1 | 1.9 | 0.632 |
| Streptococci viridans group | 2 | 3.3 | 1 | 1.9 | 0.632 |
| Klebsiella pneumoniae | 2 | 3.3 | 0 | 0 | 0.179 |
| Acinetobacter baumannii | 1 | 1.6 | 0 | 0 | 0.345 |
| Escherichia coli | 1 | 1.6 | 0 | 0 | 0.345 |
| Enterobacter cloacae | 1 | 1.6 | 0 | 0 | 0.345 |
| Staphylococcus aureus | 0 | 0 | 1 | 1.9 | 0.286 |
| Atypical pathogen | 21 | 34.4 | 31 | 57.4 | 0.013 |
| Mycoplasma pneumoniae | 15 | 24.6 | 28 | 51.9 | 0.003 |
| Chlamydia pneumoniae | 6 | 9.8 | 9 | 16.7 | 0.278 |
| Fungus | 4 | 6.6 | 0 | 0 | 0.055 |
| Candida albicans | 2 | 3.3 | 0 | 0 | 0.179 |
| Lodderomyces elongisporus | 1 | 1.6 | 0 | 0 | 0.345 |
| Candida parapsilosis | 1 | 1.6 | 0 | 0 | 0.345 |
| Virus | 3 | 4.9 | 6 | 9.8 | 0.217 |
| Influenza A virus | 1 | 1.6 | 0 | 0 | 0.345 |
| Rhinovirus | 1 | 1.6 | 3 | 5.6 | 0.253 |
| Human adenovirus | 1 | 1.6 | 1 | 1.9 | 0.931 |
| Bocavirus | 0 | 0 | 2 | 3.7 | 0.124 |
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p< 0.05.
3.4. Imaging Features of Chest and Nasopharynx
In this study, chest computed tomography were completed for all of the children. The incidence of emphysema, bronchiectasis, mosaic attenuation, interstitial inflammation, and sinusitis was significantly increased in the PCD group. Although there were no significant differences between the two groups, the incidence of mucous plugging, pulmonary consolidation, atelectasis, and pleural effusion in the PCD group showed an increasing trend (Figure 3. The lobe most commonly infected in acute lung infections in patients with PCD was the right middle lobe, while in the control group it was the right lower lobe. The probability of right middle lobe infection in children with PCD was significantly higher than in children with pneumonia (Figure 4). Comparison CT images of the two groups and typical imaging features of children with PCD, such as emphysema, bronchiectasis, mosaic attenuation, and interstitial inflammation, have been given in Figure 5.
Figure 3.
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Imaging features in the two groups. p< 0.05.
Figure 4.
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Lobe involvement in the two groups. p< 0.05.
Figure 5.
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Comparison CT scan of the two groups and typical imaging features of PCD. (a) CT shows patchy pulmonary shadows in a control-group patient, especially in the left lobes. (b) Chest CT of a patient with PCD shows the bronchiectasis and compensated emphysema; the red arrow points to signs of pulmonary emphysema. (c) Chest CT of a patient with PCD shows interstitial inflammation. (d) Chest CT shows mosaic attenuation in a patient with PCD. (e) Nasopharynx CT shows sinusitis in a control-group patient. (f) Nasopharynx CT shows sinusitis in a patient with PCD. The red arrow in (e,f) points to signs of nasosinusitis.
4. Discussion
This study assessed the clinical characteristics and immune responses during acute pulmonary infections in children with PCD.
In our cohort, 6.6% of children with PCD exhibited congenital heart disease (CHD), a similar frequency to that reported in previous studies. Properly working motile nodal cilia are in charge of generating a leftward flow pattern at the embryonic node, breaking the pattern of symmetry and permitting the appropriate development of internal organs, including a properly arranged, asymmetric cardiopulmonary circulation system . Genetically, a higher frequency of organ laterality defects, and occasionally congenital heart disease, is linked to mutations encoding for ODA proteins, ODA plus IDA proteins, and IDA proteins with microtubule disorganization defects . Moreover, many cilia-transduced cell signaling pathways, such as Shh, Wnt, Pdgf, and Tgfβ-BMP, are known to be crucial to cardiovascular development, and their disruption may contribute to the etiology of CHD . A cardiac evaluation may be a good recommendation for all children with PCD. The children with PCD in our study were 5.60 years old on average, and their lungs showed mild obstructive and restrictive dysfunctions. Causes of early reduced lung function in PCD include reversible etiologies, such as transient atelectasis and sputum plug obstruction, and irreversible etiologies, such as lung remodeling following repeated severe lung infections throughout development . Another important finding in our study was that a considerable proportion of clinical biopsies with electron microscopic findings showed insufficient cilia for TEM analysis; the reasons for this could be insufficient transverse sections, inadequate quality samples, or atypic polymorphous post-infection abnormalities, which appear to be a problem for the diagnosis of PCD [32,33,34]. The characteristic radiographic findings in patients with PCD included bronchiectasis, which is due to structural changes caused by recurrent respiratory infections in children. This finding is consistent with earlier studies [35,36].
We also found that the levels of neutrophils in children with PCD were lower than those in the control group children. The innate immune system is the initial line of defense against invading microbial pathogens, and plays an essential role in the early phases of infection . After pathogens enter the body, airway epithelial cells or local macrophages secrete chemokines to recruit neutrophils and other immune cells from the blood to the damaged area . At the local injury site, damage was exacerbated by the airway epithelial cells and macrophages, which released chemokines, and the enrichment of the neutrophils . Neutrophils are the first immune cells to arrive at the site of infection, and their antimicrobial response is critical in the early stages of the inflammatory process. In our study, the decreased count and percentage of neutrophils in the blood reflected the impaired immune response of patients with PCD. Previous in vitro studies have demonstrated that blood neutrophils from patients with PCD show reduced chemotaxis to chemokines [21,22,40,41]. The chemotactic differentials in response to chemokines, including leukotriene B4, complement 5a, and N-formylmethionyl-leucyl-phenylalanine, in neutrophils from patients with PCD were substantially lower than the comparable levels in neutrophils from patients without chronic diseases . Patients with PCD are prone to recurrent respiratory tract infections, which are mostly thought to be caused by ciliary dysfunction; however, these findings suggest that a weakened neutrophil immune response may also contribute. This conclusion corresponds to our results that show that children with PCD had significantly lower levels of neutrophils than the control group children. Neutrophils have a cytoplasmic microtubule system that, if abnormal and dysfunctional, can also lead to recurrent infections . Our previous study found that DNAH5 mutations not only lead to impaired ciliary function, but also cause a reduced immune response . We speculate that genes encoding neutrophil microstructures may overlap with PCD-related genes; more research is needed to explore and verify this. Improving neutrophil function may be a target for the future treatment of PCD. Additionally, the monocyte counts between the two groups showed no significant differences. Monocyte function is crucial for our understanding of the immune response in PCD patients. Upon stimulation with bacterial products, monocytes from PCD patients produced significantly higher levels of pro-inflammatory cytokines and chemokines compared to healthy individuals . However, Walter et al. also described the normal chemotactic migration of monocytes from PCD patients . Our study shows that peripheral blood monocytes in PCD patients are within the normal range, reflecting the normal chemotactic function of monocytes, which supports the findings of earlier studies. Moreover, in this study, patients with PCD had a higher percentage of eosinophils. Eosinophils are cells of the innate immune system, and are known to increase during specific immune responses, including parasite infections and allergic diseases . Our research reveals that a considerable proportion of patients with PCD have coexistent asthma, which has also been reported in previous studies . However, the relationship between PCD and asthma is still unclear.
Our results also found that fibrinogen and CRP levels in patients with PCD with acute respiratory infections were lower than those in control patients, which has not been reported in other studies. CRP is an acute-phase inflammatory protein that is mainly synthesized in liver cells. In the presence of calcium, CRPs can bind to polysaccharides in microbes and activate C1q to initiate the classical complement pathway of innate immunity . CRPs are crucial to the inflammatory process and the host’s response to infection, involving the complement pathway, apoptosis, and phagocytosis [47,48,49]. Another indicator, fibrinogen, including thrombin and plasminogen, is demonstrated to be involved in the early stages of the innate immune system’s response, quickly walling off and eliminating encroaching invaders . Fibrinogen can also potently drive acute and reparative inflammatory pathways that affect tissue damage, remodeling, and repair . The decreased levels of fibrinogen and CRPs in patients with PCD may mean a decrease in innate immunity and resistance to infection, which may contribute to the increased tissue damage of these patients. This suggests that in clinical practice, fibrinogen and CRP levels may not be suitable indicators of the severity of early infection in patients with PCD.
The main pathogens that infected children with PCD were Mycoplasma pneumoniae (24.6%) and Haemophilus influenzae (19.7%), while the main pathogen infecting the control group children was Mycoplasma pneumoniae (51.9%). Mycoplasma pneumoniae infection is the most common pathogenic infection in children [52,53,54]. Interestingly, children with PCD are less susceptible to Mycoplasma pneumonia infections compared to healthy children. Mycoplasma pneumonia invades the respiratory tract, slides and moves to locate in the crypts of ciliated cells, and then adheres to receptors on the surface of respiratory epithelial cells in order to resist ciliary clearance and phagocyte engulfment, which is required for mycoplasma pneumonia to cause disease . No studies have yet explored the relationship between PCD and mycoplasma pneumonia. We speculate that the lack of susceptibility in PCD patients may be caused by impaired ciliary function, which makes it difficult for mycoplasma to localize and diffuse. However, more research is needed to confirm this hypothesis. Additionally, it is noteworthy that in our study, the proportion of Haemophilus influenzae in patients with PCD with respiratory infections was high. The higher proportion of Haemophilus influenza infection in patients with PCD may be connected to the dysfunction of ciliary clearance in these patients. One explanation for the prevalence of Haemophilus influenzae is the development of biofilm, which cannot be effectively cleared due to the dysfunction of the ciliary [56,57]. Another explanation is the low airway NO levels in patients with PCD, which play a role in regulating the metabolic activity of these respiratory bacteria, influencing their susceptibility to antibiotics [56,57].
In contrast to previous studies on patients with PCD, we explored the relationship between PCD and immune responses during a pneumonia episode. However, we acknowledge several limitations regarding the present study. First, our study’s retrospective design may increase the potential for bias, and its relatively small sample from a single center means that our results should be interpreted with caution. Second, we did not analyze other inflammatory markers such as interleukin-6, interleukin-10, or tumor necrosis factor. Additionally, we did not do longitudinal studies on inflammatory indicators. We recommend larger, multi-center investigations to confirm and expand upon our findings.
5. Conclusions
Our studies showed that the neutrophil, fibrinogen, and CRP levels in children with PCD complicated by acute pneumonia infections were lower than those in the control group children, which may be related to impaired immune function and increased tissue damage. Inflammatory indicators such as fibrinogen and CRPs may not be suitable for guiding judgments on the severity of an early infection in PCD patients. The findings regarding significantly altered immune indicators greatly contribute to our understanding of the pathophysiology of this entity and may potentially provide a new perspective on PCD treatment in the future.
Author Contributions
L.C. and H.L. conceived and designed the analysis; R.Z., Y.L. (Yan Li) and T.C. extracted and checked the data; D.L., Y.L. (Yue Liao), R.Z. and W.Y. performed the analysis; D.L. wrote the manuscript; and W.Y., L.C. and H.L. reviewed and edited the manuscript. All authors have read and agreed to the published version of the manuscript.
Institutional Review Board Statement
This study was performed in line with the principles of the Declaration of Helsinki. Approval was granted by the Ethics Committee of the West China Second University Hospital (approval number: 2022YFS076, approval date: 25 April 2022).
Informed Consent Statement
Informed consent was obtained from all participants or their legal guardians.
Data Availability Statement
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare no conflict of interest.
Funding Statement
This study was partially supported by the grant 22ZDZX0014 from Science & Technology department of Sichuan Province, the clinical development grant of West China Second University Hospital of Sichuan University (KL118), the Fundamental Research Funds for the Central Universities (SCU2022D022).
Footnotes
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The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
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/Sleep Disorders / Sleep Apnea
STOP-Bang Score for Obstructive Sleep Apnea
Updated July 15, 2025
Written by Danielle Pacheco
Danielle Pacheco
Contributing Writer
Danielle is originally from Vancouver, BC, where she has spent many hours staring at her ceiling trying to fall asleep. Danielle studied the science of sleep with a degree in psychology at the University of British Columbia
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|Medically Reviewed by Abhinav Singh, MD, MPH, FAASM
Abhinav Singh, MD, MPH, FAASM
Sleep Medicine Physician
Dr. Abhinav Singh, board certified in Sleep Medicine and Internal Medicine, is the Medical Director of the Indiana Sleep Center, which is accredited by the American Academy of Sleep Medicine. He is also a Clinical Assistant Professor at Marian University College of Medicine in Indianapolis, where he developed and teaches a Sleep Medicine rotation. Dr. Singh’s research and clinical practice focuses on sleep disorders, including excessive daytime sleepiness, narcolepsy, sleep apnea, chronic snoring, insomnia, and sleep education.
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What Is the STOP-Bang Questionnaire?
What Do STOP-Bang Scores Mean?
When Is the STOP-Bang Test Used?
Is the STOP-Bang Questionnaire a Reliable Tool to Diagnose Obstructive Sleep Apnea?
Can the STOP-Bang Questionnaire Diagnose Other Types of Sleep Apnea?
Obstructive sleep apnea (OSA) is a sleep disorder in which the airway is partially or completely blocked multiple times during the night, interfering with breathing. As it causes repetitive disruptions in breathing and sleep, OSA has been linked to a higher risk of heart attacks, diabetes, and other serious conditions, as well as ongoing tiredness.
The number of people with OSA is increasing, but many people do not realize they have the disorder, because symptoms primarily appear during sleep. Some researchers estimate that moderate to severe OSA may go undiagnosed and untreated in 80% of people with the condition. To identify people who may benefit from obstructive sleep apnea testing, researchers have developed a simple eight-question survey called the STOP-Bang Questionnaire.
What Is the STOP-Bang Questionnaire?
The STOP-Bang Questionnaire is intended to give physicians an easy-to-use tool to identify people who might have obstructive sleep apnea. The questionnaire consists of eight yes-or-no questions based on the major risk factors for OSA. The name STOP-Bang is an acronym for the first letter of each symptom or physical attribute often associated with OSA:
Snoring:This question assesses whether or not you snore loudly enough to bother a bed partner.
Tiredness:This symptom involves feeling daytime tiredness, which may include falling asleep during daily tasks.
Observed Apnea:If a sleep partner has noticed that you stop breathing or gasp for air as you sleep, this can be a sign of OSA.
Pressure:High blood pressure is also a symptom.
BMI: Physicians look for a body mass index that is higher than 35.
Age: Those who are older than 50 are at higher risk for OSA.
Neck Circumference: Physicians measure your neck circumference. A measurement greater than 16 inches is considered a risk factor.
Gender: Males are considered to be more likely to have OSA.
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What Do STOP-Bang Scores Mean?
When filling out the STOP-Bang questionnaire, a person receives one point for each symptom or risk factor, for a maximum of eight points. In general, the higher a person scores on the questionnaire, the greater risk they face of having moderate or severe OSA. Studies have also found that higher STOP-Bang scores are associated with more severe OSATrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source.
A STOP-Bang score of 2 or less is considered low risk, and a score of 5 or more is high risk for having either moderate or severe OSA. For people who score 3 or 4, doctors may need to perform further assessment to determine how likely they are to have OSA.
When Is the STOP-Bang Test Used?
Doctors often use the STOP-Bang Questionnaire when they suspect a patient they are seeing might be at risk of having OSA. The STOP-Bang test can help doctors decide which people to prioritize for polysomnography, the sleep study used to diagnose OSA. First developed to screen for OSA in people about to undergo elective surgeryTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source, the STOP-Bang Questionnaire has since been used for a wide variety of people and is considered an acceptable screening tool when used in sleep clinics or the general public.
Screening for OSA is important in a surgical context, because people with OSA have a higher risk of complications. For this reason, it is important that anesthesiologists and medical staff know if a person about to undergo surgeryTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source has undiagnosed OSA, so they can provide additional care. An estimated 70% of people who undergo weight loss surgery have OSA. The STOP-Bang Questionnaire can identify people who need extra attention during and after surgery, so staff minimize the risks posed by surgery, anesthetic agents, and medications prescribed after surgery.
Is the STOP-Bang Questionnaire a Reliable Tool to Diagnose Obstructive Sleep Apnea?
STOP-Bang is an effective tool for assessing specific risk factors and ruling out OSA. Those who have more risk factors should be further evaluated.
Independent researchers have found the STOP-Bang Questionnaire useful as a screen for OSA in adults with Down syndromeTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source, people with type 2 diabetesTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source, pregnant people with obesityTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source, and adults over 40Trusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source. Findings from BrazilianTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source and SwedishTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source studies suggest that the questionnaire may be highly accurate for predicting OSA in people who receive specific scores, but less accurate for other scores.
In an analysis of over 100 research studies, researchers found that the STOP-Bang questionnaire could more accurately predict who has mild, moderate, and severe OSATrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source than three other questionnaire tools. The test may have a high rate for false positives, however, leading to unnecessary healthcare costsTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source if doctors send too many patients for sleep tests based on the STOP-Bang results. The accuracy of the test may vary depending on characteristics of the people taking it and whether they have any other conditions.
The STOP-Bang Questionnaire performs less well for certain groups of people, such as veterans and those with kidney failure. One of the items on the test, neck circumference, could turn out differently depending on how it is measured, which would also affect scores.
In the interest of keeping the test simple, each item is awarded an equal point, even though some of the risk factors are more significant than others. For example, a person with a high BMI or a thicker neck circumference is more likely to have OSA than someone who is merely older than 50 years. These details explain why the test is not perfect. However, the STOP-Bang test is a useful first step that may indicate whether it is worth conducting a more specific test.
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STOP-Bang Adaptations
Researchers have adapted details of the STOP-Bang, so it better screens different groups of people.
For example, a Brazilian study found that a simplified version with a different BMI cutoff pointTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source performed well for older adults. Researchers have pointed out that for women, the gender questionTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source automatically guarantees a lower score, so at-risk women may need to use a lower threshold when determining their OSA risk. Used together with several other measures, the STOP-Bang score can help predict a second heart-related event in a person who has been admitted to hospital for a heart attackTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source.
Researchers have also experimented with giving more weight to certain questions on the test. For example, under one system, people who score a 3 or a 4 but whose test results reflect a yes for BMI, male gender, or a larger neck circumference would be classed as high-risk.
Can the STOP-Bang Questionnaire Diagnose Other Types of Sleep Apnea?
The STOP-Bang Questionnaire was developed for detection of OSA, and the overwhelming majority of studies have focused on its use for OSA. As opposed to OSA, lapses in breathing during central sleep apneaTrusted SourceMedline Plus MedlinePlus is an online health information resource for patients and their families and friends.View Source are due to problems in brain signaling instead of physical obstruction of the airway. Generally, STOP-Bang studies exclude central apneasTrusted SourceNational Library of Medicine, Biotech Information The National Center for Biotechnology Information advances science and health by providing access to biomedical and genomic information.View Source, or cessations in breathing due to causes other than an obstructed airway, in their analyses. As a result, we do not know how well or if the STOP-Bang Questionnaire is a useful screening for central sleep apnea.
The STOP-Bang Questionnaire is a simple and validated screening tool for OSA. Interpretation and further intervention after a positive score should involve a discussion with an experienced clinician.
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### About Our Editorial Team
Written By
Danielle Pacheco, Contributing Writer
Danielle is originally from Vancouver, BC, where she has spent many hours staring at her ceiling trying to fall asleep. Danielle studied the science of sleep with a degree in psychology at the University of British Columbia
Medically Reviewed by
Abhinav Singh, MD, MPH, FAASM, Sleep Medicine Physician MD
Dr. Abhinav Singh, board certified in Sleep Medicine and Internal Medicine, is the Medical Director of the Indiana Sleep Center, which is accredited by the American Academy of Sleep Medicine. He is also a Clinical Assistant Professor at Marian University College of Medicine in Indianapolis, where he developed and teaches a Sleep Medicine rotation. Dr. Singh’s research and clinical practice focuses on sleep disorders, including excessive daytime sleepiness, narcolepsy, sleep apnea, chronic snoring, insomnia, and sleep education.
Learn more about our Editorial Team
References
16 Sources
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Chung, F., Liao, P., & Farney, R. (2015). Correlation between the STOP-Bang score and the severity of obstructive sleep apnea. Anesthesiology, 122(6), 1436–1437.
2. Nagappa, M., Wong, J., Singh, M., Wong, D. T., & Chung, F. (2017). An update on the various practical applications of the STOP-Bang questionnaire in anesthesia, surgery, and perioperative medicine. Current Opinion in Anaesthesiology, 30(1), 118–125.
3. Fernandez-Bustamante, A., Bartels, K., Clavijo, C., Scott, B. K., Kacmar, R., Bullard, K., Moss, A., Henderson, W., Juarez-Colunga, E., & Jameson, L. (2017). Preoperatively screened obstructive sleep apnea is associated with worse postoperative outcomes than previously diagnosed obstructive sleep apnea. Anesthesia and Analgesia, 125(2), 593–602.
4. De Carvalho, A. A., Amorim, F. F., Santana, L. A., de Almeida, K. J., Santana, A. N., & Neves, F. A. (2020). STOP-Bang questionnaire should be used in all adults with Down Syndrome to screen for moderate to severe obstructive sleep apnea. PLoS One, 15(5), e0232596.
5. Teng, Y., Wang, S., Wang, N., & Muhuyati (2018). STOP-Bang questionnaire screening for obstructive sleep apnea among Chinese patients with type 2 diabetes mellitus. Archives of Medical Science: AMS, 14(5), 971–978.
6. Pearson, F., Batterham, A. M., & Cope, S. (2019). The STOP-Bang questionnaire as a screening tool for obstructive sleep apnea in pregnancy. Journal of Clinical Sleep Medicine: JCSM: Official Publication of the American Academy of Sleep Medicine, 15(5), 705–710.
7. Silva, G. E., Vana, K. D., Goodwin, J. L., Sherrill, D. L., & Quan, S. F. (2011). Identification of patients with sleep disordered breathing: Comparing the four-variable screening tool, STOP, STOP-Bang, and Epworth Sleepiness Scales. Journal of Clinical Sleep Medicine: JCSM: Official Publication of the American Academy of Sleep Medicine, 7(5), 467–472.
8. Neves Junior, J. A., Fernandes, A. P., Tardelli, M. A., Yamashita, A. M., Moura, S. M., Tufik, S., & da Silva, H. C. (2020). Cutoff points in STOP-Bang questionnaire for obstructive sleep apnea. Arquivos de Neuro-Psiquiatria, 78(9), 561–569.
9. Christensson, E., Franklin, K. A., Sahlin, C., Palm, A., Ulfberg, J., Eriksson, L. I., Lindberg, E., Hagel, E., & Jonsson Fagerlund, M. (2018). Can STOP-Bang and pulse oximetry detect and exclude obstructive sleep apnea? Anesthesia and Analgesia, 127(3), 736–743.
10. Chiu, H. Y., Chen, P. Y., Chuang, L. P., Chen, N. H., Tu, Y. K., Hsieh, Y. J., Wang, Y. C., & Guilleminault, C. (2017). Diagnostic accuracy of the Berlin questionnaire, STOP-BANG, STOP, and Epworth sleepiness scale in detecting obstructive sleep apnea: A bivariate meta-analysis. Sleep Medicine Reviews, 36, 57–70.
11. Abumuamar, A. M., Dorian, P., Newman, D., & Shapiro, C. M. (2018). The STOP-BANG questionnaire shows an insufficient specificity for detecting obstructive sleep apnea in patients with atrial fibrillation. Journal of Sleep Research, 27(6), e12702.
12. Martins, E. F., Martinez, D., Cortes, A. L., Nascimento, N., & Brendler, J. (2020). Exploring the STOP-BANG questionnaire for obstructive sleep apnea screening in seniors. Journal of Clinical Sleep Medicine: JCSM: Official Publication of the American Academy of Sleep Medicine, 16(2), 199–206.
13. Orbea, C., Lloyd, R. M., Faubion, S. S., Miller, V. M., Mara, K. C., & Kapoor, E. (2020). Predictive ability and reliability of the STOP-BANG questionnaire in screening for obstructive sleep apnea in midlife women. Maturitas, 135, 1–5.
14. Calvillo-Argüelles, O., Sierra-Fernández, C. R., Padilla-Ibarra, J., Rodriguez-Zanella, H., Balderas-Muñoz, K., Arias-Mendoza, M. A., Martínez-Sánchez, C., Selmen-Chattaj, S., Dominguez-Mendez, B. E., van der Harst, P., & Juarez-Orozco, L. E. (2020). Integrating the STOP-BANG score and clinical data to predict cardiovascular events after infarction: A machine learning study. Chest, 158(4), 1669–1679.
15. A.D.A.M. Medical Encyclopedia. (2019, July 15). Central sleep apnea. MedlinePlus., Retrieved August 29, 2021, from
16. Farney, R. J., Walker, B. S., Farney, R. M., Snow, G. L., & Walker, J. M. (2011). The STOP-Bang equivalent model and prediction of severity of obstructive sleep apnea: Relation to polysomnographic measurements of the apnea/hypopnea index. Journal of Clinical Sleep Medicine: JCSM: Official Publication of the American Academy of Sleep Medicine, 7(5), 459–65B.
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3432 | https://www.quora.com/Why-do-we-take-the-principal-branch-of-sin-inverse-as-pi-2-and-pi-2 | Why do we take the principal branch of sin inverse as -pi/2 and pi/2? - Quora
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Why do we take the principal branch of sin inverse as -pi/2 and pi/2?
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The principal branch of the inverse sine function, denoted as sin−1(x)sin−1(x) or arcsin(x)arcsin(x), is defined to have a specific range to ensure that it is a function (i.e., it has only one output for each input). Here are the reasons for choosing the range [−π 2,π 2][−π 2,π 2]:
Function Definition: A function must have a single output for each input. The sine function is periodic and takes on the same values at multiple points. To define an inverse sine function, we restrict the domain of the sine function to a range where it is one-to-one.
Monotonicity: The sine function is monotonically in
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The principal branch of the inverse sine function, denoted as sin−1(x)sin−1(x) or arcsin(x)arcsin(x), is defined to have a specific range to ensure that it is a function (i.e., it has only one output for each input). Here are the reasons for choosing the range [−π 2,π 2][−π 2,π 2]:
Function Definition: A function must have a single output for each input. The sine function is periodic and takes on the same values at multiple points. To define an inverse sine function, we restrict the domain of the sine function to a range where it is one-to-one.
Monotonicity: The sine function is monotonically increasing on the interval [−π 2,π 2][−π 2,π 2]. This means that within this interval, the sine function does not decrease, which guarantees that each value of y y in the range [−1,1][−1,1] corresponds to exactly one angle x x in the interval [−π 2,π 2][−π 2,π 2].
Range of Sine: The sine function achieves all values from −1−1 to 1 1 as x x varies from −π 2−π 2 to π 2 π 2. Thus, this interval is sufficient to cover the entire range of the inverse sine function.
Convenience and Convention: The choice of [−π 2,π 2][−π 2,π 2] is also conventional in mathematics and provides a consistent way to express angles in the context of trigonometric functions.
Summary
In summary, the principal branch of arcsin(x)arcsin(x) is defined as taking values in the interval [−π 2,π 2][−π 2,π 2] to ensure that it is a well-defined function (one-to-one), covers the full range of sine values from −1−1 to 1 1, and adheres to mathematical conventions.
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Him
Author has 1.5K answers and 478.7K answer views
·3y
[0,pi/2] is the most commonly used set of values for the sin(x) function. It is the one that we are most familiar with. So we include that in the range of the arcsine function.
Now, this set doesn't cover all the values in the range of sine, so we need to go further. There are two ways we can go:-
From pi/2 to pi
From 0 to -pi/2
The first one doesn't bring us any new values, so we choose the second one.
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Related questions
More answers below
Why does sin inverse x+cos inverse x=pi÷2?
Given that cos x = -12/13, what is sin 2x, if pi < x < 3 pi/2?
Is it just a convention that we use the range (-pi/2, pi/2) for an inverse sine function?
How do you prove that sin(π/7)cos(π/14)tan(3 π/14)⋅(2 cos(π/7)−1)=√7 4 sin(π/7)cos(π/14)tan(3 π/14)·(2 cos(π/7)−1)=7 4 ?
Why isn’t the principle branch of sin inverse that is not from pi by 2 to 3pi by 2?
Maratratha
8y
arcsin() is the inverse function of Sin().arcsin() is a multivalued function Which Spans the complex plane .So the principle branch should have every Real Solution of sin() .So -pi/2 and pi/2 is taken
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Bernard Leak
Firmware Developer (2008–present) · Author has 5.8K answers and 5M answer views
·8y
Originally Answered: Why do we take the principal branch of sin inverse as -Pi/2 and Pi/2? ·
Because it includes 0, and is continous on the largest possible interval around 0. Since that’s all you need (the entire image of the sine function is in the domain of the inverse function so defined), you can stop there. If you have to pick one branch, it’s surely good for it to be well-behaved like this around some value. 0 isn't compulsory, but why not?
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Max Sklar
MS in Mathematics, University of Massachusetts, Amherst (Graduated 2005) · Upvoted by
Luís Sequeira
, PhD Mathematics, University of Lisbon (2001) · Author has 1.9K answers and 3.8M answer views
·6y
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Why is arcsine only between (pi/2, -pi/2)?
Thanks for the A2A
The arcsine function is the result of the attempt to find an inverse to the sine function. Given a value y y where −1≤y≤1−1≤y≤1. The arcsine tells you the angle θ θ where sin θ=y sinθ=y
The problem is the sine function is not one-to-one. Pick any value between -1 and 1 and there are an infinite number of angles θ θ for which sin(θ)sin(θ) has that value. The sine function has no inverse function. However we can restrict the sine down to a domain so that is one two one. Now, it turns out there is some choice in how to define that restriction. But mathematicians have decid
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Thanks for the A2A
The arcsine function is the result of the attempt to find an inverse to the sine function. Given a value y y where −1≤y≤1−1≤y≤1. The arcsine tells you the angle θ θ where sin θ=y sinθ=y
The problem is the sine function is not one-to-one. Pick any value between -1 and 1 and there are an infinite number of angles θ θ for which sin(θ)sin(θ) has that value. The sine function has no inverse function. However we can restrict the sine down to a domain so that is one two one. Now, it turns out there is some choice in how to define that restriction. But mathematicians have decided they would like the restriction to include 0.
In the image below you can see the restriction demonstrated in two ways. On the left you can see that every output value for the sine function is represented in the restriction −π 2≤θ≤π 2−π 2≤θ≤π 2.
Personally, I like the unit circle (on the right) for illustrating the inverse functions. The unit circle is the circle with radius 1 centered at the origin. As you increase your angle you are traveling round and round the circle. You can see the pattern repeats itself every trip around the circle, every 2 π 2 π radians. The value of the sine function can be defined as the y y-coordinate of the point on the unit circle corresponding to the circle. You can see that sin θ sinθ takes on values between -1 and 1. In fact, you see all of those values as you travel from the bottom of the circle to the top. One of the angles that represent the bottom of the circle is θ=−π 2 θ=−π 2. By paying attention to the y y-coordinate, we can see that the values of sine range from -1 to 1 for −π 2≤θ≤π 2−π 2≤θ≤π 2. Now follow along the yellow path above and observe the following.
The restriction of the sine function takes an angle −π 2≤θ≤π 2−π 2≤θ≤π 2 and gives as output the y y-coordinate of the corresponding point on the circle. This will be a number between -1 and 1.
The arcsine function is the inverse of the restriction. It takes as input a y y-coordinate −1≤y≤1−1≤y≤1 . It gives as output the angle that corresponds to the point (in the yellow path) that has that y y-coordinate.
Why is arcsine only between [−π/2,π/2][−π/2,π/2]?
The arcsine function only gives angles in yellow path. By convention mathematicians have chosen the angles −π 2≤θ≤π 2−π 2≤θ≤π 2 as the representatives of the points on the path.
There is some arbitrary choice to this. They could have chosen the angles you get by spinning around the circle many times and landing back in the yellow path. For example they could have chosen 3 π 2≤θ≤5 π 2 3 π 2≤θ≤5 π 2.
They could have also chosen the non-yellow path. They could have chosen to restrict the sine function to π 2≤θ≤3 π 2 π 2≤θ≤3 π 2.
But as I said they wanted to keep the focal point on θ=0 θ=0.
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How can one prove that π/2∫0(ln sin x)3 d x=−1 8 π 3 ln 2−1 2 π(ln 2)3−3 4 π ζ(3)∫0 π/2(lnsinx)3 d x=−1 8 π 3 ln2−1 2 π(ln2)3−3 4 π ζ(3)?
What is the solution for 2∣sin x∣=1 2∣sinx∣=1 for −π<x<π−π<x<π?
How do you prove π>4√2−√2 π>4 2−2 ?
Why is arcsine only between (pi/2, -pi/2)?
Can sin 2 π 7 sin2 π 7 have an exact value?
Roman Andronov
Solving technical problems is my day job. · Upvoted by
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Why does ∫∞−∞sin x x d x=π∫−∞∞sinx x d x=π hold?
Because the underlying mathematics works out that way.
In addition to the excellent Laplace transform-based answer by Dr. Sittinger.
I. For Tinkerers At Heart (and math majors :O)
Since the given integrand is a function that is even, we consider that given integral over the [0,+∞)[0,+∞) interval:
I=+∞∫−∞sin(x)x d x=2+∞∫0 sin(x)x d x=2 J(1)(1)I=∫−∞+∞sin(x)x d x=2∫0+∞sin(x)x d x=2 J
Since the given integrand conveniently hops over the origin due to the remarkable limit:
lim x→0 sin(x)x=1 lim x→0 sin(x)x=1
and then hits a zero
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Because the underlying mathematics works out that way.
In addition to the excellent Laplace transform-based answer by Dr. Sittinger.
I. For Tinkerers At Heart (and math majors :O)
Since the given integrand is a function that is even, we consider that given integral over the [0,+∞)[0,+∞) interval:
I=+∞∫−∞sin(x)x d x=2+∞∫0 sin(x)x d x=2 J(1)(1)I=∫−∞+∞sin(x)x d x=2∫0+∞sin(x)x d x=2 J
Since the given integrand conveniently hops over the origin due to the remarkable limit:
lim x→0 sin(x)x=1 lim x→0 sin(x)x=1
and then hits a zero on every (negative and positive) multiple of π π after that, we partition the entire interval [0,+∞)[0,+∞) into subintervals of equal length π/2 π/2 (Fig. 1):
Consequently, for the integral J J such that:
J=+∞∫0 sin(x)x d x J=∫0+∞sin(x)x d x
we shall have (since we also proved prior that that integral does converge over the interval specified):
J=+∞∑n=0(n+1)π 2∫n π 2 sin(x)x d x J=∑n=0+∞∫n π 2(n+1)π 2 sin(x)x d x
The given integral is a function that is pleasantly continuous, we just took that function apart and then we put it back together piece-meal over the said smaller intervals of length π/2 π/2.
Partition these intervals further into the (orange) even intervals that correspond to the even n n s representable as n=2 k n=2 k, where k k varies starting from 0 0, and the (blue) odd intervals that correspond to the odd n n s representable as n=2 k−1 n=2 k−1, where k k varies starting from 1 1.
Over the said (orange) even intervals make the following substitution:
n=2 k:x=k π+t,d x=d t n=2 k:x=k π+t,d x=d t
Using the trigonometric identity for the sine of a sum of two angles:
sin(x±y)=sin(x)⋅cos(y)±cos(x)⋅sin(y)sin(x±y)=sin(x)⋅cos(y)±cos(x)⋅sin(y)
we have for the sin(x)sin(x):
sin(x)=sin(k π+t)=(−1)k sin(t)sin(x)=sin(k π+t)=(−1)k sin(t)
and we also see that the dummy summation variables n n and k k in the corresponding boundaries of integration vanish altogether because for n=2 k n=2 k and for x=k π+t x=k π+t when x=n π/2 x=n π/2 then k π+t=k π k π+t=k π, implying that t=0 t=0, and when x=(2 n+1)π/2 x=(2 n+1)π/2 then k π+t=(2 k+1)π/2 k π+t=(2 k+1)π/2, implying that t=π/2 t=π/2:
(−1)k π 2∫0 sin(t)t+k π d t(2)(2)(−1)k∫0 π 2 sin(t)t+k π d t
where k k varies starting from 0 0.
Likewise, over the said (blue) odd intervals make the following substitution:
n=2 k−1:x=k π−t,d x=−d t n=2 k−1:x=k π−t,d x=−d t
Using the above trigonometric identity for the sine of a difference of two angles, we have for the sin(x)sin(x):
sin(x)=sin(k π−t)=−(−1)k sin(t)sin(x)=sin(k π−t)=−(−1)k sin(t)
and we also see that the dummy summation variables n n and k k in the corresponding boundaries of integration vanish altogether as well because for n=2 k−1 n=2 k−1 and for x=k π−t x=k π−t when x=n π/2 x=n π/2 then k π−t=(2 k−1)π/2 k π−t=(2 k−1)π/2, implying that t=π/2 t=π/2, and when x=(2 n+1)π/2 x=(2 n+1)π/2 then k π−t=(2 k−1+1)π/2 k π−t=(2 k−1+1)π/2, implying that t=0 t=0:
(−1)k π 2∫0 sin(t)t−k π d t(3)(3)(−1)k∫0 π 2 sin(t)t−k π d t
where k k varies starting from 1 1: k=1,2,3,…k=1,2,3,… and where we moved the −− sign in front of the above integral associated with flipping the boundaries of integration to follow their natural order from 0 0 to π/2 π/2 into the denominator of the integrand.
If we want to roll the resultant series starting from k=1 k=1 then in the above even series we have to cast out the summand that corresponds to k=0 k=0:
π 2∫0 sin(t)t d t∫0 π 2 sin(t)t d t
Then, putting these even and odd series together, we have for J J:
J=π 2∫0 sin(t)t d t+J=∫0 π 2 sin(t)t d t+
++∞∑k=1 π 2∫0(−1)k[1 t+k π+1 t−k π]sin(t)d t+∑k=1+∞∫0 π 2(−1)k[1 t+k π+1 t−k π]sin(t)d t
Over the interval 0⩽t⩽π/2 0⩽t⩽π/2 the following series:
+∞∑k=1(−1)k[1 t+k π+1 t−k π]sin(t)∑k=1+∞(−1)k[1 t+k π+1 t−k π]sin(t)
converges uniformly with respect to t t because that series is dominated by, say, the following convergent series:
1 π∑1 n 2−1 4 1 π∑1 n 2−1 4
Thus, in this case it is permissible to move the summation sign through the integral sign:
J=π 2∫0 sin(t)⋅(1 t++∞∑k=1(−1)k[1 t+k π+1 t−k π])d t J=∫0 π 2 sin(t)⋅(1 t+∑k=1+∞(−1)k[1 t+k π+1 t−k π])d t
Lastly, here on Quora we deduced the following result many a time:
1 t++∞∑k=1(−1)k[1 t+k π+1 t−k π]=1 sin(t)1 t+∑k=1+∞(−1)k[1 t+k π+1 t−k π]=1 sin(t)
See this Quora answer for a sample deduction.
Thus, we arrive at this wonderfully happy and happily wonderful cancellation that reduces this scary-looking integral into a triviality:
J=π 2∫0 sin(t)⋅1 sin(t)d t=π 2∫0 d t=π 2 J=∫0 π 2 sin(t)⋅1 sin(t)d t=∫0 π 2 d t=π 2
Officially:
J=+∞∫0 sin(x)x d x=π 2 J=∫0+∞sin(x)x d x=π 2
Consequently, the two copies of J J produce π π on the one hand and the integral in question, on the other hand:
I=+∞∫−∞sin(x)x d x=2 J=2⋅π 2=π I=∫−∞+∞sin(x)x d x=2 J=2⋅π 2=π
which is what was required to prove. □◻
II. The Fast And The Furious (for math majors and engineers :O)
Here we introduce the so-called factor of convergence e−k x e−k x into the works for k⩾0 k⩾0, planning to take the k→0+k→0+ limit of the result and we also introduce another parameter, a a, as a scaled version of the variable of the sine function:
I(k,a)=+∞∫0 e−k x sin(a x)x d x,k,a⩾0(4)(4)I(k,a)=∫0+∞e−k x sin(a x)x d x,k,a⩾0
Again, justifying the right to differentiate under the integral sign, we differentiate both sides of (4) with respect to a a once, witnessing another happy cancellation of x x s in the result:
d I d a=+∞∫0 e−k x cos(a x)d x(5)(5)d I d a=∫0+∞e−k x cos(a x)d x
But the integral shown in (5) is a previously solved problem:
∫e−k x cos(a x)d x=e−k x a sin(a x)−k cos(a x)k 2+b 2+C∫e−k x cos(a x)d x=e−k x a sin(a x)−k cos(a x)k 2+b 2+C
Moreover, the integral:
+∞∫0 e−k x cos(a x)d x∫0+∞e−k x cos(a x)d x
converges uniformly with respect to the parameter a a because it is dominated by the following convergent integral:
+∞∫0 e−k x d x=1 k∫0+∞e−k x d x=1 k
whose value does not depend on a a.
Consequently:
+∞∫0 e−k x cos(a x)d x=k k 2+a 2∫0+∞e−k x cos(a x)d x=k k 2+a 2
and:
d I d a=k k 2+a 2 d I d a=k k 2+a 2
implying that:
I(k,a)=+∞∫0 e−k x sin(a x)x d x=arctan(a k)(6)(6)I(k,a)=∫0+∞e−k x sin(a x)x d x=arctan(a k)
because when a=0 a=0 then in (4) I(k,a=0)=0 I(k,a=0)=0 also.
Finally, we take the k→0+k→0+ limit of both sides of (6):
I 0(k=0,a)=lim k→0+arctan(a k)=arctan(+∞)=π 2 I 0(k=0,a)=lim k→0+arctan(a k)=arctan(+∞)=π 2
meaning that:
I+=+∞∫0 sin(a x)x d x=π 2,a>0 I+=∫0+∞sin(a x)x d x=π 2,a>0
Note that quite remarkably the above integral does not depend on the value of the parameter a a - it does depend on the sign of a a only:
I−=+∞∫0 sin(a x)x d x=−π 2,a<0 I−=∫0+∞sin(a x)x d x=−π 2,a<0
and when a=0 a=0 then the value of this integral is, clearly, 0 0.
In particular, when a=1 a=1 we have:
J=+∞∫0 sin(x)x d x=π 2 J=∫0+∞sin(x)x d x=π 2
and consequently:
+∞∫−∞sin(x)x d x=π∫−∞+∞sin(x)x d x=π
in agreement with the earlier deduction. □◻
For more information on and ideas about problem-solving in mathematics, physics and computer science please visit my YouTube channel ProbLemma.
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Anthony Madden
Writer for Betterbuck
·Updated Aug 15
What are the weirdest mistakes people make on the internet right now?
Here are a couple of the worst mistakes I’ve seen people make:
Not using an ad blocker
If you aren’t using an ad blocker yet, you definitely should be.
A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load.
No more YouTube ads, no more banner ads, no more pop-up ads, etc.
Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options.
Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your pr
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Here are a couple of the worst mistakes I’ve seen people make:
Not using an ad blocker
If you aren’t using an ad blocker yet, you definitely should be.
A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load.
No more YouTube ads, no more banner ads, no more pop-up ads, etc.
Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options.
Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your privacy, which is nice.
More often than not, it saves even more than 50% on load times - here’s a test I ran:
Using an ad blocker saved a whopping 6.5+ seconds of load time.
Here’s a link to Total Adblock, if you’re interested.
Not getting paid for your screentime
Apps like Freecash will pay you to test new games on your phone.
Some testers get paid as much as £270/game. Here are a few examples right now (from Freecash's website):
You don't need any kind of prior experience or degree or anything: all you need is a smartphone (Android or IOS).
If you're scrolling on your phone anyway, why not get paid for it?
I've used Freecash in the past - it’s solid. (They also gave me a £3 bonus instantly when I installed my first game, which was cool).
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David Joyce
Professor Emeritus of Mathematics at Clark University · Upvoted by
Terry Moore
, M.Sc. Mathematics, University of Southampton (1968) and
Michael Jørgensen
, PhD in mathematics · Author has 9.9K answers and 68.4M answer views
·Updated Feb 2
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Can we write sin sin as sin(x)=(x)(x−π)(x+π)(x−2 π)(x+2 π)(x−3 π)(x+3 π)⋯sin(x)=(x)(x−π)(x+π)(x−2 π)(x+2 π)(x−3 π)(x+3 π)⋯?
The Euler sine product is very close to what you have, but not the same.
sin x=x(1−x π)(1+x π)(1+x 2 π)(1−x 2 π)⋯sinx=x(1−x π)(1+x π)(1+x 2 π)(1−x 2 π)⋯
It’s usually written
sin x=x(1−x 2 π 2)(1−x 2 2 2 π 2)⋯sinx=x(1−x 2 π 2)(1−x 2 2 2 π 2)⋯
=x∞∏n=1(1−x 2 n 2 π 2)=x∏n=1∞(1−x 2 n 2 π 2)
Euler probably started with what you noted, realized it wouldn’t work because that product approaches infinity, and figured out a way to save the basic concept that sine should be a p
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The Euler sine product is very close to what you have, but not the same.
sin x=x(1−x π)(1+x π)(1+x 2 π)(1−x 2 π)⋯sinx=x(1−x π)(1+x π)(1+x 2 π)(1−x 2 π)⋯
It’s usually written
sin x=x(1−x 2 π 2)(1−x 2 2 2 π 2)⋯sinx=x(1−x 2 π 2)(1−x 2 2 2 π 2)⋯
=x∞∏n=1(1−x 2 n 2 π 2)=x∏n=1∞(1−x 2 n 2 π 2)
Euler probably started with what you noted, realized it wouldn’t work because that product approaches infinity, and figured out a way to save the basic concept that sine should be a product related to its zeros.
Here are the partial products graphed.
The pink line is y=x.y=x. (The vertical axis is squeezed to show the graphs better. The lavender curve is y=x(1−x 2/π 2).y=x(1−x 2/π 2). The light blue is y=x(1−x 2/π 2)(1−x 2/(2 2 π 2).y=x(1−x 2/π 2)(1−x 2/(2 2 π 2). And the three next partial products are also graphed.
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Peter Butcher
Former Former Professional Engineer, Teacher, Tutor · Author has 1.4K answers and 4.6M answer views
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Can you explain why sin (x+Pi/2) = -sin x?
No! but I can explain why Sin(x+ Pi/2) DOESN’T = Sin(x)
Consider Graph 1 (below) Y = Sin(x)
Note that:
Sin(x) = 0 when x = 0
Sin(x) = 0 when x = pi
Sin(x) = 1 when x = pi/2
Sin(x) =-1 when x = 3pi/2
Now consider Graph 2 (below) where Sin(x)is in Black and Sin(x + pi/2) is in purple.
In Graph 2, note that:
Sin(x+pi/2) = Sin(0) = 1
Sin(x+pi/2) = Sin(pi/2) = 0
Sin(pi) = Sin(x + pi/2)
Continue Reading
No! but I can explain why Sin(x+ Pi/2) DOESN’T = Sin(x)
Consider Graph 1 (below) Y = Sin(x)
Note that:
Sin(x) = 0 when x = 0
Sin(x) = 0 when x = pi
Sin(x) = 1 when x = pi/2
Sin(x) =-1 when x = 3pi/2
Now consider Graph 2 (below) where Sin(x)is in Black and Sin(x + pi/2) is in purple.
In Graph 2, note that:
Sin(x+pi/2) = Sin(0) = 1
Sin(x+pi/2) = Sin(pi/2) = 0
Sin(pi) = Sin(x + pi/2) = Sin(pi) = -1
and most important Sin(x+pi/2) = Cos(x) NOT -Sin(x)
Finally con...
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Snahendu Majumder
I am a class 12 student. · Author has 85 answers and 114.8K answer views
·4y
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Why sin (π-π/4) = sinπ/4?
Sin(nπ/2+A )= Sin A only if n is even
and Sin (nπ/2+A) = Cos A if n is odd.
Also , value of sin is positive in 1st and 2nd quadrant.
The argument in the sin function i.e π-π/4 lies in the 2nd quadrant. Hence sin is positive. Also n is equal to 2 which is even. So sin changes to sin.
Continue Reading
Sin(nπ/2+A )= Sin A only if n is even
and Sin (nπ/2+A) = Cos A if n is odd.
Also , value of sin is positive in 1st and 2nd quadrant.
The argument in the sin function i.e π-π/4 lies in the 2nd quadrant. Hence sin is positive. Also n is equal to 2 which is even. So sin changes to sin.
Upvote ·
9 3
Terry Moore
M.Sc. in Mathematics, University of Southampton (Graduated 1968) · Author has 16.6K answers and 29.4M answer views
·3y
Related
If the unit circle has a radius 1, and sin = opposite/hypotenuse, why do we represent it with Pi/6?
If the unit circle has a radius 1, and sin = opposite/hypotenuse, why do we represent it with Pi/6?
This is like asking “If pigs can’t fly, and elephants don’t have wings, why are there cows on the Moon?”
Let’s take your question apart:
A unit circle has radius 1, that’s what “unit” means.
More precisely, we can use any unit of length we like. If your unit is a metre, then count that as 1, if your unit is a foot, then count that as 1. (In the latter case you probably call a “metre” a “meter”, thus confusing a unit of measurement with a measuring instrument—but that’s OK because you’re confused an
Continue Reading
If the unit circle has a radius 1, and sin = opposite/hypotenuse, why do we represent it with Pi/6?
This is like asking “If pigs can’t fly, and elephants don’t have wings, why are there cows on the Moon?”
Let’s take your question apart:
A unit circle has radius 1, that’s what “unit” means.
More precisely, we can use any unit of length we like. If your unit is a metre, then count that as 1, if your unit is a foot, then count that as 1. (In the latter case you probably call a “metre” a “meter”, thus confusing a unit of measurement with a measuring instrument—but that’s OK because you’re confused anyway).
2. The late Cardinal Sin would not have agreed that sin = opposite/hypotenuse.
If you mean the sine of an angle, you have to specify the angle.
3. You haven’t said what “it” is. If “it” refers to an angle of π/6 π/6 we represent it by π/6 π/6.
Measurement of an angle has nothing to do with trigonometrical functions. An angle may be measured in many different ways. In radians (which is the only respectable angle measure in pure mathematics) an angle is an arc length of a unit circle. Half way around the circle, the arc length is π π. So 1/12 1/12 of the way around the circle, the arc length is π/6 π/6.
The sine of that angle happens to be 1/2 1/2, but that’s incidental.
Cardinal Sin would have been appalled.
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find the formula of trinomial expansion
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$\begingroup$
I wonder as if there exist a equivalent forumla to newton binomial
$$(x+y)^n=\sum_{k=0}^{n} {n\choose k} x^{n-k}y^k$$
for three coefficients $(a+b+c)^n$ ?
binomial-theorem
multinomial-coefficients
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edited Sep 4, 2020 at 13:18
V.G
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asked Dec 4, 2013 at 19:16
MarkMark
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$\endgroup$
2
2
$\begingroup$ Please see the Multinomial Theorem. $\endgroup$
André Nicolas
– André Nicolas
2013-12-04 19:18:21 +00:00
Commented Dec 4, 2013 at 19:18
3
$\begingroup$ There is an extra $n$ in your formula and the summation range is wrong. $\endgroup$
user65203
– user65203
2016-06-13 14:49:09 +00:00
Commented Jun 13, 2016 at 14:49
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2 Answers 2
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$\begingroup$
The expansion is given by $$(a+b+c)^n = \sum_{i,j,k} {n \choose i,j,k}\, a^i \, b^j \, c^k $$ where $n$ is a nonnegative integer and the sum is taken over all combinations of nonnegative indices $i, j$, and $k$ such that $i + j + k = n$. The trinomial coefficients are given by $$ {n \choose i,j,k} = \frac{n!}{i!\,j!\,k!} \,.$$ This formula is a special case of the multinomial formula.
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answered Dec 4, 2013 at 20:24
alexjoalexjo
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$\begingroup$
When expanding the product, you pick one of $a,b,c$ from every factor, and get at term $a^ib^jc^k$ where $i+j+k=n$. You can scramble the $n$ factors in $n!$ ways, but as scrambling identical letters makes no difference, the factors are actually repeated $\dfrac{n!}{i!j!k!}$ times.
Hence
$$\sum_{i,j,k\ge0,i+j+k=n}\frac{n!}{i!j!k!}a^ib^jc^k.$$
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answered Jun 13, 2016 at 15:04
user65203user65203
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| | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- --- | | | | | | | | | | | | Home > Pre-Algebra calculators > Divisibility rule example | | | | | | | | | | | --- --- --- --- | | | | | Divisibility rule of 57 example ( Enter your problem ) ( Enter your problem ) | | 1. Formula 2. Divisibility rule of 2 example 3. Divisibility rule of 3 example 4. Divisibility rule of 4 example 5. Divisibility rule of 5 example 6. Divisibility rule of 6 example 7. Divisibility rule of 7 example 8. Divisibility rule of 8 example 9. Divisibility rule of 9 example 10. Divisibility rule of 10 example 11. Divisibility rule of 11 example 12. Divisibility rule of 12 example 13. Divisibility rule of 13 example 14. Divisibility rule of 14 example 15. Divisibility rule of 15 example 16. Divisibility rule of 16 example 17. Divisibility rule of 17 example 18. Divisibility rule of 18 example 19. Divisibility rule of 19 example 20. Divisibility rule of 20 example 21. 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Divisibility rule of 58 example (Next example) | --- 57. Divisibility rule of 57 example --- --- 1. Check whether 15732 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 - color{red}{2} xx 17 = 1573 -34 = 1539 153color{red}{9}=>153 - color{red}{9} xx 17 = 153 -153 = 0 Here 0 is divisible by 57. :. 15732 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 15732 is 1+5+7+3+2=18, which is divisible by 3. :. 15732 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 + color{red}{2} xx 2 = 1573 +4 = 1577 157color{red}{7}=>157 + color{red}{7} xx 2 = 157 +14 = 171 17color{red}{1}=>17 + color{red}{1} xx 2 = 17 +2 = 19 Here 19 is divisible by 19. :. 15732 is divisible by 19. 15732 is divisible by 3 and 19. :. 15732 is divisible by 57. --- 2. Check whether 18069 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 - color{red}{9} xx 17 = 1806 -153 = 1653 165color{red}{3}=>165 - color{red}{3} xx 17 = 165 -51 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 18069 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 18069 is 1+8+0+6+9=24, which is divisible by 3. :. 18069 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 + color{red}{9} xx 2 = 1806 +18 = 1824 182color{red}{4}=>182 + color{red}{4} xx 2 = 182 +8 = 190 19color{red}{0}=>19 + color{red}{0} xx 2 = 19 = 19 Here 19 is divisible by 19. :. 18069 is divisible by 19. 18069 is divisible by 3 and 19. :. 18069 is divisible by 57. --- 3. Check whether 20634 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 - color{red}{4} xx 17 = 2063 -68 = 1995 199color{red}{5}=>199 - color{red}{5} xx 17 = 199 -85 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 20634 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 20634 is 2+0+6+3+4=15, which is divisible by 3. :. 20634 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 + color{red}{4} xx 2 = 2063 +8 = 2071 207color{red}{1}=>207 + color{red}{1} xx 2 = 207 +2 = 209 20color{red}{9}=>20 + color{red}{9} xx 2 = 20 +18 = 38 Here 38 is divisible by 19. :. 20634 is divisible by 19. 20634 is divisible by 3 and 19. :. 20634 is divisible by 57. --- 4. Check whether 16428 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 - color{red}{8} xx 17 = 1642 -136 = 1506 150color{red}{6}=>150 - color{red}{6} xx 17 = 150 -102 = 48 Here 48 is not divisible by 57. :. 16428 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 16428 is 1+6+4+2+8=21, which is divisible by 3. :. 16428 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 + color{red}{8} xx 2 = 1642 +16 = 1658 165color{red}{8}=>165 + color{red}{8} xx 2 = 165 +16 = 181 18color{red}{1}=>18 + color{red}{1} xx 2 = 18 +2 = 20 Here 20 is not divisible by 19. :. 16428 is not divisible by 19. Hence 16428 is also not divisible by 57. --- 5. Check whether 19795 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1979color{red}{5}=>1979 - color{red}{5} xx 17 = 1979 -85 = 1894 189color{red}{4}=>189 - color{red}{4} xx 17 = 189 -68 = 121 12color{red}{1}=>12 - color{red}{1} xx 17 = 12 -17 = -5 Here -5 is not divisible by 57. :. 19795 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 19795 is 1+9+7+9+5=31, which is not divisible by 3. :. 19795 is not divisible by 3. Hence 19795 is also not divisible by 57. --- 6. Check whether 21256 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2125color{red}{6}=>2125 - color{red}{6} xx 17 = 2125 -102 = 2023 202color{red}{3}=>202 - color{red}{3} xx 17 = 202 -51 = 151 15color{red}{1}=>15 - color{red}{1} xx 17 = 15 -17 = -2 Here -2 is not divisible by 57. :. 21256 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 21256 is 2+1+2+5+6=16, which is not divisible by 3. :. 21256 is not divisible by 3. Hence 21256 is also not divisible by 57. --- --- This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then Submit Here --- | | | --- | | 56. Divisibility rule of 56 example (Previous example) | 58. Divisibility rule of 58 example (Next example) | | | | | --- Share this solution or page with your friends. | | | | | | | | | | |
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| | | | | | | | | | | Home > Pre-Algebra calculators > Divisibility rule example | | | | | | | | | | | --- --- --- --- | | | | | Divisibility rule of 57 example ( Enter your problem ) ( Enter your problem ) | | 1. Formula 2. Divisibility rule of 2 example 3. Divisibility rule of 3 example 4. Divisibility rule of 4 example 5. Divisibility rule of 5 example 6. Divisibility rule of 6 example 7. Divisibility rule of 7 example 8. Divisibility rule of 8 example 9. Divisibility rule of 9 example 10. Divisibility rule of 10 example 11. Divisibility rule of 11 example 12. Divisibility rule of 12 example 13. Divisibility rule of 13 example 14. Divisibility rule of 14 example 15. Divisibility rule of 15 example 16. Divisibility rule of 16 example 17. Divisibility rule of 17 example 18. Divisibility rule of 18 example 19. Divisibility rule of 19 example 20. Divisibility rule of 20 example 21. Divisibility rule of 21 example 22. Divisibility rule of 22 example 23. 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Check whether 15732 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 - color{red}{2} xx 17 = 1573 -34 = 1539 153color{red}{9}=>153 - color{red}{9} xx 17 = 153 -153 = 0 Here 0 is divisible by 57. :. 15732 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 15732 is 1+5+7+3+2=18, which is divisible by 3. :. 15732 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 + color{red}{2} xx 2 = 1573 +4 = 1577 157color{red}{7}=>157 + color{red}{7} xx 2 = 157 +14 = 171 17color{red}{1}=>17 + color{red}{1} xx 2 = 17 +2 = 19 Here 19 is divisible by 19. :. 15732 is divisible by 19. 15732 is divisible by 3 and 19. :. 15732 is divisible by 57. --- 2. Check whether 18069 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 - color{red}{9} xx 17 = 1806 -153 = 1653 165color{red}{3}=>165 - color{red}{3} xx 17 = 165 -51 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 18069 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 18069 is 1+8+0+6+9=24, which is divisible by 3. :. 18069 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 + color{red}{9} xx 2 = 1806 +18 = 1824 182color{red}{4}=>182 + color{red}{4} xx 2 = 182 +8 = 190 19color{red}{0}=>19 + color{red}{0} xx 2 = 19 = 19 Here 19 is divisible by 19. :. 18069 is divisible by 19. 18069 is divisible by 3 and 19. :. 18069 is divisible by 57. --- 3. Check whether 20634 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 - color{red}{4} xx 17 = 2063 -68 = 1995 199color{red}{5}=>199 - color{red}{5} xx 17 = 199 -85 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 20634 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 20634 is 2+0+6+3+4=15, which is divisible by 3. :. 20634 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 + color{red}{4} xx 2 = 2063 +8 = 2071 207color{red}{1}=>207 + color{red}{1} xx 2 = 207 +2 = 209 20color{red}{9}=>20 + color{red}{9} xx 2 = 20 +18 = 38 Here 38 is divisible by 19. :. 20634 is divisible by 19. 20634 is divisible by 3 and 19. :. 20634 is divisible by 57. --- 4. Check whether 16428 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 - color{red}{8} xx 17 = 1642 -136 = 1506 150color{red}{6}=>150 - color{red}{6} xx 17 = 150 -102 = 48 Here 48 is not divisible by 57. :. 16428 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 16428 is 1+6+4+2+8=21, which is divisible by 3. :. 16428 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 + color{red}{8} xx 2 = 1642 +16 = 1658 165color{red}{8}=>165 + color{red}{8} xx 2 = 165 +16 = 181 18color{red}{1}=>18 + color{red}{1} xx 2 = 18 +2 = 20 Here 20 is not divisible by 19. :. 16428 is not divisible by 19. Hence 16428 is also not divisible by 57. --- 5. Check whether 19795 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1979color{red}{5}=>1979 - color{red}{5} xx 17 = 1979 -85 = 1894 189color{red}{4}=>189 - color{red}{4} xx 17 = 189 -68 = 121 12color{red}{1}=>12 - color{red}{1} xx 17 = 12 -17 = -5 Here -5 is not divisible by 57. :. 19795 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 19795 is 1+9+7+9+5=31, which is not divisible by 3. :. 19795 is not divisible by 3. Hence 19795 is also not divisible by 57. --- 6. Check whether 21256 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2125color{red}{6}=>2125 - color{red}{6} xx 17 = 2125 -102 = 2023 202color{red}{3}=>202 - color{red}{3} xx 17 = 202 -51 = 151 15color{red}{1}=>15 - color{red}{1} xx 17 = 15 -17 = -2 Here -2 is not divisible by 57. :. 21256 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 21256 is 2+1+2+5+6=16, which is not divisible by 3. :. 21256 is not divisible by 3. Hence 21256 is also not divisible by 57. --- --- This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then Submit Here --- | | | --- | | 56. Divisibility rule of 56 example (Previous example) | 58. Divisibility rule of 58 example (Next example) | | | | | --- Share this solution or page with your friends. | | | | | | | | | |
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| | | | | | | | | | --- --- --- --- | | | | | Divisibility rule of 57 example ( Enter your problem ) ( Enter your problem ) | | 1. Formula 2. Divisibility rule of 2 example 3. Divisibility rule of 3 example 4. Divisibility rule of 4 example 5. Divisibility rule of 5 example 6. Divisibility rule of 6 example 7. Divisibility rule of 7 example 8. Divisibility rule of 8 example 9. Divisibility rule of 9 example 10. Divisibility rule of 10 example 11. Divisibility rule of 11 example 12. Divisibility rule of 12 example 13. Divisibility rule of 13 example 14. Divisibility rule of 14 example 15. Divisibility rule of 15 example 16. Divisibility rule of 16 example 17. Divisibility rule of 17 example 18. Divisibility rule of 18 example 19. Divisibility rule of 19 example 20. Divisibility rule of 20 example 21. Divisibility rule of 21 example 22. Divisibility rule of 22 example 23. Divisibility rule of 23 example 24. Divisibility rule of 24 example 25. Divisibility rule of 25 example 26. 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Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 - color{red}{2} xx 17 = 1573 -34 = 1539 153color{red}{9}=>153 - color{red}{9} xx 17 = 153 -153 = 0 Here 0 is divisible by 57. :. 15732 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 15732 is 1+5+7+3+2=18, which is divisible by 3. :. 15732 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 + color{red}{2} xx 2 = 1573 +4 = 1577 157color{red}{7}=>157 + color{red}{7} xx 2 = 157 +14 = 171 17color{red}{1}=>17 + color{red}{1} xx 2 = 17 +2 = 19 Here 19 is divisible by 19. :. 15732 is divisible by 19. 15732 is divisible by 3 and 19. :. 15732 is divisible by 57. --- 2. Check whether 18069 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 - color{red}{9} xx 17 = 1806 -153 = 1653 165color{red}{3}=>165 - color{red}{3} xx 17 = 165 -51 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 18069 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 18069 is 1+8+0+6+9=24, which is divisible by 3. :. 18069 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 + color{red}{9} xx 2 = 1806 +18 = 1824 182color{red}{4}=>182 + color{red}{4} xx 2 = 182 +8 = 190 19color{red}{0}=>19 + color{red}{0} xx 2 = 19 = 19 Here 19 is divisible by 19. :. 18069 is divisible by 19. 18069 is divisible by 3 and 19. :. 18069 is divisible by 57. --- 3. Check whether 20634 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 - color{red}{4} xx 17 = 2063 -68 = 1995 199color{red}{5}=>199 - color{red}{5} xx 17 = 199 -85 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 20634 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 20634 is 2+0+6+3+4=15, which is divisible by 3. :. 20634 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 + color{red}{4} xx 2 = 2063 +8 = 2071 207color{red}{1}=>207 + color{red}{1} xx 2 = 207 +2 = 209 20color{red}{9}=>20 + color{red}{9} xx 2 = 20 +18 = 38 Here 38 is divisible by 19. :. 20634 is divisible by 19. 20634 is divisible by 3 and 19. :. 20634 is divisible by 57. --- 4. Check whether 16428 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 - color{red}{8} xx 17 = 1642 -136 = 1506 150color{red}{6}=>150 - color{red}{6} xx 17 = 150 -102 = 48 Here 48 is not divisible by 57. :. 16428 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 16428 is 1+6+4+2+8=21, which is divisible by 3. :. 16428 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 + color{red}{8} xx 2 = 1642 +16 = 1658 165color{red}{8}=>165 + color{red}{8} xx 2 = 165 +16 = 181 18color{red}{1}=>18 + color{red}{1} xx 2 = 18 +2 = 20 Here 20 is not divisible by 19. :. 16428 is not divisible by 19. Hence 16428 is also not divisible by 57. --- 5. Check whether 19795 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1979color{red}{5}=>1979 - color{red}{5} xx 17 = 1979 -85 = 1894 189color{red}{4}=>189 - color{red}{4} xx 17 = 189 -68 = 121 12color{red}{1}=>12 - color{red}{1} xx 17 = 12 -17 = -5 Here -5 is not divisible by 57. :. 19795 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 19795 is 1+9+7+9+5=31, which is not divisible by 3. :. 19795 is not divisible by 3. Hence 19795 is also not divisible by 57. --- 6. Check whether 21256 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2125color{red}{6}=>2125 - color{red}{6} xx 17 = 2125 -102 = 2023 202color{red}{3}=>202 - color{red}{3} xx 17 = 202 -51 = 151 15color{red}{1}=>15 - color{red}{1} xx 17 = 15 -17 = -2 Here -2 is not divisible by 57. :. 21256 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 21256 is 2+1+2+5+6=16, which is not divisible by 3. :. 21256 is not divisible by 3. Hence 21256 is also not divisible by 57. --- --- This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then Submit Here --- | | | --- | | 56. Divisibility rule of 56 example (Previous example) | 58. Divisibility rule of 58 example (Next example) | | | |
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| | | | Divisibility rule of 57 example ( Enter your problem ) ( Enter your problem ) | | 1. Formula 2. Divisibility rule of 2 example 3. Divisibility rule of 3 example 4. Divisibility rule of 4 example 5. Divisibility rule of 5 example 6. Divisibility rule of 6 example 7. Divisibility rule of 7 example 8. Divisibility rule of 8 example 9. Divisibility rule of 9 example 10. Divisibility rule of 10 example 11. Divisibility rule of 11 example 12. Divisibility rule of 12 example 13. Divisibility rule of 13 example 14. Divisibility rule of 14 example 15. Divisibility rule of 15 example 16. Divisibility rule of 16 example 17. Divisibility rule of 17 example 18. Divisibility rule of 18 example 19. Divisibility rule of 19 example 20. Divisibility rule of 20 example 21. Divisibility rule of 21 example 22. Divisibility rule of 22 example 23. Divisibility rule of 23 example 24. Divisibility rule of 24 example 25. Divisibility rule of 25 example 26. Divisibility rule of 26 example 27. 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Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 - color{red}{2} xx 17 = 1573 -34 = 1539 153color{red}{9}=>153 - color{red}{9} xx 17 = 153 -153 = 0 Here 0 is divisible by 57. :. 15732 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 15732 is 1+5+7+3+2=18, which is divisible by 3. :. 15732 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 + color{red}{2} xx 2 = 1573 +4 = 1577 157color{red}{7}=>157 + color{red}{7} xx 2 = 157 +14 = 171 17color{red}{1}=>17 + color{red}{1} xx 2 = 17 +2 = 19 Here 19 is divisible by 19. :. 15732 is divisible by 19. 15732 is divisible by 3 and 19. :. 15732 is divisible by 57. --- 2. Check whether 18069 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 - color{red}{9} xx 17 = 1806 -153 = 1653 165color{red}{3}=>165 - color{red}{3} xx 17 = 165 -51 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 18069 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 18069 is 1+8+0+6+9=24, which is divisible by 3. :. 18069 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 + color{red}{9} xx 2 = 1806 +18 = 1824 182color{red}{4}=>182 + color{red}{4} xx 2 = 182 +8 = 190 19color{red}{0}=>19 + color{red}{0} xx 2 = 19 = 19 Here 19 is divisible by 19. :. 18069 is divisible by 19. 18069 is divisible by 3 and 19. :. 18069 is divisible by 57. --- 3. Check whether 20634 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 - color{red}{4} xx 17 = 2063 -68 = 1995 199color{red}{5}=>199 - color{red}{5} xx 17 = 199 -85 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 20634 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 20634 is 2+0+6+3+4=15, which is divisible by 3. :. 20634 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 + color{red}{4} xx 2 = 2063 +8 = 2071 207color{red}{1}=>207 + color{red}{1} xx 2 = 207 +2 = 209 20color{red}{9}=>20 + color{red}{9} xx 2 = 20 +18 = 38 Here 38 is divisible by 19. :. 20634 is divisible by 19. 20634 is divisible by 3 and 19. :. 20634 is divisible by 57. --- 4. Check whether 16428 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 - color{red}{8} xx 17 = 1642 -136 = 1506 150color{red}{6}=>150 - color{red}{6} xx 17 = 150 -102 = 48 Here 48 is not divisible by 57. :. 16428 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 16428 is 1+6+4+2+8=21, which is divisible by 3. :. 16428 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 + color{red}{8} xx 2 = 1642 +16 = 1658 165color{red}{8}=>165 + color{red}{8} xx 2 = 165 +16 = 181 18color{red}{1}=>18 + color{red}{1} xx 2 = 18 +2 = 20 Here 20 is not divisible by 19. :. 16428 is not divisible by 19. Hence 16428 is also not divisible by 57. --- 5. Check whether 19795 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1979color{red}{5}=>1979 - color{red}{5} xx 17 = 1979 -85 = 1894 189color{red}{4}=>189 - color{red}{4} xx 17 = 189 -68 = 121 12color{red}{1}=>12 - color{red}{1} xx 17 = 12 -17 = -5 Here -5 is not divisible by 57. :. 19795 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 19795 is 1+9+7+9+5=31, which is not divisible by 3. :. 19795 is not divisible by 3. Hence 19795 is also not divisible by 57. --- 6. Check whether 21256 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2125color{red}{6}=>2125 - color{red}{6} xx 17 = 2125 -102 = 2023 202color{red}{3}=>202 - color{red}{3} xx 17 = 202 -51 = 151 15color{red}{1}=>15 - color{red}{1} xx 17 = 15 -17 = -2 Here -2 is not divisible by 57. :. 21256 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 21256 is 2+1+2+5+6=16, which is not divisible by 3. :. 21256 is not divisible by 3. Hence 21256 is also not divisible by 57. --- --- This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then Submit Here --- | | | --- | | 56. Divisibility rule of 56 example (Previous example) | 58. Divisibility rule of 58 example (Next example) | | |
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| | --- | | 56. Divisibility rule of 56 example (Previous example) | 58. Divisibility rule of 58 example (Next example) | --- 57. Divisibility rule of 57 example --- --- 1. Check whether 15732 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 - color{red}{2} xx 17 = 1573 -34 = 1539 153color{red}{9}=>153 - color{red}{9} xx 17 = 153 -153 = 0 Here 0 is divisible by 57. :. 15732 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 15732 is 1+5+7+3+2=18, which is divisible by 3. :. 15732 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1573color{red}{2}=>1573 + color{red}{2} xx 2 = 1573 +4 = 1577 157color{red}{7}=>157 + color{red}{7} xx 2 = 157 +14 = 171 17color{red}{1}=>17 + color{red}{1} xx 2 = 17 +2 = 19 Here 19 is divisible by 19. :. 15732 is divisible by 19. 15732 is divisible by 3 and 19. :. 15732 is divisible by 57. --- 2. Check whether 18069 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 - color{red}{9} xx 17 = 1806 -153 = 1653 165color{red}{3}=>165 - color{red}{3} xx 17 = 165 -51 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 18069 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 18069 is 1+8+0+6+9=24, which is divisible by 3. :. 18069 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1806color{red}{9}=>1806 + color{red}{9} xx 2 = 1806 +18 = 1824 182color{red}{4}=>182 + color{red}{4} xx 2 = 182 +8 = 190 19color{red}{0}=>19 + color{red}{0} xx 2 = 19 = 19 Here 19 is divisible by 19. :. 18069 is divisible by 19. 18069 is divisible by 3 and 19. :. 18069 is divisible by 57. --- 3. Check whether 20634 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 - color{red}{4} xx 17 = 2063 -68 = 1995 199color{red}{5}=>199 - color{red}{5} xx 17 = 199 -85 = 114 11color{red}{4}=>11 - color{red}{4} xx 17 = 11 -68 = -57 Here -57 is divisible by 57. :. 20634 is divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 20634 is 2+0+6+3+4=15, which is divisible by 3. :. 20634 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 2063color{red}{4}=>2063 + color{red}{4} xx 2 = 2063 +8 = 2071 207color{red}{1}=>207 + color{red}{1} xx 2 = 207 +2 = 209 20color{red}{9}=>20 + color{red}{9} xx 2 = 20 +18 = 38 Here 38 is divisible by 19. :. 20634 is divisible by 19. 20634 is divisible by 3 and 19. :. 20634 is divisible by 57. --- 4. Check whether 16428 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 - color{red}{8} xx 17 = 1642 -136 = 1506 150color{red}{6}=>150 - color{red}{6} xx 17 = 150 -102 = 48 Here 48 is not divisible by 57. :. 16428 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 16428 is 1+6+4+2+8=21, which is divisible by 3. :. 16428 is divisible by 3. Divisibility rule of 19 : 2 times the last digit and add it to the rest of the number. If the answer is divisible by 19, then number is also divisible by 19. (Apply this rule to the answer again if necessary) 1642color{red}{8}=>1642 + color{red}{8} xx 2 = 1642 +16 = 1658 165color{red}{8}=>165 + color{red}{8} xx 2 = 165 +16 = 181 18color{red}{1}=>18 + color{red}{1} xx 2 = 18 +2 = 20 Here 20 is not divisible by 19. :. 16428 is not divisible by 19. Hence 16428 is also not divisible by 57. --- 5. Check whether 19795 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 1979color{red}{5}=>1979 - color{red}{5} xx 17 = 1979 -85 = 1894 189color{red}{4}=>189 - color{red}{4} xx 17 = 189 -68 = 121 12color{red}{1}=>12 - color{red}{1} xx 17 = 12 -17 = -5 Here -5 is not divisible by 57. :. 19795 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 19795 is 1+9+7+9+5=31, which is not divisible by 3. :. 19795 is not divisible by 3. Hence 19795 is also not divisible by 57. --- 6. Check whether 21256 is divisible by 57 or not? Solution: Divisibility rule of 57 : 17 times the last digit and subtract it from the rest of the number. If the answer is divisible by 57, then number is also divisible by 57. (Apply this rule to the answer again if necessary) 2125color{red}{6}=>2125 - color{red}{6} xx 17 = 2125 -102 = 2023 202color{red}{3}=>202 - color{red}{3} xx 17 = 202 -51 = 151 15color{red}{1}=>15 - color{red}{1} xx 17 = 15 -17 = -2 Here -2 is not divisible by 57. :. 21256 is not divisible by 57. --- Method-2 : Divisibility rule of 57 : If number is divisible by 3 and 19, then number is also divisible by 57. Divisibility rule of 3 : The sum of the digits is divisible by 3, then number is also divisible by 3. Sum of digits of 21256 is 2+1+2+5+6=16, which is not divisible by 3. :. 21256 is not divisible by 3. Hence 21256 is also not divisible by 57. --- --- This material is intended as a summary. Use your textbook for detail explanation. Any bug, improvement, feedback then Submit Here --- | | | --- | | 56. 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3435 | https://www.youtube.com/watch?v=0Z5KypeHzgU | Symmetry Identities
Paul Elliott-Magwood
70 subscribers
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Posted: 24 Oct 2013
The Symmetry Identities come from the graphs of the sine and cosine functions. We can use them to simplify trigonometric expressions that have multipliers of -1 inside the angle.
Transcript:
hello in this video we're going to talk about the Symmetry identities these are identities that come out of the symmetry of the graphs of yal sin x and y cosine of x so we're start off drawing those graphs there's y equal sin x and there's y equal cosine of x now I'd like to you to think for a minute about graph transformation let's say we reflect y sinx about the xais we get this graph here it's a reflection about the x-axis this is yal sinx but notice if we reflect the blue graph about the Y AIS we get the exact same green graph this is also y = sinx clearly these are the same graph what does that tell us if we get the same graph when we do those two Transformations it means that these two algebraic expressions must be the same so we get that y equals sorry that sin Theta is equal to negative sin Theta let's see what kind of information we get from reflecting the cosine graph well take a look that if we reflect the cosine graph about the Y AIS we get the exact same graph this is also y = cosine ofx again these two graphs are the same and that means that cosine of x is equal to cosine ofx so we get those two symmetry identities and they're true for any angle at all now notice the difference for the S one if you have a negative um one multiplier inside the brackets you can bring it out front for the cosine one if you have a negative 1 multiplier inside the brackets you can just drop it two very different behaviors now because of this symmetry if reflecting about the x or y AIS gives us the same graph we call that an odd function so sinine of X is an example of an odd function if we reflect about the y- axis and we get the exact same graph we call that sort of function an even function now for example y x^2 is also an even function for an odd function y = x cubed is also an odd function let's apply these symmetry identities so let's say we've got this expression here and I just like to simplify it so that all my angles are X well I can use a symmetry identity here s of negx is sinx and I can use one here cosine ofx is just cos x pardon me and we can continue to simplify sorry I have a bit of a cold right now all right now let's uh figure out a symmetry identity for tangent is it an even or odd function or maybe neither well we know the tan of something is the sign of that thing over the cosine of that angle it's by the tangent identity from last video from the Symmetry identities we have S of theta is negative sin Theta cosine of negative Theta is cos Theta and negative sin Theta over cos Theta well that's tan Theta so we see that the tangent function is an odd function okay and again that last step we're just using the tangent identity again but in Reverse so we'll talk a lot more about identities in class this week we'll see a bunch more um and we'll see how they all fit together to help us solve equations prove new identities and simplify expressions thank you |
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Convert base-2 binary number string to int
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I'd simply like to convert a base-2 binary number string into an int, something like this:
```
'11111111'.fromBinaryToInt()
255
```
Is there a way to do this in Python?
python
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edited Aug 28, 2016 at 9:48
wtanaka.com
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asked Jan 19, 2012 at 15:01
Naftuli KayNaftuli Kay
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While it doesn't really matter, a binary string typically means a string containing actual binary data (a byte contains two hexadecimal digits, ie "\x00" is a null byte).
– trevorKirkby
Commented
May 3, 2014 at 18:25
Just to mention it: the other way around it goes like '{0:08b}'.format(65) (or f'{65:08b}').
– TNT
Commented
Oct 10, 2020 at 12:58
2
it is uint rather than int. Please change the title
– Gideon Kogan
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Dec 25, 2022 at 21:09
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You use the built-in int() function, and pass it the base of the input number, i.e. 2 for a binary number:
```
int('11111111', 2)
255
```
Here is documentation for Python 2, and for Python 3.
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edited Jun 10, 2021 at 8:19
answered Jan 19, 2012 at 15:02
unwindunwind
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In case someone is looking for the opposite: bin(255) -> '0b11111111'. See this answer for additional details.
– Akseli Palén
Commented
Mar 13, 2013 at 23:29
9
It should be noted that this only works for unsigned binary integers. For signed integers, the conversion options are a mess.
– Fake Name
Commented
Nov 1, 2014 at 13:30
5
How to do this in python 3?
– Saras Arya
Commented
Feb 27, 2015 at 5:45
1
And note that in an interactive REPL session (as suggested by the >>> prompt), you don't need to use print at all. The OP's hypothetical example didn't. So it really should be identical in Python 2 and 3.
– John Y
Commented
Jul 12, 2016 at 22:36
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To add to @AkseliPalén comment: bin (255)[2:]
– Timo
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Sep 20, 2020 at 20:10
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Just type 0b11111111 in python interactive interface:
```
0b11111111
255
```
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answered Jan 27, 2015 at 4:00
lengxuehxlengxuehx
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Another way to do this is by using the bitstring module:
```
from bitstring import BitArray
b = BitArray(bin='11111111')
b.uint
255
```
Note that the unsigned integer (uint) is different from the signed integer (int):
```
b.int
-1
```
Your question is really asking for the unsigned integer representation; this is an important distinction.
The bitstring module isn't a requirement, but it has lots of performant methods for turning input into and from bits into other forms, as well as manipulating them.
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edited Feb 15, 2023 at 16:14
answered Jan 19, 2012 at 15:06
Alex ReynoldsAlex Reynolds
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Using int with base is the right way to go. I used to do this before I found int takes base also. It is basically a reduce applied on a list comprehension of the primitive way of converting binary to decimal ( e.g. 110 = 20 0 + 2 1 1 + 2 2 1)
```
add = lambda x,y : x + y
reduce(add, [int(x) 2 y for x, y in zip(list(binstr), range(len(binstr) - 1, -1, -1))])
```
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answered May 8, 2013 at 13:04
Saurabh HiraniSaurabh Hirani
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Instead of defining add = lambda x, y: x + y, int.__add__ can be provided to reduce. E.g. reduce(int.__add__, ...)
– Jordan Jambazov
Commented
Aug 28, 2016 at 10:46
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Here's another concise way to do it not mentioned in any of the above answers:
```
eval('0b' + '11111111')
255
```
Admittedly, it's probably not very fast, and it's a very very bad idea if the string is coming from something you don't have control over that could be malicious (such as user input), but for completeness' sake, it does work.
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answered Oct 27, 2021 at 2:33
Zachary BarbanellZachary Barbanell
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This might be fine for hard-coded constants, but there are multiple reasons why using eval is considered bad practice, especially when you would try to parse user input. See this post, for example: stackoverflow.com/questions/1832940/…
– larsschwegmann
Commented
Dec 3, 2021 at 8:47
I love this answer, really creative
– collinsmarra
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Dec 17, 2021 at 4:09
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If you are using python3.6 or later you can use f-string to do the
conversion:
Binary to decimal:
```
print(f'{0b1011010:#0}')
90
bin_2_decimal = int(f'{0b1011010:#0}')
bin_2_decimal
90
```
binary to octal hexa and etc.
```
f'{0b1011010:#o}'
'0o132' # octal
f'{0b1011010:#x}'
'0x5a' # hexadecimal
f'{0b1011010:#0}'
'90' # decimal
```
Pay attention to 2 piece of information separated by colon.
In this way, you can convert between {binary, octal, hexadecimal, decimal} to {binary, octal, hexadecimal, decimal} by changing right side of colon[:]
```
:#b -> converts to binary
:#o -> converts to octal
:#x -> converts to hexadecimal
:#0 -> converts to decimal as above example
```
Try changing left side of colon to have octal/hexadecimal/decimal.
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answered Oct 31, 2019 at 0:28
Robert RanjanRobert Ranjan
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For large matrix (105 rows and up) it is better to use a vectorized matmult. Pass in all rows and cols in one shot. It is extremely fast. There is no looping in python here. I originally designed it for converting many binary columns like 0/1 for like 10 different genre columns in MovieLens into a single integer for each example row.
```
def BitsToIntAFast(bits):
m,n = bits.shape
a = 2np.arange(n)[::-1] # -1 reverses array of powers of 2 of same length as bits
return bits @ a
```
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answered Jan 13, 2020 at 18:19
Geoffrey AndersonGeoffrey Anderson
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If you wanna know what is happening behind the scene, then here you go.
```
class Binary():
def init(self, binNumber):
self._binNumber = binNumber
self._binNumber = self._binNumber[::-1]
self._binNumber = list(self._binNumber)
self._x =
self._count = 1
self._change = 2
self._amount = 0
print(self._ToNumber(self._binNumber))
def _ToNumber(self, number):
self._number = number
for i in range (1, len (self._number)):
self._total = self._count self._change
self._count = self._total
self._x.append(self._count)
self._deep = zip(self._number, self._x)
for self._k, self._v in self._deep:
if self._k == '1':
self._amount += self._v
return self._amount
mo = Binary('101111110')
```
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edited Mar 15, 2021 at 15:01
mkrieger1
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answered Nov 16, 2016 at 7:10
Mohammad MahjoubMohammad Mahjoub
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A recursive Python implementation:
```
def int2bin(n):
return int2bin(n >> 1) + [n & 1] if n > 1 else
```
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edited Mar 23, 2018 at 4:40
Pang
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answered Mar 23, 2018 at 4:32
Ludovic TrottierLudovic Trottier
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For the record to go back and forth in basic python3:
```
a = 10
bin(a)
'0b1010'
int(bin(a), 2)
10
eval(bin(a))
10
```
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answered Feb 16, 2021 at 8:53
ClementWalterClementWalter
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3437 | https://mathworld.wolfram.com/LaguerrePolynomial.html | Laguerre Polynomial
The Laguerre polynomials are solutions to the Laguerre differential equation with . They are illustrated above for and , 2, ..., 5, and implemented in the Wolfram Language as LaguerreL[n, x].
The first few Laguerre polynomials are
| |
| (1) |
| (2) |
| (3) |
| (4) |
When ordered from smallest to largest powers and with the denominators factored out, the triangle of nonzero coefficients is 1; , 1; 2, , 1; , 18, 1; 24, , ... (OEIS A021009). The leading denominators are 1, , 2, , 24, , 720, , 40320, , 3628800, ... (OEIS A000142).
The Laguerre polynomials are given by the sum
| |
| (5) |
where is a binomial coefficient.
The Rodrigues representation for the Laguerre polynomials is
| |
| (6) |
and the generating function for Laguerre polynomials is
| |
| (7) |
| (8) |
A contour integral that is commonly taken as the definition of the Laguerre polynomial is given by
| |
| (9) |
where the contour encloses the origin but not the point (Arfken 1985, pp. 416 and 722).
The Laguerre polynomials satisfy the recurrence relations
| |
| (10) |
(Petkovšek et al. 1996) and
| |
| (11) |
Solutions to the associated Laguerre differential equation with and an integer are called associated Laguerre polynomials (Arfken 1985, p. 726) or, in older literature, Sonine polynomials (Sonine 1880, p. 41; Whittaker and Watson 1990, p. 352).
See also
Associated Laguerre Polynomial, Laguerre Differential Equation, Orthogonal Polynomials
Related Wolfram sites
Explore with Wolfram|Alpha
More things to try:
associated Laguerre polynomial 3 0
Laguerre polynomial order 1
Laguerre polynomial order 5
References
Abramowitz, M. and Stegun, I. A. (Eds.). "Orthogonal Polynomials." Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.Andrews, G. E.; Askey, R.; and Roy, R. "Laguerre Polynomials." §6.2 in Special Functions. Cambridge, England: Cambridge University Press, pp. 282-293, 1999.Arfken, G. "Laguerre Functions." §13.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 721-731, 1985.Chebyshev, P. L. "Sur le développement des fonctions à une seule variable." Bull. Ph.-Math., Acad. Imp. Sc. St. Pétersbourg 1, 193-200, 1859.Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea, pp. 499-508, 1987.Iyanaga, S. and Kawada, Y. (Eds.). "Laguerre Functions." Appendix A, Table 20.VI in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980.Koekoek, R. and Swarttouw, R. F. "Laguerre." §1.11 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its -Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 47-49, 1998.Laguerre, E. de. "Sur l'intégrale ." Bull. Soc. math. France 7, 72-81, 1879. Reprinted in Oeuvres, Vol. 1. New York: Chelsea, pp. 428-437, 1971.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 61-62, 1996. S. "The Laguerre Polynomials." §3.1 i The Umbral Calculus. New York: Academic Press, pp. 108-113, 1984.Rota, G.-C.; Kahaner, D.; Odlyzko, A. "Laguerre Polynomials." §11 in "On the Foundations of Combinatorial Theory. VIII: Finite Operator Calculus." J. Math. Anal. Appl. 42, 684-760, 1973.Sansone, G. "Expansions in Laguerre and Hermite Series." Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295-385, 1991.Sloane, N. J. A. Sequences A000142/M1675 and A021009 in "The On-Line Encyclopedia of Integer Sequences."Sonine, N. J. "Sur les fonctions cylindriques et le développement des fonctions continues en séries." Math. Ann. 16, 1-80, 1880.Spanier, J. and Oldham, K. B. "The Laguerre Polynomials ." Ch. 23 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 209-216, 1987.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.Whittaker, E. T. and Watson, G. N. Ch. 16, Ex. 8 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 352, 1990.
Referenced on Wolfram|Alpha
Laguerre Polynomial
Cite this as:
Weisstein, Eric W. "Laguerre Polynomial." From MathWorld--A Wolfram Resource.
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3438 | https://math.stackexchange.com/questions/4756878/parametric-form-of-parabola | Stack Exchange Network
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Parametric form of Parabola [closed]
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-1
$\begingroup$
Recently I was going through co-ordinate geometry. While studying about parabola I encountered something called "parameteric coordinates". In my textbook some parameteric coordinates were given [y²=4ax : (at², 2at)] for specific equations of parabola, but nothing related to how these coordinates are defined was there.
Searched a lot but cant find anything convincing, one of many posts said it's done by defining x and y in terms of an other variable where x(t)=t itself and y(t)=ax(t)² => y=at² from standard eqn of parabola.
Well, it does not make any sense as (t,at²) isn't any parameteric coordinate.
I am really confused (to the extent where I can no longer differentiate between maths and biology) Please someone help me.
I need an elaborate answer ASAP. (I am no mathemagician please be clear and don't use heavy terminology)
Thanks
conic-sections
coordinate-systems
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edited Aug 23, 2023 at 15:25
Gopal KaushikGopal Kaushik
asked Aug 22, 2023 at 13:11
Gopal KaushikGopal Kaushik
2933 bronze badges
$\endgroup$
3
$\begingroup$ Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. $\endgroup$
José Carlos Santos
– José Carlos Santos
2023-08-22 13:14:50 +00:00
Commented Aug 22, 2023 at 13:14
$\begingroup$ I've added an answer but realise I've made some assumptions about exactly what your question is. My main suggestion would be to use something like geogebra or desmos and play around with the sort of curves you can get. $\endgroup$
Chris Lewis
– Chris Lewis
2023-08-22 14:33:12 +00:00
Commented Aug 22, 2023 at 14:33
$\begingroup$ en.wikipedia.org/wiki/Parametric_equation $\endgroup$
Intelligenti pauca
– Intelligenti pauca
2023-08-22 15:20:46 +00:00
Commented Aug 22, 2023 at 15:20
Add a comment |
1 Answer 1
Reset to default
1
$\begingroup$
In the example you gave for a parabola, $t$ is the parameter. It looks like you have the right understanding of how parametric coordinates work (ie each point on a curve is given by two functions of a parameter; in your parabola example, if you put in, say, $t=3$, you get the point $(9a,6a)$). However, I don't think a parabola is the best example of why these can be helpful, even among conics.
Let's start with a circle with radius $r$ centred at $(0,0)$. The Cartesian equation for this is $x^2+y^2=r^2$. But this is an implicit formula; you have to do some calculation to get $y$ in terms of $x$. OK; rearrange and we get $$y=\sqrt{r^2-x^2}$$ ...and plotting this, we get a semicircular arc. In fact, for every $-r, we need two corresponding $y$ values; we can write this as $$y=\pm\sqrt{r^2-x^2}$$ but it doesn't change the fact that we can't write $y$ as a function of $x$ (technically, a function needs to just have one output for a given input).
Here's the parametric form for the same circle: $$x=r\cos{t},\quad y=r\sin{t}$$
Hopefully you can see why this works. Now, for every value of $t$, we get exactly one point on the circle, which makes this form far easier to work with (in certain applications - eg physics of circular motion; in fact, one of the common uses for parametric coordinates is in modelling motion, where the parameter is often time).
Now, in this case, $t$ corresponds exactly to the angle the line joining the point on the circle to $O$ makes with the positive $x$-axis; but this isn't a necessary property of the parameterisation. We could just as well say $$x=r\cos{2t+5},\quad y=r\sin{2t+5}$$ and get the same circle.
Similarly, it's easy to parameterise an ellipse: $$x=a\cos{t},\quad y=b\sin{t}$$
(just scaling the circle) or a hyperbola: $$x=a\cosh{t},\quad y=b\sinh{t}$$
The similarities between these forms are another reason it can be useful to use parametric coordinates for conics (and other curves).
In case you're not familiar with $\cosh$ and $\sinh$, another form for the hyperbola is $$x=a\sec{t},\quad y=b\tan{t}$$
Another advantage of parametric coordinates is it is relatively easy to shift and rotate curves by using matrix transformations (not every ellipse has axes parallel to the coordinate axes, after all).
I hope that's useful - let me know if anything isn't clear!
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answered Aug 22, 2023 at 14:30
Chris LewisChris Lewis
3,64111 gold badge66 silver badges1111 bronze badges
$\endgroup$
4
$\begingroup$ Ok but could please explain how parameteric coordinates for parabola are derived. $\endgroup$
Gopal Kaushik
– Gopal Kaushik
2023-08-23 05:06:51 +00:00
Commented Aug 23, 2023 at 5:06
$\begingroup$ Like for a circle it can be derived by simply using trigonometry. I need to know how did someone come up with the result that coordinates of parabola (say, y²=4ax) be written as (at²,2at). $\endgroup$
Gopal Kaushik
– Gopal Kaushik
2023-08-23 05:16:56 +00:00
Commented Aug 23, 2023 at 5:16
1
$\begingroup$ "How did someone come up with it" is a hard question to answer. Why it works is that from the parametric form, $y^2 = 4a^2 t^2$ and $4ax=4a^2 t^2$. As in the above answer, the parameterisation is not unique - you could simply have $y=u$ and $x=\frac{u^2}{4a}$. But the form you had is more informative - it tells you something about the scaling of that parabola as you change the constant $a$. $\endgroup$
Chris Lewis
– Chris Lewis
2023-08-23 08:58:58 +00:00
Commented Aug 23, 2023 at 8:58
$\begingroup$ By the way, the title of your question mentions conics. If you're only interested in the parabola you might want to edit that. $\endgroup$
Chris Lewis
– Chris Lewis
2023-08-23 09:00:19 +00:00
Commented Aug 23, 2023 at 9:00
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3439 | https://onlinelibrary.wiley.com/doi/abs/10.1111/evo.12597 | A generation‐time effect on the rate of molecular evolution in bacteria - Weller - 2015 - Evolution - Wiley Online Library
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Evolution
Volume 69, Issue 3 pp. 643-652
ORIGINAL ARTICLE
A generation-time effect on the rate of molecular evolution in bacteria
Cory Weller,
Cory Weller
Department of Biology, University of Virginia, Charlottesville, Virginia, 22904
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Martin Wu,
Martin Wu
mw4yv@virginia.edu
Department of Biology, University of Virginia, Charlottesville, Virginia, 22904
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Cory Weller,
Cory Weller
Department of Biology, University of Virginia, Charlottesville, Virginia, 22904
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Martin Wu,
Martin Wu
mw4yv@virginia.edu
Department of Biology, University of Virginia, Charlottesville, Virginia, 22904
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First published: 07 January 2015
Citations: 60
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Abstract
Molecular evolutionary rate varies significantly among species and a strict global molecular clock has been rejected across the tree of life. Generation time is one primary life-history trait that influences the molecular evolutionary rate. Theory predicts that organisms with shorter generation times evolve faster because of the accumulation of more DNA replication errors per unit time. Although the generation-time effect has been demonstrated consistently in plants and animals, the evidence of its existence in bacteria is lacking. The bacterial phylum Firmicutes offers an excellent system for testing generation-time effect because some of its members can enter a dormant, nonreproductive endospore state in response to harsh environmental conditions. It follows that spore-forming bacteria would—with their longer generation times—evolve more slowly than their nonspore-forming relatives. It is therefore surprising that a previous study found no generation-time effect in Firmicutes. Using a phylogenetic comparative approach and leveraging on a large number of Firmicutes genomes, we found sporulation significantly reduces the genome-wide spontaneous DNA mutation rate and protein evolutionary rate. Contrary to the previous study, our results provide strong evidence that the evolutionary rates of bacteria, like those of plants and animals, are influenced by generation time.
Supporting Information
Disclaimer: Supplementary materials have been peer-reviewed but not copyedited.
| Filename | Description |
--- |
| evo12597-sup-0001-SupInfo.zip112.3 KB | Figure S1. The number of sporulation genes plotted against the number of protein coding genes in the genomes. Table S1. Phylogenetic profiles of sporulation genes in genomes of 200 Firmicutes representatives. Table S2. List of molecular evolutionary rates and codon bias index for each of the 197 Firmicutes species analyzed in this study. Table S3. Log-likelihood and AIC values of different clock models. Table S4. Independent contrast analysis of protein evolutionary rates of 200 Firmicutes representatives. Table S5. Independent contrast analysis of evolutionary rates in 137 Firmicutes representatives after removing 60 species not positively identified as spore-forming in Galperin et al. (2012). Table S6. Independent contrast analysis of evolutionary rates in 112 Firmicutes representatives after removing three contrast nodes with less than 80% bootstrap support from the 197-species dataset. |
Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.
LITERATURE CITED
Abecasis, A. B., M. Serrano, R. Alves, L. Quintais, J. B. Pereira-Leal, and A. O. Henriques. 2013. A genomic signature and the identification of new sporulation genes. J. Bacteriol. 195: 2101–2115. 10.1128/JB.02110-12 CASPubMedWeb of Science®Google Scholar
Afkar, E., J. Lisak, C. Saltikov, P. Basu, R. S. Oremland, and J. F. Stolz. 2003. The respiratory arsenate reductase from Bacillus selenitireducens strain MLS10. FEMS Microbiol. Lett. 226: 107–112. 10.1016/S0378-1097(03)00609-8 CASPubMedWeb of Science®Google Scholar
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Associate Editor: A. D. Visser
Handling Editor: T. Lenormand
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Volume 69, Issue 3
March 2015
Pages 643-652
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© 2015 The Author(s).
Keywords
Evolutionary rate
Firmicutes
sporulation
Publication History
Issue Online: 16 March 2015
Version of Record online: 05 February 2015
Accepted manuscript online: 07 January 2015
Manuscript accepted: 17 December 2014
Manuscript received: 13 September 2014
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3440 | https://web.mit.edu/ceder/publications/Percolation.pdf | Percolation Theory Dr. Kim Christensen Blackett Laboratory Imperial College London Prince Consort Road SW7 2BW London United Kingdom October 9, 2002 Aim The aim of the percolation theory course is to provide a challenging and stimulating introduction to a selection of topics within modern theoretical condensed matter physics.
Percolation theory is the simplest model displaying a phase transition. The analytic solutions to 1d and mean-field percolation are presented. While percolation cannot be solved exactly for intermediate dimensions, the model enables the reader to become familiar with important concepts such as fractals, scaling, and renormalisation group theory in a very intuitive way.
The text is accompanied by exercises with solutions and visual interactive simulations for the percolation theory model to allow the readers to experience the behaviour, in the spirit ”seeing is be-lieving”. The animations can be downloaded via the URL I greatly appriciate the suggestions and comments provided by Nicholas Moloney and Ole Peters without whom, the text would have been incomprehensible and flooded with mistakes. However, if you still are able to find any misprints, misspellings and mistakes in the notes, I would be very grateful if you would report those to k.christensen@ic.ac.uk.
1 Contents 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 1.3 Percolation in 1d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 1.4 Percolation in the Bethe Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 1.5 Cluster Number Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 1.6 Cluster Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 1.6.1 Cluster Radius and Fractal Dimension . . . . . . . . . . . . . . . . . . . . . .
22 1.6.2 Finite Boxing of Percolating Clusters . . . . . . . . . . . . . . . . . . . . . . .
24 1.6.3 Mass of the Percolating Cluster . . . . . . . . . . . . . . . . . . . . . . . . . .
25 1.7 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27 1.8 Finite-size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29 1.9 Real space renormalisation in percolation theory . . . . . . . . . . . . . . . . . . . .
31 1.9.1 Renormalisation group transformation in 1d.
. . . . . . . . . . . . . . . . . .
34 1.9.2 Renormalisation group transformation on 2d triangular lattice. . . . . . . . .
35 1.9.3 Renormalisation group transformation on 2d square lattice of bond percolation. 36 1.9.4 Why is the renormalisation group transformation not exact?
. . . . . . . . .
37 2 1.1 Introduction Percolation theory is the simplest not exactly solved model displaying a phase transition. Often, the insight into the percolation theory problem facilitates the understanding of many other physical systems. Moreover, the concept of fractals, which is intimately related to the percolation theory problem, is of general interest as it pops up more or less everywhere in Nature. The knowledge of percolation, fractals, and scaling are of immense importance theoretically in such diverse fields as biology, physics, and geophysics and also of practical importance in e.g. oil recovery. We will begin gently by developing a basic understanding of percolation theory, providing a natural introduction to the concept of scaling and renormalisation group theory.
1.2 Preliminaries Let P(A) denote the probability for an event A and P(A1 ∩A2) the joint probability for event A1 and A2.
Definition 1 Two events A1 and A2 are independent ⇔P(A1 ∩A2) = P(A1)P(A2).
Definition 2 More generally, we define n ≥3 events A1, A2, . . . , An to be mutually independent if P(A1 ∩A2 ∩· · · ∩An) = P(A1)P(A2) · · · P(An) and if any subcollection containing at least two but fewer than n events are mutually independent.
Let each site in a lattice be occupied at random with probability p, that is, each site is occupied (with probability p) or empty (with probability 1−p) independent of the status (empty or occupied) of any of the other sites in the lattice. We call p the occupation probability or the concentration.
Definition 3 A cluster is a group of nearest neighbouring occupied sites.
Percolation theory deals with the numbers and properties of the clusters formed when sites are occupied with probability p, see Fig. (1.1).
Figure 1.1: Percolation in 2d square lattice of linear size L = 5. Sites are occupied with probability p. In the lattice above, we have one cluster of size 7, a cluster of size 3 and two clusters of size 1 (isolated sites).
Definition 4 The cluster number ns(p) denotes the number of s-clusters per lattice site.
The (average) number of clusters of size s in a hypercubic lattice of linear size L is Ldns(p), d being the dimensionality of the lattice. Defining the cluster number per lattice site as opposed to the total number of s-clusters in the lattice ensures that the quantity will be independent of the lattice size L.
3 For finite lattices L < ∞, it is intuitively clear, that if the occupation probability p is small, there is only a very tiny chance of having a cluster percolating between two opposite boundaries (i.e., in 2d, from top-to-bottom or from left-to-right). For p close to 1, we almost certainly will have a cluster percolating through the system. In Fig. 1.2, sites in 2d square lattices are occupied at random with increasing occupation probability p. The occupied sites are shown in gray while the sites belonging to the largest cluster are shown in black. Unoccupied sites are white. Note that for p ≈0.59, a percolating cluster appears for the first time.
Figure 1.2: Percolation in 2d square lattices with system size L × L = 150× 150. Occupation prob-ability p = 0.45, 0.55, 0.59, 0.65, and 0.75, respectively. Notice, that the largest cluster percolates through the lattice from top to bottom in this example when p ≥0.59.
Definition 5 The percolation threshold pc is the concentration (occupation probability) p at which an infinite cluster appears for the first time in an infinite lattice.
Note, that pc is defined with respect to an infinite lattice, that is, in the limit of L →∞. Table (1.1) lists the percolation threshold in various lattices and dimensions.
Exercise 1 Why is pc not well defined in a finite lattice?
1.3 Percolation in 1d We will consider the percolation problem in 1d where it can be solved analytically. Many of the characteristic features encountered in higher dimensions are present in 1d as well, if we know 4 Lattice # nn Site percolation Bond percolation 1d 2 1 1 2d Honeycomb 3 0.6962 1 −2 sin(π/18) ≈0.65271 2d Square 4 0.592746 1/2 2d Triangular 6 1/2 2 sin(π/18) ≈0.34729 3d Diamond 4 0.43 0.388 3d Simple cubic 6 0.3116 0.2488 3d BCC 8 0.246 0.1803 3d FCC 12 0.198 0.119 4d Hypercubic 8 0.197 0.1601 5d Hypercubic 10 0.141 0.1182 6d Hypercubic 12 0.107 0.0942 7d Hypercubic 14 0.089 0.0787 Bethe lattice z 1/(z-1) 1/(z-1) Table 1.1: The percolation threshold for the site percolation problem is given in column 3 for various lattices in various dimensions. Column 2 lists the number of nearest-neighbours (nn), also known as the coordination number. Within a given dimension, the percolation threshold decrease with increasing number of nearest-neighbours. The site percolation problem has a counterpart called the bond percolation problem: In a lattice, each bond between neighbouring lattice sites can be occupied (open) with probability p and empty (closed) with probability (1−p). A cluster is a group of connected occupied (open) bonds. NB: In all cases, a cluster is defined as a group of nearest neighbouring occupied sites (bonds). Note that the percolation threshold for the site-percolation on high-dimensional hypercubic lattices, where loops become irrelevant, approaches that of the Bethe lattice 1/(z −1), if we substitute the coordination number z with 2d.
where and how to look. Thus the 1d case serves as a transparent window into the world of phase transitions, scaling, scaling relations, and renormalisation group theory.
Imagine a 1d lattice with an infinite number of sites of equal spacing arranged in a line. Each site has a probability p of being occupied, and consequently 1 −p of being empty (not occupied).
These are the only two states possible, see Fig. 1.3.
Figure 1.3: Percolation in a 1d lattice. Sites are occupied with probability p. The crosses are empty sites, the solid circles are occupied sites. In the part of the infinite 1d lattice shown above, there is one cluster of size 5, one cluster of size 2, and three clusters of size 1.
What is of interest to us now and in future discussions in higher dimensions is the occupation probability at which an infinite cluster is obtained for the first time. A percolating cluster in 1d spans from −∞to +∞. Clearly, in 1d this can only be achieved if all sites are occupied, that is, the percolation threshold pc = 1, as a single empty site would prevent a cluster to percolate.
A precise “mathematical” derivation of pc = 1 in 1d goes as follows.
Definition 6 Let Π(p, L) denote the probability that a lattice of linear size L percolates at concen-tration p.
5 Combining the two definitions (5) and (6), we have, lim L→∞Π(p, L) = ( 0 for p < pc 1 for p ≥pc.
Consider a finite 1d lattice of size L where each site is occupied with probability p. As the events of occupying sites are independent, all sites are occupied with probability Π(p, L) = pL, see Fig. (1.4), and lim L→∞Π(p, L) = lim L→∞pL = ( 0 for p < 1 1 for p = 1, implying pc = 1.
0.0 0.5 1.0 Occupation probability p 0.0 0.5 1.0 Π(p,L) L = 1 L = 2 L = 3 L = 4 L = 5 L = 10 L = 20 L = 50 L = 100 Figure 1.4: The probability of a 1d lattice of linear size L to percolation at occupation probability p. In the limit L →∞, Π(p, L) converges to a discontinous step function.
Let us consider the clusters formed in a 1d lattice. A cluster of size s is formed when s sites are occupied next to one another bounded by two empty sites, see Fig. 1.3. When L →∞, we can ignore the effects of the boundary sites of the lattice and the probability of an arbitrary site (occupied or not) being, say, the left hand side (LHS) of an s-cluster is ns(p) = (1 −p)ps(1 −p) = (1 −p)2ps.
(1.1) This expression is obtained from the assumption that the occupancy of each site is independent of the state of any other site. If this was not the case then it would be much more complicated.
Note that, since all sites have equal probability of being occupied (or empty), the probability that an arbitrary site is part of an s-cluster is s times the probability of it being the LHS of the cluster.
We can re-write the cluster number Eq.(1.1) for 1d percolation as ns(p) = (1 −p)2ps = (1 −p)2 exp(ln(ps)) = (1 −p)2 exp(s ln(p)) = (pc −p)2 exp(−s sξ ) (1.2) with the definition of a cutoffcluster size or characteristic cluster size sξ = −1 ln(p) = −1 ln(pc −(pc −p)) → 1 pc −p = (pc −p)−1 for p →pc, (1.3) 6 where we, to obtain the limit, have used pc = 1 and the Taylor expansion ln(1 −x) = −x −1 2x2 −1 3x3 −· · · ≈−x where the last approximation is valid for x →0, see Fig. 1.5.
10 0 10 1 10 2 10 3 Cluster size s 10 −10 10 −8 10 −6 10 −4 10 −2 10 0 ns(p) p = 0.7 p = 0.9 p = 0.95 p = 0.99 0 0 1 10 100 s(pc − p) = s/sξ 10 −6 10 −4 10 −2 10 0 s 2ns(p) p = 0.7 p = 0.9 p = 0.95 p = 0.99 0.0 0.2 0.4 0.6 0.8 1.0 Occupation probability p 0 100 200 300 400 500 sξ −1/ln(p) 1/(pc − p) 10 −4 10 −3 10 −2 10 −1 10 0 pc − p 10 −1 10 0 10 1 10 2 10 3 10 4 sξ −1/ln(p) 1/(pc − p) Figure 1.5: Percolation in 1d. (a) The cluster number distribution ns(p) = (pc −p)2 exp(−s sξ ) for various values of p approaching pc = 1. The vertical lines indicate the cutoffcluster size sξ(p). (b) By plotting s2ns(p) versus s/sξ ≈s(pc −p), all the data collapses onto a function x2 exp(−x). (c) The characteristic cluster size sξ = − 1 ln(p) diverges when p →pc = 1. (d) In the limit p →pc = 1, sξ ∝(pc −p)−1.
Exercise 2 Verify the Taylor expansion of ln(1 −x) around x = 0 given above.
Thus the cutoffcluster size sξ diverges for p →pc as a power law in the distance from the critical occupation probability pc, see Fig. 1.5. The divergence of the cutoffcluster size when p →pc is also seen in higher dimensions where, however, another numerical exponent will describe the divergence.
Thus it is natural to introduce a symbol for the exponent.
Definition 7 The critical exponent σ is defined by sξ ∝|pc −p|−1 σ for p →pc.
(1.4) 7 In 1d percolation theory, σ = 1 and log sξ = constant −1 σ log |pc −p|, see Fig. 1.5.
Let us continue our journey into 1d percolation theory.
For p < pc we can state that the probability that an arbitrary site belongs to any (finite) cluster is simply the probability p of it being occupied. Since the probability that an arbitrary sites belongs to an s-cluster is given by sns(p), we arrive at ∞ X s=1 sns(p) = p for p < pc.
(1.5) Using the formula for summing a geometric series, we can satisfy those who prefer a rigorous mathematical proof: ∞ X s=1 sns(p) = ∞ X s=1 s(1 −p)2ps = (1 −p)2 ∞ X s=1 pd(ps) dp = (1 −p)2p d dp ∞ X s=1 ps !
= (1 −p)2p d dp p 1 −p = p.
How large on average is a cluster or, equivalently, how large is a cluster on average to which an occupied site belongs? The probability that a site is occupied is p. The probability that an arbitrary site belongs to an s-cluster is sns(p). Thus the probability ws that the cluster to which an occupied sites belongs contains s sites is ws = sns(p) p = sns(p) P∞ s=1 sns(p).
Thus, the mean cluster size or average cluster size S(p) is given by S(p) = ∞ X s=1 sws = ∞ X s=1 s2ns(p) P∞ s=1 ns(p)s = 1 p(1 −p)2 ∞ X s=1 s2ps = 1 p(1 −p)2 p d dp 2 ∞ X s=1 ps !
, where the operator p d dp 2 = p d dp p d dp ̸= p2 d2 dp2 .
Using the formula for summing a geometric series and the operator p d dp twice, we finally arrive at S(p) = 1 + p 1 −p = pc + p pc −p (1.6) where the last equality follows because in 1d the critical occupation probability pc = 1.
8 Exercise 3 Derive the result in Eq.(1.6) for the mean cluster size in 1d.
We thus see that the mean cluster size diverges for p →p− c , where the minus sign signifies that we are approaching pc from below, which is what we intuitively expect if considering an infinite lattice.
It is not possible to approach pc from above in 1d as pc = 1. This is actually the main difference between 1d and higher dimensions, where, pc < 1 and we can approach pc from both below and above.
In order to investigate in detail how the mean cluster size diverges, when taking the limit p →p− c , we note that the numerator in Eq.(1.6) approaches 2pc, so S(p) = pc + p pc −p → 2pc pc −p ∝(pc −p)−1 for p →p− c .
(1.7) Thus in 1d, the mean cluster diverges like a power law in the quantity (pc −p) when p →pc, see Fig. 1.6. The same phenomenon will be encountered in higher dimensions.
Definition 8 The critical exponent γ is defined by S(p) ∝|pc −p|−γ for p →pc.
(1.8) In 1d percolation theory, γ = 1.
10 −4 10 −3 10 −2 10 −1 10 0 Occupation probability p 10 −1 10 0 10 1 10 2 10 3 10 4 Average cluster size S (1 + p)/(1 − p) = (pc + p)/(pc − p) 10 −4 10 −3 10 −2 10 −1 10 0 pc − p 10 −1 10 0 10 1 10 2 10 3 10 4 Average cluster size S (1 + p)/(1 − p) = (pc + p)/(pc − p) Figure 1.6: Percolation in 1d. The average cluster size S(p) = (1 + p)/(1 −p) = (pc + p)/(pc −p) diverges when p →pc = 1. In the limit p →pc = 1, S(p) →2/(pc −p).
Definition 9 The correlation function or pair connectivity g(r) is the probability that a site at position r from an occupied site belongs to the same finite cluster.
Note this definition excludes the contribution from the infinite cluster. That need not worry us in 1d, where all clusters are finite if p < pc = 1. Let r = |r|. Clearly, g(r = 0) = 1, since the site is occupied by definition. In 1d, for a site at position r to be occupied and belong to the same (finite) cluster, this site and the (r −1) intermediate sites must be occupied, leaving g(r) = pr, for all p, which can also be expressed in the form g(r) = exp(ln(pr)) = exp(r ln(p)) = exp −r ξ , (1.9) 9 where ξ = − 1 ln(p) = −1 ln(pc −(pc −p)) → 1 (pc −p) = (pc −p)−1 for p →pc = 1, (1.10) where we use the expansion ln(1 −x) ≈−x for small x. The quantity ξ is called the correlation length which diverges for p →pc. The same phenomenon will be encountered in higher dimensions.
Note that in 1d we have sξ = ξ which is why we haven’t bothered displaying a figure for ξ, as it would be identical to Fig. 1.5. However, this identity will not be true in higher dimensions, where we will find sξ ∝ξD, where D is the fractal dimension, but more about this later.
Definition 10 The critical exponent ν is defined by ξ ∝|pc −p|−ν for p →pc.
(1.11) In 1d percolation theory, ν = 1.
By summing over all possible lattice sites r of the correlation function, the mean cluster size can be shown to be X r g(r) = S(p).
(1.12) In 1d, this sum is straight forward identifying r with r = 0, ±1, ±2, . . ., see Problem 2, where you will also discover, that the sum rule is valid in all dimensions d.
The general pattern for the exact solutions of the 1d percolation problem is that certain quan-tities, such as the cutoffcluster size sξ, the mean cluster size S(p), and the correlation length ξ diverge at the percolation threshold. The divergence can be described by simple power laws of the distance from the critical occupation probability |∆p| = |pc −p|, e.g. ξ ∝(pc −p)−1, at least asymptotically close to pc where ∆p is small. The same phenomena will be encountered in higher dimensions even though we cannot obtain exact analytic solutions.
1.4 Percolation in the Bethe Lattice The percolation problem can be solved analytically in d = 1 and d = ∞. The infinitely dimensional case is synonymous with the Bethe lattice, a special lattice where each site has z neighbouring sites, such that each branch gives rise to z −1 other branches, see Fig. 1.7.
Figure 1.7: The Bethe lattice with z = 3. Each site has three neighbours. Each branch contains z −1 = 2 subbranches.
10 The 1d case is effectively a Bethe lattice with z = 2. Why does the Bethe lattice correspond to the spatial dimension d = ∞you might rightly ask! Well, in a hypercubic lattice, (a) the number of surface sites relative to the total number of sites approaches a constant when d →∞and (b) there are no closed loops when d →∞. The Bethe lattice has both these properties.
(a) Let g denote the generation, that is, the distance from a “centre site”. Note, however, that in an infinite Bethe lattice, all sites are equivalent, so the notion of a “centre site” is not to be taken literally. In the figure above, the first “ring” of three sites belong to generation g = 1, the second “ring” of six sites belongs to the second generation and so on. The total number of sites in a Bethe lattice consisting of g generations is Total no. sites = 1 + 3 · (1 + 2 + · · · + 2g−1) = 1 + 3 · 1 −2g 1 −2 = 3 · 2g −2, while the number of surface sites is 3 · 2g−1. Thus No. of surface sites Total no. of sites = 3 · 2g−1 3 · 2g −2 →1 2 for g →∞.
and the surface/volume tends to a constant.
Exercise 4 Show, that for a general Bethe lattice with coordination number z No. of surface sites Total no. of sites →z −2 z −1 for g →∞.
In a hypercubic lattice of linear size L, the surface is proportional to the volume only when d →∞: the surface in d dimensions is proportional to Ld−1 while the volume is proportional to Ld leaving Surface ∝Volume d−1 d = Volume1−1 d , that is, the surface is proportional to the volume if d →∞.
(b) There are no closed loops in a Bethe lattice. Starting from the “centre site” going outwards, one will never return to the starting point. In a hypercubic lattice, the chance (probability) of having a loop approaches zero as the dimension d →∞: As an example, let us place four particles in a chain in a hypercubic lattice with dimension d. When the first particle has been placed, there are 2d nearest neighbour sites, where the second particle can be placed.
However, for the third and fourth particle, there are only 2d −1 possible sites, implying a total no. of different chains 2d · (2d −1)2. Calculating the number of ways to place four particles in a loop, we arrive at 2d · (2d −2) · 1, that is, the probability of having a four loop No. loops Total no. chains = 2d · (2d −2) · 1 2d · (2d −1)2 = (2d −2) (2d −1)2 →0 for d →∞.
For d = ∞, there is no chance of having a closed loop, which is intuitively clear, isn’t it?
What is the critical occupation probability in a Bethe lattice, that is, at which occupation probability does an infinite cluster (path) appear for the first time in an infinite Bethe lattice?
Starting from a “centre site” and going outwards, we encounter (z −1) new neighbours. Thus, on average, we have p(z −1) new occupied sites on which we can continue the path. The critical occupation probability is determined by the equation pc(z −1) = 1 ⇔pc = 1 z −1.
11 Notice, that for z = 2, the 1d percolation problem, we recover pc = 1. For z = 3, e.g., pc = 1 2. Note that in a hypercubic lattice with d →∞one would expect pc →1/(2d −1) in order to be able to continue walking along a path of occupied sites.
Definition 11 The strength of the infinite cluster P(p) is the probability that an arbitrary site belongs to the infinite cluster.
Exercise 5 Discuss why, obviously, P(p) = 0 for p < pc in an infinite lattice.
In order to calculate P(p) for p > pc in the Bethe lattice, we introduce the quantity Q as the probability that an arbitrary site is NOT connected to infinity through a fixed branch originating at this site. Restricting ourselves to the lattice with z = 3 and using basic probability theory, we have that the strength P(p) of the infinite network, that is, the probability that an arbitraryly selected site is connected to infinity by occupied sites is P(p) = (Prob. site is occupied) × (Prob. at least ONE branch lead to infinity) = p(1 −Q3).
Next, we need to determine Q, the probability that a branch originating at that site does not lead to infinity. This depends on whether or not the neighbouring site is occupied. We find Q = (Prob. site is empty) + (Prob. site is occupied) × (Prob. no subbranch leads to infinity) = (1 −p) + pQ2.
We have relied on the fact that all sites in a Bethe lattice are equivalent, so Q is also the probability that a subbranch is not connected to infinity. The quadratic equation can easily be solved: Q = 1 ± p (2p −1)2 2p = ( 1 1−p p .
Exercise 6 Show that, in a general Bethe lattice where each site has z neighbours (Problem 4), Q = ( 1 1 − 2p(z−1)−2 p(z−1)(z−2).
When p < pc, there are no infinite clusters, by definition, so with probability 1 there is no connection to infinity. Thus Q = 1 is the trivial solution associated with P(p) = 0 for p < pc. The non-trivial solution Q = 1−p p is associated with p > pc, leaving P(p) = 0 for p < pc p 1 − 1−p p 3 for p ≥pc.
Using a Taylor expansion for P(p) around p = pc = 1 2, it can be shown that (see Problem 4) P(p) ∝(p −pc) for p →p+ c .
Definition 12 The critical exponent β is defined by P(p) ∝(p −pc)β for p →p+ c .
(1.13) 12 0.0 0.2 0.4 0.6 0.8 1.0 Occupation probability p 0.0 0.2 0.4 0.6 0.8 1.0 Strength P 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 p − pc 10 −4 10 −3 10 −2 10 −1 10 0 Strength P Analytic P(p) p − pc = 0.0025 P(p) = 6 (p − pc) Figure 1.8: The strength P(p) of the infinite cluster in a Bethe lattice with z = 3. When p →p+ c , the strength P(p) →6(p −pc) implying β = 1 in a Bethe lattice.
In the Bethe lattice, β = 1, see Fig. 1.8.
The strength P(p) measures how large a fraction of the sites in the lattice belongs to the infinite cluster (in an infinite lattice) and is known as the order parameter. The phenomenon that the order parameter (strength) becomes non-zero for p > pc is known as a phase transition and p = pc is often called a critical point or critical occupation probability.
Exercise 7 What is the strength P(p) in the 1d percolation problem?
What is the mean cluster size S(p) to which an occupied site (e.g. the “centre site”) belongs? Let T be the mean cluster size in one branch.
S(p) = average cluster size to which the origin belongs = (contribution from origin) + (contributions from the 3 branches) = 1 + 3T.
Thus, we have to determine T. Again, as all sites are equivalent in an infinite Bethe lattice, T will also denote the mean cluster size in the subbranch, so we can argue as follows: If the neighbour to the “centre site” is empty, probability (1 −p), there is no contribution to T from this branch. If the neighbour to the “centre site” is occupied (probability p), it contributes its own mass (unity) to the cluster, and adds the mass T for each of its z −1 = 2 subbranches. Thus, T = (1 −p) · 0 + p · (1 + 2T) ⇔T = p 1 −2p for p < pc.
Therefore, the mean cluster size S(p) is given by S(p) = 1 + 3T = 1 + p (1 −2p) = 1 + p 2(1 2 −p) = 1 + p 2(pc −p) = Γ2(pc −p)−1, (1.14) where Γ2 = 1+p 2 is the amplitude, and the critical exponent γ = 1 in the Bethe lattice. This is the exact result for a mean cluster of size S(p) and we notice that it diverges for p →pc.
An occupied site either belongs to the infinite cluster or a finite cluster. Thus, the generalisation of Eq.(1.5) is valid for all p P(p) + ∞ X s=1 sns(p) = p ∀p.
(1.15) 13 The sum runs over all finite s and excludes the infinite cluster. Of course, Eq.(1.15) reduces to Eq.(1.5) for p < pc since the strength P(p) is identical zero in this range.
Exercise 8 Reflect upon the identity Eq.(1.15) for p = pc. What is the sum P∞ s=1 sns(pc)? Is your answer also valid in d = 1?
Critical phenomena occur in a variety of systems. The Bethe lattice approximation for percolation theory is somewhat analogous to the mean-field approximation for magnetic systems dislaying a phase transition, or the Van der Waals equation for a fluid. In all cases, the order parameter becomes non-zero above (or below) the critical point. In a magnetic system, the order parameter is the spontaneous magnetisation per spin m, which is zero above the critical temperature (Curie temperature) Tc and non-zero for T < Tc as the system enters an ordered state. We will see later, that m ∝(Tc −T)β. In the case of a fluid, the order parameter is identified as the the difference between liquid and vapour density which becomes nonzero as T < Tc. The analogy between mean-field thermal critical phenomena and percolation is not complete since the critical exponent β for the order parameter is 1 2 for thermal phase transitions and unity for percolation, but nevertheless, the qualitative similarities are clear.
In both the thermal phase transitions mentioned above and percolation, the order parameter goes to zero continuously as one approaches the critical point. Such phase transitions are called continuous phase transitions or second-order phase transitions.
If instead the order parameter jumps to zero, one has a first order phase transition. Such transitions can occur in more complicated percolation theory situations, like “bootstrap percolation” where on a square lattice a site remains occupied only if three or four of its neighbours are still occupied .
1.5 Cluster Number Distribution Unfortunately, it is not possible to obtain an exact form for the cluster number ns(p) in d > 1 or the Bethe lattice since, unlike in 1d, there are a very large number of different ways in which clusters can arrange themselves. Even for relatively small cluster numbers in 2d we run into difficulties.
Exact enumeration of small cluster “animals” has been tabulated. In a 2d square lattice a computer has counted a total of 68, 557, 762, 666, 345, 165, 410, 168, 738 different cluster configurations with s = 46 . Nevertheless, we are still able to account for the general scaling behaviours of certain quantities that characterise the clusters.
Definition 13 The perimeter or external boundary t of a cluster is the number of empty nearest neighbours of the cluster.
Using the concept of a perimeter, we are able write down the general expression for the cluster number ns(p) as ns(p) = X t gs,tps(1 −p)t, (1.16) where we have introduced gs,t, the number of different lattice configurations with size s and perime-ter t.
In 1d, all clusters have two perimeter sites and thus gs,t = ( 1 for t = 2 0 otherwise implying ns(p) = ps(1 −p)2.
Exercise 9 Consider clusters of size s = 3 in 2d on a square lattice. Verify, that gs,t is only non-zero for t = 7 or t = 8 and that gs=3,t=7 = 4 and gs=3,t=8 = 2, implying ns=3(p) = 4p3(1 − p)7 + 2p3(1 −p)8.
14 In a Bethe lattice, however, we can show that there is a unique perimeter for a given cluster size s.
Exercise 10 Show that in a Bethe lattice t = 2 + s(z −2).
Now we apply this formula to our general result for cluster numbers (1.16) to get: ns(p) = X t gs,tps(1 −p)t = gs,2+s(z−2)ps(1 −p)2+s(z−2).
(1.17) For simplicity, from here on we shall work with the z = 3 case rather than the general Bethe lattice. For z = 3 we have pc = 1 2. As we have said, determining gs,2+s for large s is difficult, but we can avoid having to calculate gs,2+s by considering the ratio ns(p) ns(pc) = (1 −p) (1 −pc) 2 p pc (1 −p) (1 −pc) s = (1 −p) (1 −pc) 2 exp s ln p pc (1 −p) (1 −pc) .
(1.18) In order to proceed, define the function f(p) = p pc (1 −p) (1 −pc) = p 1 2 (1 −p) (1 −1 2) = 4p −4p2 for z = 3.
Exercise 11 Make a Taylor expansion of the function f(p) = 4p −4p2 around p = 1 2 = pc and verify f(p) = 1 −4(p −pc)2.
Substituting the expression for f(p) given in Exercise 11, we find ns(p) ns(pc) = (1 −p) (1 −pc) 2 exp(−s sξ ) (1.19) with sξ = −1 ln 1 −4 p −1 2 2 ∝ 1 4 p −1 2 −2 for p →1 2 ∝ |p −pc|−1 σ .
(1.20) In the Bethe lattice, σ = 1 2.
We now have a very simple exponential decay for the ratio of cluster numbers. This is to say that large clusters with s ≫sξ are very rare indeed. This explains why sξ is known as the cutoff cluster size. In order to investigate the cluster number distribution in more detail, we have to consider the term ns(pc) appearing in the denominator.
From the analysis on the average cluster size, we know that, S(p) = ∞ X s=1 s2ns(p) P∞ s=1 sns(p) →∞ for p →pc.
(1.21) However, the denominator remains finite at p = pc (why?) so the numerator must diverge for p = pc, that is, ∞ X s=1 s2ns(pc) = ∞.
15 Generally, we have that ∞ X s=1 sa = ( convergent for a < −1 divergent for a ≥−1.
Thus a power-law decay ns(pc) ∝s−τ ⇒ ∞ X s=1 s2ns(pc) = ∞ X s=1 s2−τ would imply a divergence of the average cluster size if the critical exponent τ ≤3.
However, see Eq.(1.15) and problem (8), we also need pc = P∞ s=1 sns(pc) ∝P∞ s=1 s1−τ < ∞ implying that τ > 2. Therefore, we arrive at a general form for the cluster number distribution in the Bethe lattice: ns(p) ∝s−τ exp(−s sξ ), for s ≫1 (1.22) with sξ ∝|pc −p|−1 σ for p →pc, (1.23) where σ = 1 2 and 2 < τ ≤3.
Actually, we can determine τ as it is not independent of the critical exponents γ and σ deter-mined previously. Let us evaluate the average cluster size S(p) assuming Eq.(1.22) is valid for all s. Neglecting the denominator which for p →pc will approach pc and thus be a constant we find S(p) ∝ ∞ X s=1 s2ns(p) ∝ ∞ X s=1 s2−τ exp(−s sξ ) ≈ Z ∞ 1 s2−τ exp(−s sξ ) ds = Z ∞ 1/sξ (zsξ)2−τ exp(−z)sξ dz with z = s/sξ = s3−τ ξ Z ∞ 1/sξ z2−τ exp(−z)dz = s3−τ ξ Z ∞ 0 z2−τ exp(−z)dz for p →pc where sξ →∞ = s3−τ ξ Γ(3 −τ) (1.24) where Γ is the Gamma function Γ(x) = Z ∞ 0 zx−1 exp(−z) dz.
Exercise 12 Show that the Gamma function Γ defined by the integral above satisfies the recursion relation Γ(x + 1) = xΓ(x) and deduce that Γ(x + 1) = x! for x = 0, 1, 2, . . . where 0! = 1.
Using sξ ∝(p −pc)−2, we see that S(p) ∝(p −pc)2τ−6 ∝(p −pc)−1, (1.25) where we have used Eq.(1.14). Thus we must require that 2τ −6 = −1 ⇔τ = 5 2.
(1.26) 16 In summary, for the Bethe lattice ns(p) ∝ s−5 2 exp(−s sξ ) s ≫1 sξ ∝ |p −pc|−2 for p →pc.
The above procedure is a general way of deriving a relationship between critical exponents, as we shall see shortly.
10 0 10 2 10 4 10 6 Cluster size s 10 −15 10 −10 10 −5 10 0 ns(p) p = 0.25 p = 0.425 p = 0.475 p = 0.49 p = 0.4975 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 s 4(pc − p) 2 = s/sξ 10 −15 10 −13 10 −11 10 −9 10 −7 10 −5 10 −3 10 −1 10 1 (1−pc) 2/(1−p) 2s τns(p) p = 0.25 p = 0.425 p = 0.475 p = 0.49 p = 0.4975 Figure 1.9: (a) The cluster number distribution ns(p) = h (1−p) (1−pc) i2s−τ exp(−s sξ ), see Eq.(1.19) for different values of p approaching pc = 1/(z −1) = 1/2. The vertical lines indicate the cutoffcluster size sξ(p) for the various p values. The cluster number distribution is well approximated by a power law for s ≪sξ while is decays rapidly for s ≫sξ. (b) By plotting h(1−pc) (1−p) i2sτns(p) versus s/sξ ≈s · 4(pc −p)2, all the data collapses onto the scaling function e−x.
Now, we postulate a scaling ansatz (form) valid for all p and large s Ansatz 1 ns(p) ∝ s−τ exp(−s sξ ), s ≫1, sξ ∝ |p −pc|−1 σ p →pc, (1.27) or equivalently ns(p) ∝s−τ exp(−s|p −pc| 1 σ ) s ≫1 and p →pc.
(1.28) Note the role of sξ as a cutoffcluster size, where ns(p) is characterised by ns(p) ∝ ( s−τ for s ≪sξ decays rapidly for s ≫sξ, (1.29) representing the crossover from a behaviour of “critical clusters” (power-law distributed) to that of non-critical clusters. Note also that ns(pc) = s−τ as sξ = ∞at p = pc.
We can, however, immediately identify two problems (a third problem will surface shortly) with the scaling ansatz 1.
17 (1) The scaling ansatz 1 fails to describe the 1d case. We found previously that ns(p) = (1 −p)2 exp(−s sξ ) = (pc −p)2 exp(−s sξ ), which is not a special case of our ansatz 1 since, instead of a power of s, we have (pc −p)2 in front of the exponential. You might be very tempted to say that τ = 0, but that does actually not solve the problem, as we shall see below.
(2) For fixed s ns(p) = ∞ X t=1 gs,tps(1 −p)t (1.30) is a polynomial in p with a finite number of terms. Thus all derivatives of ns(p) with respect to p will remain finite for all p. However, using the scaling ansatz 1 for p > pc ns(p) = As−τ exp(−s(p −pc) 1 σ ) we find, keeping s constant dns(p) dp = As−τ exp(−s(p −pc) 1 σ ) 1 σ (p −pc) 1 σ −1 (1.31) and by differentiating once more d2ns(p) dp2 = As−τ " exp(−s(p −pc) 1 σ ) 1 σ (p −pc) 1 σ −1 2 + exp(−s(p −pc) 1 σ ) 1 σ 1 σ −1 (p −pc) 1 σ −2 # .
In 2d, e.g., σ = 0.4 ⇔1 σ −2 = 0.5, so both the first and second derivative of ns(p) w.r.t. p remain finite as p →pc. The third derivative of ns(p), however, will contain a term with the factor (p −pc) 1 σ −3 which will diverge as p →pc contrary to the derivatives of Eq.(1.30).
We shall later resolve these two problems, so there is no need for you to worry too much at the moment.
Let us return to the question of deriving scaling relations assuming the scaling ansatz above for the cluster number distribution. Following the previous calculation of the average cluster size S(p) and noting that the Gamma function is just a number, we find S(p) ∝s3−τ ξ ∝|p −pc| τ−3 σ , for p →pc, (1.32) as sξ ∝|p −pc|−1 σ for p →pc.
(1.33) However, by definition S(p) ∝|p −pc|−γ for p →pc, (1.34) implying the scaling relation γ = 3 −τ σ .
(1.35) We notice again that to ensure the divergence of S(p) for p →pc we must have τ < 3 in order to leave γ > 0. Also, we recover problem (1) mentioned above because in 1d percolation theory γ = 1 and σ = 1 implying τ = 2. However, there is no s−τ term (yet!) in our expression for the 1d cluster number distribution ns(p) and obviously the suggested “solution” of assuming τ = 0 is not valid either.
18 Now we focus our attention on the percolation strength P(p), starting from Eq.(1.15): P(p) = p − ∞ X s=1 sns(p) = pc − ∞ X s=1 sns(p) + (p −pc) = ∞ X s=1 sns(pc) − ∞ X s=1 sns(p) + O(p −pc) (O(p −pc) = term of order(p −pc)) = ∞ X s=1 s1−τ − ∞ X s=1 s1−τ exp(−s sξ ) + O(p −pc) using the scaling ansatz 1 ∝ ∞ X s=1 s1−τ ( 1 −exp(−s sξ ) ) + O(p −pc).
∝ ∞ X s=1 s1−τ ( 1 −exp(−s sξ ) ) ∆s, (1.36) where we have dropped the term of order (p−pc) and introduced ∆s = 1. When p →pc, the cutoff cluster size sξ →∞. Thus the term {1−exp(−s sξ )} ≈0 for small s and the main contribution to the integral will come from large s of the order of sξ. To solve (1.36) we need to replace the summation by an integral.
Going from a discrete sum to a continuous sum does not, of course, create a mathematical identity, unless one takes certain limits. However, when the main contribution from the sum is for large s, it is legitimate to approximate the sum by an integral P ∆s ≈ R ds, as we are only interested in the the general scaling behaviour of the system. Thus P(p) ∝ Z ∞ 1 s1−τ ( 1 −exp(−s sξ ) ) ds.
(1.37) One might proceed by integration by parts. You will then discover that only one of the terms are non-zero. However, we will take another approach. We differentiate Eq.(1.37) w.r.t. 1/sξ to find dP(p) d( 1 sξ ) ∝ Z ∞ 1 s2−τ exp(−s sξ ) ds ∝ s3−τ ξ Z ∞ 1/sξ z2−τ exp(−z) dz substituting z = s/sξ.
As we are ultimately interested in the limit p →pc where the cutoffcluster size diverges, we replace the lower limit 1/sξ with zero and thus P(p) ∝ 1 sξ !τ−3 (1.38) so integrating we find P(p) = c1 + c2 1 sξ !τ−2 (1.39) At p = pc, sξ = ∞. But, P(pc) = 0 implies that c1 = 0. So, P(p) ∝ c2 1 sξ !τ−2 , ∝ |p −pc| τ−2 σ .
(1.40) 19 Hence, we have derived a new scaling relation β = τ −2 σ .
(1.41) By definition P(p) = ( 0 p ≤pc (p −pc)β p →p+ c .
(1.42) Now we find a third problem with our ansatz 1.
(3) In the derivation above we did not assume p > pc. Thus the ansatz 1 also predicts an infinite cluster for p < pc, which is of course not correct.
Reflecting upon the three problems, we realise that they can all be resolved, as we shall see shortly, by avoiding the argument s|p−pc| 1 σ in the scaling ansatz. Thus let us assume the following general scaling form Ansatz 2 ns(p) = q0s−τf[q1(p −pc)sσ] s ≫1 and p →pc.
(1.43) The scaling ansatz 2 has the following properties: • q0 and q1 are proportionality factors and pc the critical occupation probability.
They all depend on the lattice details. They are non-universal.
• τ, σ are the critical scaling indices which are independent of p, pc and the lattice structure.
They are universal.
• f, the scaling function, is independent of p, pc and detailed lattice structure. It is univer-sal. The precise form of the scaling function f = f(z) has to be determined by computer simulations or other numerical methods since the function f is not predicted by the scaling assumption. Typically, however f(z) = ( constant |z| ≪1 ⇔|p −pc|sσ ≪1 ⇔s ≪|p −pc|−1 σ ∝sξ decays fast |z| ≫1 ⇔s ≫sξ.
(1.44) that is, the scaling function f(z) reaches a constant value (except in 1d percolation theory) for |z| ≪1 ⇔s ≪sξ, and decays rather fast for |z| ≫1 ⇔s ≫sξ. Thus, the role of sξ as a cutoffand crossover size is maintained. Later, we shall look more carefully at the crossover properties.
• τ, σ and f depend on dimensionality.
• By plotting 1 q0sτns(p) versus (p−pc)q1sσ, all the data will fall on the universal curve outlining the scaling function f, see Fig. 1.9. This phenomenon is known as data collapse.
Problems 7 and 8 highlights some of the points above.
Let us check whether the general scaling ansatz 2 really resolves the three problems associated with the former scaling ansatz 1.
20 (1) The 1d cluster number distribution ns(p) = (pc −p)2 exp(−s sξ ) ≈ (pc −p)2 exp(−s(pc −p)) for p →pc = s−2(s(pc −p))2 exp(−s(pc −p)) = s−2f[sσ(pc −p)] (1.45) which is of the form of the general scaling ansatz 2 with τ = 2 and σ = 1 and f(x) = x2 exp(−x). Thus the scaling relation Eq.(1.35) is indeed fulfilled. Note, however, that unlike in higher dimensions, the scaling function increases like x2 for x ≪1 but that it indeed decays rapidly for x ≫1, see Fig. (1.5).
(2) Keeping s constant we find ∂kns(p) ∂pk = q0s−τf (k)[q1(p −pc)sσ]qk 1sσk which is not divergent for p →pc assuming that f is analytic, that is, all derivatives of f(z) with respect to z are finite everywhere, also at z = 0.
(3) Let us calculate the strength of the infinite cluster.
−P(p) = ∞ X s=1 sns(p) −p = ∞ X s=1 sns(p) −pc + (pc −p) = ∞ X s=1 [sns(p) −sns(pc)] + O(p −pc) using P sns(pc) = pc ≈ Z ∞ 1 q0s1−τ (f[q1(p −pc)sσ] −f) ds Now substitute z = q1(p −pc)sσ. Notice that if p > pc the argument z is positive while for p < pc the argument is negative.
The cluster size s, however, is always positive so s = |z| 1 σ (q1|p −pc|)−1 σ implying that ds = 1 σ|z| 1 σ −1(q1|p −pc|)−1 σ dz with dz positive. Thus we find −P(p) = Z q0|z| 1−τ σ (q1|p −pc|) τ−1 σ [f(z) −f(0)] 1 σ|z| 1 σ −1(q1|p −pc|)−1 σ dz = q0(q1|p −pc|) τ−2 σ Z |z| 2−τ σ −1[f(z) −f(0)] 1 σ dz.
(1.46) Now it is important to realise that the limits of the integral will depend on the value of p since z = q1(p−pc)sσ. If p > pc, the upper limit will be ∞(lower limit p−pc approaches 0 as p →p+ c ) while for p < pc, the lower limit will be −∞(upper limit approaching 0 as p →p− c ).
Thus if Z |z| 2−τ σ −1[f(z) −f(0)] 1 σ dz = ( constant ̸= 0 for p > pc 0 for p < pc (1.47) we have solved the third problem as well, implying P(p) ∝ ( (p −pc)β for p →p+ c 0 for p < pc 21 with β = τ −2 σ .
Exercise 13 Discuss the qualitative behaviour of the scaling function f(z) as a function of z = q1(p −pc)sσ in order to satisfy the constraint given in Eq.(1.47).
Problem 8 will ask you to calculate the average cluster size S(p) assuming the general scaling ansatz 2.
1.6 Cluster Structure Till now we have only considered the distribution of cluster sizes and have come up with some useful scaling laws and scaling relations. Now we turn our attention to the geometry of the clusters. We shall do so by mainly studying the fractal geometry, which contains information about the density of the clusters on different length scales.
1.6.1 Cluster Radius and Fractal Dimension Let ri denote the position of the ith occupied site in a cluster of size s.
Definition 14 The centre of mass of an s-cluster rcm = 1 s Ps i=1 ri.
Definition 15 The radius of gyration Rs of a given s-cluster is defined by R2 s = average square distance to the centre of mass = ⟨|ri −rcm|2⟩ = 1 s s X i=1 |ri −rcm|2 We want to prove that R2 s = 1 2 average square distance between two cluster sites = 1 2 1 s2 X ij |ri −rj|2.
(1.48) By definition, the lefthand side of Eq.(1.48), R2 s = 1 s s X i=1 |ri −rcm|2 = 1 s s X i=1 (ri · ri + rcm · rcm −2 ri · rcm) = 1 s s X i=1 ri · ri + 1 s s X i=1 rcm · rcm −2 s s X i=1 ri · rcm = 1 s s X i=1 ri · ri + rcm · rcm −2 (1 s s X i=1 ri) · rcm = 1 s s X i=1 ri · ri −rcm · rcm.
22 Let us rewrite the righthand side of Eq.(1.48) R2 s = 1 2 1 s2 X ij |ri −rj|2 = 1 2 1 s2 s X i=1 s X j=1 (ri · ri + rj · rj −2 ri · rj) = 1 2 1 s2 s X i=1 s ri · ri + s X j=1 s rj · rj −2 s X i=1 ri !
· s X j=1 rj = 1 s s X i=1 ri · ri −1 s2 (s rcm) · (s rcm) = 1 s s X i=1 ri · ri −rcm · rcm.
The radius of gyration is a more useful quantity to work with since in many situations (like in Polymer Science) we have to deal with more complicated structures than straight lines, squares or spheres.
We also remember that the correlation function g(r) is defined as the probability that a site at a position r from an occupied site belongs to the same finite cluster. We have the identity X r g(r) = S(p) (1.49) because the sum is the average number of sites to which an occupied site is connected, see Problem 2.
Definition 16 The correlation length ξ is defined as ξ2 = P r r2g(r) P r g(r) .
(1.50) The correlation length represents some average distance of two sites belonging to the same cluster.
For a given s cluster, 2R2 s is the average squared distance between two cluster sites.
The probability of a site being part of an s cluster is sns(p), and it will be connected to s sites (if we include self-connection). The corresponding average over 2R2 s gives the squared correlation length ξ2 = P s 2R2 ss2ns(p) P s s2ns(p) .
(1.51) The correlation length ξ is the upper cutoffof the radius of those clusters which contribute to the mean cluster size near the percolation threshold. It is expected that ξ diverges as p →pc, like ξ ∝|p −pc|−ν.
(1.52) We have introduced a new critical exponent, ν, and we wish to determine how it is related to τ and σ. For 2d percolation, plausible but not rigorous arguments give ν = 4 3, in excellent agreement with numerical results. In 3d, ν is somewhat smaller than 0.9, whilst for Bethe lattices one has ν = 1 2, analogous to the mean-field theories for thermal phase transitions.
As we have already seen, many quantities diverge at the percolation threshold. Most of these involve sums over all cluster sizes s; their main contribution comes from s of the order sξ ∝ |p−pc|−1 σ . Now we see that the correlation length ξ, which is also one of these quantities, is simply the radius of those clusters which contribute the most to divergence. This effect is the foundation of scaling theory. There is one and only one length ξ dominating the critical behaviour in an infinite lattice.
23 1.6.2 Finite Boxing of Percolating Clusters We now want to find out how the Rs varies with s at the percolation threshold p = pc where ξ = ∞.
Let M(L) denote the mass of the percolating cluster within linear distance L. If the percolating cluster was a compact object, then it would be of the form M(L) ∝L2. However, at p = pc, the percolating cluster is a fractal object. Figure 1.10 displays the size of the largest cluster S∞ Figure 1.10: The size of the largest cluster S∞at p = pc as a function of the lattice size L.
(the percolating cluster) as a function of lattice size L in a double-logarithmic plot. The data are consistent with the existence of a fractal dimension D such that M(L) ∝LD.
(1.53) For the 2d case D = 91 48 < 2 and in 1d, D = 1.
Having obtained the relationship in (1.53) it is natural to assume s ∝RD s for s ≫1 at p = pc.
(1.54) Thus, Rs ∝s 1 D for p →pc.
(1.55) Inserting Eq.(1.55) into Eq.(1.51) we have ξ2 ∝ P s 2s2+ 2 D ns(p) P s s2ns(p) ∝ |p −pc|τ−3−(2) Dσ |p −pc|τ−1−2 σ ∝ |p −pc|− 2 Dσ ∝ |p −pc|−2ν, (1.56) where we have used ∞ X s=1 skns(p) ∝|p −pc| τ−1−k σ .
24 Therefore we end up with the scaling relation ν = 1 Dσ ⇔D = 1 νσ.
(1.57) Note that the behaviour at p = pc and the behaviour for s ≪sξ, or Rs ≪ξ are indistinguishable.
Therefore, ξ is the crossover length between a critical and noncritical behaviour. On length scales much less than ξ we have ns ∝s−τ and Rs ∝s 1 D but for length scales much larger than ξ the scaleless behaviour disappears. Note also, that sξ ∝ξD ∝|p −pc|−νD ∝|p −pc|−1 σ confirming once again the scaling relation above between D, ν and σ.
If, however, p ≫pc, the fractal dimension D is expected to be equal to the Eucledian dimension d of the lattice. So, D = d for p > pc.
(1.58) Clusters above pc are not fractals but ‘normal’ objects with D = d, provided s ≫sξ or equivalently Rs ≫ξ. Thus sξ or ξ sets the cluster size and linear scale where there is a crossover from fractal to non-fractal behaviour.
Exponent 1d 2d 3d 4d 5d 6d Bethe α 1 −2/3 -0.62 -0.72 -0.86 -1 -1 β 0 5/36 0.41 0.64 0.84 1 1 γ 1 43/18 1.80 1.44 1.18 1 1 ν 1 4/3 0.88 0.68 0.57 1/2 1/2 σ 1 36/91 0.45 0.48 0.49 1/2 1/2 τ 2 187/91 2.18 2.31 2.41 5/2 5/2 D(p = pc) 1 91/48 2.53 3.06 3.54 4 4 Table 1.2: The critical exponents for the percolation theory problems in dimensions d = 1, 2, 3, 4, 5, 6 and in the Bethe lattice.
1.6.3 Mass of the Percolating Cluster Now we wish to see how the scaling relation of the mass M(L) changes for the two different cases.
(1) For L ≪ξ: (always the case at p = pc, where ξ = ∞) The percolating cluster appears fractal, implying M(L) ∝LD.
(2) For L ≫ξ: The cluster appear homogeneous M(L) = (No. of lattice sites) × (Prob. site belongs to percolating cluster) = LdP(p) = Ld(p −pc)β = Ldξ−β ν since ξ ∝|p −pc|−ν.
(1.59) Now consider L ≈ξ, that is, we match the observations above by substituting L with ξ: M(L) ∝LD ∝ξD, (1.60) 25 and M(L) ∝Ldξ−β ν ∝ξd−β ν .
(1.61) Therefore, we can state that D = d −β ν , (1.62) which is know as a hyperscaling relation because the Euclidean dimension d enters in the scaling relation.
Since the percolating cluster has a constant density for L ≫ξ, it is natural to divide the system into boxes of linear size ξ. In d dimensions, the total volume Ld will be divided into L ξ d boxes.
Since the cluster inside each of these boxes of size ξd has a mass of order ξD, the total mass of the cluster is given by M(L, ξ) = L ξ d ξD = ξD−dLd, (1.63) which is, of course, equivalent to P(p)Ld.
In summary, we have M(L, ξ) ∝ LD L ≪ξ ξD L ξ d L ≫ξ (1.64) which can be written in an alternative form of M(L, ξ) = LDm(L ξ ), (1.65) where the scaling function m(x) = ( constant x = L ξ ≪1 xd−D x = L ξ ≫1.
(1.66) The argument x of the scaling function is a dimensionless number, namely the ratio of the two length scales L ξ . No other length scales play a role. The scaling function describes a crossover from fractal behaviour at length scales L much smaller than the correlation length ξ to uniform behaviour at length scales L much larger than the correlation length ξ. In Nature, there are many examples of crossover phenomena from one type of behaviour to another. Take as an example the table in front of you. On large length scales, the surface of the table is smooth, but going to very small length scales, the surface will become very rough and probably fractal. The mountain range is yet another example. Far away, i.e., on large length scales, it looks pretty smooth, but, as you are well aware, if you are in the middle of the mountain range, on small length scales, it is pretty rough and rugged.
Exercise 14 Give one or two more examples of such crossover phenomena in Nature.
The density has the form ρ(L, ξ) = M(L, ξ) Ld ∝ ( LD−d L ≪ξ ξD−d L ≫ξ = LD−d˜ ρ(L ξ ), (1.67) with a scaling function ˜ ρ(x) = ( constant x = L ξ ≪1 xd−D x = L ξ ≫1.
(1.68) To be aware that the density actually decreases with the length scale when L < ξ can be of immense practical importance. Imagine that you have just been hired by an oil company and you were given a sample of dimensions 0.1×0.1×0.1 m3 of the porous medium where the oil resides and 26 Figure 1.11: The density of sites belonging to the percolating cluster within a region of linear size L. In region I, the ratio L/ξ ≪1 and in region II, L/ξ ≫1. Region III is related to the fact, that the lattices used are finite.
had to estimate how much oil the company could expect to recover in the oil field of dimension, say, 100 × 100 × 100 km3. The percolation model is used as a model for the distribution of oil and gas inside porous rock in oil reservoirs. The empty sites model hard rock while the occupied sites are pores filled with oil. The average concentration of oil is p. Now, if p < pc there are only finite clusters, so let us assume p > pc. The strength P(p) will be the probability of drilling into a percolating cluster, but how much oil can you recover? You would head for an immediate dismissal if you argued as follows: Measure the density of oil in the sample ρ(L1 = 0.1m). The mass of oil to be recovered is M = ρ(L1 = 0.1m) · (100000m)3. The argument is wrong simply because the density of oil at length scales L2 = 100 km is not given by the density of oil at length scales L1 = 0.1 m, see Fig. (1.11). The proper way to argue would be the following: measure the density of oil in the sample ρ(L1 = 0.1m) = CLD−d 1 . The density of oil at the reservoir length scale ρ(L2 = 100km) = CLD−d 2 with the same constant C. Taking the ratio we find ρ(L2 = 100km) = L2 L1 D−d ρ(L1 = 0.1m) = 106D−d ρ(L1 = 0.1m) ≈ 1.5 · 10−3ρ(L1 = 0.1m), since in 3d D −d ≈2.53−3 = −0.47, see table 1.2. The former estimate is wrong by a factor 1000.
1.7 Fractals The percolating cluster at p = pc is an example of a random fractal. When p ̸= pc large clusters appear fractal on length scales up to the correlation length ξ ∝|p −pc|−ν, roughly speaking. In mathematics, we can also encounter deterministic fractals as for example the Sierpinski carpet and the Sierpinski gasket.
For a geometrical fractal, the mass scales with the linear size raised to the fractal dimension M(L) = LD ⇔D = log M(L) log L .
27 Figure 1.12: Two examples of deterministic fractals. (a) The Sierpinski carpet. The first three levels in the algorithm for constructing the Sierpinski carpet are shown. The linear scale is enlarged by a factor 3 with the condition that an occupied square (white) is replaced with with 3 × 3 squares in which the centre square is empty (grey) while an empty square (grey) is replaced with 3 × 3 empty squares. (b) The Sierpinski gasket. A similar algorithm generates the Sierpinski gasket.
Let n denote the iteration number. By inspection of the Sierpinski carpet, M = 8n and L = 3n, implying D = log 8n log 3n = log 8 log 3 ≈1.893.
In the case of the Sierpinski gasket, M = 3n and L = 2n, yielding D = log 3 log 2 ≈1.585.
Note that if the length scale is rescaled by a factor b, then M(L b ) = L b D = LD bD = M(L) bD so that the mass is reduced by a factor bD. Using this observation as the characteristic of fractal behaviour for random fractals we can easily find, say, the scaling of the mass of the percolating cluster M(L, ξ) as a function of L and ξ. Let us rescale the length scales by a factor b, that is, L →L b and ξ →ξ b (no other length scales are present). Thus M(L, ξ) = bDM(L b , ξ b) = bDbDM( L b2 , ξ b2 ) .
.
.
= bDlM(L bl , ξ bl ).
(1.69) 28 If we are at the critical point p = pc, ξ ∝|p−pc|−ν = ∞, but then ξ bl = ∞∀l. If we stop the above renormalisation procedure when bl = L we find M(L, ξ = ∞) = blD M(L bl , ξ bl = ∞) = LDM(1, ∞) = LD.
(1.70) This proves that self-similarity is mathematically expressed by power-law behaviour. Now, assume p > pc with ξ ∝|p −pc|−ν < ∞with M(L, ξ) = bDlM(L bl , ξ bl ).
We now have to consider two cases depending on the ratio L/ξ.
(a) If L/ξ ≪1 ⇔L ≪ξ (which is allways the case at p = pc) we stop the iteration process when we reach the smallest length scale bl = L and we “recover” M(L, ξ) = LDM(1, ξ L) = LDM(1, ∞) ∝LD.
(b) If L/ξ ≫1 ⇔L ≫ξ, we stop the iteration process when bl = ξ and we find M(L, ξ) = ξDM(L ξ , 1) ∝ξD L ξ d = ξD−dLd, since for a uniform system M( L ξ , 1) ∝ L ξ d.
Exercise 15 Discuss why the system looks uniform when ξ/bl = 1.
1.8 Finite-size scaling Let us consider the percolation problem for p ̸= pc where the correlation length is finite ξ ∝|p −pc|−ν ⇒|p −pc| ∝ξ−1 ν .
The strength of the infinite cluster for p > pc can then be expressed in terms of the finite correlation length P(p) ∝(p −pc)β ∝ξ−β ν and similarly for the average cluster size S(p) ∝|p −pc|−γ ∝ξ γ ν .
These are the results in infinite lattices with L = ∞. Now, what happens at finite lattice sizes when L < ∞? Well, nothing happens as long as L ≫ξ. The only relevant length scale is the 29 correlation length ξ. However, when L ≪ξ, the length scale will be set by L. This we have already encountered before, we just didn’t stress that point. Better late that never: P(L, ξ) = ρ(L, ξ) = M(L, ξ) Ld = ( LD−d L ≪ξ ξD−d L ≫ξ.
= ( L−β ν L ≪ξ ξ−β ν L ≫ξ.
= ξ−β ν f(L ξ ) where the scaling function f(L ξ ) = L ξ −β ν L ξ ≪1 constant L ξ ≫1.
0.0 0.2 0.4 0.6 0.8 1.0 Occupation probability P 0.0 0.2 0.4 0.6 0.8 1.0 Strength P L = 25 L = 50 L = 100 L = 200 Indicates pc 0.0 0.2 0.4 0.6 0.8 1.0 Occupation probability p 0 10 20 Average cluster size S L = 25 L = 50 L = 100 L = 200 Indicates pc Figure 1.13: (a) The strength P(p) measured in a 2d square lattice for finite system sizes as a function of occupation probability p. When p is far away from pc, L ≫ξ and the strength is determined by ξ−β ν . However, with p closer to pc, the finite-size effect is seen and P(p) decays with system size as L−β ν . (b) The average cluster size S(p) measured in a 2d square lattice for finite system sizes L as a function of occupation probability p. When p is far away from pc, L ≫ξ and the average cluster size is determined by the correlation length ξ γ ν . However, with p closer to pc, the finite-size effect is seen and S(p) increases with system size as L γ ν .
Similarly for the average cluster size, we would expect S(L, ξ) ∝ ( L γ ν L ≪ξ ξ γ ν L ≫ξ.
Generally, if we have a quantity X ∝|p −pc|−χ ∝ξ χ ν for L ≫ξ 30 then we would expect X(L, ξ) ∝ ( L χ ν L ≪ξ ξ χ ν L ≫ξ.
= ξ χ ν x1(L ξ ) (1.71) or, in terms of occupation probabilities X(L, p) = |p −pc|−χx2(L 1 ν (p −pc)).
The important message we get is that by studying finite-size scaling, that is, studying X as a function of (finite) system sizes L at p = pc, we can extract the exponent χ ν .
1.9 Real space renormalisation in percolation theory Imagine a percolation lattice with n sites. The total number of different microscopic configurations is 2n as a site can be in one of two states – occupied with probability p or empty with probability 1 −p. Table (1.3) shows the number of microscopic configurations in a two dimensional square lattice of linear size L.
L 1 2 3 4 5 6 7 8 9 10 17 100 n 1 4 9 16 25 36 49 64 81 100 289 10000 2n 2 16 512 65536 3 · 107 7 · 1010 6 · 1014 2 · 1019 2 · 1024 1030 1087 103000 Table 1.3: The number of microscopic configurations 2n as a function the linear size L of a two-dimensional square lattice where n = L2, the number of sites in the lattice. For lattice size L = 17, the number exceeds the estimated number of atoms in the universe ≈1081.
In principle, if we wanted to calculate the average of a quantity ⟨A⟩(say the average cluster size), we should evaluate the quantity Ai associated with each configuration i and then weight the quantity Ai with the probability pi (which would be of the form pm(1 −p)(n−m), where m is the number of occupied sites in that particulate configuration) of the system to be in that microscopic state, that is, ⟨A⟩= P i piAi P i pi = X i piAi.
(1.72) This procedure, however, soon becomes an impossible task due to the numbers of different states involved. With today’s computing power one might do, say, a 7 × 7 system, but an 8 × 8 system would take approximately 40 years on a tera flop (1012 calculations per second) while the time required to do a 10 × 10 system would exceed the age of the universe which is approximately 15 · 109 years.
Real space renormalisation is a coarse graining procedure where we systematically reduce the number of degrees of freedom (sites) in order to break a large problem down into a sequence of smaller and more manageable stages by eliminating fluctuations on scales less than a given length scale b. The effect will be to reveal the large scale behaviour of the system.
Before we introduce the real space renormalisation method, let us make some observations related to the phase transition in a percolation model.
(a) The empty state at occupation probability p = 0 is self-similar with an associated correlation length ξ = 0.
31 (b) The fully occupied state at occupation probability p = 1 is self-similar. The associated corre-lation length ξ = 0. The ordered state has an order parameter P(p = 1) = 1.
(c) The states at the critical occupation probability p = pc are also self-similar and the correlation length ξ = ∞. The infinite cluster is fractal and looks alike on all length scales and we have clusters of all sizes from s = 1 to s = ∞.
Loosely speaking, the correlation length ξ sets the scale of the largest fluctuation from the “averaged” state. For p ̸= pc, the correlation length is identified with the linear size of the cutoff cluster size sξ.
When p < pc the upper cutoffof the linear size of the fluctuations away from the empty state is given by ξ and when p approaches zero, ξ approaches zero. At p = 0, there are no clusters left at all and ξ = 0.
When p > pc, the correlation length can also be identified with the upper cutoffin the linear size of the largest holes in the percolating (spanning) cluster which indicates how far away you are from the fully occupied state. When p approaches one, ξ approaches zero. At p = 1, there are no holes left and ξ = 0.
At the phase transition point p = pc, the correlation length is infinite. Thus there are fluctua-tions from the smallest length scale up to infinity.
The real space renormalisation technique is based on the so-called block site (spin) technique in-troduced by Leo Kadanoffand later formalised into the renormalisation group method by Kenneth Wilson, who received the Nobel price in 1982 for “his theory for critical phenomena in connection with phase transitions”, see, for example, the website on physics laureates The renormalisation group method has three basic steps: 1. Divide the lattice into blocks of linear size b (in terms of the lattice constant) with each block containing a few sites (spin).
2. Next, the coarse graining procedure takes place. The sites in the blocks are averaged in some way (to be specified more precisely shortly) and the entire block is replaced by a single su-per site (spin) which is occupied with a probability according to the renormalisation group transformation p′ = Rb(p).
In the combined procedure 1 and 2, one should keep the symmetry of the original lattice such that we can repeat the coarse graining procedure again. The result of these two operations are to create a new lattice whose fundamental spacing is b times as large as the original lattice.
3. Restore original lattice constant by rescaling the length scales by the factor b.
These 3 steps define a renormalisation group transformation Rb, where the number of degrees of freedom is reduced drastically. Note that it is called a group because of the property that when applying the transformation to a configuration {si}, Rb2(Rb1({si})) = Rb1b2({si}) (1.73) but it is of course not a group in a strictly mathematical sense because no inverse transformation exists since we are reducing the number of degree of freedom.
The effect of the coarse graining procedure in step 2 is to eliminate from the system all fluctu-ations whose scale is smaller than the block size b. Any small scale fluctuations of the sites over a range of less than b lattice units will be smeared out. This is somewhat similar to the process taking place if you view the system through a lens out of focus – all the smaller features are blurred but the large scale features are unaffected.
Let us clarify the 3 steps by looking at a simple example. Consider a 2d square lattice with 9 × 9 sites, that is, 81 degrees of freedom and a total of 281 = 2 · 1024 different configurations.
32 Figure 1.14: The three renormalisation group steps illustrated in a 2d square lattice with b = 3.
The majority rule defines the renormalisation group transformation.
1. Divide the square lattice up into small 3 × 3 blocks with a total of 9 sites (b = 3.) 2. We now have to specify the course graining procedure which will define the renormalisation group transformation. One possibility would be to use the so-called majority rule which states that if a majority of the sites in the block is occupied, then the super site should be occupied, otherwise empty. The probability, that a majority of sites are occupied in the 3× 3 block then defines the renormalisation group transformation p′ = Rb(p) = p9 + K9,1p8(1 −p) + K9,2p7(1 −p)2 + K9,3p6(1 −p)3 + K9,4p5(1 −p)4, where K9,k is the number of different ways to place k empty sites into the cells of 9 sites.
The super site should be occupied with probability p′ = Rb(p) and empty with probability 1 −p′ = 1 −Rb(p).
3. Rescale the lattice of super sites by a factor 3 to restore original lattice constant.
Let ξ denote the correlation length in the original lattice. Assume we are close to pc. Then ξ = constant |p −pc|−ν (1.74) We denote the correlation length in the new renormalised lattice by ξ′. Since the length scales have been rescaled by a factor b (step 3) ξ′ = ξ b ⇔ (1.75) constant |Rb(p) −pc|−ν = constant |p −pc|−ν b where Rb(p) is the renormalisation group transformation that determines the new occupation prob-ability in the rescaled lattice associated with the new (smaller as b > 1) correlation length.
Simple rearranging implies that |Rb(p) −pc| |p −pc| −ν = 1 b (1.76) so the critical exponent ν is given by ν = log b log |Rb(p)−pc| |p−pc| .
(1.77) The basic idea is that at the critical point pc we have self-similarity, that is, ξ′ = ξ.
(1.78) 33 0.0 0.2 0.4 0.6 0.8 1.0 Occupation probability p 0 2000 Corraletion length ξ Figure 1.15: When renormalising the length scales by a factor b, the correlation length decreases unless it was 0 or ∞initially. Associated with a decrease in the correlation length is a flow in occupation probability space either towards p = 0, if initially p < pc or towards p = 1, if initially p > pc. The fixed points of the renormalisation group transformation are associated with the trivial self-similar states at p = 0 or p = 1 or the nontrivial self-similar states at p = pc.
This equation can only be consistent with Eq.(1.76) if ξ = ( ∞ 0.
(1.79) This is intuitively clear as ξ = 0 is associated with either p = 0 (empty lattice) or p = 1 (fully occu-pied lattice) the two trivially self-similar cases and ξ = ∞with the critical point pc. Furthermore, we can also conclude that ξ = constant |p −pc|−ν = ξ′ = constant |Rb(p) −pc|−ν (1.80) which implies that the phase transition ξ = ∞is identified with (one of) the fixed points p⋆of the renormalisation group transformation, that is, the solutions of the equation Rb(p⋆) = p⋆.
(1.81) Thus |Rb(p) −pc| |p −pc| = |Rb(p) −Rb(pc)| |p −pc| = dRb(pc) dp for p →pc (1.82) leaving, see Eq.(1.77) ν = log b log dRb(pc) dp (1.83) 1.9.1 Renormalisation group transformation in 1d.
Consider a renormalisation group transformation in 1d. Take a 1d lattice where each site is occupied with probability p.
Divide the lattice into blocks with b sites.
Let the renormalisation group 34 transformation be determined by the probability of having a spanning cluster. The probability of having a spanning cluster in a block of b sites is Rb(p) = pb.
The fixed point equation can easily be solved Rb(p⋆) = p⋆b = p⋆⇔p⋆= ( 0 1.
If we start with an empty lattice p = 0, the renormalised lattice will also be empty, associated with the fixed point p⋆= 0. If we start with a fully occupied lattice p = 1, the renormalised lattice will also be fully occupied since all the blocks contain a percolating cluster, associated with the fixed point p⋆= 1. However, if we start with a lattice containing some empty sites p < 1, the renormalised lattice will contain even more empty sites because p′ = Rb(p) = pb < p. Repeating the renormalisation procedure will gradually bring the renormalised occupation probability towards zero. Clearly, p⋆= 0 is associated with ξ = 0, while p⋆= 1 is associated with ξ = ∞and must thus be identified with the critical occupation probability pc.
Furthermore, in order to calculate the correlation length exponent ν we need to take the deriva-tive of the renormalisation group transformation evaluated at p⋆= 1.
dRb(p) dp |p⋆=1 = bpb−1|p⋆=1 = b, implying ν = log b log dRb(p) dp |p⋆=1 = 1, so in 1d the renormalisation group transformation is exact.
1.9.2 Renormalisation group transformation on 2d triangular lattice.
Divide the lattice into triangular cells containing three sites each. Let the probability for having a spanning cluster define the renormalisation group transformation. Thus Rb(p) = p3 + 3p2(1 −p) = 3p2 −2p3.
(1.84) The fixed point equation Rb(p⋆) = 3p⋆2 −2p⋆3 = p⋆⇔p⋆= 0 1 1/2.
The two trivial fixed point is associated with the self-similar states of an empty lattice and fully occupied lattice. The unstable fixed point p⋆= 1/2 will be associated with the nontrivial self-similar states at pc. The critical exponent ν is ν = log b log dRb(p) dp |p⋆=1/2 = log √ 3 log(6p −6p2)|p⋆=1/2 = log √ 3 log 3/2 = 1.355.
The exact values are pc = 1/2 and ν = 4/3 so the renormalisation group transformation does a good job.
35 Figure 1.16: Renormalisation group method on a 2d square lattice according to the majority rule for 3 × 3 blocks. Shown are part of the original lattice L = 729 and the renormalised lattices L = 243 (part of), L = 81, 27, and 3. (a) Initial occupation probability p < p⋆in the original lattice and the flow is towards the empty lattice. (b) Initial occupation probability p > p⋆in the original lattice and the flow is towards the fully occupied lattice. (c) Initial occupation probability p = p⋆in the original lattice. This is a fixed point for the renormalisation group transformation and consequently there is no flow. It looks like itself on all length scales.
1.9.3 Renormalisation group transformation on 2d square lattice of bond per-colation.
Consider the block defined in Fig. (1.17) which, after the coarse graining procedure, is replaced by two superbonds. The renormalisation group transformation is p′ = Rb(p) = probability to have a spanning cluster in horizontal direction = p5 + p4(1 −p) + 4p4(1 −p) + 2p3(1 −p)2 + 2p3(1 −p)2 + 4p3(1 −p)2 + 2p2(1 −p)3 = 2p5 −5p4 + 2p3 + 2p2 (1.85) with the fixed point equation Rb(p⋆) = 2p⋆5 −5p⋆4 + 2p⋆3 + 2p⋆2 = p⋆⇔p⋆= 0 1 1/2.
36 Figure 1.17: Renormalisation group method of 2d square lattice with bond percolation according to the spanning cluster rule. (a) and (b) After the coarse graining procedure, the block is replaced with two super bonds AC and AG. (c) The various configurations with a horizontal spanning cluster.
The two dangling bonds DG and EH need not be considered as they do not affect the probability of having a spanning cluster from left to right.
Again, the two trivial fixed points are associated with the self-similar states of an empty lattice and fully occupied lattice. The unstable fixed point p⋆= 1/2 will be associated with the nontrivial self-similar states at pc. For bond percolation in 2d, pc = 1/2, see Table 1.1. The critical exponent ν is ν = log b log dRb(p) dp |p⋆=1/2 = log 2 log 13/8 = 1.428, which should be compared to the analytical result of ν = 4/3 in 2d. Thus the renormalisation group transformation defined above for bond percoaltion in 2d results in an exact prediction of pc and a good estimate of ν.
1.9.4 Why is the renormalisation group transformation not exact?
There are two significant sources of error which is the reason for not obtaining the exact results (1) Two sites that were connected in the original lattice can be disconnected and vica versa.
(2) The renormalised system is not really a true percolation system because the probabilities of having bonds between supersites are no longer independent. As we shall see later, this is the equivalent of introducing next-nearest-neighbour couplings in the renormalisation group procedure in the 2d Ising model. Thus we should really introduce new parameters but we choose here to truncate the problem to only one parameter, the occupation probability p.
37 Glossary Taken from The New Physics ed. Paul Davies, Cambridge University Press.
Coarse-graining An operation implementing some form of spatial averaging which smoothes out relatively small length scale configurational structure while preserving the larger length-scale struc-tures.
Correlation length The correlation length ξ gives a measure of the typical length scale over which fluctuations of one microscopic variable are correlated with the fluctuations of another. In percola-tion theory, it is the typical cluster diameter of the clusters sξ which give the main contribution to the divergence of the second (and higher) moments of the cluster distribution. Close to a critical point ξ ∝|p −pc|−ν for p →pc.
Critical phenomena The phenomena which occur in the neighbourhood of a continuous (second order) phase transition, characterised by very long correlation lengths.
Critical exponents (or indices) Near a ctitical point, physical quantities are often proportional to a power of another quantity, such as the distance from the critical point |p −pc|. The power that occurs known as a critical exponent (index).
Critical point A point in a phase diagram, where the correlation length associated with the physical system is, in principle, infinite.
Fractal geometry Generalisation of Euclidean geometry suitable for describing irregular and frag-mented patterns such as the percolating cluster at p = pc. A noninteger fractal dimension D can frequently be associated with such patterns.
Order parameter A variable such as the strength of the infinite cluster in percolation theory (or the magnetisation in an Ising model) used to describe the degree of order in a phase above (below) its critical point. In a continuous phase transition (second order phase transition), the order parameter goes continuously to zero as the critical point is approached from above (below).
Phase transition A change of state such as occurs in the boiling or freezing of a liquid, or in the change between ferromagnetic and paramagnetic states of a magnetic solid. An abrupt change, characterised by a jump in an order parameter is known as first order; a change in which the order parameter evolves smoothly to or from zero is called continuous or second order.
Renormalisation group In statistical physics, the renormalisation group method systematically implement some form of coarse-graining operation (e.g. rescaling length scales with a factor b: L →L b , ξ →ξ b) to expose the character of the large-scale phenomena, in systems where many length scales are important.
Scale invariance A physical system is said to exhibit scale-invariance if it remains unchanged (in a statistical sense) by a coarse-graining operation.
Universality The phenomena whereby many microscopically quite different physical systems ex-hibits critical point behaviour, with quantitatively identical features, such as critical exponents. In percolation theory e.g. the critical exponents do not depend on the microscopic details of the lattice but only on the Euclidean dimension d.
38 Bibliography D. Stauffer and A. Aharony, Introduction to Percolation Theory, (Taylor & Francis, 1994).
J. Adler, Physica A, 171, 453 (1991).
Iwan Jensen, J. Stat. Phys., 102, 865 (2000).
39 |
3441 | https://www.chegg.com/homework-help/questions-and-answers/solve-following-differential-equation-using-procedure-separating-variables-dy-dt-5t-y-q184631231 | Solved Solve the following differential equation, using the | Chegg.com
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3442 | https://geometry.flippedmath.com/112-chords-and-arcs.html | | | | | |
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| Home List of Lessons Semester 1 + Unit 1 Tools for Geometry > - 1.1 Points, Lines, and Planes - 1.2 Measuring Segments - 1.3 Measuring Angles - 1.4 Addition Postulate - 1.5 Angle Pairs - Unit 1 Review - Unit 1 Algebra Review + Unit 2: Reasoning and Proof > - 2.1 Reasoning and Proof - 2.2 Intro to Proofs - 2.3 More with Proofs - Unit 2 Review + Unit 3 Parallel and Perpendicular Lines > - 3.1 Lines and Angles - 3.2 Properties of Parallel Lines - 3.3 Proving Lines Parallel - 3.4 Parallel Lines and Triangles - 3.5 Equations of Lines in the Coordinate Plane - 3.6 Slopes of Parallel and Perpendicular Lines - Unit 3 Review - Unit 3 Algebra Review + Unit 4 Triangle Congruence > - 4.1 Triangles - 4.2 SSS and SAS - 4.3 ASA and AAS - 4.4 CPCTC and HL - Unit 4 Review - Unit 4 Algebra Review + Unit 5 Quadrilaterals and Polygons > - 5.1 Polygon Angles - 5.2 Parallelogram Properties - 5.3 Conditions for Parallelograms - 5.4 Rhombuses and Rectangles - 5.5 Conditions for Rhombuses and Rectangles - 5.6 Kites and Trapezoids - Unit 5 Review - Unit 5 Algebra Skillz Review + Unit 6 Similar Figures > - 6.1 Similar Figures - 6.2 Prove Triangles Similar - 6.3 Side Splitter Theorem - Unit 6 Review - Unit 6 Algebra Review + Semester Exam Review Semester 2 + Unit 7 Right Triangles > - 7.1 Pythagorean Theorem and Its Converse - 7.2 Special Right Triangles I - 7.3 Special Right Triangles II - 7.4 Trig Ratios - 7.5 Inverse Trig Functions - Unit 7 Review - Unit 7 Algebra Review + Unit 8 Transformations > - 8.1 Transformations - 8.2 Reflections - 8.3 Rotations - 8 Review + Unit 9 Area of Polygons > - 9.1 Parallelograms and Triangles - 9.2 Trapezoid, Kites, Rhombi - 9.3 Regular Polygons - 9.4 Circles and Arcs - 9.5 Sectors and Segments - Unit 9 Review - Unit 9 Algebra Review + Unit 10 Surface Area/Volume > - 10.1 SA of Prisms and Cylinders - 10.2 SA of Pyramids and Cones - 10.3 Volume of Prisms and Cylinders - 10.4 Volume of Pyramids and Cones - 10.5 SA and Volume of Spheres - Unit 10 Review - Unit 10 Algebra Review + Unit 11 Circles > - 11.1 Tangents to Circles - 11.2 Chords and Arcs - 11.3 Intercepted Arcs - 11.4 Secants and Tangents - Unit 11 Review + Unit 12 Probability > - 12.1 Introduction to Probability - 12.2 More Probability - Unit 12 Review - Unit 12 Algebra Review + Semester 2 Exam Review Teacher Resources FlippedMath.com |
11.2 Chords and Arcs
G.3.3: Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle.
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3443 | https://blog.csdn.net/weixin_47187147/article/details/123346478 | 高数 | 定理及性质证明 | 开 闭区间上连续函数的性质及证明
一、有界性定理
函数的上界和下界的绝对值不一定相等。
函数在某区间上不是有界就是无界，二者必属其一；
要证明f(x)在X上有界，必须找到一个M>0，使任意x属于X都有 |f(x)|<=M；要证明f(x)在X上无界，只需要找到一个数列{xn}存在于X,使f(xn) n趋于∞，f(xn)趋于∞
外界函数有界，复合函数必有界。
函数有界，从几何意义看就是图形被框定在两条平行于x轴的直线之间，不会跑出去；从代数意义看，就是函数值不会趋于正无穷大，也不会趋于负无穷大；当时并不意味着有极限，比如y=sinx，被框定在y=±1这两条直线之间，x→∞时，sinx游走于[-1,+1]之间。
二、最大最小值定理
三、零点存在定理
四、介值定理
五、反函数连续性定理
证明见闭区间上连续函数性质的证明 - 道客巴巴
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夏天的清晨:
块、盘块、磁盘块都是一个东西，多个盘块可以组成一个簇，供操作系统在逻辑上进行分配，这里说多个相邻的扇区可以形成一个簇，其实这里的簇也是盘块，王道书上写了在Linux系统中称为块（盘块），在windows上称为簇，而真正的簇其实就是多个盘块的组合
山东林更新:
因为1/x
Gfddyv:
为什么王道书上说ACC是不可见寄存器阿，到底那个是对的
2401_88144795:
可以打印吗，写的真的好好啊，想打印下来学习
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这么多背得完吗
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3444 | https://chem.libretexts.org/Courses/University_of_Arkansas_Little_Rock/Chem_1402%3A_General_Chemistry_1_(Belford)/Text/8%3A_Bonding_and_Molecular_Structure/8.3%3A_Resonance | Skip to main content
8.3: Resonance
Last updated
: Jun 17, 2020
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8.2: Covalent Bonding and Lewis Structures
8.4: Formal Charge
Page ID
: 52850
Robert Belford
University of Arkansas at Little Rock
( \newcommand{\kernel}{\mathrm{null}\,})
Introduction
One of the postulates of the Lewis Dot Structure for representing molecules is that a bond is the result of a pair of electrons being shared between two different nuclei, and as such, can be represented as a line between the two nuclei (the letters that represent the elements involved). But what if the electrons are shared between more than two nuclei? When this happens, there is no one Lewis Dot Structure that accurately describes the molecule. When this happens you need to draw resonance structures, none of which accurately describe the bonds, with the real structure sort of being the average of all the resonance structures. In the next Chapter we will look at the types of bonds in molecules and learn that there are two fundamentally different types of bonds, bonds and bonds. The sigma bonds can be described with Lewis dot structures as they represent bonding electrons shared between two nuclei, but sometimes the bonds have electrons that are shared over more than two nuclei, in which case no single Lewis dot structure can accurately describe the bond. For example the organic solvent benzene (C6H6) has 6 electrons that are shared by all 6 carbons and can form a ring circuit, as depicted in video , and as we shall see, this requires two resonance forms of the Lewis dot structure.
NOTE: Resonance structures represent different ways of placing electrons on the atoms in a molecule's Lewis dot structure. They do not describe different molecules and all resonance structures have the same connectivity. If you change the connectivity, you change the molecule, and that is not a resonance structure. In section 8.5 we will cover drawing Lewis dot structures and you will learn how to recognize if they exist while you are drawing them. But before we draw them, we want to define and describe them. So the following examples should help you identify and describe resonance structures.
Lets start with a look at the Lewis Dot Structure of Ozone
Illustration of Resonance: Ozone
Ozone (O3) is an allotrope of oxygen, with diatomic oxygen (O2) being the most common form of oxygen. Ozone is a very reactive form of oxygen that has detrimental health effects (which is why ozone alerts are posted along the highways of cities), but it also reacts with ultraviolet radiation in the upper atmosphere and thus shields the biosphere from the harmful effects of UV radiation. There are two ways of drawing the Lewis dot structure of ozone
That is, from Figure there is a choice of between which two atoms the double bond goes, oxygen B & C (left) or A & B (right), which is also demonstrated in the video .
Figure shows another way of drawing ozone that indicates the electrons of the second bond in the double bond are actually shared over all three oxygen atoms
The problem with Figure is that it does not tell you how many electrons there are (you can't count dots and bars).
Resonance Structures vs. Isomers
It is important to denote the difference between resonance structures and isomers. Resonance structures are not real molecules, but a shortcoming of the postulates that Lewis dot structures are based on, that is, a covalent bond is formed when electrons are shared by two atoms, and thus can be represented by a line between the atoms. In the case of ozone above, the bonding electrons are shared by three atoms, and so no one Lewis dot structure could accurately represent ozone. Isomers are real molecules, Figure shows two different isomers, hydrogen cyanide (HCN) and hydrogen isocyanide (HNC) and a chemical reaction can occur where these molecules interconvert into each other.
Note
A & B are not two different molecules but different ways of writing the same molecule, they are resonance structures.
A & B are two different molecules that can interconvert between each other and form an equilibrium mixture.
Bond Order
Bond order is a way of describing the magnitude of a bond; a single bond has a bond order of 1, double bond 2 and triple bond 3. An individual Lewis dot structure can only have integer bond orders.
Note
For bonds between identical nuclei, the higher the bond order the stronger the bond and the shorter the bond length.
If there are resonance structures, the bond order of the bonds in the resonant structure is the average bond order, which can be determined from any individual resonant structure, and it need not be an integer value.
Bond Order = 1: Single Bond
Bond Order = 2: Double Bond
Bond Order = 3: Triple Bond
Bond Order = 1.5: Combination of Single and Double Bond
Bond Order = 1.33: Combination of Two Single and one Double Bond
Example
What is the bond order of the oxygen-oxygen bond in ozone.
Solution
Both bonds in Ozone have a bond order of 1.5 and are the same length, which is longer than a double bond (because the double bond is a stronger bond) and shorter than a single bond (because a single bond is a weaker bond).
Nitrate Ion
The Lewis dot structure of nitrate has three resonance structures. Because it is an ion, it must be in brackets and the charge shown outside the Lewis dot structure.
If we were to merge the Lewis dot structure into one Figure like we did in fig. 8.3.4 for ozone
Exercise
What is the bond order for the nitrogen=oxygen bond in nitrate?
Answer
So the bond order of nitrogen is 1.33, that is, it is 2/3 single bond and 1/3 a double bond.
Exercise
Would you expect the nitrogen-oxygen bond length of nitrate to be closer to the N-O single bond or the N=O double bond?
Answer
: It is closer to the single bond in length because a bond order of 1.33 is closer to 1 than it is to 2.
Benzene
Benzene (C6H6) is a ring structure that is common in organic compounds known as aromatics and is represented in video at the beginning of this Chapter. There are two resonance structures for benzene as indicated in Figure .
Note, as carbon is in the second period of the periodic table there are no d-orbitals and so carbon can not have an expanded octet, nor is it in group IA, IB or IC, which allow it to have less than an octet. Thus carbon can be considered to follow the octet rule and needs 8 valence electrons in its Lewis dot structure. This allows us to remove the hydrogens from the Lewis dot structure and implicitly infer them. That is, each carbon needs 8 electrons and if the drawing omits 2, we assume there is a hydrogen present.
Figures and are two ways of drawing benzene and it is important that students become familiar with the convention of Figure . The reason this convention is so common is that the bonds to hydrogen are terminal, in the sense that hydrogen can only have one bond, and this convention of not drawing the hydrogen is common in chemistry, especially organic chemistry.
Exercise
What is the bond order of the carbon-carbon bond in benzene?.
Do resonance structures always add up to non-integer bond orders?
Answer a
Answer b
: FALSE, look at carbon dioxide, the next example.
Carbon Dioxide
The Lewis dot structure of carbon dioxide is shown in Figure .
In the above Figure we see the second and third resonance structures average out to the first, and so the average of all the resonance structures is a double bond. Thus it is common to write carbon dioxide as having two double bonds, and that resonance structure is the correct structure of carbon dioxide.
Exercise
Calculate the bond order of carbon dioxide from the Lewis dot structures on the left and right of Figure .
Answer
: For the structure on the left:
For the structure on the right:
From exercise it is clear that no matter which resonance structure you use, the bond order is the same. Now even though carbon dioxide has two double bonds, if you are asked to draw it's Lewis dot structure, you should include the two with the single and triple bonds, and this will be covered in section 8.5.
Summary
So what we have seen is that one of the shortcoming of Lewis dot structures comes from the concept that a line represents a bond between two nuclei, but sometimes this is false and the bond is between more than two nuclei. In that case you need to write multiple Lewis dot structures, the resonance structures, and use an arrow with two heads to indicate that each of the individual structures are resonance structure, and the true molecule is a combination of them all. If, as in the case of carbon dioxide, all the resonance structures average out to one of the resonance structures, then it can be treated as the real structure, and you have an integer bond order. If not, the real structure is none of the resonance structures, and the bond order is of non-integer order, often indicated with a dotted line.
It must be emphasized that resonance structures do not show the true bond order and thus do not represent the true structure of the molecule, which results from the combination of all the resonance structures.
8.2: Covalent Bonding and Lewis Structures
8.4: Formal Charge |
3445 | https://www.zhihu.com/question/41195119 | 年均增长率公式？ - 知乎
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开方数到底是n还是n-1，为什么感觉前后矛盾冲突 [图片] [图片]显示全部
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哈哈，这个其实不矛盾，题主只要把握住核心公式即可。
B=A(1+m)n B=A(1+m)^{n}
其中m为所求的增长率，n为计算增长率的期间。
之所以出现n-1的问题，是混用了字母n，两处字母n所表示的含义是不同的。
公式中的n表示的是计算增长率的期间，如计算2001~2004年间的GPD增长率，计算期间为2001~2002、2002~2003、2003~2004，n=3；
而上文出现的n-1中的n，表示的是2001~2004年的自然年个数，2001、2002、2003、2004，n=4，n-1=3，实质是一样的。
发布于 2016-03-09 14:12
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开方数是n还是（n-1）是根据所计算的时段进行确定的，更确切地说，是根据起算点和截止时间点之间，总共经历过多少个完整年份进行确定的。
eg1：计算2001年末至2005年末之间的年均复合增长率，总共经历了4个完整的年份，则开方数取4；
eg2：计算2001年初至2005年末之间的年均复合增长率，总共经历了5个完整的年份，则开方数取5。
可以画时间轴进行辅助理解。
发布于 2020-06-17 03:40
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年均增量：表示的是n年间增量的绝对平均值。年均增量 =（本期-基期）/n
年均增长率：也叫复合增长率，表示的是n年间的年平均增速。
公式：（1+r）^n = 本期/基期
具体在行测考题中运用，主要有几类：
一、比较类题目不一定非要真算出来——有时比较增长量/基期即可
1953年末全国16-59岁劳动年龄人口为3.10亿人，1964年末为3.53亿人，1982年、1990年、2000年和2010年四次人口普查数据显示，我国劳动年龄人口分别为5.67亿人、6.99亿人、8.08亿人和9.16亿人。在2012年末我国劳动年龄人口总量达到峰值9.22亿人后增量由正转负，2018年末为8.97亿人。
（20苏A-117）用V1982-1990、V1990-2000、V2000-2010分别表示四次人口普查间的全国劳动年龄人口年均增长率，下列关系正确的是
A、V1982-1990＞V1990-2000＞V2000-2010
B、V1982-1990＞V2000-2010＞V1990-2000
C、V1990-2000＞V1982-1990＞V2000-2010
D、V1990-2000＞V2000-2010＞V1982-1990
先比较三个阶段增长量/基期的比值，注意V1982-1990是8年，其余两者是10年。
很明显，1.32/5.67＞1.09/6.99（分子大而分母小，且V1982-1990仅用8年）
1.09/6.99＞1.08/8.08，则A
注意此题年均增长率的年份间隔不相同，且此题较为简单，但若年份小者总增长率小呢？需要进一步比较，值得注意。
二、可以用近似公式来算
其中A是基期值，B是现期值，n是年份差。
注意：利用上述公式算出的年均增长率略大于实际值（由后面二项式公式可以理解）
（19深圳-94）2012—2017年，深圳市进口总额的年增长率约为（ ）。
A、－2.8% B、－6.3% C、－10.0% D、－13.2%
由图表（2012年进口总额1954亿美元、2017年进口总额1697亿美元），可得2017年比2012年增长了（1697-1954）/1954=-257/1954≈-13.1%、年均增长率≈-13.1%/5，选A
三、利用二项式定理巧算一些看上去很复杂的题
可以记住一些常见平方数：
11~30^2: 121 144 169 196 225 256 289 324 361 400
441 484 529 576 625 676 729 784 841 900
1~10^3: 1 8 27 64 125 216 343 512 729 1000
2015年C省稻谷产量53.2万吨、2018年C省稻谷产量121.9万吨
(19黑边境-120）根据资料，下列说法正确的是：
D.2015～2018年，C省稻谷产量的年均复合增长率约为31.8%
（1+R）^3=121.9/53.2≈2.3，
（1+R）^3≈1+3R+3R^2 代入R=31.8% 较为接近，说法正确，选D
再让我们看下这道题：
（18赣-119）2013-2017年移动宽带用户数的年均增长率为:
A.28% B.29% C.30% D.31%
由图形材料可知：2013年移动宽带用户数为40161，2017年移动宽带用户数为113152。
可能光看选项，大家会觉得出题人好变态，就是逼人死算。
但我却觉得，出得非常好，很典型，也很有区分度。
为什么这么说？
我是这么做的：
分三步
1、实际先假设是30% 算得4年混合增长倍数是2.85
2、13年到17年的倍数是2.83
3、其他选项 误差远大于此 则C
然后有同学说“这样也很费时间诶”
实际上，这几个步骤，我基本没动笔，40秒不到即解出。
有人觉得我在装逼吹牛逼，其实我一般只是懒得打字解释而已
因为打字真的很麻烦啊。。。比如这道题
首先，要知道一个定理——二项式定理
(a+b)^n=C(n,0)a^n+C(n,1)a^(n-1)×b+C(n,2)a^(n-2)×b^2+...+C(n,n)b^n
1、假设是30%，则（1+r）^4=[(1+r）^2]^2
1.3^2=1.69（20以内平方数如能熟悉，此处无需计算）
1.69^2=(1.7-0.01)^2≈1.7^2-2×0.01×1.7=2.89(亦是无需计算）-0.035≈2.85
（注：也可以直接利用二项式得出，不过这个需要动笔，所以我是拆成两步，但心算方便）
2、113152÷40161≈113152÷40000≈2.83
（为何可以如此估算，因为相对误差；
如果其他数字怎么办？还是利用相对误差）
3、其他选项 误差远大于此，为什么？（这一步，对于熟悉的同学，实际可省略）
因为2.85＞2.83 最可能的是1.29
1.28和1.31可以直接排除
直接利用四次二项式定理
（1.3-0.01）^4≈1.3^4-4×0.01×1.3^3约等于1.3^4-0.06
实际上，这三步我都没有动笔计算，只是分别写下了几个数字：1.69,2.85,2.83
其余的步骤，都是在心里分析，用时不过20秒。
为什么有的同学觉得难？
因为他们只看到选项的接近，
但没有看到题目的题境
——在二项式定理的放大下，看似1%的差距，实际上是比6%还大的误差。
还是那句话，资料分析，是分析，不是硬算。
（19赣法检-109）2014-2018年R&D经费支出的年均增长率为：
A、10.9% B、10.4% C、9.9% D、8.9%
（1+R）^4=19657/13016≈1.51，
假设R'=10%，1+C(4，1)R+C(4，2)R^2=1+0.4+0.06=1.46
4（R-R'）=1.51-1.46=0.05
（R-R'）=0.012 故R约为11.2% 最接近为A
（19联A鄂-119）在2017年马拉松运动年度产业总规模的基础上，从2018年开始，每年大约需要平均增长多少才能实现中国田径协会设置的2020年马拉松运动产业规模目标？
A.15% B.20% C.25% D.30%
由第二段（2017年年度产业总规模达700亿元。中国田径协会设置的发展目标是到2020年，马拉松运动产业规模达到1200亿元），
（1+r）^3=1+3r+3r^2+r^3=1200/700=1+5/7=1.71+
若r=25%，则（1+r）^3＞1+3r＞1.75
若r=20%，（1+r）^3≈1+3×20%+3×4%=1.72
故B
四、72/115法则
B/A=2时，r=72/n%，B/A=3时，r=115/n%。
比如18（1+r）^4=37，因为37/18=2.06，非常接近2，72/4=18， 所以r=18%就可以了，大一点点，也可以。这里n越大，越精确。
又比如“交通事故的数量每年递增7%。”——倍增时间为70÷7=10（年），也就是“交通事故的数量每10年翻一倍。”
有空仔细找下相关例题
五、代入法
如果记忆力很强，可以考虑此法，不是很推荐。
更多干货见此：
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3447 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3705765/ | Levetiracetam versus phenytoin for seizure prophylaxis in severe traumatic brain injury - PMC
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As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
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Neurosurg Focus
. Author manuscript; available in PMC: 2013 Jul 9.
Published in final edited form as: Neurosurg Focus. 2008 Oct;25(4):E3. doi: 10.3171/FOC.2008.25.10.E3
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Levetiracetam versus phenytoin for seizure prophylaxis in severe traumatic brain injury
Kristen E Jones
Kristen E Jones, M.D.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
Find articles by Kristen E Jones
1, Ava M Puccio
Ava M Puccio, R.N., Ph.D.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Kathy J Harshman
Kathy J Harshman, R.N., B.S.N.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Bonnie Falcione
Bonnie Falcione, Pharm.D.
2 Department of Pharmacology, University of Pittsburgh, Pennsylvania
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2, Neal Benedict
Neal Benedict, Pharm.D.
2 Department of Pharmacology, University of Pittsburgh, Pennsylvania
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2, Brian T Jankowitz
Brian T Jankowitz, M.D.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Martina Stippler
Martina Stippler, M.D.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Michael Fischer
Michael Fischer, B.S.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Erin K Sauber-Schatz
Erin K Sauber-Schatz, M.P.H.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Anthony Fabio
Anthony Fabio, Ph.D., M.P.H.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1, Joseph M Darby
Joseph M Darby, M.D.
3 Critical Care Medicine, University of Pittsburgh, Pennsylvania
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3, David O Okonkwo
David O Okonkwo, M.D., Ph.D.
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
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1
Author information
Copyright and License information
1 Department of Neurological Surgery, University of Pittsburgh Medical Center, Pennsylvania
2 Department of Pharmacology, University of Pittsburgh, Pennsylvania
3 Critical Care Medicine, University of Pittsburgh, Pennsylvania
✉
Address correspondence to: David O. Okonkwo, M.D., Ph.D., Department of Neurological Surgery, University of Pittsburgh, 200 Lothrop Street, Suite B-400, Pittsburgh, Pennsylvania 15213. okonkwodo@upmc.edu
PMC Copyright notice
PMCID: PMC3705765 NIHMSID: NIHMS484002 PMID: 18828701
The publisher's version of this article is available at Neurosurg Focus
Abstract
Object
Current standard of care for patients with severe traumatic brain injury (TBI) is prophylactic treatment with phenytoin for 7 days to decrease the risk of early posttraumatic seizures. Phenytoin alters drug metabolism, induces fever, and requires therapeutic-level monitoring. Alternatively, levetiracetam (Keppra) does not require serum monitoring or have significant pharmacokinetic interactions. In the current study, the authors compare the EEG findings in patients receiving phenytoin with those receiving levetiracetam monotherapy for seizure prophylaxis following severe TBI.
Methods
Data were prospectively collected in 32 cases in which patients received levetiracetam for the first 7 days after severe TBI and compared with data from a historical cohort of 41 cases in which patients received phenytoin monotherapy. Patients underwent 1-hour electroencephalographic (EEG) monitoring if they displayed persistent coma, decreased mental status, or clinical signs of seizures. The EEG results were grouped into normal and abnormal findings, with abnormal EEG findings further categorized as seizure activity or seizure tendency.
Results
Fifteen of 32 patients in the levetiracetam group warranted EEG monitoring. In 7 of these 15 cases the results were normal and in 8 abnormal; 1 patient had seizure activity, whereas 7 had seizure tendency. Twelve of 41 patients in the phenytoin group received EEG monitoring, with all results being normal. Patients treated with levetiracetam and phenytoin had equivalent incidence of seizure activity (p = 0.556). Patients receiving levetiracetam had a higher incidence of abnormal EEG findings (p = 0.003).
Conclusions
Levetiracetam is as effective as phenytoin in preventing early posttraumatic seizures but is associated with an increased seizure tendency on EEG analysis.
Keywords: antiepileptic drug, posttraumatic seizure, seizure prophylaxis, traumatic brain injury
Current standard practice for patients with severe TBI is prophylactic treatment with phenytoin for 7 days after injury to decrease the risk of early posttraumatic seizures.3,11 Compared with placebo, phenytoin administration is associated with a relative risk of 0.25 for the occurrence of seizures during this time period (95% CI 0.11–0.57).14 However, phenytoin therapy after the initial 7-day period has not been shown to reduce the development of late seizures nor has prophylaxis against early seizures been shown to minimize morbidity or mortality rates associated with severe TBI.1,14
Phenytoin has a well-described side-effect profile that includes severe cutaneous hypersensitivity reactions and induction of the hepatic cytochrome P450 system, causing significant drug-drug interactions.5,10 Additionally, phenytoin usage requires close monitoring to maintain a narrow therapeutic window and has been described as causing fever and decreased levels of consciousness, particularly concerning in patients with TBI. For these reasons, alternative AED therapy has been sought. Valproate and carbamazepine have been investigated for usage in TBI but have similar side-effect profiles and require serum monitoring.4,13
Levetiracetam (Keppra, UCB, Inc.) is a non–enzyme-inducing AED that does not require serum level monitoring or induce fever or cutaneous hypersensitivity reactions and is not known to have significant pharmacokinetic interactions.7 In November 2006, FDA approval of the intravenous form of levetiracetam created an attractive option for antiseizure prophylaxis. Szaflarski et al.12 retrospectively analyzed levetiracetam use in a neuroscience ICU population (including patients with TBI, tumor, subarachnoid hemorrhage, stroke, and/or infection), and concluded that levetiracetam monotherapy was associated with lower complication rates and shorter ICU stays than treatment with other AEDs. To date, there are no published data investigating the efficacy of seizure prevention with levetiracetam compared with standard treatment with phenytoin in patients with severe TBI.
In the current study, we evaluated the occurrence of early posttraumatic seizure activity recorded by electroencephalography in patients with severe TBI treated with phenytoin versus levetiracetam as seizure prophylaxis. We hypothesized that levetiracetam would be equivalent to phenytoin in preventing early seizures in severe TBI. Secondarily, we hypothesized that in terms of abnormal EEG findings there would be no difference between patients treated with levetiracetam and those treated with phenytoin.
Methods
Under an innovative off-label investigational use policy approved by the Pharmacy and Therapeutics Committee at the University of Pittsburgh Medical Center, we initiated a protocol of intravenous levetiracetam monotherapy for early seizure prophylaxis in patients with severe TBI, defined by a postresuscitation GCS score of 3–8. Levetiracetam therapy was initiated within 24 hours of injury, either in the emergency department or upon admission to the NeuroTrauma ICU.
From November 2006 to December 2007, 32 consecutive patients with severe TBI were admitted and received levetiracetam 500 mg IV every 12 hours for the first 7 days after traumatic injury. We then compared this prospective cohort to a historical cohort of patients from our severe TBI database in which patients received phenytoin for 7 days after trauma. From July 2005 to June 2006, 41 patients with severe TBI received IV phenytoin therapy within 24 hours of traumatic injury.
In our institution, all cases of severe TBI are managed in accordance with a standard protocol. Patients undergo an EEG examination if there is a suspicion of a seizure on the basis of mental status changes or persistent coma or if clinical seizure activity has occurred. In the current study, only patients who received an EEG examination were included in the analysis. Fifteen (46.9%) of the 32 patients in the levetiracetam cohort warranted EEG testing to assess for seizures given persistent coma, change in mental status, or clinical seizure activity. Twelve (29.3%) of the 41 patients in the phenytoin cohort required an EEG examination to investigate for seizures. Propofol is the sedative of choice in our TBI protocol and patients were weaned from propofol as tolerated at the time of EEG recordings per institutional protocol.
To assess the efficacy of seizure prevention in the 2 cohorts, individual EEG recordings were reviewed by an attending neurologist specializing in electroencephalography. The EEG findings for each patient were then stratified as normal or abnormal based on the presence of focal abnormal waveforms. Abnormal EEG findings were further classified into the following 3 categories: status epilepticus, seizure activity, or seizure tendency. The electroencephalograms categorized as demonstrating seizure tendency exhibited epileptiform activity of intermittent sharp waves or periodic lateralized epileptiform discharges without capturing electrographic seizures. When a patient underwent more than one EEG examination within the 7-day monitoring period, the most abnormal result was used for categorical stratification for that patient.
The CT-based classification of head injury of Marshall et al.6 was used to define intracranial pathology on admission head CTs for each patient undergoing EEG monitoring. Glasgow Outcome Scale scores were obtained at 3 and 6 months posttrauma when possible.
Descriptive statistics, chi-square tests, and Mann–Whitney U-tests were used to compare the phenytoin and levetiracetam cohorts with respect to gender, age, and GCS score at admission. The Fisher exact test was used to determine whether there were significant differences between the 2 cohorts with respect to abnormal EEG findings, seizure activity, and seizure tendency with epileptiform activity. Analyses were performed in SPSS 15.0 for Windows (SPSS, Inc.). Statistical analysis of GOS scores obtained at 3 and 6 months postinjury was performed to monitor functional TBI outcomes.
Results
Univariate analysis was performed on possible confounders to test whether the differences seen between the drugs could be attributed to other factors. There were no significant differences in age, sex, or admission GCS scores between the cohorts (Table 1). Indication for obtaining an EEG, Marshall CT Classification, CT findings, and surgical interventions are outlined in Table 2. The median Marshall CT scores were the same in the 2 cohorts, and there was no statistically significant between-groups difference with respect to the CT findings or frequency of craniotomy for evacuation of mass lesions.
TABLE 1.
Patient demographics—gender, age, and admission GCS stratified by cohort
| | Cohort | | | |
:---: :---:
| | | | | |
| Variable | Phenytoin | Levetiracetam | Chi-Square | df | p-Value |
| no. of patients | 12 | 15 | | | |
| sex | | | | | |
| F | 3 (25) | 4 (27) | 0.010 | 1 | 0.922 |
| M | 9 (75) | 11 (73) | | | |
| age in yrs | | | | | |
| ≤25 | 4 (33) | 4 (27) | 1.89 | 3 | 0.596 |
| 26–35 | 2 (16) | 6 (40) | | | |
| 36–45 | 3 (25) | 2 (13) | | | |
| >46 | 3 (25) | 3 (20) | | | |
| GCS Score | | | | | |
| 6–8 | 5 (42) | 10 (67) | 1.688 | 1 | 0.194 |
| 3–5 | 7 (58) | 5 (33) | | | |
Open in a new tab
TABLE 2.
Reason for EEG monitoring, imaging findings, and GOS scores in patients in the levetiracetam and phenytoin cohorts
| Cohort & Study ID | Reason for EEG† | CT Score‡ | EDH | SDH | ICH | SAH | Evac of Mass Lesion | 3-Mo GOS Score§ | 6-Mo GOS Score |
:---: :---: :---: :---: :---: |
| levetiracetam | | | | | | | | | |
| K5 | A | III | | | | x | — | 1 | 1 |
| K7 | A | II | x | | x | | — | 4 | 5 |
| K9 | B | V | | x | | | PTD 0 | 3 | 3 |
| K11 | C | III | | | x | x | PTD 0 | 3 | 3 |
| K12 | A | I | | | | | — | 4 | 4 |
| K13 | A | III | | | x | x | — | 1 | 1 |
| K15 | C | III | | x | x | x | — | 1 | 1 |
| K16 | A | V | x | | x | x | PTD 0 | 2 | 2 |
| K18 | A | IV | | x | x | x | — | 3 | 5 |
| K20 | B | V | | x | | | PTD 1 | 3 | 3 |
| K23 | A | II | | x | x | x | — | 3 | 3 |
| K28 | A | V | | x | | x | PTD 0 | 1 | 1 |
| K29 | B | V | x | | | | PTD 0 | 4 | — |
| K30 | A | III | x | | x | x | — | 1 | 1 |
| K32 | A | V | x | | | | PTD 0 | 3 | 3 |
| |
| median | | III | | | | | | 3 | 3 |
| |
| phenytoin | | | | | | | | | |
| D5 | A | V | | x | x | x | PTD 0 | 3 | 4 |
| D7 | C | III | | | | x | — | 1 | 1 |
| D10 | A | III | | | x | x | — | 1 | 1 |
| D18 | A | II | | | x | x | — | 5 | 5 |
| D19 | A | V | x | x | x | x | PTD 0 | 4 | 4 |
| D20 | A | V | | x | x | x | PTD 1 | 3 | 3 |
| D25 | A | V | | x | x | | PTD 0 | 3 | 3 |
| D29 | C | III | | | x | x | — | 2 | 2 |
| D33 | A | I | | | | | — | 2 | 3 |
| D34 | A | III | | x | x | x | — | 1 | 1 |
| D39 | C | II | | | x | x | — | 2 | — |
| D41 | B | V | | x | x | x | PTD 0 | 3 | 3 |
| |
| median | | III | | | | | | 3 | 3 |
Open in a new tab
EDH = epidural hematoma; Evac = surgical evacuation; SDH = subdural hematoma; ICH = intracerebral hematoma; PTD = post-trauma day; SAH = subarachnoid hemorrhage;×= present; — = procedure not performed or data not obtained.
†
Reason for EEG monitoring: A = persistent comatose state; B = decrease in level of consciousness; C = suspicion for clinical seizure activity.
‡
Classification according to Marshall et al., 1992: I = no visible intracranial abnormality; II = cisterns present, midline shift 0–5 mm, no mass lesion > 25 cm 3; III = cisterns compressed or absent, midline shift 0–5 mm, no mass lesion > 25 cm3; IV = midline shift > 5 mm, no mass lesion > 25 cm 3; V = evacuated mass lesion, any lesion surgically evacuated.
§
Glasgow Outcome Scale scoring: 1 = dead; 2 = vegetative state; 3 = severe disability; 4 = moderate disability; 5 = good recovery.
Four of the patients in the levetiracetam cohort underwent 2 EEG studies, for a total of 19 EEG examinations performed in the 15 patients treated with levetiractam. Seven (46.7%) of these 15 had normal EEG findings while 8 (53.3%) had abnormal EEGs (Table 3). The abnormal EEG findings revealed no status epilepticus, although 1 (12.5%) of the 8 patients with abnormal EEG findings had seizure activity, and 7 (87.5%) had seizure tendency with abnormal waveforms (Table 3).
TABLE 3.
Electroencephalographic findings stratified by cohort
| | Phenytoin | Levetiracetam | |
:---: :---: |
| | | | |
| EEG Finding | Yes | No | Yes | No | p Value |
| seizure tendency† | 0 | 12 | 7 | 8 | 0.007 |
| seizure activity | 0 | 12 | 1 | 14 | 0.556 |
| abnormal EEG | 0 | 12 | 8 | 7 | 0.003 |
Open in a new tab
Values represent numbers of patients unless otherwise indicated.
†
Seizure tendency included epileptiform activity of intermittent sharp waves or periodic lateralized epileptiform discharges without electrographic seizures.
Four of the patients in the phenytoin cohort underwent 2 EEG studies and 1 underwent 3; thus a total of 19 EEG examinations were performed in the 12 patients treated with phenytoin. All 12 patients had normal findings (Table 3). The Fisher exact test showed a significant difference between the occurrence of abnormal EEG findings (seizure or seizure tendency with epileptiform activity) in the levetiracetam versus phenytoin cohorts (p = 0.003), but there was no significant difference between the finding of seizures on electroencephalograms between the 12 phenytoin and 15 levetiracetam patients who underwent EEG monitoring (p = 0.556).
Although the study was not powered to detect outcome differences, GOS scores were collected at 3 and 6 months post injury (Table 2). We dichotomized the GOS scores to poor and good outcome (1–3 = poor; 4–5 = good). With respect to GOS scores at 3 and 6 months, there were no statistically significant differences between patients treated with levetiracetam and those treated with phenytoin. Specifically, the chi-square statistic was 0.049 for GOS score at 3 months (p = 0.825) and 0.115 for GOS score at 6 months (p = 0.734). No patients were lost to follow-up at 3 months; one patient from each group was lost to follow-up at 6 months.
Discussion
The present study indicates that levetiracetam mono-therapy in the first 7 days following severe TBI is associated with an increased seizure tendency and increased epileptiform activity on electroencephalograms compared with phenytoin. The rates of seizure activity were equivalent in patients treated with levetiracetam and those treated with phenytoin. The implications of increased seizure tendency and epileptiform activity require further study.
Traumatic brain injury occurs in approximately 1.5 million people in the US each year.9 One important complication of TBI is seizure. Seizure risk after TBI is related to injury severity. In one study,1 seizures developed during the first year postinjury in < 1% of patients with mild TBI and in 6% of patients with severe TBI. Temkin et al.14 found a 2-year seizure rate of 21% in patients with severe TBI. Early seizures, defined as occurring within the first 7 days posttrauma, have been shown to increase intracranial pressure episodically, thereby increasing the risk for cerebral ischemia.15 Late seizures, developing after the initial 7-day period, are associated with a worse functional outcome measured by the GOS.2 Early seizure prophylaxis, however, does not influence occurrence of late seizures.
Ronne-Engstrom and Winkler,2 in study of a series of cases involving continuous EEG recordings in patients with unspecified severity of TBI, reported high-frequency bursts of epileptiform activity that developed into electrographic seizures in 12 (66.7%) of 18 patients.8 The current study is limited by lack of 24-hour EEG monitoring, raising the possibility of the occurrence of seizures not captured by 1-hour EEG monitoring. Patients in our study treated with levetiracetam whose EEG studies showed seizure tendency may have had uncaptured waveform deterioration into electrographic seizures. The current study is limited further by a small sample size and by the retrospective, historical cohort design.
The strength of the current findings lies in the use of actual EEG data to evaluate brain and seizure activity in relation to the use of AEDs as seizure prophylaxis following severe TBI. We restricted the analysis to patients in whom EEG monitoring was performed because overt clinical seizures themselves are relatively rare events and subject to observer bias.
The current study highlights the fact that clinical seizures are difficult to identify through observation or physical examination in the early stages after severe TBI. Both the TBI itself and sedative and neuromuscular blockade agents used in intensive care management of severe TBI may mask seizure activity. Therefore, any prospective study intended to test the utility of levetiracetam versus phenytoin for seizure prophylaxis after severe TBI should include routine use of EEG monitoring to discern abnormal EEG patterns and between-group differences in seizure frequency.
Since levetiracetam does not require loading doses or monitoring of drug levels and lacks significant drug-drug interaction, it is an appealing alternative therapy in the prevention of early posttraumatic seizures. On the basis of the results of the current study, we urge caution against widespread practice changes in converting to levetiracetam as monotherapy for seizure prophylaxis following severe TBI. Prospective studies investigating levetiracetam in TBI are ongoing, although the results may require further scrutiny if objective data, like EEG recordings, are not included.
Conclusions
Our data indicate that levetiracetam is as effective as phenytoin in preventing early posttraumatic seizures. Nevertheless, levetiracetam monotherapy was associated with increased frequency of abnormal EEG findings. Further study is necessary to determine the clinical implication of this finding as well as to determine whether these patients exhibit a higher incidence of late posttraumatic seizures. Prospective studies comparing levetiracetam and phenytoin among subsets of patients with mild, moderate, and severe TBI are indicated. Because prophylaxis against early seizures has not been shown either to reduce the incidence of late posttraumatic seizures or influence outcome, future prospective studies should also include a placebo group to assess the possibility that early seizure prophylaxis following TBI exposes patients to unnecessary morbidities of AEDs without benefits of enhancing neurological recovery.
Acknowledgments
Disclosure
This research was performed at the University of Pittsburgh Medical Center and was supported by the University of Pittsburgh Brain Trauma Research Center (NIH P50NS30318).
Abbreviations used in this paper
AED
antiepileptic drug
EEG
electroencephalographic
GCS
Glasgow Coma Scale
GOS
Glasgow Outcome Scale
ICU
intensive care unit
TBI
traumatic brain injury
Footnotes
Disclaimer
The authors report no conflict of interest concerning the materials or methods used in this study or the findings specified in this paper.
References
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3448 | https://ocw.mit.edu/courses/6-1200j-mathematics-for-computer-science-spring-2024/pages/syllabus/ | Published Time: Thu, 24 Jul 2025 19:04:11 GMT
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Concern
[edit]
The article says:
Example 2:Ordered Data Set: 7, 15, 36, 39, 40, 41 Q1 = (36+15)/2 = 25.5 Q2 = (39+36)/2 = 37.5 Q3 = (40+39)/2 = 39.5
Now unless I've misunderstood, shouldn't Q1 be 15 and Q3 be 40? The median cuts the 6 member data set into two 3 member data sets. The median of a 3 member data set is the item in the middle. 80.177.129.25111:33, 11 September 2006 (UTC)[reply]
Example 1
[edit]
This is flat out wrong. Q1=15, the median is 40, and Q3=43. How could they see the correct median (Q2), but screw up the other quartiles? The number of variable is either odd or even: if it's an odd number, there is no need to do any averaging to find the quartiles. If it's even, you need the averages of three sets of two numbers. I could see this error happening if we were dealing with say 117 variables but 11? Sheesh.
Examples, discrete case
[edit]
As the article stands ggk , the above posts have been taken into account. However, the two examples in the article - and also the 2nd post above - seem to indicate there are two cases, n odd and n even. I believe there are four cases (depending on n modulus 4):
2, 6, 10, 14, ... observations:
Example with 10 observations: 11,13,16,17,19;22,23,27,28,30: Q1=16, M=(19+22)/2, Q3=27 (mean involved in median only)
3, 7, 11, 15, ... observations:
Example with 7 observations: 11,13,16,17,19,22,23: Q1=13, M=17, Q3=22 (no means involved)
4, 8, 12, 16, ... observations:
Example with 8 observations: 11,13;16,17;19,22;23,27: Q1=(13+16)/2, M=(17+19)/2, Q3=(22+23)/2 (all means!)
5, 9, 13, 17, ... observations:
Example with 9 observations: 11,13;16,17,19,22,23;27,28: Q1=(13+16)/2, M=19, Q3=(23+27)/2 (means involved in quartiles only)
In other words:
With an odd number of observations, the median is the middle observation, and the quartiles are the medians of the lower resp. upper half of the observations, omitting the middle one.
With an even number of observations, the median is the mean of the two middle observations, and the quartiles are the medians of the lower resp. upper half of the observations.
Finding the median of half the observations, one may again have to consider either an odd or an even number of observations; hence the four cases above.
Grouped data
[edit]
Honestly I do not understand everything in the article; the answer to the following question may be hidden in there. But how do you find quartiles for grouped data? (I know the answer, I think, involving a cumulative frequency graph, but I have discovered that some advocate strange variants of "my" methods, and I donøt understand why.)--Niels Ø22:44, 27 November 2006 (UTC)[reply]
Erroneous statement
[edit]
I have removed an erroneous statement saying that in applied work the quartiles are the intervals between the quartile points. This is a widespread error and I'd like to keep it out out Wikipedia.
Blaise (talk)
Explicit rule
[edit]
I added an example of an explicit rule for computing quartile values (there is no uniform agreement on this). I this way the reader can work out the examples for himself. I also added an example where the quartile values are not data points. 84.196.107.23507:13, 10 March 2007 (UTC)[reply]
Invalid Example That doesn't follow the rule
[edit]
I think the title is enough to explain everything, not to mention all the discussions above. The first example is clearly inconsistent with what is mentioned to be the rule of finding quartiles (lower and upper). Please resolve this. --Freiddie12:56, 1 April 2007 (UTC)[reply]
Article should cover all methods
[edit]
The article should cover all the competing definitions of the sample quartile, and compare and contrast them. The article notes that there is no universal agreement on how to choose the values, but then goes on to give a formula for how to choose them, without reconciling the contradiction. From quantile, there are nine distinct ways of calculating sample quantiles. Unless some of them are redundant for quartiles, the article should explain all of them.--Srleffler (talk) 04:53, 20 March 2010 (UTC)[reply]
Quarter
[edit]
Does the link for Quarter belong? --Zzo38 (talk) 06:20, 19 February 2011 (UTC)[reply]
No, it's a disambiguation page. +mt21:13, 19 February 2011 (UTC)[reply]Which is what I thought. I noticed that, too. But I want to know what might be the purpose that whoever put that there, I don't know that, please. --Zzo38 (talk) 00:16, 7 March 2011 (UTC)[reply]Ah, I see what your after now. I removed the link since I don't see how it is helpful. Thanks for pointing that out. +mt01:35, 7 March 2011 (UTC)[reply]
"In epidemiology, the four ranges defined by the three values are discussed here." - huh?
[edit]
The sentence "In epidemiology, the four ranges defined by the three values are discussed here." has been in the article lead for a very long time, but it doesn't make sense. Quartiles in epidemiology are discussed neither in Quartile, nor in epidemiology.
-- Dandv(talk|contribs)08:54, 8 September 2011 (UTC)[reply]
That version had been incorrectly changed from a more correct earlier version. I have replaced it by a slightly more expanded version of what it was trying to say. Unfortunately the rest of the article doesn't cover both of the possible meanings of "quartile". Melcombe (talk) 09:13, 9 September 2011 (UTC)[reply]
Applying the percentile formula
[edit]
When one applies the percentilie formula, the number obtained should be rounded UP: L=1.2 becomes 2 not 1 as the section says. The percentile formula should be enlisted as one of the methods not standing separately. All the three methods mentioned, give sometimes slightly different estimates for the quartiles, depending on N mod 4, as one user indicated before. — Preceding unsigned comment added by 205.178.108.254 (talk) 17:54, 28 January 2012 (UTC)[reply]
There is a reference given for the formulation stated. Does the formulation stated agree with the reference? If you want to state a diffeent formulation, provide a reference for it. Melcombe (talk) 21:00, 28 January 2012 (UTC)[reply]The reference is the quantile article on wiki: .It's in the description and also in the table at the end describing all used estimates for quantiles. The method to round DOWN the L number corresponds to the first formula in the table with [h-1/2] all the other formulas use predominantly [h+1/2] which is rounding UP. Rounding L UP is much more common because it is more natural when the discrete distribution is modeled with a step-wise continuous one. — Preceding unsigned comment added by 205.178.104.178 (talk) 15:10, 30 January 2012 (UTC)[reply]Other Wikipedia articles are not considered reliable sources: see Wikipedia:Reliable sources. If that article does contain a reliable source it could be copied over. Otherwise find a reliable source that can be referenced. Melcombe (talk) 18:05, 30 January 2012 (UTC)[reply]The reliable sourse is right there in the quantile article, citation I see that one is already cited in this article. In that case you'll have no trouble constructing an acceptable addition to the article. Melcombe (talk) 22:21, 1 February 2012 (UTC)[reply]
"There is no universal agreement on choosing the quartile values"
[edit]
For continuous probability distributions there is only one definition. For distributions with gaps between the allowed values, like discrete distributions, there is no agreement because when a quartile falls in a gap, you can position it anywhere in that gap.
The article should point out that the lack of universal agreement on quartiles is for distributions with gaps, like discrete distributions. — Preceding unsigned comment added by 205.178.104.178 (talk) 15:34, 30 January 2012 (UTC)[reply]
Examples
[edit]
As another commenter pointed out, the different definitions of quartiles for discrete distributions differ depending on N mod 4 (the remainder when the number of datums N is divided by 4). The primordial examples, without being trivial, are for:
data = {1,2,3,4} (this is N mod 4 = 0)
data = {1,2,3,4,5} (N mod 4 = 1}
data = {1,2,3,4,5,6} (N mod 4 = 2}
data = {1,2,3,4,5,6,7} {N mod 4 = 3}
Comparing the quartiles for these data lists will reveal all the possible scenarios in which the quartile definitions in the article differ.
The way the quartiles are calculated according to the chosen quartile definition, depends only on N. The actual values obtained depends on the actual data. Choosing numbers like 1,2,3 ... for the data emphasizes that point and hints directly to the reader how a quartile was calculated. For example a quartile of 1.5 would hint that this quartile is in the middle between the first and second number of the list. — Preceding unsigned comment added by 205.178.104.178 (talk) 15:48, 30 January 2012 (UTC)[reply]
Example 2 Method 2
[edit]
Is it just me or is that messed up? Basically, if the median is the average of the two middle numbers, then Methods 1 and 2 should be exactly the same, should they not? That is what I see from my reading of the rules. And, that would mean that the answer for Example 2 Method 2 would be 15, 37.5, 40, just as with Method 1. NumberTheorist (talk) 17:27, 9 May 2012 (UTC)[reply]
1-pass calculation method
[edit]
The calculation methods mentioned in the article are 2-pass methods (first sort all data, then calculate quartile). This method is not feasable for huge amounts of data, which will e.g. not fit into memory (e.g. median age of all human beings). Apparently there are as well 1-pass methods. talks about a 1-pass algorithm based on the piecewise-parabolic (P2) algorithm developed by Jain and Chlamtac (1985). I'd like to see an explanation of such a 1-pass algorithm in the article. --Sebastian.Dietrich (talk) 17:08, 18 January 2013 (UTC)[reply]
Please write a version for Simple English Wikipedia
[edit]
I would appreciate if someone competent (about both quartiles and writing) would compose a version of this article for Simple English Wikipedia so that I might grasp the quartile concept. Perhaps if I read this unclear existing version by sliding my finger slowly along the words and moving my lips, I might eventually come to understand it, but life is short, so – fuck it. —O'Dea (talk) 13:53, 27 May 2013 (UTC)[reply]
Outliers is a separate topic discussed elsewhere
[edit]
There is no good reason for discussing outliers on this page. In particular,
the page makes recommendation about action on finding outliers; this is broadly inappropriate in a discussion about quartiles as well as being inconsistent with Wikipedia policy.
There are few inline citations in teh outliers section;
Outliers are discussed in far more detail on the 'outliers' page;
The outlier criteria given here are commonly used in box plots and need not be repeated here.
I suggest reducing the Outliers section to a comment to the effect that quartiles are used in some outlier screening checks with a reference to those sections. SLR Ellison (talk) 11:59, 1 June 2018 (UTC)[reply]
Method 3 description
[edit]
On method 3: "This always gives the arithmetic mean of Methods 1 and 2"
Method 3 only gives the arithmetic mean of method 1 and method 2 in the 4n+3 case, not the 4n+1 case.
Why isn't there a description of quartile excel functions like quartile.inc function (lowest from 0 and highest from 4) and quartile.exc function (return of a sp. value excluding quartiles 0 and 4). —Preceding unsigned comment added by 174.4.26.61 (talk) 19:21, 7 November 2018 (UTC)[reply]
unclear terminology
[edit]
As of this writing, the article intro says "The first quartile (Q1) is defined as the middle number between the smallest number (minimum) and the median of the data set."
I think both "middle number" is too-ambiguous terminology here, for uninitiated readers. I'm sure that there are students reading this as meaning "the middle of the subrange" instead of "the median of the subrange".
I'm not sure the best terminology here, but I propose replacing "middle number" with something else.
25th (empirical) quartile? Do you mean 1st quartile or 25th percentile? —Preceding unsigned comment added by Haruhiko Okumura (talk • contribs) 02:30, 1 January 2022 (UTC)[reply]
Method 3 makes no sense
[edit]
The method 3 point #2 and #3 say, "If there are (4n+1) data points", and "If there are (4n+3) data points".
However, there are always exactly "n" data points per definition, so there is no way either apply.
Merudo77 (talk) 18:32, 30 January 2022 (UTC)[reply]
Does excel/google really use Method 3 and 4?
[edit]
When I calculate it by hand, it looks like these tools use Method 1 and 2. 45.22.247.39 (talk) 16:56, 19 March 2024 (UTC)[reply]
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20: Electrochemistry
Map: Chemistry - The Central Science (Brown et al.)
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20.1: Oxidation States and Redox Reactions
21790
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Contents
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4. Map: Chemistry - The Central Science (Brown et al.)
5. 20: Electrochemistry
6. 20.1: Oxidation States and Redox Reactions
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Front Matter
1: Introduction - Matter and Measurement
2: Atoms, Molecules, and Ions
3: Stoichiometry- Chemical Formulas and Equations
4: Reactions in Aqueous Solution
5: Thermochemistry
6: Electronic Structure of Atoms
7: Periodic Properties of the Elements
8: Basic Concepts of Chemical Bonding
9: Molecular Geometry and Bonding Theories
10: Gases
11: Liquids and Intermolecular Forces
12: Solids and Modern Materials
13: Properties of Solutions
14: Chemical Kinetics
15: Chemical Equilibrium
16: Acid–Base Equilibria
17: Additional Aspects of Aqueous Equilibria
18: Chemistry of the Environment
19: Chemical Thermodynamics
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22: Chemistry of the Nonmetals
23: Chemistry of Coordination Chemistry
24: Chemistry of Life- Organic and Biological Chemistry
Back Matter
20.1: Oxidation States and Redox Reactions
Last updated Jul 7, 2023
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20: Electrochemistry
20.2: Balanced Oxidation-Reduction Equations
Page ID 21790
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Oxidation States (Numbers)
1. Example 20.1.1
1. Solutions
Oxidation Numbers and Nomenclature
Oxidation-Reduction Reaction Examples
Example 20.1.2: Identifying Oxidized and Reduced Elements
1. Solution
Combination Reactions
Example 20.1.3: Combination Reaction
Decomposition Reactions
Example 20.1.4: Decomposition Reaction
Calculation
Single Replacement Reactions
Example 20.1.5: Single Replacement Reaction
Explanation
Double Replacement Reactions
Example 20.1.6: Double Replacement Reaction
Solution
Combustion Reactions
Disproportionation Reactions
Example 20.1.7: Disproportionation Reaction
Explanation
Summary
References
Contributors and Attributions
Electron transfer is one of the most basic processes that can happen in chemistry. It simply involves the movement of an electron from one atom to another. Many important biological processes rely on electron transfer, as do key industrial transformations used to make valuable products. In biology, for example, electron transfer plays a central role in respiration and the harvesting of energy from glucose, as well as the storage of energy during photosynthesis. In society, electron transfer has been used to obtain metals from ores since the dawn of civilization.
Oxidation States (Numbers)
Oxidation state is a useful tool for keeping track of electron transfers. It is most commonly used in dealing with metals and especially with transition metals. Unlike metals from the first two columns of the periodic table, such as sodium or magnesium, transition metals can often transfer different numbers of electrons, leading to different metal ions (e.g., sodium is generally found as NaA+ and magnesium is almost always MgA 2+, but manganese could be MnA 2+, MnA 3+, and so on, as far as MnA 7+). Oxidation state is a number assigned to an element in a compound according to some rules. This number enables us to describe oxidation-reduction reactions, and balancing redox chemical reactions. When a covalent bond forms between two atoms with different electronegativities the shared electrons in the bond lie closer to the more electronegative atom:
For hydrochloric acid, the hydrogen is positive while the chlorine is negative. The dipole starts at the hydrogen and points towards the chlorine.
The oxidation state of an atom is the charge that results when the electrons in a covalent bond are assigned to the more electronegative atom and is the charge an atom would possess if the bonding were ionic. In HCl (above) the oxidation number for the hydrogen would be +1 and that of the Cl would be -1.
Example 20.1.1
Determine which element is oxidized and which element is reduced in the following reactions (be sure to include the Oxidation State of each):
Zn+2HA+ZnA 2++HA 2
2 Al+3 CuA 2+2 AlA 3++3 Cu
COA 3A 2−+2HA+COA 2+HA 2O
Solutions
Zn is oxidized (Oxidation number: 0 → +2); HA+ is reduced (Oxidation number: +1 → 0)
Al is oxidized (Oxidation number: 0 → +3); CuA 2+ is reduced (+2 → 0)
This is not a redox reaction because each element has the same oxidation number in both reactants and products: O= -2, H= +1, C= +4.
An atom is oxidized if its oxidation number increases, and an atom is reduced if its oxidation number decreases. The atom that is oxidized is the reducing agent, and the atom that is reduced is the oxidizing agent. (Note: the oxidizing and reducing agents can be the same element or compound).
Oxidation Numbers and Nomenclature
Compounds of the alkali (oxidation number +1) and alkaline earth metals (oxidation number +2) are typically ionic in nature. Compounds of metals with higher oxidation numbers (e.g., tin +4) tend to form molecular compounds
In ionic and covalent molecular compounds usually the less electronegative element is given first.
In ionic compounds the names are given which refer to the oxidation (ionic) state
In molecular compounds the names are given which refer to the number of molecules present in the compound
Figure 20.1.1: Example of nomenclature based on oxidation states.| Ionic | Molecular |
--- |
| MgH 2 | magnesium hydride | H 2 S | dihydrogen sulfide |
| FeF 2 | iron(II) fluoride | OF 2 | oxygen difluoride |
| Mn 2 O 3 | manganese(III) oxide | Cl 2 O 3 | dichlorine trioxide |
An oxidation-reduction (redox) reaction is a type of chemical reaction that involves a transfer of electrons between two species. An oxidation-reduction reaction is any chemical reaction in which the oxidation number of a molecule, atom, or ion changes by gaining or losing an electron. Redox reactions are common and vital to some of the basic functions of life, including photosynthesis, respiration, combustion, and corrosion or rusting.
Oxidation-Reduction Reaction Examples
Redox reactions are comprised of two parts, a reduced half and an oxidized half, that always occur together. The reduced half gains electrons and the oxidation number decreases, while the oxidized half loses electrons and the oxidation number increases. Simple ways to remember this include the mnemonic devices OIL RIG, meaning "oxidation is loss" and "reduction is gain," and LEO says GER, meaning "loss of e- = oxidation" and "gain of e- = reduced." There is no net change in the number of electrons in a redox reaction. Those given off in the oxidation half reaction are taken up by another species in the reduction half reaction.
The two species that exchange electrons in a redox reaction are given special names. The ion or molecule that accepts electrons is called the oxidizing agent; by accepting electrons it causes the oxidation of another species. Conversely, the species that donates electrons is called the reducing agent; when the reaction occurs, it reduces the other species. In other words, what is oxidized is the reducing agent and what is reduced is the oxidizing agent. (Note: the oxidizing and reducing agents can be the same element or compound, as in disproportionation reactions).
Figure 20.1.1: A thermite reaction taking place on a cast iron skillet. A thermite reaction, using about 110 g of the mixture, taking place. The cast-iron skillet was destroyed in the process. (CC BY-SA 2.5 generic; Schuyler S via Wikipedia).
A good example of a redox reaction is the thermite reaction, in which iron atoms in ferric oxide lose (or give up) O atoms to Al atoms, producing AlA 2OA 3 (Figure 20.1.1).
FeA 2OA 3(s)+2Al(s)AlA 2OA 3(s)+2Fe(l)
Another example of the redox reaction (although less dangerous) is the reaction between zinc and copper sulfate.
Zn+CuSOA 4ZnSOA 4+Cu
Example 20.1.2: Identifying Oxidized and Reduced Elements
Determine what is oxidized and what is reduced in the following reaction.
Zn+2HA+ZnA 2++HA 2
Solution
The oxidation state of HA+ changes from +1 to 0, and the oxidation state of Zn changes from 0 to +2. Hence, Zn is oxidized and acts as the reducing agent.
The oxidation state of HA+ changes from +1 to 0, and the oxidation state of Zn changes from 0 to +2. Hence, HA+ ion is reduced and acts as the oxidizing agent.
Combination Reactions
Combination reactions are among the simplest redox reactions and, as the name suggests, involves "combining" elements to form a chemical compound. As usual, oxidation and reduction occur together. The general equation for a combination reaction is given below:
A+BAB
Example 20.1.3: Combination Reaction
Equation: HA 2+OA 2HA 2O
Calculation: 0 + 0 → (2)(+1) + (-2) = 0
Explanation:
In this equation both HA 2 and OA 2 are free elements; following Rule #1, their Oxidation States are 0. The product is HA 2O, which has a total Oxidation State of 0. According to Rule #6, the Oxidation State of oxygen is usually -2. Therefore, the Oxidation State of H in HA 2O must be +1.
Decomposition Reactions
A decomposition reaction is the reverse of a combination reaction, the breakdown of a chemical compound into individual elements:
AB→A+B
Example 20.1.4: Decomposition Reaction
Identify the oxidation state of the products and reactant in the decomposition of water:
HA 2OHA 2+OA 2
Calculation
(2)(+1)+(−2)=0→0+0
In this reaction, water is "decomposed" into hydrogen and oxygen. As in the previous example the H 2 O has a total Oxidation State of 0; thus, according to Rule #6 the Oxidation State of oxygen is usually -2, so the Oxidation State of hydrogen in H 2 O must be +1.
Single Replacement Reactions
A single replacement reaction involves the "replacing" of an element in the reactants with another element in the products:
[\ce{A + BC \rightarrow AB + C} \nonumber ]
Example 20.1.5: Single Replacement Reaction
Chlorine gas is a great oxidizing agent and will replace bromide ions from sodium bromide salt.
ClA 2(g)+Na Br―(s)NaCl―(s)+BrA 2(l)
Explanation
In this equation, Br is replaced with Cl, and the Cl atoms in ClA 2 are reduced, while the Br ion in NaBr is oxidized.
Double Replacement Reactions
A double replacement reaction is similar to a single replacement reaction, but involves "replacing" two elements in the reactants, with two in the products:
[\ce{AB + CD \rightarrow AD + CB} \nonumber ]
Example 20.1.6: Double Replacement Reaction
The reaction of gaseous hydrogen chloride and iron oxide is a double replacement reaction. Write the expected reaction for this chemistry equation.
Solution
FeA 2OA 3+6 HCl2 FeClA 3+3HA 2O
In this equation, Fe and H trade places, and oxygen and chlorine trade places.
Combustion Reactions
Combustion reactions almost always involve oxygen in the form of OA 2, and are almost always exothermic, meaning they produce heat. Chemical reactions that give off light and heat and light are colloquially referred to as "burning."
CA xHA y+OA 2COA 2+HA 2O
Although combustion reactions typically involve redox reactions with a chemical being oxidized by oxygen, many chemicals "burn" in other environments. For example, both titanium and magnesium burn in nitrogen as well:
2Ti(s)+NA 2(g)2TiN(s)
3Mg(s)+NA 2(g)MgA 3NA 2(s)
Moreover, chemicals can be oxidized by other chemicals than oxygen, such as ClA 2 or FA 2; these processes are also considered combustion reactions
Disproportionation Reactions
A single substance can be both oxidized and reduced in some redox reactions. These are known as disproportionation reactions, with the following general equation:
2 AAA+n+AA−n
where n is the number of electrons transferred. Disproportionation reactions do not need begin with neutral molecules, and can involve more than two species with differing oxidation states (but rarely).
Example 20.1.7: Disproportionation Reaction
Disproportionation reactions have some practical significance in everyday life, including the reaction of hydrogen peroxide, HA 2OA 2 poured over a cut. This is a decomposition reaction of hydrogen peroxide (catalyzed by the catalaseenzyme) that produces oxygen and water. Oxygen is present in all parts of the chemical equation and as a result it is both oxidized and reduced. The reaction is as follows:
2HA 2OA 2(aq)2HA 2O(l)+OA 2(g)
Explanation
On the reactant side, H has an Oxidation State of +1 and O has an Oxidation State of -1, which changes to -2 for the product HA 2O (oxygen is reduced), and 0 in the product OA 2 (oxygen is oxidized).
Redox Reactions: Redox Reactions(opens in new window) [youtu.be]
Summary
Oxidation signifies a loss of electrons and reduction signifies a gain of electrons. Balancing redox reactions is an important step that changes in neutral, basic, and acidic solutions. The types of redox reactions: Combination and decomposition, Displacement reactions (single and double), Combustion, Disproportionation. The oxidizing agent undergoes reduction and the reducing agent undergoes oxidation.
References
Petrucci, et al. General Chemistry: Principles & Modern Applications. 9th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2007.
Sadava, et al. Life: The Science of Biology. 8th ed. New York, NY. W.H. Freeman and Company, 2007
Contributors and Attributions
Christopher Spohrer (UCD), Christina Breitenbuecher (UCD), Luvleen Brar (UCD)
Chris P Schaller, Ph.D., (College of Saint Benedict / Saint John's University)
20.1: Oxidation States and Redox Reactions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by LibreTexts.
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20: Electrochemistry
20.2: Balanced Oxidation-Reduction Equations
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3451 | https://www.youtube.com/watch?v=mDaeyiHLY40 | [SL]Solving Logarithmic Equations with Linear and Quadratic Arguments
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Transcript:
welcome to help your matcom in this video we're gonna look at examples of solving logarithmic equations when checking your work is actually part of the problem and I say it's part of the problem and not just like an optional step because if we look at a logarithmic equation we have log base B of X is equal to Y we may recall that the argument which is usually where the variables found in logarithmic equations the argument must be positive so X must be greater than zero and with some logarithmic equations we might end up with values that would provide a negative argument which is not a thing that is allowed so we need to make sure that we're checking all possible solutions and checking to make sure they don't actually provide a negative argument so in our first two examples let's see we need to combine into one logarithm because we have two logarithms and that's that's not good since they have the same base there's no base given so that's assumed to be base 10 we can combine them using multiplication this is same base multiplication so we can write this as log base 10 and then we would multiply the arguments two times five D is ten D is equal to three now what you can do if you want is you can put this in exponential form that we ten cubed is equal to 10 times D 10 cubed is 1000 equals 10 D divide both sides by 10 and we end up with 100 is equal to d now we need to plug in and make sure that D really is the solution so if I plug in D up here that would be 5 times 100 which would be 500 so we would have log of 500 plus log of 2 equals 3 that's okay because we have a positive argument here and we have a positive argument here so we're good to go and it's confirmed that D in fact does equal 100 that's when we need to check is immediately at the beginning of the problem that's where we would see some sort of issue let's see we might see one here so we have log base 2 of x plus 2 plus log base 2 of x equals 3 we want to get this to one logarithm since they're separated using addition that means we want to multiply the arguments so this will be log base 2 of x times X times plus x times - which will be x squared plus 2x so we multiply that means I need to distribute X 2x + 2 - and this equals 3 now that I have just one single log now I can put this in exponential form that would be 2 cubed equals x squared plus 2x since this is a quadratic equation I do want to set it equal to 0 I couldn't use completing the square but I think this is gonna be factorable usually these are so I just want to see if it's factorable I'll set it equal to 0 2 cubed is 8 so I'm going to subtract 8 from both sides and we get 0 equals x squared plus 2x minus 8 target product is 8 target's um sorry target product is negative 8 target sum is 2 this is going to factor into X plus 4 times X minus 2 now that we have two factors whose product is zero one of the factors must be 0 if this is the one that equals 0 then x equals negative 4 if this is the one that equals 0 then we have x equals 2 we need to check both of these to see if they are in fact legitimate solutions so we'll start with negative 4 since that was the first one I found and when I go to plug it in I get log base 2 of negative 4 plus 2 plus log base 2 of negative 4 I can stop right here there's a negative argument right there not good it gets thrown out it was not actually a solution okay done with that one and you can see what would happen here once you go to combine the logarithms the negative times the negative will turn that argument positive but there's a block from the beginning because negative 4 was never going to be an allowable value for X if we just check 2 we get log base 2 of 2 plus 2 plus log base 2 of 2 equals 3 this one looks good because here we have log base 2 of 4 plus log base 2 of 2 yeah I think this one's gonna be solid we'll just finish this log base 2 of 4 that's 2 because it's asking what power of 2 is 4 that's the second power and log base 2 of 2 is 1 and 1 plus 2 plus 1 does equal 3 so we have one into this and it's x equals to that x equals negative 4 got thrown out two more examples here this one's good to go we're ready to put it straight into exponential form that would be 3 to the 0 equals a squared minus 2 3 to the 0 is 1 so we get 1 equals a squared minus 2 from here it's a quadratic it's it's up to you in this case because there's no middle term so maybe you want to add 2 to both sides normally with quadratics you want we want to set it equal to 0 but since we don't have just an a term we can do this and take the square root of both sides and then we get plus or minus the square root of 3 to get the a squared to undo that we would take the square root so I end up with the positive or negative square root of 3 is equal to a if we want to we can approximate I don't think we're gonna want to do that I think we're just gonna want to see if these are in fact both solutions so then from here we would want to check let's start with the negative square root of 3 so is it okay that log base 3 and then we have negative root 3 squared minus 2 equals 0 well this ends up becoming positive 3 so that's okay 3 minus 2 is positive 1 and we end up with log base 3 of 1 equals 0 that's totally fine and then I notice if we plug in positive square root of 3 we will also end up with a 3 here so both are legitimate solutions so here we can say a is equal to negative square root of 3 or positive square root of 3 in the next example and perhaps the final yes in the final example again we're good to go in terms of there's only one single logarithm so we can put it right in exponential form this would be written as 4 to the 1 equals 6w minus 24 and now this is a nice linear equation so with linear equations we want to move everything away from the variable we'll start by adding 24 to both sides this gives us 28 equals 6w now to get W by itself we'll divide both sides by 6 6 doesn't go evenly into 28 but they are both divisible by 2 that would be 14 over 3 which is equal to W so you could either leave it as 14 or three or you could convert it into a mixed number three goes in fourteen four whole times with two thirds left over so either way is fine we should check our work and make sure we don't end up with a negative argument so let's check I'm going to go back to the improper fraction I already put a box around it I am sure that this is gonna work apparently okay I'm gonna check down here because there's more space this is log base four of six times fourteen over three minus 24 does that equal 1 this would be 1 2 this would be 28 minus 24 so we would have log base 4 of a positive argument and we're good to go and log base 4 4 does in fact equal 1 so that's good thing I was sure because that was the answer thank you for stopping by |
3452 | https://www.freemathhelp.com/feliz-isosceles/ | Isosceles Triangles - Free Math Help
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There are many types of triangles in the world of geometry. There is a special triangle called an isosceles triangle. In an isosceles triangle, the base angles have the same degree measure and are, as a result, equal (congruent). Similarly, if two angles of a triangle have equal measure, then the sides opposite those angles are the same length. The easiest way to define an isosceles triangle is that it has two equal sides.
In an isosceles triangle, we have two sides called the legs and a third side called the base. The angle located opposite the base is called the vertex.
Sample A:
The vertex angle B of isosceles triangle ABC is 120 degrees. Find the degree measure of each base angle.
Solution:
(1) Let x = the measure of each base angle.
(2) Set up an equation and solve for x.
base angle + base angle + 120 degrees = 180 degrees
x + x + 120 degrees = 180 degrees
2x + 120 = 180
2x = 180 - 120
2x = 60
x = 60/2
x = 30
Each base angle of triangle ABC measures 30 degrees.
Sample B:
In isosceles triangle RST, angle S is the vertex angle. Base angles R and T both measure 64 degrees. Find the degree measure of the vertex angle S.
Solution:
(1) Let x = measure of vertex angle S.
(2) Set up an equation and solve for x.
base angle + base angle + vertex angle S = 180 degrees
64 degrees + 64 degrees + x = 180 degrees
128 + x = 180
x = 180 - 128
x = 52
The measure of vertex angle S in triangle RST is 52 degrees.
Sample C:
The degree measure of a base angle of isosceles triangle XYZ exceeds three times the degrees measure of the vertex Y by 60. Find the degree measure of the vertex angle Y. Notice that it's hard to draw a picture without knowing which angles are largest.
We need to make an equation out of this problem, so let's figure out what it's trying to tell us. First we read "The degree measure of a base angle", so let's start with X=
Our equation so far: X=
Now we see "exceeds three times... Y... by 60", which means 3Y + 60.
Our equation now: X = 3Y + 60
Since we know that X = Z because it is an isosceles triangle, then we can solve for the measures of all the angles.
base angle + base angle + vertex = 180
X + Y + Z = 180
(3Y + 60) + Y+ (3Y + 60) = 180
7Y + 120 = 180
7Y = 60
Y = 60/7
Y = 8.57 degrees
The vertex angle Y of triangle XYZ equals 8.57 degrees.
Lesson provided by Mr. Feliz
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3453 | https://www.wyzant.com/resources/answers/559169/how-do-i-prove-this-statement-using-the-axioms-of-integer-arithmetic-if-a-0 | WYZANT TUTORING
Jay T.
How do I prove this statement using the axioms of integer arithmetic: If a < 0, b > 0, then ab < 0?
1 Expert Answer
Paul M. answered • 12/08/18
BS in Mathematics, MD
I think the following lines will suffice as proof:
a<0 => -a>0 by the exclusion,if a is not equal to 0 then either a>0 or -a>0
-ab >0 (because the product of 2 numbers both > 0 is > 0)
and then ab<0 for the same reason as in the first line
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3455 | https://www.geeksforgeeks.org/java/overriding-tostring-method-in-java/ | Overriding toString() Method in Java - GeeksforGeeks
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Overriding toString() Method in Java
Last Updated : 23 Jul, 2025
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Java being object-oriented only deals with classes and objects so do if we do require any computation we use the help of object/s corresponding to the class. It is the most frequent method of Java been used to get a string representation of an object. Now you must be wondering that till now they were not using the same but getting string representation or in short output on the console while using System.out.print. It is because this method was getting automatically called when the print statement is written. So this method is overridden in order to return the values of the object which is showcased below via examples.
Example 1:
Java
```Java
// file name: Main.java
class Complex {
private double re, im;
public Complex(double re, double im) {
this.re = re;
this.im = im;
}
}
// Driver class to test the Complex class
public class Main {
public static void main(String[] args) {
Complex c1 = new Complex(10, 15);
System.out.println(c1);
}
}
```
// file name: Main.java
class Complex { private double re, im; public Complex(double re, double im) { this.re = re; this.im = im; }} // Driver class to test the Complex class public class Main { public static void main(String[] args) { Complex c1 = new Complex(10, 15); System.out.println(c1); }}
OutputComplex@214c265e
Output Explanation:The output is the class name, then 'at' sign, and at the endhashCodeof the object. All classes in Java inherit from the Object class, directly or indirectly (See point 1 of this). The Object class has some basic methods like clone(), toString(), equals(),.. etc. The default toString() method in Object prints "class name @ hash code". We can override the toString() method in our class to print proper output. For example, in the following code toString() is overridden to print the "Real + i Imag"form.
Example 2:
Java
```Java
// Java Program to illustrate Overriding
// toString() Method
// Class 1
public class GFG {
// Main driver method
public static void main(String[] args) {
// Creating object of class1
// inside main() method
Complex c1 = new Complex(10, 15);
// Printing the complex number
System.out.println(c1);
}
}
// Class 2
// Helper class
class Complex {
// Attributes of a complex number
private double re, im;
// Constructor to initialize a complex number
// Default
// public Complex() {
// this.re = 0;
// this.im = 0;
// }
// Constructor 2: Parameterized
public Complex(double re, double im) {
// This keyword refers to
// current complex number
this.re = re;
this.im = im;
}
// Getters
public double getReal() {
return this.re;
}
public double getImaginary() {
return this.im ;
}
// Setters
public void setReal(double re) {
this.re = re;
}
public void setImaginary(double im) {
this.im = im;
}
// Overriding toString() method of String class
@Override
public String toString() {
return this.re + " + " + this.im + "i";
}
}
```
// Java Program to illustrate Overriding
// toString() Method// Class 1 public class GFG { // Main driver method public static void main(String[] args) { // Creating object of class1 // inside main() method Complex c1 = new Complex(10, 15); // Printing the complex number System.out.println(c1); }}// Class 2// Helper class class Complex { // Attributes of a complex number private double re, im; // Constructor to initialize a complex number // Default // public Complex() { // this.re = 0; // this.im = 0; // } // Constructor 2: Parameterized public Complex(double re, double im) { // This keyword refers to // current complex number this.re = re; this.im = im; } // Getters public double getReal() { return this.re; } public double getImaginary() { return this.im ; } // Setters public void setReal(double re) { this.re = re; } public void setImaginary(double im) { this.im = im; } // Overriding toString() method of String class @Override public String toString() { return this.re + " + " + this.im + "i"; }}
Output10.0 + 15.0i
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3456 | https://www.youtube.com/watch?v=EVuRJxVyme4 | Solve the Differential Equation y' - 5y = 0
The Math Sorcerer
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Solve the Differential Equation y' - 5y = 0
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Transcript:
hello in this problem we're going to solve this very simple differential equation and we're actually going to solve it in a very simple way so it's y prime minus 5y equals 0. so this is a first order linear differential equation so you can solve it using the method that you learn when you solve you know first order linear differential equations we're going to solve it using another method so recall that when you're trying to solve higher order des the first step is to write down the characteristic or auxiliary equation basically you write down a polynomial so because this is the first derivative we write down r to the first power so just r and then here y is the zeroth derivative so we don't write down the y and we set this equal to zero and we solve so here we get r equals five and so that's the solution to this characteristic or auxiliary equation because we have a single real root the answer is of the form y equals c e to the r x where r is the root so in this case it's y equals c e to the 5x and that would be the solution to the differential equations so you can actually do it this way which is really quite nice normally you apply this to higher order des but i wanted to make this video to show you that you can apply it to lower order ones as well for example if you had y prime plus two y equals zero same thing you would get r plus two equals zero r equals negative two and then boom you've got the answer c equals e to the negative 2x so very quick way to solve very simple differential equations that you learn in a differential equations class i hope this video has been helpful good luck |
3457 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3955166/ | Huntington’s disease: underlying molecular mechanisms and emerging concepts - PMC
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Trends Biochem Sci
. Author manuscript; available in PMC: 2014 Aug 1.
Published in final edited form as: Trends Biochem Sci. 2013 Jun 12;38(8):378–385. doi: 10.1016/j.tibs.2013.05.003
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Huntington’s disease: underlying molecular mechanisms and emerging concepts
John Labbadia
John Labbadia
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, Richard I Morimoto
Richard I Morimoto
1 Department of Molecular Biosciences, Rice Institute for Biomedical Research, Northwestern University, Evanston, IL 60208, USA
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1 Department of Molecular Biosciences, Rice Institute for Biomedical Research, Northwestern University, Evanston, IL 60208, USA
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Corresponding author: Richard I. Morimoto Department of Molecular Biosciences, Rice Institute for Biomedical Research, 2205 Tech Drive, Hogan 2-100, Northwestern University, Evanston, IL 60208 Phone: 847-491-3340 Fax: 847-491-4461 r-morimoto@northwestern.edu
Issue date 2013 Aug.
© 2013 Elsevier Ltd. All rights reserved.
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PMCID: PMC3955166 NIHMSID: NIHMS494125 PMID: 23768628
The publisher's version of this article is available at Trends Biochem Sci
Abstract
Huntington’s disease (HD) is a progressive neurodegenerative disorder for which no disease modifying treatments exist. Many molecular changes and cellular consequences that underlie HD are observed in other neurological disorders suggesting that common pathological mechanisms and pathways may exist. Recent findings have enhanced our understanding of the way cells regulate and respond to expanded polyglutamine proteins such as mutant huntingtin. These studies demonstrate that in addition to effects on folding, aggregation, and clearance pathways, a general transcriptional mechanism also dictates the expression of polyglutamine proteins. Here we summarize the key pathways and networks that are important in HD in the context of recent therapeutic advances and highlight how their interplay may be of relevance to other protein folding disorders.
Keywords: Huntington’s disease, molecular mechanisms, therapeutic approaches
Huntington’s disease: one gene many consequences
Huntington’s disease (HD) is an autosomal dominant neurodegenerative disorder that affects approximately 5 – 10 individuals per 100,000. Individuals typically suffer from progressive motor and cognitive impairments, loss of self and spatial awareness, depression, dementia, and increased anxiety over the course of 10 – 20 years before death. Currently, treatment is limited to suppressing chorea, the involuntary, irregular movements of the arms and legs that accompanies HD, and battling the mood altering aspects of the disorder with no disease modifying treatments available .
At the molecular level, HD is caused by a CAG tri-nucleotide repeat expansion within exon 1 of the HTT gene. In affected individuals, the number of CAG repeats expands from the normal population range (between 16 and 20 repeats) to > 35 repeats [1, 2]. This gives rise to an elongated polyglutamine tract at the amino terminus of the translated huntingtin (HTT) protein that is associated with protein aggregation and a gain-of-function toxicity .
Mutant huntingtin (mHTT) is highly aggregation prone and the formation of cytoplasmic aggregates and nuclear inclusions throughout the brain is one of the most striking hallmarks of HD [4, 5]. Polyglutamine inclusions contain highly ordered amyloid fibres with high β-sheet content and low detergent solubility; they also sequester numerous other proteins, including factors important for transcription and protein quality control, suggesting that their presence is deleterious to cellular function and contributes to a complex loss-of-function phenotype . Several lines of evidence implicate small oligomeric forms of mHTT as the most toxic species and propose that the formation of large inclusions may represent an alternative coping strategy in which mHTT is partitioned into a less pervasive structure . Aggregate formation is a complex multi-step process in which mHTT monomers assemble into a range of intermediate oligomeric species before inclusions are formed. This process is influenced by the amino acid sequences flanking the polyglutamine stretch, post-translational modifications of mHTT, and levels of molecular chaperones [8-12]. The spectrum of oligomeric conformations adopted by mHTT has made it challenging to understand the pathogenic role of each species as mHTT monomers, oligomers, and large inclusions can co-exist and disrupt multiple cellular pathways and influence disease progression. Additionally, extracellular polyglutamine aggregates can be internalised by cells to promote polyglutamine aggregation. This raises the intriguing possibility of mHTT spreading between cells and regions during disease progression .
Despite its monogenic nature, HD pathogenesis is incredibly complex. The HTT interactome is comprised of proteins involved in transcription, DNA maintenance, cell cycle regulation, cellular organization, protein transport, energy metabolism, cell signalling, and protein homeostasis (proteostasis) . Given this diversity of molecular interactions, it is unsurprising that wide-scale destabilization of the proteome and subsequent disruption of multiple cellular processes occurs in the presence of mHTT (Figure 1).
Figure 1. Major cellular pathways disrupted in Huntington’s disease.
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A diverse array of cellular processes is disturbed by the presence of mHTT. Here we depict a neuron and demonstrate the major sites of molecular disruption caused by the presence of mHTT. Roman numerals indicate which zoomed-in views (surrounding boxes) correspond to particular areas/processes in the cell. We provide a simplified over-view of the pathogenic pathways in HD discussed in our review, specifically, (i) transcriptional dysregulation of basal and inducible gene expression, (ii) impaired protein degradation, (iii) altered protein folding, (iv) disrupted synaptic signalling, and (v) perturbed energy metabolism through altered mitochondrial maintenance and localization. Pathway dysfunction can arise from direct or indirect interference of key components by soluble, oligomeric and/or aggregated mHTT. While we represent each pathway as an insular process, in reality, disruptions in one pathway likely influence HD pathogenesis (at least in part) by exacerbating dysfunction of other key events.
Recent advances in our understanding of mHTT synthesis, processing, aggregation and toxicity have suggested a number of therapeutic approaches, several of which have shown some promise against HD. Furthermore, despite being caused by unrelated proteins with distinct interactomes and unique expression patterns, other polyglutamine disorders, Alzheimer’s disease (AD), Parkinson’s disease (PD) and Amyotrophic lateral sclerosis (ALS) all share characteristics with HD (Box 1), suggesting that common genetic modifiers of neurodegeneration exist and could be targeted as a potential panacea for neurological disorders [3, 6, 15]. Here, we highlight recent advances in HD research and address how these findings might further our understanding of other neurodegenerative diseases.
Box 1. Protein conformational disease.
HD is one of nine inherited neurodegenerative disorders caused by an expansion of glutamine residues in the causative protein, the others being spinocerebellar ataxias (SCA) 1, 2, 3, 6, 7, and 17, spinobulbar muscular atrophy (SBMA), and dentatorubral-pallidoluysian atrophy (DRPLA) . Toxicity in these disorders stems primarily from a gain-of-function conferred by the polyglutamine stretch, the pathogenic length of which is disease-specific. All nine disorders arise from aberrant protein folding as a result of the polyglutamine expansion and can therefore be thought of as protein conformational diseases . Interestingly, other neurodegenerative diseases such as Alzheimer’s disease (AD), Parkinson’s disease (PD), and Amyotrophic lateral sclerosis (ALS) are also characterized by the presence of misfolded and aggregated proteins, specifically extracellular amyloid beta plaques and intracellular tau tangles in AD, alpha synuclein lewy bodies in PD, and SOD-1 or TDP-43 aggregates in ALS [3, 6]. Despite significant differences in pathology, these disorders share similarities with HD such as late onset of disease, neuronal dysregulation, altered energy metabolism and global changes in gene expression. This suggests that the chronic expression of misfolded proteins may cause progressive neuronal toxicity through common mechanisms and pathways.
Altered neural circuitry underlies cognitive and molecular abnormalities in HD
Perturbed neuronal activity underlies the cognitive and physical decline observed in HD patients [16, 17]. The expression of genes important for calcium homeostasis, neuronal differentiation, neuronal survival and neurotransmission are reduced early in HD . In addition, mHTT dramatically impairs neurotransmitter release at pre-synaptic junctions by physically impeding axonal transport and by reducing the efficiency with which synapse-bound cargo can be loaded onto microtubules [18, 19].
While the presence of mHTT is deleterious to many neuronal sub-types, medium spiny neurons (MSNs) of the striatum exhibit enhanced vulnerability. This observation has been attributed to a number of factors, including reduced neurotrophin availability, MSN-specific SUMOylation of mHTT mediated by the small GTP binding protein Rhes, and glutamate receptor mediated excitotoxic cell death (Box 2) [20-22].
Box 2. Glutamatergic excitotoxicity.
Glutamate is the most abundant excitatory neurotransmitter in the central nervous system and is associated with synaptic plasticity and learning. Glutamate binds to and activates NMDA receptors, which results in calcium influx to the cell. Normally this interaction is transient, however, an excess of extracellular glutamate can lead to continuous stimulation of NMDA receptors and neuronal death in a process termed excitotoxicity [16, 83]. Excitotoxic cell death was one of the first mechanisms proposed to explain the selective vulnerability of medium sized spiny neurons (MSNs) in HD . The first evidence supporting this claim came from the development of similar biochemical and behavioural symptoms in animals injected with the NMDAR agonist quinolinic acid . In contrast to the relatively spared interneurons, MSNs express high levels of NMDARs, perhaps providing some explanation for the increased susceptibility of these cells to mHTT . Given that excitotoxicity has also been described as a feature of AD, PD, and ALS, it is possible that excitotoxicity is central to many neurological disorders.
Glutamatergic excitotoxicity of MSNs through aberrant N-methyl D-aspartate receptor (NMDAR) activity is proposed to be a major component of HD. NMDAR activity can be modified by neuroactive metabolites of the tryptophan degradation pathway, particularly kynurenic acid (KA) (an NMDA antagonist) and quinolinic acid (QA) (an NMDA agonist) [16, 23]. The abundance of these molecules in the bloodstream is regulated by the enzyme kynurenine-3-monoxoygenase (KMO), the inhibition of which leads to elevated levels of the neuroprotective metabolite KA in the brain. Manipulation of the tryptophan pathway or KMO activity has been shown to abrogate disease in diverse models of HD, AD, and PD [23-25]. Importantly, the novel KMO inhibitor JM6 successfully suppresses disease in rodent models of HD and AD without entering the brain, thereby providing support that modifying metabolites in the blood can markedly influence neurological dysfunction and that brain permeability may not be a prerequisite for small molecule treatments of HD .
Alternatively, excitotoxicity can be directly suppressed with small molecule NMDAR antagonists. Treatment with one such molecule, Memantine, has contrasting effects on HD progression in mice . At high doses, Memantine inhibits both synaptic and extra-synaptic NMDAR activity and exacerbates disease progression, whereas at low doses that selectively inhibit extra-synaptic NMDAR activity it can suppress mHTT toxicity . The suppression of mHTT toxicity via low dose treatment with Memantine is proposed to occur through alterations to the cellular protein quality control machinery and supports observations from Caenorhabditis elegans (C. elegans) that neuronal signalling regulates proteostasis and polyglutamine toxicity throughout the organism [20, 26-28].
Together, these findings highlight the opposing roles of synaptic and extra-synaptic NMDAR activity in HD and suggest that changes in neuronal circuitry could result in non-autonomous cellular dysfunction in neuronal and non-neuronal cells as a result of altered protein homeostasis. This could explain recent reports that striatal projection neurons present in neural grafts die when transplanted into HD patients even though the grafted cells do not express mHTT. However, other cell non-autonomous mechanisms may also explain these observations . While it is clear that changes in brain physiology and neuronal circuitry underlie behavioural abnormalities in HD, the precise molecular mechanisms responsible for cellular dysregulation are more nebulous. Here, we attempt to summarize the major molecular pathways linked to neuronal dysfunction in HD.
Mitochondrial dysfunction and impaired energy metabolism
Numerous observations support a role for mitochondrial dysregulation in HD pathogenesis. For instance, mHTT associates with the mitochondrial outer membrane and leads to an impairment of electron transport chain (ETC) complexes II and III, an observation that correlates with depletion of the intra-cellular ATP pool and increased reactive oxygen species (ROS) . The mitochondrial tricarboxylic acid (TCA) cycle enzyme aconitase is particularly susceptible to superoxide-mediated inactivation suggesting that generation of ROS through disruption of the ETC may further restrict ATP production through inhibition of the TCA cycle .
Another example of mitochondrial dysregulation in HD involves mitochondrial trafficking. Retrograde and anterograde mitochondrial trafficking along axons is impeded by mHTT leading to disruption of mitochondrial maintenance and reduced deposition of mitochondria at sites with high energy demand such as synapses . Furthermore, mHTT has been shown to impair PPAR-γ co-activator-1α (PGC-1α) mediated expression of genes that regulate mitochondrial biogenesis [32-34]. Early mitochondrial fragmentation has also been reported in HD and has recently been proposed to occur through the GTPase dynamin related protein-1 (DRP-1). In support of this, reducing DRP-1 GTPase activity restores aberrant mitochondrial fission, mitochondrial transport, and improves phenotype in HD mice . These observations highlight the potential significance of perturbed mitochondrial function in HD and suggest a mechanism by which mHTT causes neuronal dysfunction by disrupting energy metabolism and promoting oxidative damage.
Perturbations in mitochondrial maintenance, localization, and activity have also been reported for ALS, AD, and PD [36, 37]. This is particularly intriguing as a deficit in cellular energy could have far reaching consequences, not only in terms of neuronal signalling, but for maintenance of a functional proteome in general, as a progressive depletion of ATP levels could impede core activities of the proteostasis and transcriptional networks.
Transcriptional dysregulation in HD
The expression of mHTT has global effects on the transcriptome suggesting that transcriptional dysregulation is a key feature of HD pathogenesis . mHTT interacts with, and disrupts, major components of the general transcriptional machinery, affecting both general promoter accessibility and recruitment of RNA polymerase II . Studies in pre-symptomatic HD brains have shown that soluble mHTT oligomers interact with and impede the function of specificity protein 1 (SP1), TATA box binding protein (TBP), the TFIID subunit TAFII130, the RAP30 subunit of the TFIIF complex, and the CAAT box transcription factor NF-Y, all of which are important mediators of general promoter accessibility and transcription initiation [39-43].
The expression of mHTT also disrupts the activity of histone acetyl transferases (HATs), such as CBP/p300 and p300/CBP associated factor (PCAF), which results in histone hypoacetylation and increased heterochromatin formation . Strategies that utilise histone deacetylase (HDAC) inhibitors to correct transcriptional dysregulation by restoring or enhancing histone acetylation have been shown to ameliorate mHTT toxicity in flies and mice, thereby supporting a central role for transcriptional dysregulation in HD . However, because of the broad action of many HDAC inhibitors and the promiscuous nature of HDAC activity, the precise mechanisms by which these molecules influence mHTT toxicity remain unclear . Yet, genetic studies in flies and worms suggest HDACs 1 and 3 are required for mHTT toxicity and could be the primary targets of HDAC inhibitors [46, 47].
Impaired protein homeostasis in neurodegenerative disease
Although aberrant neuronal signalling, energy production, and gene expression underlie the molecular basis of HD, ultimately cell function is dictated by the functional properties of the proteome. Therefore, to fully describe HD it is essential that we understand and integrate how the dynamic properties of the proteome are re-organized upon expression of mHTT.
Under normal conditions, proteome integrity is maintained by the proteostasis network (PN), the main effectors of which are molecular chaperones and clearance machineries (Box 3) . Intriguingly, chronic expression of expanded polyglutamine peptides results in an age-dependent collapse of proteostasis as evidenced by increased aggregation and mislocalization of meta-stable proteins [49, 50]. Recent proteomic analysis of the mHTT interactome has revealed that members of the heat shock protein 90kDa (HSP90), TCP-1 ring complex (TRiC), HSP70, and DNAJ chaperone families all associate with mHTT . Moreover, levels of HSP70 and DNAJ chaperones are progressively reduced in brain tissues of HD mice through a combination of sequestration and transcriptional dysregulation [43, 51]. Proteostasis collapse also occurs in the presence of mutant superoxide dismutase-1 (SOD1), an aggregation prone protein that is the primary cause of familial ALS, and expression of synthetic amyloid forming peptides suggesting that proteostasis collapse may be a general feature of protein folding disorders [52, 53]. These observations support a model where the chronic expression of aggregation-prone proteins, such as mHTT, titrates chaperones away from clients and leads to global disruption of the proteome.
Box 3. The proteostasis network.
The transition from nascent polypeptide to functional tertiary structure is an incredible challenge in the context of the intracellular milieu. Errors in translation coupled with the disordered structure of newly synthesized peptides promote inappropriate intra- and inter-molecular interactions within the cell that can lead to protein mislocalization, aggregation, cell dysfunction, and death; therefore, the ability to efficiently maintain proteostasis is essential . Proteome integrity is maintained through the concerted action of molecular chaperones, which are proteins that facilitate the folding of nascent polypeptide chains, recognise and re-fold misfolded proteins, disassemble protein aggregates, and direct clients to distinct sub-cellular locations . In addition, irrevocably damaged or misfolded proteins must be selectively degraded as and when required. This clearance is achieved via two major pathways, the ubiquitin proteasome system (UPS) and autophagy. Old or irreversibly damaged or misfolded proteins are recognised by chaperone/co-chaperone complexes, poly-ubiquitylated by E3 ligases, and transported to the proteasome, a large multi-subunit complex that proteolytically degrades ubiquitylated substrates . In contrast, bulkier cargo, such as protein aggregates or organelles, are too large to pass through the proteasomal pore. Instead, they are sequestered by the formation of an autophagosome, which is then transported along microtubules and fused with the lysosome where the cargo is degraded [62, 84]. Through these mechanisms, cells successfully balance protein synthesis, folding, trafficking, and degradation to maintain proteostasis. The network of proteins involved in this is collectively referred to as the proteostasis network (PN) [48, 84].
The PN is able to maintain proteome integrity in response to fluctuating intra- and extracellular conditions. However, more extreme conditions that cause acute, wide-scale disruption of protein folding (e.g. elevated temperature, altered pH, increased reactive oxygen species) can place demands on the PN that cannot be met. In response to these insults, the composition of the PN is dramatically altered by the activation of stress response pathways such as the heat shock response (HSR), unfolded protein response (UPR) and oxidative stress response (OSR) . These pathways are regulated by the transcription factors HSF1 (HSR), DAF-16/FOXO3a (HSR and OSR), SKN-1/NRF2 (OSR), ATF-6, PERK and XBP1 (UPR). When activated, these stress transcription factors act in a concerted manner and lead to the up-regulation of molecular chaperones and other pro-survival genes, thereby enhancing protein folding capacity and promoting cell survival.
In support of this hypothesis, restoration or enhancement of protein folding capacity through chaperone over-expression or enhancement of chaperone gene regulatory pathways suppresses mHTT toxicity in multiple models of HD . These effects have been attributed to the suppression of aggregate formation, enhanced mHTT degradation, and the partitioning of mHTT into less toxic structures . Further support comes from a screen of HSP70 and DNAJ chaperones for suppressors of polyglutamine aggregation in mammalian cells that identified the DNAJB sub-class of molecular chaperones (particularly DNAJB2a, B6b, and B8) as potent inhibitors of mHTT aggregation, with DNAJB2a over-expression also found to suppress aggregation in HD mice [55, 56]. DNAJ chaperones are generally considered to be co-chaperones for the main effector of protein re-folding, HSP70; however, DNJB6b and DNAJB8 appear to suppress polyglutamine aggregation independent of HSP70 suggesting that mHTT can be targeted by several different molecular chaperone machines, each of which could have distinct effects on the folding and aggregation state of mHTT.
The notion that novel folding pathways can influence polyglutamine toxicity is further supported by recent findings that the gene moag-4 (encoding a small protein of unknown function), identified in a C. elegans RNAi screen for modifiers of aggregation (MOAG), influences polyglutamine aggregation independently of the proteasome or autophagy and without activation of stress response pathways or up-regulation of molecular chaperones . While a precise role for moag-4 in the absence of disease causing proteins is unknown, the presence of moag-4 appears to promote a conformational change in polyglutamine monomers that facilitates their assembly into large aggregate species via the formation of compact oligomeric structures . The involvement of moag-4 in polyglutamine toxicity is conserved across species, as RNAi of the human orthologues of moag-4, SERF1A and SERF2, suppresses mHTT aggregation and toxicity in neuronal cells. Furthermore, moag-4 deletion in worms also suppresses the toxicity of amyloid-β (Aβ) and α-synuclein, two aggregation prone proteins central to AD and PD pathogenesis respectively, suggesting that moag-4 could have a role in multiple neurodegenerative disorders
These observations demonstrate that numerous protein quality control pathways can suppress mHTT toxicity and that a number of novel protein quality control mechanisms can be targeted for therapeutic gain.
Impaired protein degradation pathways in HD
The ubiquitin proteasome system (UPS) (Box 3) has been a focus of study in HD since mHTT inclusions were first identified as ubiquitin-positive . An accumulation of ubiquitin chains occurs in brain tissue from HD patients and HD mice ; however, the mechanism by which mHTT causes disruption of the UPS is unclear. Recent findings suggest that the accumulation of ubiquitin chains in HD is not a result of direct proteasome inhibition by mHTT oligomers. Rather, it appears that imbalance of the UPS arises due to an overwhelming of the PN by mHTT, which in turn leads to increased levels of improperly folded clients. As a consequence, the abundance of poly-ubiquitylated proteins within the cell increases causing a “queue” of poly-ubiquitylated proteins that overloads the proteasome. This is further enhanced by reduced trafficking of ubiquitylated clients as a consequence of molecular chaperone sequestration .
In addition, while autophagosome formation and lysosomal fusion appear to be unaffected by the expression of mHTT, recent evidence suggests that the engulfment of cytosolic cargo (particularly organelles) by autophagosomes, is inefficient in HD, possibly due to aberrant interactions between p62, ubiquitin chains, and mHTT . UPS impairment and defects in autophagy are thought to contribute to AD, PD, and ALS suggesting that a loss of protein degradation pathways could also be a central feature of neurodegeneration [58, 62].
Aberrant activation of stress responses in polyglutamine disease
Cytoprotective stress responses (Box 3) are crucial determinants of life- and health-span that must be controlled with exquisite precision to maintain cellular health and prevent disease. Recent evidence suggests that the dysregulation of stress response transcription factors could contribute to neurodegeneration.
The ability to effectively initiate the heat shock response (HSR) (Box 3) is compromised in mouse and cell models of HD. This correlates with reduced occupancy of HSF1 at the promoters of chaperone genes following stress, but also reflects genome-wide changes in HSF-1 DNA binding, likely due to changes in HSF1 expression, and/or altered chromatin architecture [63-65]. Dysregulation of the HSR has also recently been described in models of other polyglutamine/amyloid disorders [42, 53] whilst activation of HSF1 has been shown to suppress mHTT, mutant ataxin-3 (another polyglutamine disease protein that is the cause of spinocerebellar ataxia type 3 (SCA-3)), and Aβ aggregation and toxicity in cells, flies, worms, and mice .
Likewise, the activity of the metabolic stress factor, DAF-16/FOXO3a, is also impaired in HD mice, possibly through dysregulation of the deacetylase SIRT1 . Over-expression of SIRT1 increases DAF-16/FOXO3a activity and improves phenotype in HD mice, possibly by reducing oxidative stress [66, 67]. The activity of SIRT1 is also involved in the regulation of HSF-1 by enhancing DNA binding activity, therefore it may be of interest to ascertain whether increased HSF-1 activity also contributes to SIRT1 mediated neuroprotection in HD mice . Furthermore, reduced insulin signalling (which reduces polyglutamine toxicity in C. elegans in an HSF1- and DAF-16-dependent manner), through reduced levels of insulin receptor substrate-2 (IRS2) or insulin-like growth factor-1 (IGF-1), improves disease phenotypes in mouse models of HD and AD respectively [69, 70].
These observations, coupled with our existing knowledge that stress transcription factors are prominent modifiers of lifespan and proteostasis, support a model in which the progressive loss or compromise of stress response pathways renders neurons increasingly vulnerable to transient environmental insults and to the chronic presence of mHTT or other aggregation-prone proteins. Early activation of these pathways has been shown to ameliorate disease progression in multiple models of neurodegenerative disease suggesting that small molecules that can activate stress transcription factors may be an effective strategy for the treatment of HD. However, it remains unclear whether disease progression could stymie the long term efficacy of stress pathway activation .
Small molecule regulators of proteostasis as therapeutics for neurodegenerative disease
While genetic approaches have proven invaluable to identify the pathways that modify HD, translation of these findings to the clinic will likely require small molecule pharmacological agents that can selectively modify disease progression. For example, small molecules that reduce excitotoxicity or enhance histone acetylation have shown promise in mouse models of HD [20, 23, 45]. However, these approaches attempt to rectify the harmful consequences of mHTT rather than targeting the early events associated with mHTT expression. A more effective approach will likely target the causative agent itself through reduced expression, enhanced re-folding, and/or increased degradation of mHTT.
One promising approach is the pharmacological activation of HSF1 leading to increased protein folding capacity through up-regulation of multiple chaperones. This has been achieved with molecules that inhibit HSP90 (a negative regulator of HSF1) and suppress mHTT aggregation and toxicity in a variety of disease models . While these studies represent an important proof-of-principle, long-term inhibition of HSP90 is likely to be detrimental. Recent screens have identified new classes of HSF1-activating molecules that act independently of HSP90. For instance, the small molecule HSF1A was identified using a yeast strain engineered to express human HSF1 and was found to suppress toxicity in cell and fly models of HD and SCA-3 .
More recently, a ~1,000,000 compound screen for novel small molecule activators of HSF1 identified a barbituric acid-like compound (F1) that restores proteostasis in cell and worm models of protein conformational disease . Intriguingly, suppression of polyglutamine toxicity was achieved despite only modest induction of gene expression suggesting that subtle changes to the PN may be sufficient to achieve therapeutic benefit in HD . Pharmacological activation of HSF1 suppresses toxicity in numerous models of neurodegenerative disease, suggesting that augmenting the PN via activation of HSF1 may be a common strategy for treatment of neurological disorders .
Suppressing the generation of huntingtin fragments could significantly influence HD pathogenesis
While approaches for re-folding or clearance of mHTT have proven successful in multiple disease models, the ability to specifically reduce intracellular levels of mHTT could be of great benefit, either alone or in combination with other approaches.
Full length mHTT is processed into an array of fragments that exhibit toxicity when expressed; this is particularly well demonstrated with small N-terminal fragments of mHTT containing the polyglutamine stretch [74-76]. Therefore, the ability to prevent the generation of these fragments could prevent disease progression. Initial efforts suggested that mHTT fragments generated by caspase-6 are the primary pathogenic species in HD . However, these results have been inconsistently observed, with evidence that caspase-6-derived mHTT fragments may undergo further proteolysis to smaller, more toxic, N-terminal fragments, perhaps through the action of matrix metalloproteinases [76, 77]. Other recent findings have added a new perspective by demonstrating that N-terminal mHTT fragments may also be generated by aberrant splicing . This suggests that N-terminal fragments may not be generated solely through proteolysis and that understanding the mechanistic basis of these observations could be of great relevance to treating HD.
Targeting mutant huntingtin expression for therapeutic gain in HD
Seminal observations using a conditional mouse model of HD demonstrated that mHTT aggregation and toxicity can be reversed when mHTT expression is arrested . These observations revealed that the intrinsic protein quality control mechanisms of cells can reverse the toxic effects of polyglutamine expression and that mHTT inclusions and oligomers can be cleared. More recent work now demonstrates that anti-sense oligonucleotides (ASO) or single–stranded siRNAs (ss-siRNA) (Box 4) delivered to the central nervous system of HD mice can reduce levels of mHTT protein with little to no effect on the levels of wild type huntingtin [80, 81]. Furthermore, ASO treatment of HD mice results in a pronounced reduction in aggregate load with a concomitant improvement in motor coordination and survival . These RNAi based silencing approaches have also been shown to be effective in mouse models of SCA, ALS, and Parkinson’s disease.
Box 4. Methods for silencing toxic proteins.
Oligonucleotides that reduce the expression of target genes are used extensively in studies of gene function. Two prominent classes of gene silencing molecules are antisense oligonucleotides (ASOs) and small interfering RNAs (siRNAs) . ASOs are single-stranded oligonucleotides that engage target RNA sequences and prevent message translation through RNase H-dependent and -independent pathways. By contrast, siRNAs are generally expressed as double stranded RNAs that silence target mRNAs through the RNA induced silencing complex (RISC) . Whilst siRNAs can be expressed as single stranded forms (ss-siRNA), these have significantly reduced potency compared to their double stranded counterparts. However, recent studies to deduce the impact of chemical modifications on ss-siRNA stability and potency have revealed that a 5′-(E)-vinylphosphate modification of the 5′-phosphate group in the oligonucleotide results in enhanced ss-siRNA activity . This has numerous advantages for treatment of disease as ss-siRNAs are suggested to have reduced off-target effects and lower toxicity .
While these findings are encouraging, they represent treatment protocols at an early-stage of development, as they could have significantly debilitating effects on the lifestyles of affected individuals. In particular, the transient nature of these effects would demand patients to suffer repeat infusions throughout their life. Therefore, the ultimate aim must be to derive less-invasive and less-time-consuming methods of treatment through the administration of drugs that can selectively reduce mHTT expression. While this goal is ambitious, recent findings suggest that this may be possible .
As well as general mechanisms of folding and clearance, new findings suggest that polyglutamine proteins also share a common mode of expression. A genetic screen for modifiers of polyglutamine toxicity in yeast has revealed that the conserved transcription elongation factor Spt4 is required for the selective expression of long polyglutamine stretches . Remarkably, a loss of Spt4 results in a dramatic reduction in mHTT expression, aggregation, and toxicity without obvious gross changes to the transcriptome. These findings suggest that it may be possible to pharmacologically target SPT4 or other factors and selectively reduce the expression of polyglutamine disease proteins .
Concluding remarks
In this review, we have highlighted the plethora of genetic pathways that underlie pathogenesis in HD and likely also modulate disease progression in other neurodegenerative disorders. We propose that the chronic exposure of cells to misfolded and aggregation-prone proteins exerts a demand on the PN that cannot be met, in particular as the system ages, resulting in wide-scale cellular disruption and disease. If this hypothesis is correct, an attractive therapeutic approach for HD may be to develop small molecule cocktails that enhance the cellular protein folding capacity whilst simultaneously reducing mHTT expression. As we continue to understand the precise mechanisms that govern neurodegeneration, it may be possible to develop strategies that act as a panacea for protein conformational disorders.
Highlights.
Spotlight on key advances in our understanding of molecular dysfunction and HD
Description of novel pathways influencing intracellular levels of mutant huntingtin
Exciting new therapeutic strategies involving gene silencing and small molecules
Emphasis on molecular similarities between HD and other neurological disorders
Footnotes
Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Abstract
Huntington’s disease: one gene many consequences
Altered neural circuitry underlies cognitive and molecular abnormalities in HD
Mitochondrial dysfunction and impaired energy metabolism
Transcriptional dysregulation in HD
Impaired protein homeostasis in neurodegenerative disease
Impaired protein degradation pathways in HD
Aberrant activation of stress responses in polyglutamine disease
Small molecule regulators of proteostasis as therapeutics for neurodegenerative disease
Suppressing the generation of huntingtin fragments could significantly influence HD pathogenesis
Targeting mutant huntingtin expression for therapeutic gain in HD
Concluding remarks
Footnotes
References
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Contents
Preface to Biology
Unit 1: The Chemistry of Life
Chapter 1. The Study of Life
1.1 The Science of Biology
1.2 Themes and Concepts of Biology
Chapter 2: The Chemical Foundation of Life
2.1 Atoms, Isotopes, Ions, and Molecules: The Building Blocks
2.2 Water
2.3 Carbon
Chapter 3. Biological Macromolecules
3.1 Synthesis of Biological Macromolecules
3.2 Carbohydrates
3.3 Lipids
3.4 Proteins
3.5 Nucleic Acids
Unit 2: The Cell
Chapter 4. Cell Structure
4.1 Studying Cells
4.2 Prokaryotic Cells
4.3 Eukaryotic Cells
4.4 The Endomembrane System and Proteins
4.5 The Cytoskeleton
4.6 Connections between Cells and Cellular Activities
Chapter 5. Structure and Function of Plasma Membranes
5.1 Components and Structure
5.2 Passive Transport
5.3 Active Transport
5.4 Bulk Transport
Chapter 6. Metabolism
6.1 Energy and Metabolism
6.2 Potential, Kinetic, Free, and Activation Energy
6.3 The Laws of Thermodynamics
6.4 ATP: Adenosine Triphosphate
6.5 Enzymes
Chapter 7. Cellular Respiration
7.1 Energy in Living Systems
7.2 Glycolysis
7.3 Oxidation of Pyruvate and the Citric Acid Cycle
7.4 Oxidative Phosphorylation
7.5 Metabolism without Oxygen
7.6 Connections of Carbohydrate, Protein, and Lipid Metabolic Pathways
7.7 Regulation of Cellular Respiration
Chapter 8. Photosynthesis
8.1 Overview of Photosynthesis
8.2 The Light-Dependent Reactions of Photosynthesis
8.3 Using Light Energy to Make Organic Molecules
Chapter 9. Cell Communication
9.1 Signaling Molecules and Cellular Receptors
9.2 Propagation of the Signal
9.3 Response to the Signal
9.4 Signaling in Single-Celled Organisms
Chapter 10. Cell Reproduction
10.1 Cell Division
10.2 The Cell Cycle
10.3 Control of the Cell Cycle
10.4 Cancer and the Cell Cycle
10.5 Prokaryotic Cell Division
Unit 3: Genetics
Chapter 11. Meiosis and Sexual Reproduction
11.1 The Process of Meiosis
11.2 Sexual Reproduction
Chapter 12. Mendel's Experiments and Heredity
12.1 Mendel’s Experiments and the Laws of Probability
12.2 Characteristics and Traits
12.3 Laws of Inheritance
Chapter 13. Modern Understandings of Inheritance
13.1 Chromosomal Theory and Genetic Linkage
13.2 Chromosomal Basis of Inherited Disorders
Chapter 14. DNA Structure & Functions
14.1 Historical Basis of Modern Understanding
14.2 DNA Structure and Sequencing
14.3 Basics of DNA Replication
14.4 DNA Replication in Prokaryotes
14.5 DNA Replication in Eukaryotes
14.6 DNA Repair
Chapter 15. Genes and Proteins
15.1 The Genetic Code
15.2 Prokaryotic Transcription
15.3 Eukaryotic Transcription
15.4 RNA Processing in Eukaryotes
15.5 Ribosomes and Protein Synthesis
Chapter 16. Gene Expression
16.1 Regulation of Gene Expression
16.2 Prokaryotic Gene Regulation
16.3 Eukaryotic Epigenetic Gene Regulation
16.4 Eukaryotic Transcription Gene Regulation
16.5 Eukaryotic Post-transcriptional Gene Regulation
16.6 Eukaryotic Translational and Post-translational Gene Regulation
16.7 Cancer and Gene Regulation
Chapter 17. Biotechnology and Genomics
17.1 Biotechnology
17.2 Mapping Genomes
17.3 Whole-Genome Sequencing
17.4 Applying Genomics
17.5 Genomics and Proteomics
Unit 4: Evolutionary Processes
18.1 Understanding Evolution
Testing
Reproductive Development and Structure (HUGH TESTING)
MATH: INTEGRALS (HUGH TESTING)
Math out of the box (HUGH TESTING)
OpenStax Biology
6.5 Enzymes
Learning Objectives
By the end of this section, you will be able to:
Describe the role of enzymes in metabolic pathways
Explain how enzymes function as molecular catalysts
Discuss enzyme regulation by various factors
A substance that helps a chemical reaction to occur is a catalyst, and the special molecules that catalyze biochemical reactions are called enzymes. Almost all enzymes are proteins, made up of chains of amino acids, and they perform the critical task of lowering the activation energies of chemical reactions inside the cell. Enzymes do this by binding to the reactant molecules, and holding them in such a way as to make the chemical bond-breaking and bond-forming processes take place more readily. It is important to remember that enzymes don’t change the ∆G of a reaction. In other words, they don’t change whether a reaction is exergonic (spontaneous) or endergonic. This is because they don’t change the free energy of the reactants or products. They only reduce the activation energy required to reach the transition state (Figure).
Enzymes lower the activation energy of the reaction but do not change the free energy of the reaction.
Enzyme Active Site and Substrate Specificity
The chemical reactants to which an enzyme binds are the enzyme’s substrates. There may be one or more substrates, depending on the particular chemical reaction. In some reactions, a single-reactant substrate is broken down into multiple products. In others, two substrates may come together to create one larger molecule. Two reactants might also enter a reaction, both become modified, and leave the reaction as two products. The location within the enzyme where the substrate binds is called the enzyme’s active site. The active site is where the “action” happens, so to speak. Since enzymes are proteins, there is a unique combination of amino acid residues (also called side chains, or R groups) within the active site. Each residue is characterized by different properties. Residues can be large or small, weakly acidic or basic, hydrophilic or hydrophobic, positively or negatively charged, or neutral. The unique combination of amino acid residues, their positions, sequences, structures, and properties, creates a very specific chemical environment within the active site. This specific environment is suited to bind, albeit briefly, to a specific chemical substrate (or substrates). Due to this jigsaw puzzle-like match between an enzyme and its substrates (which adapts to find the best fit between the transition state and the active site), enzymes are known for their specificity. The “best fit” results from the shape and the amino acid functional group’s attraction to the substrate. There is a specifically matched enzyme for each substrate and, thus, for each chemical reaction; however, there is flexibility as well.
The fact that active sites are so perfectly suited to provide specific environmental conditions also means that they are subject to influences by the local environment. It is true that increasing the environmental temperature generally increases reaction rates, enzyme-catalyzed or otherwise. However, increasing or decreasing the temperature outside of an optimal range can affect chemical bonds within the active site in such a way that they are less well suited to bind substrates. High temperatures will eventually cause enzymes, like other biological molecules, to denature, a process that changes the natural properties of a substance. Likewise, the pH of the local environment can also affect enzyme function. Active site amino acid residues have their own acidic or basic properties that are optimal for catalysis. These residues are sensitive to changes in pH that can impair the way substrate molecules bind. Enzymes are suited to function best within a certain pH range, and, as with temperature, extreme pH values (acidic or basic) of the environment can cause enzymes to denature.
Induced Fit and Enzyme Function
For many years, scientists thought that enzyme-substrate binding took place in a simple “lock-and-key” fashion. This model asserted that the enzyme and substrate fit together perfectly in one instantaneous step. However, current research supports a more refined view called induced fit (Figure). The induced-fit model expands upon the lock-and-key model by describing a more dynamic interaction between enzyme and substrate. As the enzyme and substrate come together, their interaction causes a mild shift in the enzyme’s structure that confirms an ideal binding arrangement between the enzyme and the transition state of the substrate. This ideal binding maximizes the enzyme’s ability to catalyze its reaction.
LINK TO LEARNING
View an animation of induced fit at this website.
When an enzyme binds its substrate, an enzyme-substrate complex is formed. This complex lowers the activation energy of the reaction and promotes its rapid progression in one of many ways. On a basic level, enzymes promote chemical reactions that involve more than one substrate by bringing the substrates together in an optimal orientation. The appropriate region (atoms and bonds) of one molecule is juxtaposed to the appropriate region of the other molecule with which it must react. Another way in which enzymes promote the reaction of their substrates is by creating an optimal environment within the active site for the reaction to occur. Certain chemical reactions might proceed best in a slightly acidic or non-polar environment. The chemical properties that emerge from the particular arrangement of amino acid residues within an active site create the perfect environment for an enzyme’s specific substrates to react.
You’ve learned that the activation energy required for many reactions includes the energy involved in manipulating or slightly contorting chemical bonds so that they can easily break and allow others to reform. Enzymatic action can aid this process. The enzyme-substrate complex can lower the activation energy by contorting substrate molecules in such a way as to facilitate bond-breaking, helping to reach the transition state. Finally, enzymes can also lower activation energies by taking part in the chemical reaction itself. The amino acid residues can provide certain ions or chemical groups that actually form covalent bonds with substrate molecules as a necessary step of the reaction process. In these cases, it is important to remember that the enzyme will always return to its original state at the completion of the reaction. One of the hallmark properties of enzymes is that they remain ultimately unchanged by the reactions they catalyze. After an enzyme is done catalyzing a reaction, it releases its product(s).
According to the induced-fit model, both enzyme and substrate undergo dynamic conformational changes upon binding. The enzyme contorts the substrate into its transition state, thereby increasing the rate of the reaction.
Control of Metabolism Through Enzyme Regulation
It would seem ideal to have a scenario in which all of the enzymes encoded in an organism’s genome existed in abundant supply and functioned optimally under all cellular conditions, in all cells, at all times. In reality, this is far from the case. A variety of mechanisms ensure that this does not happen. Cellular needs and conditions vary from cell to cell, and change within individual cells over time. The required enzymes and energetic demands of stomach cells are different from those of fat storage cells, skin cells, blood cells, and nerve cells. Furthermore, a digestive cell works much harder to process and break down nutrients during the time that closely follows a meal compared with many hours after a meal. As these cellular demands and conditions vary, so do the amounts and functionality of different enzymes.
Since the rates of biochemical reactions are controlled by activation energy, and enzymes lower and determine activation energies for chemical reactions, the relative amounts and functioning of the variety of enzymes within a cell ultimately determine which reactions will proceed and at which rates. This determination is tightly controlled. In certain cellular environments, enzyme activity is partly controlled by environmental factors, like pH and temperature. There are other mechanisms through which cells control the activity of enzymes and determine the rates at which various biochemical reactions will occur.
Regulation of Enzymes by Molecules
Enzymes can be regulated in ways that either promote or reduce their activity. There are many different kinds of molecules that inhibit or promote enzyme function, and various mechanisms exist for doing so. In some cases of enzyme inhibition, for example, an inhibitor molecule is similar enough to a substrate that it can bind to the active site and simply block the substrate from binding. When this happens, the enzyme is inhibited through competitive inhibition, because an inhibitor molecule competes with the substrate for active site binding (Figure). On the other hand, in noncompetitive inhibition, an inhibitor molecule binds to the enzyme in a location other than an allosteric site and still manages to block substrate binding to the active site.
Competitive and noncompetitive inhibition affect the rate of reaction differently. Competitive inhibitors affect the initial rate but do not affect the maximal rate, whereas noncompetitive inhibitors affect the maximal rate.
Some inhibitor molecules bind to enzymes in a location where their binding induces a conformational change that reduces the affinity of the enzyme for its substrate. This type of inhibition is called allosteric inhibition (Figure). Most allosterically regulated enzymes are made up of more than one polypeptide, meaning that they have more than one protein subunit. When an allosteric inhibitor binds to an enzyme, all active sites on the protein subunits are changed slightly such that they bind their substrates with less efficiency. There are allosteric activators as well as inhibitors. Allosteric activators bind to locations on an enzyme away from the active site, inducing a conformational change that increases the affinity of the enzyme’s active site(s) for its substrate(s).
Allosteric inhibitors modify the active site of the enzyme so that substrate binding is reduced or prevented. In contrast, allosteric activators modify the active site of the enzyme so that the affinity for the substrate increases.
EVERYDAY CONNECTION
Have you ever wondered how pharmaceutical drugs are developed? (credit: Deborah Austin)
Drug Discovery by Looking for Inhibitors of Key Enzymes in Specific Pathways Enzymes are key components of metabolic pathways. Understanding how enzymes work and how they can be regulated is a key principle behind the development of many of the pharmaceutical drugs (Figure) on the market today. Biologists working in this field collaborate with other scientists, usually chemists, to design drugs.
Consider statins for example—which is the name given to the class of drugs that reduces cholesterol levels. These compounds are essentially inhibitors of the enzyme HMG-CoA reductase. HMG-CoA reductase is the enzyme that synthesizes cholesterol from lipids in the body. By inhibiting this enzyme, the levels of cholesterol synthesized in the body can be reduced. Similarly, acetaminophen, popularly marketed under the brand name Tylenol, is an inhibitor of the enzyme cyclooxygenase. While it is effective in providing relief from fever and inflammation (pain), its mechanism of action is still not completely understood.
How are drugs developed? One of the first challenges in drug development is identifying the specific molecule that the drug is intended to target. In the case of statins, HMG-CoA reductase is the drug target. Drug targets are identified through painstaking research in the laboratory. Identifying the target alone is not sufficient; scientists also need to know how the target acts inside the cell and which reactions go awry in the case of disease. Once the target and the pathway are identified, then the actual process of drug design begins. During this stage, chemists and biologists work together to design and synthesize molecules that can either block or activate a particular reaction. However, this is only the beginning: both if and when a drug prototype is successful in performing its function, then it must undergo many tests from in vitro experiments to clinical trials before it can get FDA approval to be on the market.
Many enzymes don’t work optimally, or even at all, unless bound to other specific non-protein helper molecules, either temporarily through ionic or hydrogen bonds or permanently through stronger covalent bonds. Two types of helper molecules are cofactors and coenzymes. Binding to these molecules promotes optimal conformation and function for their respective enzymes. Cofactors are inorganic ions such as iron (Fe++) and magnesium (Mg++). One example of an enzyme that requires a metal ion as a cofactor is the enzyme that builds DNA molecules, DNA polymerase, which requires bound zinc ion (Zn++) to function. Coenzymes are organic helper molecules, with a basic atomic structure made up of carbon and hydrogen, which are required for enzyme action. The most common sources of coenzymes are dietary vitamins (Figure). Some vitamins are precursors to coenzymes and others act directly as coenzymes. Vitamin C is a coenzyme for multiple enzymes that take part in building the important connective tissue component, collagen. An important step in the breakdown of glucose to yield energy is catalysis by a multi-enzyme complex called pyruvate dehydrogenase. Pyruvate dehydrogenase is a complex of several enzymes that actually requires one cofactor (a magnesium ion) and five different organic coenzymes to catalyze its specific chemical reaction. Therefore, enzyme function is, in part, regulated by an abundance of various cofactors and coenzymes, which are supplied primarily by the diets of most organisms.
Vitamins are important coenzymes or precursors of coenzymes, and are required for enzymes to function properly. Multivitamin capsules usually contain mixtures of all the vitamins at different percentages.
Enzyme Compartmentalization
In eukaryotic cells, molecules such as enzymes are usually compartmentalized into different organelles. This allows for yet another level of regulation of enzyme activity. Enzymes required only for certain cellular processes can be housed separately along with their substrates, allowing for more efficient chemical reactions. Examples of this sort of enzyme regulation based on location and proximity include the enzymes involved in the latter stages of cellular respiration, which take place exclusively in the mitochondria, and the enzymes involved in the digestion of cellular debris and foreign materials, located within lysosomes.
Feedback Inhibition in Metabolic Pathways
Molecules can regulate enzyme function in many ways. A major question remains, however: What are these molecules and where do they come from? Some are cofactors and coenzymes, ions, and organic molecules, as you’ve learned. What other molecules in the cell provide enzymatic regulation, such as allosteric modulation, and competitive and noncompetitive inhibition? The answer is that a wide variety of molecules can perform these roles. Some of these molecules include pharmaceutical and non-pharmaceutical drugs, toxins, and poisons from the environment. Perhaps the most relevant sources of enzyme regulatory molecules, with respect to cellular metabolism, are the products of the cellular metabolic reactions themselves. In a most efficient and elegant way, cells have evolved to use the products of their own reactions for feedback inhibition of enzyme activity. Feedback inhibition involves the use of a reaction product to regulate its own further production (Figure). The cell responds to the abundance of specific products by slowing down production during anabolic or catabolic reactions. Such reaction products may inhibit the enzymes that catalyzed their production through the mechanisms described above.
Metabolic pathways are a series of reactions catalyzed by multiple enzymes. Feedback inhibition, where the end product of the pathway inhibits an upstream step, is an important regulatory mechanism in cells.
The production of both amino acids and nucleotides is controlled through feedback inhibition. Additionally, ATP is an allosteric regulator of some of the enzymes involved in the catabolic breakdown of sugar, the process that produces ATP. In this way, when ATP is abundant, the cell can prevent its further production. Remember that ATP is an unstable molecule that can spontaneously dissociate into ADP. If too much ATP were present in a cell, much of it would go to waste. On the other hand, ADP serves as a positive allosteric regulator (an allosteric activator) for some of the same enzymes that are inhibited by ATP. Thus, when relative levels of ADP are high compared to ATP, the cell is triggered to produce more ATP through the catabolism of sugar.
Section Summary
Enzymes are chemical catalysts that accelerate chemical reactions at physiological temperatures by lowering their activation energy. Enzymes are usually proteins consisting of one or more polypeptide chains. Enzymes have an active site that provides a unique chemical environment, made up of certain amino acid R groups (residues). This unique environment is perfectly suited to convert particular chemical reactants for that enzyme, called substrates, into unstable intermediates called transition states. Enzymes and substrates are thought to bind with an induced fit, which means that enzymes undergo slight conformational adjustments upon substrate contact, leading to full, optimal binding. Enzymes bind to substrates and catalyze reactions in four different ways: bringing substrates together in an optimal orientation, compromising the bond structures of substrates so that bonds can be more easily broken, providing optimal environmental conditions for a reaction to occur, or participating directly in their chemical reaction by forming transient covalent bonds with the substrates.
Enzyme action must be regulated so that in a given cell at a given time, the desired reactions are being catalyzed and the undesired reactions are not. Enzymes are regulated by cellular conditions, such as temperature and pH. They are also regulated through their location within a cell, sometimes being compartmentalized so that they can only catalyze reactions under certain circumstances. Inhibition and activation of enzymes via other molecules are other important ways that enzymes are regulated. Inhibitors can act competitively, noncompetitively, or allosterically; noncompetitive inhibitors are usually allosteric. Activators can also enhance the function of enzymes allosterically. The most common method by which cells regulate the enzymes in metabolic pathways is through feedback inhibition. During feedback inhibition, the products of a metabolic pathway serve as inhibitors (usually allosteric) of one or more of the enzymes (usually the first committed enzyme of the pathway) involved in the pathway that produces them.
Review Questions
Which of the following is not true about enzymes:
They increase ∆G of reactions
They are usually made of amino acids
They lower the activation energy of chemical reactions
Each one is specific to the particular substrate(s) to which it binds
An allosteric inhibitor does which of the following?
Binds to an enzyme away from the active site and changes the conformation of the active site, increasing its affinity for substrate binding
Binds to the active site and blocks it from binding substrate
Binds to an enzyme away from the active site and changes the conformation of the active site, decreasing its affinity for the substrate
Binds directly to the active site and mimics the substrate
Which of the following analogies best describe the induced-fit model of enzyme-substrate binding?
A hug between two people
A key fitting into a lock
A square peg fitting through the square hole and a round peg fitting through the round hole of a children’s toy
The fitting together of two jigsaw puzzle pieces.
Free Response
With regard to enzymes, why are vitamins necessary for good health? Give examples.
Explain in your own words how enzyme feedback inhibition benefits a cell.
Glossary
active site specific region of the enzyme to which the substrate binds allosteric inhibition inhibition by a binding event at a site different from the active site, which induces a conformational change and reduces the affinity of the enzyme for its substrate coenzyme small organic molecule, such as a vitamin or its derivative, which is required to enhance the activity of an enzyme cofactor inorganic ion, such as iron and magnesium ions, required for optimal regulation of enzyme activity competitive inhibition type of inhibition in which the inhibitor competes with the substrate molecule by binding to the active site of the enzyme denature process that changes the natural properties of a substance feedback inhibition effect of a product of a reaction sequence to decrease its further production by inhibiting the activity of the first enzyme in the pathway that produces it induced fit dynamic fit between the enzyme and its substrate, in which both components modify their structures to allow for ideal binding substrate molecule on which the enzyme acts
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3459 | https://help2.innovyze.com/infoworksicm/Content/HTML/ICM_ILCM/Weirs.htm | You are here:
Database Items | Networks | Network Objects | Links | Weirs
Weirs (InfoWorks)
Weirs are used for overflow structures and for outlets to storage ponds.
Weir controls have the same characteristics for flow in both directions.
A weir is represented as a link of zero length, forming a head-discharge
relationship between two nodes. The boundary condition between the link
and a node is that of equal water levels. The weir crest invert level
determines when the control first comes into operation.
The weir types available in InfoWorks ICM are:
Standard Weir (Rectangular Full Width Weir) (WEIR) - thin plate weir that extends the full width
of the channel
Variable Crest Weir (VCWEIR)
a standard thin plate weir whose crest level can vary according to a
set of control rules. Used for Real Time Control simulations only
Variable Width Weir (VWWEIR)
a standard thin plate weir whose width can vary according to a set of
control rules. Used for Real Time Control simulations only
Contracted Rectangular Weir (COWEIR)
thin plate weir with a single rectangular notch that does not extend
the full width of the channel
Vee Notch Weir (VNWEIR)
thin plate weir with one or more vee shaped notches
Trapezoidal Notch Weir (TRWEIR)
thin plate weir with one or more trapezoidal notches
Broad Crested Weir (BRWEIR)
a weir with a horizontal crest. In practice, the length of the crest
is normally equal to the channel width
Gated Weir (GTWEIR) - a gated weir where the crest level can vary according to a
set of control rules. Used for Real Time Control simulations only
Flow Characteristics
The discharge over the various kinds of weir is calculated using the
equations below.
There are two flow conditions that are possible over weirs:
Free discharge: the downstream depth does not
affect the upstream depth or discharge. This occurs when the downstream
depth is below the crest level.
Drowned discharge: the downstream depth affects
the upstream depth and the discharge. This occurs when the downstream
depth is above the crest level.
Standard Thin Plate Weir
For free discharge InfoWorks ICM uses the governing model equation based
on the Kindsvater and Carter equation:
| | | | | --- | | | | Qo is the free outfall discharge Cd is the discharge coefficient g is the acceleration due to gravity B is the width of the weir Du is the upstream depth with respect to the crest |
For drowned discharge InfoWorks ICM uses the Kindsvater and Carter equation
where Dd is greater than zero:
| | | | | --- | | | (2) | where: Dd is the downstream depth with respect to the crest |
See BS3680 (Part 4A) for more information.
If the Use Villemonte equation option has been
chosen as a Simulation Parameter,
when the weir is under drowned conditions, the Villemonte formula is used
to modify the free outfall discharge as presented in the equation below:
| | | | | --- | | | (3) | where: Q is the discharge Qo is the free outfall discharge (calculated using the free discharge equation above) Dd is the downstream depth with respect to the crest Du is the upstream depth with respect to the crest |
If the water level exceeds the roof height then InfoWorks ICM will use the
lower of the flow values calculated by using the weir equations above and the orifice formula.
Contracted Thin Plate Weir
For free discharge InfoWorks ICM uses the Kindsvater and Carter equation,
in which a minimum value of critical depth over the weir crest is enforced,
and the contracted width is assumed to extend up to the channel wall height.
| | | | | --- | | | (4) | where: Qo is the free outfall discharge Du is the upstream depth with respect to the crest p is the height of the weir crest above the channel invert B is the weir width g is the acceleration due to gravity |
For drowned discharge InfoWorks ICM uses the Kindsvater and Carter equation
where Dd is greater than zero:
| | | | | --- | | | (5) | where: Dd is the downstream depth with respect to the crest |
The Villemonte formula is then used to modify the free outfall discharge.
See Equation 3 above.
Trapezoidal Notch Thin Plate (Cipolletti) Weir
Under free discharge, InfoWorks ICM uses the equation:
| | | | | --- | | | (6) | where: Qo is the free outfall discharge B is the weir width at crest level Du is the upstream depth with respect to the crest |
The crest is the level at which the weir comes into operation - in other
words the base of the trapezoidal notch(es).
The weir is assumed to be drowned if the downstream water level exceeds
the crest level. The Villemonte formula is then used to modify the free
outfall discharge. See Equation 3 above.
See BS3680 (Part 4A) for more information.
V-Notch Thin Plate Weir
Under free discharge InfoWorks ICM uses the Kindsvater and Shen formula:
| | | | | --- | | | (7) | where: Qo is the free outfall discharge g is the acceleration due to gravity Cd is the discharge coefficient. In InfoWorks ICM, this value is fixed at 0.585. This is correct for 90° notch weirs. q is the angle of the notch Du is the upstream depth with respect to the crest |
The weir is assumed to be drowned if the downstream water level exceeds
the crest level. The Villemonte formula is then used to modify the free
outfall discharge. See Equation 3 above.
See BS3680 (Part 4A) for more information.
Broad Crested Weir
InfoWorks ICM uses the standard formula for a round nosed broad crested
weir:
| | | | | --- | | | (8) | where: Qo is the free outfall discharge Cd is the discharge coefficient. This is calculated using the formula below Cv is a dimensionless coefficient allowing for the effect of approach velocity B is the width of the weir crest g is the acceleration due to gravity Du is the upstream depth with respect to the crest |
Cd is calculated
using the equation:
| | | | | --- | | | (9) | where: L is the length of the horizontal section of the crest in the direction of flow |
Cv is then calculated
in terms of Cdusing:
| | | | | --- | | | (10) | where: A is the cross sectional area of the approach channel below the water level |
When the modular ratio (the ratio above the crest level of downstream
depth to calculated upstream depth) exceeds 66% the weir is deemed to
be drowned and a nominal headloss is set.
Note
Remember that most controls in a InfoWorks ICM network
are treated as operating in the same way hydraulically for flow in both
directions. This behaviour is not appropriate for a broad crested weir,
which should only have flow in one direction. It is up to you to avoid
situations where reverse flow occurs at broad crested weirs in your network.
See BS3680
(Part 4F) for more information.
Gated Weir
Three flow regimes can be used to model for flow over a gated weir:
when the gate is lowered and the weir operates
as a flume or broad crested weir with critical depth in the body of the
structure.
when the gate is raised and critical depth occurs
at the gate crest.
when the direction of flow is from the free crest
towards the hinge when performance is similar to a weir with a triangular
profile.
These are shown on the diagrams below:
Equations
Gated Weir Parameters
If the level on both sides is below the gate crest, then the following equation is used:
| Equation | Q = 0 |
Determination of throat/gate and free/drowned flow combinations
m = Modular Limit
b = Breadth of sluice at control section (normal to the flow)
Ctc = Discharge coefficient for throat
control
Cgt = Discharge coefficient for gate control
Crev = Discharge coefficient for reverse
gate control
| | Flow is determined to be throat flow if |Qthroat| < |Qgate| and gate flow otherwise. Throat Flow: | | (1) | | | | --- | | where: otherwise: | | Gate flow: | | (2) | | | | --- | | where: otherwise: If q³30: where: otherwise: | | To determine whether flow is free flow or drowned flow: | Throat flow: free flow is declared if and drowned otherwise | (3) | | Gate flow: free flow is declared if and drowned otherwise | (4) | |
Forward flow combinations
Free weir flow - throat control
| Condition | |
| Equation | as Equation (1) |
Drowned weir flow - throat control
| Condition | |
| Equation | as Equation (1) |
Free weir flow - gate control
| Condition | |
| Equation | as Equation (2) |
Drowned weir flow - gate control
| Condition | |
| Equation | As Equation (2) |
Free weir flow - reverse gate control
| Condition | |
| Equation | | | | --- | | | | If q < 22 where: |
Drowned weir flow - reverse gate control
| Condition | |
| Equation | | | | where all terms above are the same as defined for 'Free weir flow - reverse gate control', except fgate, which is as defined in equation (2) |
Reverse flow combinations
The equations and conditions are the same as for the forward flow combinations (described above) but with
reverse flow (Q<0). Note that the orientation
of the gate specified for reverse flow combinations is relative to the crest, not
the flow direction, e.g. for 'Free weir flow - reverse gate, throat
control, reverse flow', the position of the gate is not physically
oriented the same as for 'Free weir flow - throat control'. For 'Free weir flow - throat control' and 'Free weir flow - forward gate control,
reverse flow', the gates are in
the same position, relative to the crest.
Free weir flow - reverse gate, throat
control, reverse flow
Drowned weir flow - reverse gate, throat
control, reverse flow
Free weir flow - reverse gate control,
reverse flow
Drowned weir flow - reverse gate control,
reverse flow
Free weir flow - forward gate control,
reverse flow
Drowned weir flow - forward gate control,
reverse flow
The equations used for the gated weir were
derived from a physical model. The work is described in HR Wallingford Hydraulic Studies Report EX1296.
Weir Data Fields (InfoWorks)
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InfoWorks® ICM version 2023.2 - Issued 22 May 2023
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3460 | https://www.pbs.org/wgbh/americanexperience/films/eugenics-crusade/ | WOSU TV is your local station.
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part01
Watch Trailer
Aired October 16, 2018
The Eugenics Crusade
What’s wrong with perfect?
From the Collection:
Documentaries to Watch Now
Film Description
A hybrid derived from the Greek words meaning “well” and “born,” the term eugenics was coined in 1883 by Sir Francis Galton, a British cousin to Charles Darwin, to name a new “science” through which human beings might take charge of their own evolution. The Eugenics Crusade tells the story of the unlikely –– and largely unknown –– movement that turned the fledgling scientific theory of heredity into a powerful instrument of social control. Perhaps more surprising still, American eugenics was neither the work of fanatics, nor the product of fringe science. The goal of the movement was simple and, to its disciples, laudable: to eradicate social ills by limiting the number of those considered to be genetically “unfit” –– a group that would expand to include many immigrant groups, the poor, Jews, the mentally and physically disabled, and the “morally delinquent.” At its peak in the 1920s, the movement was in every way mainstream, packaged as a progressive quest for “healthy babies.” Its doctrines were not only popular and practiced, but codified by laws that severely restricted immigration and ultimately led to the institutionalization and sterilization of tens of thousands of American citizens. Populated by figures both celebrated and obscure, The Eugenics Crusade is an often revelatory portrait of an America at once strange and eerily familiar.
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Transcript
NARRATOR: On August 18th, 1934, twenty-year-old Ann Cooper Hewitt, heiress to one of the largest fortunes in the United States, was admitted to a San Francisco hospital for an emergency appendectomy.
She later learned the surgeons not only had removed her appendix, but also a length of her fallopian tubes–-rendering her incapable of ever becoming pregnant.
The story of the "sterilized heiress" hit the papers just after the New Year in 1936, when Ann filed a half-million dollar damage claim against the surgeons and her own mother for sterilizing her without her knowledge or consent.
Ann's mother denied any wrongdoing. She'd done what she'd done for
"society's sake," she insisted, because her daughter was "feebleminded."
It was the sort of bizarre, high-society scandal that would have captured the national imagination under any circumstances. But that one word, "feebleminded," struck a familiar chord for Americans––and linked Ann's plight to a decades-old campaign to control human reproduction, known as "eugenics."
ARCHIVAL: What is the bearing of the laws of heredity upon human affairs? Eugenics provides the answers.
PAUL A. LOMBARDO, HISTORIAN: Eugenics was proposed as the scientific solution for social problems. It was a combination of hope and aspiration on one side and on the other side it was about fear and in some cases about hate.
ARCHIVAL: They are identified early, categorized feebleminded, imbecile, idiot. It would have been better by far if they had never been born.
DANIEL KEVLES, HISTORIAN: People tend to think that eugenics was a doctrine that originated with the Nazis, that it was grounded in wild claims that were far outside the scientific mainstream. Both of those impressions are fundamentally not true.
ADAM COHEN, WRITER: It was almost a mania that sort of swept through the country. And there was that kind of naïve, optimistic vision of eugenics like, “Hey, let’s all get together and make better people."
ALEXANDRA MINNA STERN, HISTORIAN: The Eugenics Movement was about having healthy children, about having a stronger society. There’s nothing wrong with that. You have to look at the underbelly of what was implemented in the name of eugenics to see what was so problematic about it.
NARRATOR: In the fall of 1902, an American biologist named Charles Benedict Davenport arrived in London on a sort of pilgrimmage.
He was thirty-six, Harvard-educated, and like many biologists of his generation, absorbed with the study of evolution. He'd been traveling in Europe with his wife, collecting seashells for research on species variation, but this was to be the highlight of the trip: a meeting with the world-renowned gentleman scientist, Sir Francis Galton.
A pioneering statistician, Galton had lived his eighty years by a single motto: "Whenever you can, count." His obsession with measurements and patterns had led him to create the world's first weather maps, established fingerprinting as a means of identification, and set data-backed parameters for the perfect cup of tea.
Charles Davenport had come to discuss another matter: Galton's work on heredity.
DR. SIDDHARTHA MUKHERJEE, WRITER: Francis Galton was a great quantifier. He liked to quantify height, hair color. You know, what is the chest size of an average man? What is the thigh length of an average man? Even things like intelligence.
JONATHAN SPIRO, HISTORIAN: Galton had a theory that talent, as he called it—what we would call intelligence—seemed to run in families. And so it quickly occurred to him, “If we can get people with high talent to mate with each other, prevent people with low talent from mating with each other, we will,
within a few generations, create this race of super men.”
NATHANIEL COMFORT, HISTORIAN: Francis Galton was borrowing ideas and kind of riffing off of the work of his half cousin Charles Darwin.
KEITH WAILOO, HISTORIAN: Darwin believed that evolution was this natural process that was inevitably leading towards what they called the “survival of the fittest.” Galton really turns that idea on its head and says, “You know, natural selection isn’t working very well. We need to do a form of selection. We need to intervene.”
NARRATOR: To name the effort, Galton had coined the term "eugenics"–-a hybrid derived from two Greek words meaning "well" and "born."
Charles Davenport believed, as Galton did, that selective breeding could transform the human race. What was needed was a scientific understanding of how heredity actually worked––and over dinner at Galton's home, Davenport declared his intention to get to the bottom of it.
NATHANIEL COMFORT, HISTORIAN: Davenport said "I’m gonna create a new kind of institution, a station for experimental evolution, not Darwinian natural selection that you just go out and observe, but can we figure out how inheritance works, can we do experiments and find the patterns of heredity?
NARRATOR: When Davenport sailed for home in December 1902, he carried with him not only a letter of recommendation signed by Galton, but also, he later wrote, "a renewed courage for the...study of evolution."
THOMAS C. LEONARD, HISTORIAN: Davenport and Galton really did imagine that the idea of improving human heredity was of almost religious significance, of profound moral importance. They also believed they themselves were qualified to breed a better race because they believed that they were the best and the brightest.
NARRATOR: Scarcely more than a year later, with funding from the Carnegie Institution, Davenport opened his research station on the north shore of Long Island, at Cold Spring Harbor.
Situated on ten acres along Oyster Bay, the place had been purpose-built for the breeding and analyzing of plants and animals––complete with sprawling garden plots, an aviary, and a half-dozen tidy enclosures housing chickens, goats, and sheep.
By mating organisms with unusual characteristics––a tailless Manx cat, or a rooster with a black comb––and then studying their offspring, generation after generation, Davenport hoped to unlock the mystery of evolution.
NATHANIEL COMFORT, HISTORIAN: Davenport wasn’t yet thinking much about humans. He was just absorbing all these different theories of heredity and trying to figure out which ones applied when and under what conditions.
NARRATOR: After scores of experiments, one theory seemed to stand out from the rest: the recently discovered work of an Austrian monk named Gregor Mendel, who'd spent a decade in the mid-nineteenth century experimenting with peas.
PAUL A. LOMBARDO, HISTORIAN: Mendel learned that there was a pattern to how pea plants passed down certain traits. And you could come up with certain ratios to predict how likely it was that a pea plant would look one way or another.
JONATHAN SPIRO, HISTORIAN: Davenport took that and ran with it. He goes about breeding all kinds of animals looking for the Mendelian ratio and in trait after trait he seems to find the Mendelian ratio.
ADAM COHEN, WRITER: Suddenly, they’re beginning to see a mathematical scientific explanation for things that had been merely conjectural before.
DR. SIDDHARTHA MUKHERJEE, WRITER: It’s becoming obvious that, in fact, there are these things called genes. These units are being transmitted from parents to their offspring and they’re giving rise to physical traits.
NARRATOR: By 1906, the work at Cold Spring Harbor had caught the attention of the press––and established Charles Davenport as a rising star in the new science of genetics.
Thanks to Mendel's laws of heredity, Davenport told one reporter, agricultural breeders could now precisely select for desirable traits––to develop a strain of protein-rich wheat or a chicken that laid more eggs. The same methods would one day lead, he predicted, to a "rapid and thorough-going improvement of the human race."
DANIEL KEVLES, HISTORIAN: Davenport was tantalized by the possibility that you could take charge of human evolution. And then along comes Mendelian genetics which seems to offer a very powerful tool. So extrapolating from the work that the breeders were doing in animals to the breeding of better human beings was a natural step.
NARRATOR: In 1909, Davenport informed his funder, the Carnegie Institution, that he'd shifted his focus from the breeding of cats and roosters to an investigation of human traits.
Having already found the Mendelian ratio in early studies of eye color and hair color––and convinced he was on the right track––he now began to collect data on a wide range of other human characteristics.
He sent out a family history questionnaire to hundreds of individuals, and solicited prisons, hospitals and educational institutions for their records.
JONATHAN SPIRO, HISTORIAN: You can’t really do breeding experiments with human beings. Aside from the ethics you just can’t live long enough to see generations and generations. So it was Davenport’s genius to realize if he could collect family pedigrees, he could trace family inheritances and try to prove that evolution works for human beings the way it works for animals.
NARRATOR: Davenport's plan was to analyze the pedigree charts for Mendelian patterns, and to identify the desirable traits human beings might encourage through careful breeding––and the undesirable ones they could breed out.
PAUL A. LOMBARDO, HISTORIAN: “Wouldn’t it be a better world if we could wipe out poverty? Wouldn’t it be a better world if we didn’t have criminals? Wouldn’t it be a better world if everyone behaved themselves? And if the reason we have poverty, and crime is something that’s determined by our genes, if we can change that and make it so that the people who have those bad traits don’t pass them down wouldn’t that be a better world?"
NARRATOR: Among Americans of Charles Davenport's class and generation, there was perhaps no word that had more currency at the turn of the century than improvement.
They had come of age in the midst of a revolution––a seismic shift
that had made the United States the most prosperous and powerful nation on earth––and, in the eyes of many, had simultaneously plunged it into chaos.
JONATHAN SPIRO, HISTORIAN: It's the beginning of the 20th century and we have rampant urbanization, rampant industrialization, rampant immigration. The old order is passing and wherever you look society seems to be deteriorating.
NARRATOR: More and more farms were giving way to factories, and the cities were overrun with newcomers, many from the countryside, many hundreds of thousands more from abroad.
NATHANIEL COMFORT, HISTORIAN: There was impure water, and the schools were awful, and the disease was rampant, and immigrants were pouring in.
DANIEL KEVLES, HISTORIAN: People were apprehensive about rapid change, about the kinds of people you saw on the streets––slums, crime, alcoholism, prostitution. Native white Protestants felt that they were losing control of American society.
NARRATOR: Determined not only to meet the new challenges, but to master them, a veritable army of educated, middle- and upper-middle class Americans had launched a crusade to remake society––to eliminate corruption, stamp out disease and vice, assimilate the immigrant and uplift the poor––all in the name of Progress.
THOMAS C. LEONARD, HISTORIAN: The world had never seen invention as powerful, and remarkable, and as influential the last three decades of the 19th century. This fueled an already existing American optimism about what can be improved and it directed it into a particular track, which was scientific improvement.
NATHANIEL COMFORT, HISTORIAN: There was a great belief in science. There was a great belief in government, in bureaucracy as a tool for solving social problems, and also a belief in collectivism, that the population needs to work together to improve society.
CHRISTINE ROSEN, WRITER: The Progressive Movement said, we can use state power and expert advice and knowledge to solve things like poverty, to solve things like alcoholism. So that was an incredibly hopeful and optimistic idea. Eugenics was part of that.
NARRATOR: When the letter from Charles Davenport arrived in 1909, at New Jersey's Vineland Training School for the feebleminded, the staff hadn't known quite what to make of it––a mere two-line note requesting hereditary information, one of hundreds Davenport had sent.
NATHANIEL COMFORT, HISTORIAN: Davenport investigated any and all traits––eye color, weight, mood, habit, temperament, diseases, anything. And then he finds this psychologist in New Jersey, and he begins to zero in on low intelligence, something known as feeblemindedness.
NARRATOR: Psychologist Henry Goddard, Vineland's Director of Research, had no family histories to share. But what he lacked in data, he more than made up for with enthusiasm. Not only was Goddard interested in the new science of heredity, he asked Davenport to guide him in making his own study of feeblemindedness.
PAUL A. LOMBARDO, HISTORIAN: Like lots of people who are working in institutions––doctors or social workers––Henry Goddard was interested in identifying the kinds of conditions that were passed down in heredity and preventing them.
NARRATOR: Henry Goddard was 42 and a one-time teacher in Quaker schools. It was, in part, an interest in education that had brought him to Vineland in 1906.
He'd spent the three years since trying to parse the many varieties of feeblemindedness, an all-too-common mental deficiency associated with anti-social behavior.
Some of Vineland's 300 inmates were violent or deranged, others unruly, still others merely slow. Hoping to improve their individual care and training, Goddard had pioneered the use of an "intelligence test," which purported to measure a person's mental abilities in relation to that of so-called "normal" people of the same age.
The scores enabled him to sort his charges into categories. To the existing classifications of "idiot" and "imbecile," which long had been used to describe debilitating mental impairment, Goddard had added a third–-a higher-functioning group he called "morons."
WENDY KLINE, HISTORIAN: That was actually a diagnostic term and not just an insult. Henry Goddard argued the high-grade moron is high functioning enough to act normal, but they’re kind of stuck in this evolutionary phase and they don’t emerge as true adults. What’s missing is moral judgment. So
Goddard constructs that term, "moron," and “mental deficiency” and “immorality” become basically interchangeable.
NARRATOR: Now, with Davenport's tutoring, Goddard began to survey the family histories of thirty-five of his students at Vineland.
What he found made him an instant believer in eugenics. Not only did
morons seem clearly to pass on their feeblemindedness to their offspring, their family trees often were rife with alcoholics, prostitutes, criminals and paupers.
As Goddard put it to the New Jersey State Conference of Charities and Corrections in 1910: "Feeblemindedness is at the root of probably two-thirds of the problems that you...have before you." The cause was "defective ancestry."
NATHANIEL COMFORT, HISTORIAN: Henry Goddard puts forward this idea that if you got rid of feeblemindedness you would get rid of all of these problems, or at least greatly reduce them. And we love explanations like that. It’s so simple—oh, it’s just feeblemindedness so let’s, you know, that’s the fix.
NARRATOR: Goddard reasoned that if the test he'd devised to better care for the feebleminded instead were used to identify them, the contagion could be halted––and future generations spared the scourges of mental deficiency.
CHRISTINE ROSEN, WRITER: Henry Goddard said “You know, it takes an expert to identify the true menace of feeblemindedness. So someone you’re sitting next to at a restaurant or in a theater could look perfectly normal to you and it only takes one feebleminded person marrying another one, even someone who’s not feebleminded, to create generations of feeblemindedness.” What it did is up the stakes of feeblemindedness by claiming that it was a hidden menace that was more difficult to pinpoint than people might think.
NARRATOR: By early 1910, Charles Davenport was convinced that certain human traits were passed down in a predictable way––and that American society could be dramatically improved if only reproduction were controlled. Anxious to spread the word, he began to lay plans for a new institution dedicated to eugenic research and education.
In February, in search of a patron, Davenport traveled to New York to lunch with Mrs. E. H. Harriman, widow of a recently-deceased railroad magnate.
CHRISTINE ROSEN, WRITER: Davenport’s pitch to Mrs. Harriman was to say, “Right now you give your money to all kinds of good organizations. They feed the poor. They clothe the poor. They do many wonderful things but it’s never ending. With eugenics we eventually won’t need your philanthropy and charity because we’ll solve the problems that right now you’re just throwing money at.” He persuaded Mrs. Harriman that the future of the country was at stake and that only a eugenic project could save it.
JONATHAN SPIRO, HISTORIAN: Charles Davenport says, "All you people who think that if we just educate the poor, if we just give them charity, if we just reform their environment even the poor can rise to our level, forget about it. That’s just sentimental hogwash. It’s not the environment that makes you what you are. It’s your genetic inheritance from your parents. So now, yes, let’s regulate the matings of human beings, eliminate the bad genes from the population, and keep the fittest genes in the gene pool."
KEITH WAILOO, HISTORIAN: By limiting the birth of people who were deemed to be unfit you were by definition enhancing the stock of human society. And so there was this social mission of really fighting dependency, fighting crime through eugenics. The idea was that eugenics would solve all of these broader social problems if enacted in a robust way.
NARRATOR: Mrs. Harriman was a great believer in the importance of proper matings––she credited her late husband's interest in horse-breeding for that––and she enthusiastically pledged to finance Davenport's eugenic enterprise.
It was, Davenport later wrote in his journal, "a red letter day for humanity!"
DR. SIDDHARTHA MUKHERJEE, WRITER: The impulse to perfect humanity is an ancient aspiration. The idea that somehow or the other that you can get the best humans by selectively breeding the best, most fit, heartiest, most beautiful. It’s an ancient desire. You find it in Sanskrit texts. You find it in Greek texts. The trouble is that only some human beings can dictate or decide what those, what the correct features might be. Who decides?
JONATHAN SPIRO, HISTORIAN: Charles Davenport thought, "By breeding a superior race of people, we can bring about the millennial kingdom on earth."
NATHANIEL COMFORT, HISTORIAN: The problem with utopias is that they set a set of aspirations that then blind you to a certain set of consequences and that, that can be dangerous.
NARRATOR: In October 1910, on a 80-acre plot adjacent to the Cold Spring Harbor campus, Charles Davenport opened the doors of his new institute.
It was a modest structure built for a grand purpose: to house hereditary information on American families and use it to guide the reproductive choices of the nation.
He called it the Eugenics Record Office.
NATHANIEL COMFORT, HISTORIAN: Eugenic ideas were very much floating around as early as 1880. But Davenport gave eugenics teeth. He was institutionalizing eugenics. He was marshaling people around a research program.
NARRATOR: Davenport already had assembled a prestigious Board of Scientific Directors, among them prominent scientists, physicians, and famed inventor Alexander Graham Bell.
Day-to-day operations, meanwhile, would be overseen by Harry Laughlin, a high school superintendent from the Midwest with a lifelong passion for poultry breeding.
CHRISTINE ROSEN, WRITER: Laughlin was a very zealous proponent of eugenics and in that sense he got along well with Davenport. They both believed in the mission and they believed in the cause.
ADAM COHEN, WRITER: For Davenport, a lot of it was about what you can do in the laboratory and how you can analyze the data. For Laughlin a lot of it was about, “Well, how are we gonna get out there in the world and change the direction of history?”
NARRATOR: To gather new disciples to the cause––and to aid in the collection of data––Laughlin and Davenport launched an academic program, which offered training in eugenic field-research techniques.
Over the course of six weeks each summer, recent college graduates––from Vassar, Harvard, Oberlin––were taught how to investigate family histories; how to conduct interviews and make eugenically useful measurements; and how to chart family pedigrees and analyze them. Then, at a salary of seventy-five dollars a month, came a year's work in the field.
Armed with the official "Trait Book," which assigned numerical codes to a broad spectrum of human characteristics, the newly-minted researchers fanned out: to study delinquents in the Juvenile Psychopathic Institute of Chicago, the insane at the New Jersey State Hospital at Matawan, albinos in Massachusetts, circus families at Coney Island, the Amish in Pennsylvania.
ADAM COHEN, WRITER: They would go into some holler in Virginia and find some family that seemed to have a lot of alcoholics and criminals, and, you know, other ne’er-do-wells, and they would go, “Ah ha. We’ve found a family with terrible genes.” They'd interview someone and they’d say, “Oh, yeah, you know, John seems a little slow but I knew his uncle and his uncle was a big drunk,” and they’d write that down, “Uncle a big drunk.” So they would come back with all this evidence of the way in which human traits were inherited.
DANIEL KEVLES, HISTORIAN: Of course the reliability of the data was not questioned even though it was based on interpretation, impression, recollection by the living members of the family with whom they spoke. No checking.
NARRATOR: Year by year––as trainees rotated out of the summer program and into positions at universities, hospitals and mental institutions––Davenport's assumptions and methods of fieldwork gained currency all across the country.
And year by year, the data accumulated. Stored in fireproof cabinets and intricately indexed, it comprised, as Scientific American noted, “a sort of inventory of the blood of the community"––and supplied grist for a multitude of books, pamphlets, lectures, and press releases regarding the danger of so-called "inferior germ-plasm."
JONATHAN SPIRO, HISTORIAN: They’re using the family records that are stored in the fireproof vault to prove that all these traits not just physical but mental traits, moral traits, are caused by genes. There’s nothing you can do about it. This is cold, hard, pure science.
NARRATOR: "Just as we have strains of scholars, of military men," Davenport told the New York Times, "we have strains of paupers, of sex offenders...strains with strong tendencies toward larceny, assault, lying, running away...The cost to society of these strains is enormous."
KEITH WAILOO, HISTORIAN: Davenport took this basic idea, applied it far more widely than it had ever been before and really promoted this probabilistic idea as if it were a deterministic one. That is to say, it’s not just likelihoods but in some ways we’re dealing with certainties. And that idea really sold.
NARRATOR: By the time the Eugenics Record Office issued its first official report in 1913, many Americans had begun to see the wisdom in eugenics.
"I agree with you that society has no business to permit degenerates to reproduce their kind," former president Theodore Roosevelt wrote Davenport that year. "It is really extraordinary that our people refuse to apply to human beings such...knowledge as every successful farmer is obliged to apply to his own stock breeding."
THOMAS C. LEONARD, HISTORIAN: If you’re going to be in the business of breeding, you’re gonna have to convince thought leaders and politicians, most especially the government, to begin a kind of unprecedented program. So they understood from the beginning that they needed to persuade those who were in a position to do something about it that it was possible, indeed, desirable.
NATHANIEL COMFORT, HISTORIAN: The eugenicists thought that people’s base interests are just self serving, selfish, right, and if you just leave them to themselves they’re gonna evolve in all these random, dumb directions.
NARRATOR: The Eugenics Record Office recommended both widespread eugenic education and aggressive government intervention: laws that would keep defectives out of the country, prohibit them from marrying, and prevent them from becoming parents by segregating them in asylums throughout their reproductive years.
Also recommended was a new and somewhat controversial surgical procedure known as "sterilization." By cutting and sealing organs involved in reproduction, both men and women could be made infertile. So far, the technique had been used primarily on criminals––particularly sex offenders––and it was thought to have a curative effect.
Harry Laughlin envisioned a broader application: as a eugenic tool that would eliminate defective germ-plasm once and for all.
ADAM COHEN, WRITER: Harry Laughlin really has this political vision of what we can do with eugenics. And he said, “In order for eugenic sterilization to really do what had to be done 15,000,000 Americans would have to go under the knife.
ALEXANDRA MINNA STERN, HISTORIAN: The idea was that eugenics was for the common good and by implementing the science of heredity, they could protect America and strengthen America.
DANIEL KEVLES, HISTORIAN: They thought of it as the beginning of a revolution. Of a religious movement. You have to start with a few converts and then you try to grow it into a bigger movement. All that seemed exciting and full of possibility and they were gonna create a new world.
NARRATOR: The eugenics movement that Charles Davenport had launched rested on Mendel's laws of inheritance, which assumed each trait was governed by one gene, and was passed down in predictable patterns.
But for all of Davenport's certainty about the gene, there remained open questions––about the gene's physical properties, and its location within the cell, and the means by which it accomplished its function.
All over the world, scientists looked to fast-breeding organisms in search of clues. Some focused their experiments on the sea urchin, which turned out a new generation each year; others, on the even speedier meal worm, with its larvae-to-larvae cycle of four months.
For zoologist Thomas Hunt Morgan, the organism of choice was the fruit fly––which was capable of reproducing in just ten days.
DR. SIDDHARTHA MUKHERJEE, WRITER: The organism breeds so quickly that Morgan is able to see things that the eugenicists cannot because he’s watching mutations move across multiple generations.
NARRATOR: By 1913, Morgan had been studying fruit flies for so long that his laboratory at Columbia University was known simply as "the Fly Room," and his assistants, "the Fly Boys."
For nearly a decade, they'd been holed up there, on the sixth floor of Schermerhorn Hall, breeding flies in half-pint milk bottles pilfered from the campus cafeteria.
Thousands upon thousands of mutants were crossed, and the results meticulously recorded: white-eyed, bristled, red-eyed, short-winged. When the data was collated, Morgan made a startling discovery: the mechanism of heredity in flies was far more complex than in Mendel's peas.
DR. SIDDHARTHA MUKHERJEE, WRITER: Gregor Mendel thought that every gene was its own unique discrete entity. Morgan showed that, in fact, that’s not the case, that, in fact, genes live, genes have a physical entity and they live in chromosomes. And because they live in chromosomes often they travel in packs.
DR. SIDDHARTHA MUKHERJEE, WRITER: Morgan's work complicates the idea of simple eugenics because you don’t just pick one thing out of one drawer, a second thing out of another drawer until you get your ideal child. It’s not so easy to pick and choose what your next generation might be.
NALONDRA NELSON, SOCIOLOGIST: It makes sense in pea plants, it makes sense in cattle, it should make sense in humans, but there were no experiments that really could support Davenport’s theory.
NARRATOR: Thomas Hunt Morgan was a believer in the transformative power of eugenics. He had served on the board at the Eugenics Record Office since it opened. But based on the lessons he'd learned in the Fly Room, it seemed clear that eugenic science, such as it was, had no business informing American laws.
"If [the eugenicists] want to do this sort of thing, well and good," Morgan wrote a friend, "but I think it is just as well for some of us to set a better standard, and not appear as participators in the show."
PAUL A. LOMBARDO, HISTORIAN: Morgan writes a letter saying, “I’m going to ask to be taken off this letterhead. I study fruit flies and I can’t figure out how their eyes work. I can’t figure out which one’s going to inherit certain kinds of wings and you seem to be saying you can understand who’s gonna inherit something as vague as criminality or pauperism. So he backed away but privately.
NARRATOR: Morgan's withdrawal from the Record Office was regrettable; but Davenport was undeterred. At this point, the eugenics movement would not be stalled by the minutae of science.
DR. SIDDHARTHA MUKHERJEE, WRITER: What genes are is a great biological and biochemical question. But there's kind of a Yankee practicality about eugenics, “Let’s get this job done,” and so they move right along.
NARRATOR: When the Panama-Pacific International Exposition opened in San Francisco, on the morning of February 20th, 1915, one hundred thousand people streamed through its turnstiles. Over the next nine months, the number would reach more than eighteen million.
Billed as "an encyclopedia of modern achievement," the fair offered a dizzying array of diversions and curiosities: a 23-minute ride over a functioning replica of the recently-completed Panama Canal; an assembly line that turned out eighteen Model T's a day; a fifty-seven-tier tower built entirely of Heinz condiment products.
ALEXANDRA MINNA STERN, HISTORIAN: The Panama-Pacific Expo was a celebration of science, efficiency, engineering. It was an opportunity for the United States to demonstrate the power of science and technology, and also a utopian vision looking towards the future.
NARRATOR: Nowhere did the future look brighter than from the Race Betterment Exhibit.
Housed in the Palace of Education, the display featured imposing plaster casts of Atlas, Venus and Apollo; a collection of medical instruments used to gauge human biological capacity; and a welter of charts, graphs, and lists that outlined the way eugenics would better the human race.
All of it was the work of Dr. John Harvey Kellogg, a fierce proponent of what he called "biologic living."
NATHANIEL COMFORT, HISTORIAN: John Harvey Kellogg was an incredibly energetic man. He was a health reformer, and a physician, and an amazing entrepreneur, and he developed all these regimens, and invented different medical instruments, had a whole dietary plan.
JONATHAN SPIRO, HISTORIAN: He was obsessed with cleanliness, with purity and he believed that the key to reforming society is to cleanse our bowels on a regular basis. He invents something called cornflakes to help cleanse your bowels. And he had a spa in Battle Creek, Michigan, and lots of eugenicists came to the Battle Creek Sanatorium to have their bowels cleansed and to talk about eugenics.
CHRISTINE ROSEN, WRITER: For Kellogg, eugenics made perfect sense. It was about health. He linked these views about heredity, which were difficult to change, with these ideas about what human beings can do to improve themselves.
THOMAS C. LEONARD, HISTORIAN: John Harvey Kellogg believed that the environment can effect the gene, that consuming alcohol or consuming meat could lead to genetic inferiority in offspring. So there was more than one way to improve heredity.
NARRATOR: At the Expo, Kellogg sought to usher his brand of eugenics onto the national stage. With assistance from Charles Davenport––who had supplied him with both data and contacts––Kellogg had organized not only the Race Betterment exhibit, but also a major eugenics conference at the Fair.
The turnout exceeded expectation––drawing reform-minded medical professionals, university presidents, conservationists, and business leaders from all over the country and across the political spectrum.
PAUL A. LOMBARDO, HISTORIAN: Eugenics had a little bit of something for everyone. So if you’re a social hygienist you’re interested in wiping out prostitution, eugenics is interested in that, too. If you’re a prohibitionist and you wanna get rid of alcohol because alcohol breaks up families. It makes men unemployable. Eugenics wants to get rid of all those things too. So it manages to match up with the concerns of many other different kinds of reforms.
KEITH WAILOO, HISTORIAN: What led people to get behind the eugenics campaign wasn’t just their ardent belief in the science or in heredity. It was a fundamentally broad and sweeping social and political agenda to try to recreate society, one might say, in their own image.
DANIEL KEVLES, HISTORIAN: They’re almost all white, they’re almost all Protestant, middle-to-upper-middle class and they tended to equate human worth with the qualities that they themselves possessed.
NARRATOR: Over five days in August, the sixty-odd conference delegates delivered talks on everything from proper toothbrushing to eugenic sterilization.
"[Unless we weed out the weaklings]," one speaker warned, "[w]e will reach a point...where...many of those born and helped to survive will be a burden to the race."
All told, the Race Betterment Conference drew an estimated ten thousand people, and generated more than a million lines of press. "Your efforts [on] behalf of eugenics are certainly beginning to bear fruit," Kellogg told Charles Davenport. "The public [is] beginning to understand better, and appreciate more."
ALEXANDRA MINNA STERN, HISTORIAN: The Panama-Pacific Expo was really a defining moment for the American Eugenics Movement. The Eugenics Movement was coalescing. It was solidifying.
ADAM COHEN, WRITER: These elites are all saying, “Yes, you know, we believe in progress and this is progress.” Eugenics gave them a way to view the world and to say, “Okay, you know, all these vague anxieties I have about the present and particularly the future, this is what the problem is. Well, let’s get to work on solving that.”
NARRATOR: In May 1917, as new converts spread the eugenic creed in cities and towns across America, a half-dozen psychologists gathered at the Vineland Training School for the feebleminded to meet with Henry Goddard, by now considered the nation's leading expert on mental deficiency.
His groundbreaking 1912 study, The Kallikak Family, had awakened the public to the menace of the feebleminded, with its true-life tale of an old-stock American, a feebleminded tavern girl, and a fateful tryst that over several generations had spawned more than a hundred mental defectives, among them one of Goddard's own patients at Vineland, a girl he'd diagnosed as a moron.
JONATHAN SPIRO, HISTORIAN: The Kallikak Family was a huge best seller for many, many years, references to the Kallikaks were in the speeches of politicians, books, scholarly journals, popular magazines. Everyone knew what the Kallikaks meant. You have to watch out who you mate with or your descendants could turn out to be feebleminded, criminals, alcoholics, and so forth.
NARRATOR: Goddard was eager to demonstrate the value of intelligence testing as a diagnostic tool––and he'd spent the years since his book's publication administering tests to scores of institutional inmates, immigrants, school children.
Now, with his colleagues, he designed a program to carry out intelligence testing on a mass scale.
Just seven weeks earlier, the United States had plunged into the First World War––and the draft ultimately would swell the army's ranks by nearly three million men. The aim of the testing program was to classify them for service, and to identify the mental defectives lurking among them.
They began in late September 1917 at Camp Lee and Camp Taylor, Camp Devens and Camp Dix. First, new recruits were sorted according to their level of literacy––and then administered one of two tests.
DANIEL KEVLES, HISTORIAN: They had one test called the Alpha Test for draftees who were literate in English and another called the Beta Test for draftees who were not literate in English or illiterate completely. One of the questions on the Alpha Test was, “The Knight engine is used in the Ford, the Pierce-Arrow, or the Lozier car?” Now, tell me, is that known to you?
NARRATOR: While the literate testers puzzled over multiple choice questions, the others attempted to draw their way out of mazes and sketch in the missing bits of simple pictures.
One Sicilian recruit, a Catholic, considered an image of a house and drew a crucifix where a chimney might be. He was marked wrong.
"It was touching to see the intense effort put into answering the questions," an Army examiner later recalled, "often by men who never before had held a pencil in their hands."
DANIEL KEVLES, HISTORIAN: The tests were by no means measures of intelligence, whatever that may mean. How well you did on them depended upon your degree of education and how many years you’d been in school, and also how attuned you were with middle-class culture.
NARRATOR: Administered to 1.7 million army personnel over the course of the conflict––officers and enlisted men, black soldiers as well as white––the tests led to a shocking conclusion.
Roughly half of the draftees were considered to be morons.
PAUL A. LOMBARDO, HISTORIAN: The Army’s experience became a headline, "America is degenerating. We have to somehow interrupt this swamp of defect.”
ADAM COHEN, WRITER: There was a movement to institutionalize more people at this time driven by eugenics. You can see how an IQ test can really grease the wheels. If you’re gonna start moving people into institutions, if you’re gonna start sterilizing them and all that you need some numbers and the IQ test provided that.
NARRATOR: By 1919, intelligence testing was a full-fledged craze.
An adapted version of the Army test, the National Intelligence Test, sold half a million copies in one year. Businesses administered mental tests to prospective employees, schools and universities evaluated their students, and ever more paupers, prostitutes, drunkards and deliquents found themselves suddenly with pencil in hand.
ADAM COHEN, WRITER: Feeblemindedness was a big fear in that era. There was a thought that there were a lot of these people out there who were deficient, who were morons and they were not only out there they were reproducing much more rapidly than other people.
DR. SIDDHARTHA MUKHERJEE, WRITER: The trouble is that in practice the word “moron” could be anyone who was not part of the, you know, the so-called social norm. So, the word “moron” begins as a scientific attempt to classify intelligence but very soon becomes usable as a means of social control.
NARRATOR: By 1920, the vast majority of those committed to institutions for the feebleminded were classified as morons.
ADAM COHEN, WRITER: To some extent, humanity’s always been about othering and about, you know, there’s us and there’s the other. The Eugenics Movement really gave this scientific, you know, punch to this idea that, “There are us and there are the others and we’re the right people. We’re the people that it’s important not only to favor now but we’re the people who have to own the future." Suddenly eugenics comes along and gives them a scientific basis for believing that.
THOMAS C. LEONARD, HISTORIAN: Eugenics is easy to accept because it preserves existing hierarchies. It didn’t seek to overturn them. What it did was lend new weight to established hierarchies.
PAUL A. LOMBARDO, HISTORIAN: I don't think there has ever been a time when people didn't think that some people simply better than others. The Eugenics Movement like a chameleon took on the colors of those attitudes which existed before the word “eugenics” was coined and certainly exist today.
NARRATOR: They'd been swarming the ports of entry since 1890––as many as a million of them a year, in flight from poverty and oppression, lured by the promise of equality and opportunity.
The Great War had staunched the flow; but with the armistice, the tap had been opened once more. By 1920, some 75,000 new immigrants were landing at Ellis Island each month.
That May, at Cold Spring Harbor, Charles Davenport penned a letter to a friend: “Can we build a wall high enough around this country," he wondered, "so as to keep out these cheaper races?”
ADAM COHEN, WRITER: Charles Davenport was born into a very fancy, old stock family. So he was someone brought up to believe that family mattered and that, you know, good qualities ran in good families like his.
CHRISTINE ROSEN, WRITER: Charles Davenport was very focused on wanting to maintain the traditional American stock. And he wasn't alone in that. There was this fear that the right sort of American wasn’t having enough children. And the race as it existed was being diluted and polluted by incoming waves of immigrants.
JONATHAN SPIRO, HISTORIAN: Most immigrants used to come from Northern and Western Europe, from the British Isles, from Germany. And then all of a sudden in the 1890's, immigrants started coming here from Eastern Europe, from Southern Europe. These are Catholics, these are Jews, these are peasants. And Davenport feels correctly that his race is losing the demographic game.
NARRATOR: On the receiving end of Davenport's letter was Madison Grant, a zealous convert to the eugenics cause with a sterling American pedigree and an abiding preoccupation with endangered species.
JONATHAN SPIRO, HISTORIAN: Madison Grant was a very wealthy lawyer. His ancestors are traced back to the original Puritan founders of the United States. Some of his ancestors signed the Declaration of Independence. He was a committed conservationist. He saved the redwoods from extinction. At some point he realized, “I’ve been spending all my time and effort trying to save our nation's flora and fauna while my own race is dying out.
JONATHAN SPIRO, HISTORIAN: When Madison Grant walks out the door of his Wall Street law office, he is accosted by thousands of foreign-speaking peasants. They don’t know and they don’t care that Madison Grant’s ancestors signed the Declaration of Independence and he is offended.
NARRATOR: Grant had sounded the alarm for old-stock Americans in 1916 with The Passing of the Great Race––a 476-page elegy for what he called "the white man par excellence."
ALEXANDRA MINNA STERN, HISTORIAN: His vision is one of America as a country wrought by great men who ventured from Europe, and an America that is facing an onslaught from the undesirable hoards from most of the rest of the world.
JONATHAN SPIRO, HISTORIAN: He invents this race called the Nordics, this tall, blond-haired, blue-eyed race. According to Grant, the Nordics are the most recently evolved of all the races. That means their genetic traits are still fragile. They’re not fully formed. And so if a blond-haired, blue-eyed Nordic mates with a more primitive race, a Mediterranean, a Jew, certainly a Negro or an Asiatic, the more primitive genes of the inferior race will actually overwhelm the superior but not yet stable genes of the Nordics.
ADAM COHEN, WRITER: So this is a threat, and threat not just, you know, “Hey, I look around the city and it looks a little different.” This is a genetic invasion.
NARRATOR: As Grant saw it, the threat from the Negro race was mostly neutralized by laws already on the books in many states, that forbid marriage between blacks and whites. The threat posed by the foreign-born, however, was at once more insidious and more pressing.
“We Americans must realize that the altruistic ideals...and the maudlin sentimentalism that has made America 'an asylum for the oppressed,' are sweeping the nation toward a racial abyss," Grant declared. "This generation must completely repudiate the proud boast of our fathers that they acknowledged no distinction in 'race, creed, or color,' or else...turn the page of history and write 'Finis America.'"
JONATHAN SPIRO, HISTORIAN: Madison Grant takes eugenics, which had hitherto been concerned only with survival of the fittest individual, and he says we need to be concerned with the survival of the fittest race. We need to preserve the Nordic race.
NARRATOR: Grant's mission in 1920 was to rally his fellow eugenicists and convince the federal government to drastically reduce immigration.
He began with a charm offensive directed at Congressman Albert Johnson, the Chairman of the House Committee on Immigration and Naturalization––inviting Johnson to New York, plying him with whiskey and cigars, and gradually persuading him of the urgent need for eugenics.
Once Johnson was in the fold, Grant suggested he bring Harry Laughlin, the superintendent of the Eugenics Record Office, to Washington D.C., to testify on the so-called "biological aspects of immigration." Johnson was so impressed with the presentation, he named Laughlin "Expert Eugenics Agent" and commissioned him to make a study of the foreign-born.
In the meantime––amid a rising anti-immigrant clamor from labor unions, social workers, conservationists––Congress curbed the influx with the Emergency Quota Act of 1921.
JONATHAN SPIRO, HISTORIAN: This was supposedly a one-year temporary measure. But in 1922, the bill is renewed for another two years and that gave Madison Grant and the eugenicists time to launch a massive propaganda campaign convincing Americans that immigration restriction must be permanent.
NARRATOR: In September 1921, at New York's American Museum of Natural History, Grant convened an international eugenics congress to whip up support for the cause.
Organized in tandem with Charles Davenport, the week-long event drew some 300 delegates from twenty-eight foreign countries. Numerous members of the Senate and House immigration commitees were in attendance, as was actress Lillian Russell––who now informed her legions of fans that the American melting pot was a catastrophe.
“If we don’t put up the bars and make them higher and stronger," she warned, "there no longer will be an America for Americans.”
JONATHAN SPIRO, HISTORIAN: There are all kinds of exhibits at the Congress showing that Negro fetuses have smaller skulls. Italians have a higher level of criminality than other people. And at the end of the Congress, the exhibits are packed up and shipped to Washington, DC where they are prominently displayed in the committee rooms so that Congressmen could not help but, consciously or not, imbibe all the latest scientific findings of eugenics.
NARRATOR: But it was Harry Laughlin's return to Capitol Hill––and the reports on his study––that convinced many on the House Committee of the perils of unchecked immigration.
ADAM COHEN, WRITER: He had numbers that purported to show that rates of insanity were different among immigrants from different countries, that certain nationalities were much more likely to have their immigrants become prison inmates. And he also argued that just biologically because we were largely a Nordic, Northern European country it was harder to assimilate immigrants from other parts of the world.
NARRATOR: Citing data from the Army intelligence tests, Laughlin claimed that foreign-born whites––and in particular, Jews––were intellectually inferior to native-born Americans, and therefore likely, over time, to diminish the intelligence of the nation.
JONATHAN SPIRO, HISTORIAN: The Jews on the Immigration Committee object. They claim correctly that the eugenicists have first come up with their theory that Jews are inferior and then found the data to back it up. But Congress is converted to the cause of eugenics. The Congressional Record is filled with Congressmen reading excerpts from The Passing of the Great Race, Madison Grant’s book, on the floor of Congress and so the Restrictionists win the day and Congress passes immigration restriction legislation.
NARRATOR: On May 26th, 1924, President Calvin Coolidge signed the restriction act into law. Madison Grant hailed it as "one of the greatest steps forward in the history of this country."
THOMAS C. LEONARD, HISTORIAN: They shut the door and reduced immigration to the United States by 97%. The door was shut and it didn’t open again for forty years. And in a very real sense this was a political policy victory for eugenics.
NARRATOR: The new policy would help the nation to remain, as one congressman said on the House floor, "the home of a great people: English-speaking—a white race with great ideals, the Christian religion, one race, one country, one destiny."
ALEXANDRA MINNA STERN, HISTORIAN: It was really a reversal of, you know, “Give us your tired and your huddled masses," and it sends a message that the open arms of Ellis Island are now closed.
NARRATOR: For many of those across the Atlantic who would pin their hopes on America in the years to come, the consequences would be dire.
ADAM COHEN, WRITER: Congress passed this law and closed the door on Jews in Eastern Europe and Germany who were trying to flee the Nazis. Otto Frank wrote to the U. S. State Department trying to get visas for his family and he wrote repeatedly, and he had connections, and he was turned down because of this law. We think about Anne Frank dying in a concentration camp because the Germans thought the Jews were genetically inferior, but to some extent Anne Frank died in a concentration camp because the US Congress believed that as well.
ARCHIVAL: [Margaret Sanger] We believe that married people who have transmissible diseases should not have children. No couple who has the disease of feeblemindedness or insanity or epilepsy should have children. Babies should not be brought into the world when the father's income is obviously inadequate to provide for its food, clothing, or shelter.
NARRATOR: On August 5th, 1926, a crowd gathered at Vassar College to hear a lecture given by Margaret Sanger, the controversial founder of the American Birth Control League.
Sanger's reputation preceeded her. In her dozen years as a crusader for contraception and family planning, she'd been denounced, jeered, and jailed repeatedly. Now, she'd undertaken a cross-country speaking tour intended to bolster her cause by linking it to eugenics.
THOMAS C. LEONARD, HISTORIAN: Margaret Sanger was laser beam focused on promoting birth control, which she saw as a liberatory agent for women. It was a hard push, reproductive rights, contraception. Her embrace of the eugenicists was a way of getting some influential and powerful allies behind her cause.
NARRATOR: "The Question of race betterment," Sanger told the Vassar audience, "is one of immediate concern and I am glad to say that...the Government has already taken certain steps to control the quality of our population through the drastic immigration laws...But while we close our gates to the so-called 'undesirables' from other countries, we make no attempt to discourage or cut down the rapid multiplication of the unfit and undesirable at home."
KEITH WAILOO, HISTORIAN: Margaret Sanger is struggling to open a conversation at a time when public discussion of birth control let alone access to birth control was illegal. But her views are fairly persistent with regard to issues of biological inferiority. You could argue that they’re strategic but the difference is not that significant from the standpoint of those listening to her words.
WENDY KLINE, HISTORIAN: Once birth control is packaged as a way of improving the human race, it seems more manageable. There were a lot of people that were on the fence that she convinced to embrace birth control because of its eugenic potential. It was being labeled a birth control activist that was truly controversial. Being a eugenicist was far more acceptable.
NARRATOR: Amid the many American enthusiasms of the 1920's––skimpy dresses, dance marathons, mahjong––breeding a better human race was perhaps the most unlikely. But by the middle years of the decade, the notion was everywhere.
Included in the curriculum at more than 350 American colleges and universities––among them Harvard, Northwestern, and the University of California at Berkeley––eugenics also was preached from pulpits, promoted on lecture circuits, and appropriated to sell everything from newfangled beauty treatments to children's toys.
Disseminated by a host of popularizers––and at times diluted, distorted, or both––the eugenic creed filtered down to the masses through magazine articles, advice manuals, even a movie called "Are You Fit to Marry?"
ADAM COHEN, WRITER: This was really something that permeated the culture. It was really a craze. It was something people were excited about.
CHRISTINE ROSEN, WRITER: Eugenics starts to trickle into mainstream popular culture in the 1920s and it says to individual Americans, “If you want your society to improve you have to marry the right person. You have to have healthy children. You have obligations to the human race and to your country.”
NARRATOR: As one newlywed confessed in a letter to his local eugenics society: "My wife and I are both extremely tall, and this...worries us as we do not wish to bring abnormally tall children into the world."
At state and county fairs across the country––in Massachusetts, Kansas, Georgia, Texas––a human stock contest known as "Fitter Families for Future Firesides" drew throngs.
Sponsored by the American Eugenics Society, a propaganda organization run by the movement's evangelists, Harry Laughlin and Madison Grant, the competition offered a primer on eugenics, disguised as wholesome family entertainment.
CHRISTINE ROSEN, WRITER: What the American Eugenics Society realized is that if you’re gonna spread a message about eugenics, you have to get people involved
in more than just reading something in a popular magazine.
JONATHAN SPIRO, HISTORIAN: Eugenics is all-encompassing creed. It’s a faith. It’s a religion. And Harry Laughlin and Madison Grant understood, we need the people to be converted of this religion so that everyone will understand, “If I am eugenically superior I cannot date and certainly cannot mate with a eugenically unfit person.
NARRATOR: Fitter Families contestants came from miles around, often dressed in their Sunday best, and submitted themselves to a rigorous three-hour inspection.
Straight, healthy teeth earned them high marks––as did musical talent, or a family history of longevity. Disease or disability––even a lame grandmother or an epileptic uncle––was a demerit.
"While the stock judges are testing the Holsteins, Jerseys, and Whitefaces in the stock pavilion," one contest organizer said, "we are judging the Joneses, Smiths, and the Johnsons."
JONATHAN SPIRO, HISTORIAN: Just as they would have a contest who had bred the best cows, who had bred the best sheep, who had bred the best children. And at the end of the state fair, the eugenic winning family, the fitter family, would be driven down the midway and wave to the people and show off their ribbons.
PAUL A. LOMBARDO, HISTORIAN: By the 1920’s, eugenics was a household word. A generation of people grows up thinking of this word as a aspiration, healthy babies, and as a warning. They’ve read it in school, they’ve heard it at church, it has become part of the consciousness of the country.
NARRATOR: So pervasive was the impulse to human improvement, even prominent African-Americans took up the theme.
W.E.B. DuBois, one of the founders of the National Association for the Advancement of Colored People, maintained that the "best" of the black race––what he called "the Talented Tenth"––was the hope for the future.
"The Negro," DuBois declared, "...must begin to...breed for brains, for efficiency, for beauty."
KEITH WAILOO, HISTORIAN: DuBois’ ideas are fundamentally about combating prejudice, but at the same time he talked about and embraced the notion that not all blacks were equally gifted and equally talented, and that the future of African Americans should hinge on the future procreation of the talented. Those ideas really are resonant with eugenic ideals of the time.
WENDY KLINE, HISTORIAN: Eugenics became a really powerful ideology because it made sense to a lot of different groups who were concerned about disparate things. Part of the draw is how science can make us better human beings, that we can engineer ourselves into being even better than we are. And viewing that as a source of progress.
NALONDRA NELSON, SOCIOLOGIST: The Eugenics Movement of the early 20th century got traction because the slogans were simple, things like “Better babies and happy families,” you know. On the face of it, you know, better babies, healthier babies, what’s not to like? It would have taken considerable effort to demonstrate to people what that simple slogan was actually hiding.
NARRATOR: In September 1924, at the Virginia Colony for the Epileptic and Feebleminded, the colony's board of directors met to discuss the case of patient 1692, a 17-year-old named Carrie Buck.
She'd been admitted to the colony several months before, at the request of her foster parents, who claimed that they could no longer "control or care for her."
ADAM COHEN, WRITER: Carrie Buck had been raised by a foster family, not a nice family. She is rented out to other people in the community to do house cleaning, and she’s pulled out of school after fifth grade even though she’s doing very well and is a perfectly good student. Then, a nephew of her foster mother rapes her and she gets pregnant and they wanna get rid of her.
NARRATOR: By the time her daughter was born, the state had labeled Buck "morally delinquent" for having given birth out of wedlock, diagnosed her a "middle grade moron," and confined her to the colony.
WENDY KLINE, HISTORIAN: Sexual delinquent, sexually immoral. These terms are intentionally vague. Immoral tendency could be that a woman had been sexually abused. It could mean she was going out late at night. It could mean she's a prostitute. If you’re morally deficient that’s evidence that you’re mentally deficient and vice versa. So the state needs to intervene.
NARRATOR: The question before the Colony's board of directors now was whether or not to sterilize Carrie Buck.
ADAM COHEN, WRITER: She lands at the Colony for Epileptics and Feebleminded right when Virginia has passed a eugenic sterilization law and the lawyer for the state hospitals really wants there to be a test before sterilizations occur. So the Superintendent of the Colony, he basically needs an inmate that he can say, “I’m gonna sterilize you,” have that person challenge the law, and then hopefully prevail against her. So he’s looking for someone and Carrie Buck checks a lot of boxes.
NARRATOR: From the board's perspective, the menace posed by Buck's own feeblemindedness was doubled by her lineage. Her mother, who was alleged to have engaged in prostitution, was likewise an inmate at the colony.
"By the laws of heredity," the board concluded, "[Carrie Buck] is the probable potential parent of socially inadequate offspring." It was recommended she be sterilized––for both her own welfare and the good of society.
Then Buck was assigned an attorney, friendly to the eugenic cause, who would appeal her sterilization––ideally, all the way to the Supreme Court of the United States.
Across the country, eugenicists would be watching––to see if the Virginia test case could create a national consensus on sterilization.
PAUL A. LOMBARDO, HISTORIAN: Sterilization was a radical procedure. Between 1915 and the mid-1920’s, you have a dozen or more states that pass laws that allowed for mandatory sterilization of people in institutions. Some of them were used actively. Many of them were just on the books but nobody was being operated on. Some of them had been struck down by state courts. So it wasn’t at all clear what was going to happen to eugenic sterilization.
ADAM COHEN, WRITER: There was a hope among eugenicists, “If we could just get a case that goes up to the Supreme Court, one ruling from the Supreme Court and suddenly we’ve got a national legal standard that eugenic sterilization is acceptable. So that became high on the wish list of the Eugenics Movement.
NARRATOR: No one was more interested in the Virginia test case than Harry Laughlin, who had spent much of the previous decade promoting sterilization as a cheap, effective way to rid the nation of what he called "the socially inadequate classes."
ADAM COHEN, WRITER: Harry Laughlin believes that to really move the needle on the national genetic pool and really improve things, sterilization was the answer.
PAUL A. LOMBARDO, HISTORIAN: Eugenical sterilization was Laughlin’s life work. He published a book in 1922, a compendium of every law that had been passed, of every case that had been brought, excruciating detail about the history of eugenical sterilization. And it became the bible for people who wanted to pass sterilization laws.
NARRATOR: It was only a matter of time before Laughlin was asked to serve as an expert witness in the case against Carrie Buck––and though he was unable to appear in person, he was more than happy to help.
PAUL A. LOMBARDO, HISTORIAN: Laughlin never met Carrie Buck. Laughlin never traveled to Virginia to see her. His testimony was read into the record of the Carrie Buck case as a deposition.
NARRATOR: The Buck family, Laughlin argued, was "mentally defective"––members of what he described as the "shiftless, ignorant, and worthless class of anti-social whites of the South." As such, Carrie was certainly likely to give birth to defective children. No doubt, with her infant daughter Vivian, she already had.
Laughlin's testimony proved persuasive. As the eugenicists hoped, first the County judge, then the state Supreme Court upheld Virginia's sterilization law.
The next––and final––ruling would come from the Supreme Court of the United States.
ADAM COHEN, WRITER: Poor Carrie Buck, there’s no weaker person perhaps who’s ever come before the Supreme Court. She is poor, and she is alone, and her mother is an inmate, and she has a lawyer that’s been chosen by her enemies to not represent her. And she’s asking the font of justice in our society, "Don’t let them forcibly operate on me so I can’t have children." And they say, "Go ahead, sterilize her."
NARRATOR: In May 1927, the court's majority opinion was rendered by the venerable Oliver Wendell Holmes, who, at 86, was widely regarded as America's most brilliant legal mind.
"It is better for all the world," Holmes wrote, "if instead of waiting to execute degenerate offspring for crime, or to let them starve for their imbecility, society can prevent those who are manifestly unfit from continuing their kind... Three generations of imbeciles are enough."
DANIEL KEVLES, HISTORIAN: Justice Holmes says, "If she is allowed to reproduce or if the Carrie Bucks of the world in general are allowed to reproduce this will be deleterious to American society. And so therefore the government has the authority to step in and in pursuit of the greater public good to suppress her individual right to reproduce."
NARRATOR: Carrie Buck was sterilized on October 19th, 1927. Less than a month afterward, she was paroled from the Colony.
Thanks to Carrie Buck, a jubilant Laughlin declared, eugenical sterilization's "experimental period," had come to an end.
Over the two decades that preceded the Carrie Buck case, only about 6,000 sterilizations had been performed nationwide. In the six years that followed it, as states across the country rushed to enact sterilization laws, that number would more than double.
PAUL A. LOMBARDO, HISTORIAN: If you look back at all the sterilization laws passed, the easiest way to sum up who their targets were is, “Round up the usual suspects.” You are generally going to be dealing with poor people, people who are part of a disfavored minority, people who were on private charity or public welfare, people who had disabilities, mental or physical, and people who were generally considered somehow on the margins of society.
KEITH WAILOO, HISTORIAN: The conceit of eugenics was that scientists understood what traits were associated with health and well-being over the long term. But hereditary science in the early 20th century was still emerging. Eugenics led the public discussion, promoted the science of human heredity in a time when hereditarian scientists were themselves developing their craft. And I think for a period of time they saw this as a positive development—society taking interest in the kind of science that they were doing. And then I think by the Twenties there’s a problem.
NARRATOR: It was the fall of 1926, and geneticist Hermann J. Muller, a former Columbia University Fly Boy, was looking for ways to speed his experiment along.
He was still working with flies––though now on his own, at the University of Texas in Austin. So far, he'd been using the technique he'd learned in the Fly Room, from Thomas Hunt Morgan: hunt for naturally arising mutations, then track across them across generations. Breed a generation, peer at its members one by one through a jewelers loup, repeat.
But at this point, Muller had lost patience.
DR. SIDDHARTHA MUKHERJEE, WRITER: It took an enormous amount of time to generate these mutants. You had to wait until you basically found one. It was a process of, of discovery. So Muller began to wonder he could actually create mutants de novo, from scratch, by doing something to the genes.
NARRATOR: One night, on a whim, Muller switched on the X-ray machine, and began irradiating male fruit flies. Once they'd been exposed, he slid them into glass bottles with a roughly equal number of female flies. Then he waited.
When the larvae began to appear on day five, it was clear the whim had worked.
DR. SIDDHARTHA MUKHERJEE, WRITER: Muller, by using the exact right dosage of X-rays, finds that he can make dozens of mutations, mutations that would have taken months or years to find. He becomes a mutant maker. He can’t do it in a predictable way. But the principle that human gene material was malleable, was changeable is an idea that Muller understands and embraces.
NARRATOR: If an insect's genes could be altered by a blast of radiation, Muller realized, human genes one day might be manipulated as well––and heredity would no longer be the prerogative, he said, of "an unreachable god playing pranks on us."
The idea of controlling human heredity had captivated Muller since his earliest days in Morgan's lab. He'd been aware of the flaws in so-called "eugenic science" for nearly as long––and his doubts about the American eugenics movement had been steadily mounting.
DR. SIDDHARTHA MUKHERJEE, WRITER: Muller began to think that you couldn’t have a Eugenics Movement without asking questions about equality. What was the criteria for judging you know a better human being than a worse human being and thereby sterilizing the, the worst human being or selectively breeding the better human being? Who would ensure that the Eugenics Movement was selecting the best features, when the, when the best features were dictated by the elites?
NARRATOR: Concerns about the eugenics movement had been raised before
––but they'd come mainly from lone voices, shouting into the wind.
Now, increasingly, hereditary scientists began to speak as one.
PAUL A. LOMBARDO, HISTORIAN: More and more scientists are realizing that heredity’s not something that you can understand simply like Mendel understood his pea plants, that some human traits are really complex and you can’t predict whether they’re gonna appear, or reappear, that some conditions that we think of as hereditary are really about social issues. Nobody really discards the idea that heredity is important but there is growing chorus of scientists who are being more careful in the way that they talk about heredity.
NARRATOR: Even the father of the intelligence test, Henry Goddard––who had done so much to stoke fears of hereditary feeblemindedness––disavowed his earlier conclusions. In particular, he regretted having coined the term "moron." With proper education, he now believed, such individuals were perfectly capable of managing their own affairs.
DANIEL KEVLES, HISTORIAN: The eugenic scientists were doing what they understood to be reliable science and it turned out that, in many cases, their science was mistaken. Science is a process. People make claims, they advance evidence for it. And then others come along who have more sophisticated understanding of the methodological problems, and they say, "Hey, prostitution may result from a woman’s having no other choice economically," or, "Alcoholism may arise from all sorts of stresses in one’s life." You don’t need genetics at all to explain these things.
NARRATOR: As the 1920's came to a close––and the Great Depression radically rearranged American society––the dogma of the eugenics movement rang ever more hollow.
ADAM COHEN, WRITER: 25% of the country’s unemployed. People’s life savings have been wiped out by both the stock market crash and the bank failures. The person who’s now on the bread line might have been a lawyer who graduated from Harvard. And this was a clear indication that poverty was not biological.
NARRATOR: When, in 1932, yet another eugenics congress convened in New York, most in the scientific community declined to attend.
JONATHAN SPIRO, HISTORIAN: They hold this conference to propagate the idea of eugenics. All the same guys are there—Madison Grant, Charles Benedict Davenport, Harry Laughlin––espousing the same ideas. Their ideas have not changed in 25 years and almost nobody comes. Because among scientists, eugenics is now viewed as the purview of a bunch of old, white cranks whom science has passed by.
NARRATOR: Improbably, Hermann Muller did turn up at the congress––though only to deliver a scathing ten-minute speech.
“There is no scientific basis for the conclusion that the socially lower classes...have...genetically inferior intellectual equipment," he insisted. "[C]ertain slum districts of our cities are veritable factories for the production of criminality among those who happen to be born in them...under these circumstances it is society, not the individual, which is the real criminal, and which stands to be judged.”
KEITH WAILOO, HISTORIAN: The problem of eugenic thinking was an utter ignorance of social causes of social problems, a tendency to overbiologize, to think through the biological lens about everything in society.
NARRATOR: Eugenics might yet perfect the human race, Muller told the audience, but only in a society "consciously organized for the common good...".
NARRATOR: In July 1933, in Germany, Adolf Hitler came to power––
ARCHIVAL: [Hitler] He who beats you, beats us...
NARRATOR: and immediately enshrined eugenics in state policy, with a law that mandated the sterilization of men and women suffering from any one of nine presumably heritable conditions.
It had been based on a model law written by Harry Laughlin.
DANIEL KEVLES, HISTORIAN: Before Hitler, there was a German eugenics movement. But it did not have a sterilization law. The sterilization law was ultimately enacted with the inspiration of what American states had been doing.
ARCHIVAL: German film on Kallikak Family
ADAM COHEN, WRITER: Harry Laughlin is corresponding with German scientists all along and encouraging them. He’s proud of the fact that when the Nazis adopt a eugenic sterilization law, it’s strongly modeled on his own law.
JONATHAN SPIRO, HISTORIAN: The United States has the reputation of being on the forefront of scientific endeavor. When Adolf Hitler was in prison, he read Madison Grant’s The Passing of the Great Race, wrote Madison Grant a fan letter saying, "This book is my bible," and when he wrote Mein Kampf, his autobiography, he said, "We Germans must emulate what the Americans are doing."
NARRATOR: Nazi officials estimated no fewer than 400,000 Germans would be sterilized––roughly 25 times the number sterilized in the United States so far.
The more zealous American eugenicists applauded the Nazi law, which applied to all people, whether institutionalized or not. As one Virginia sterilization advocate put it: "The Germans are beating us at our own game."
But for many Americans, the news from Germany was accompanied by an uncomfortable revelation.
"Many...interviewed...about the Hitler proposal expressed shock...," the Daily News reported. "They were surprised to find out [that]...twenty-seven of our forty-eight [American] states have laws permitting the performance of [sterilizations] upon the feebleminded...".
NATHANIEL COMFORT, HISTORIAN: The 1930’s was the peak of eugenic sterilization. And that was after geneticists––professional, scientific geneticists––had largely abandoned the eugenic program.
DANIEL KEVLES, HISTORIAN: There’s this trend that discredits the doctrine on which eugenic sterilization is based. At the same time, paradoxically, sterilization rates shot up in the United States––because of the Depression. It costs money to keep people in homes for the feebleminded. So if you wanna reduce the cost of keeping people, you sterilize them, and that’s what happened.
ARCHIVAL: [a group of women talking] Woman 1: I'd like to know just what sterilization is. Woman 2: So would I. Just how do they do it? Woman 3: Well, I'll tell you.
NARRATOR: As public awareness of eugenic sterilization spread, a controversial Hollywood film opened in theaters, a cautionary tale about good intentions gone dangerously wrong.
ARCHIVAL: [Alice] And do you mean they're going to stop me from having children ever? [Social worker] Exactly.
NARRATOR: Released in 1934, "Tomorrow's Children" told the story of 17-year-old Alice Mason, the sole functional member of an otherwise drunken, crippled, feebleminded family, who is slated for sterilization along with her parents and siblings...
ARCHIVAL: [Judge] Three generations of unfit are enough.
NARRATOR: ... and saved from the scalpel only by the revelation that she'd been adopted.
ARCHIVAL: [Alice] Look at me. Can't you see that I'm well and strong? And I'll be a good mother too, Judge, honest I will.
PAUL A. LOMBARDO, HISTORIAN: Tomorrow’s Children raises the question of whether or not you always get it right when you sterilize someone. How much can you really know about someone’s background? Without getting into the details of how much do we understand about genetics in 1934, it simply says, “Sometimes people make mistakes with these things and so maybe we should be more careful.
NARRATOR: Tomorrow's Children was still playing on screens across the country when, in 1935, a committee of scientists turned up at Cold Spring Harbor.
They'd been sent by the Carnegie Institution, which had sponsored the Eugenics Record Office since 1918, and had long been embarrassed by its political activities. Now, Carnegie's board of directors had ordered a review of the work being done there.
The visiting committee's report was decidedly unfavorable: from a scientific vantage, they concluded, the thousands of heredity records stored in the famed fireproof vault were useless for the study of human genetics.
ADAM COHEN, WRITER: They rightly saw that this eugenics fieldwork was largely ridiculous and was not scientific. But they also were troubled by the degree to which clearly Harry Laughlin was acting not as a scientist but as a evangelist for eugenics. And this was a clear indication that the tide was really turning against eugenics.
NARRATOR: For the movement's faithful, the message was plain: if they were going to continue to cull the unfit, they would need a new justification for it.
NARRATOR: From the moment the case of the "sterilized heiress" first hit the news, in January 1936, Americans were enthralled by it.
First, there was the girl, Ann Cooper Hewitt––a San Francisco socialite who stood to inherit two-thirds of her late father's vast estate––and her shocking claim: that her mother had had her sterilized to gain control of that inheritance.
WENDY KLINE, HISTORIAN: Ann Cooper Hewitt is sent to the hospital for an emergency appendectomy and she comes out sterilized. And when she discovers it, she is understandably horrified, and she sues both her mother and the two surgeons. She claims that her mother has done it because her father's will stipulates that if Ann should die childless the inheritance would go to her mother.
NARRATOR: Equally intriguing was the claim of the mother, Maryon, that her daughter Ann was feebleminded––a diagnosis based on an intelligence test she'd been given just hours before her sterilization.
WENDY KLINE, HISTORIAN: Ann says that she’s writhing in pain and then a woman walks in the room, and the woman starts asking her all these questions. “What’s the longest river in the United States?" and, "How many years is a presidential term?" And Ann’s reaction is, “Why are you asking me these asinine questions? What does this have to do with appendicitis? And she doesn’t answer most of the questions.
NARRATOR: Although her score identified the girl as a high-grade moron, a court-appointed psychiatrist at a preliminary hearing found her to be well read, fluent in French and Italian, and "perfectly normal in every respect."
PAUL A. LOMBARDO, HISTORIAN: The Cooper Hewitt sterilization case was one of those cases that people call “the trial of the century.” Headlines all over the country. And if you weren’t paying attention to what sterilization was by then you would have heard in that story.
WENDY KLINE, HISTORIAN: Ann Cooper Hewitt is not emblematic of the typical sterilization patient. And for that very reason she gets a lot more attention.
NARRATOR: By eugenic standards, Ann was the very definition of well-born. She was the scion of the successful: white, wealthy, seemingly sound in both body and mind. On what grounds then, those following the case may well have wondered, could her sterilization possibly be justified?
Attorney I. M. Golden, who represented the surgeons named in the suit, wondered much the same––and he decided to solicit the opinion of an expert.
In May 1936, he composed a letter to one of California's leading eugenicists, Paul Popenoe, and laid out for him the details of the case––among them, the reasons Maryon Cooper Hewitt had given for wanting her daughter sterilized.
WENDY KLINE, HISTORIAN: Maryon makes three charges about her daughter’s behavior that she sees as indicative of someone who is mentally defective. The first is that she becomes infatuated with a chauffeur. The second is that she is infatuated with men in uniform. And then finally that she has plans to run off with a Negro porter on a train. These are not people that probably Maryon believed her daughter should be associating with. Not somebody she should have children with.
NARRATOR: "In your opinion," Golden asked Popenoe, "was it proper to sterilize her... as a matter of medical and scientific procedure?"
Paul Popenoe long had been a proponent of eugenic sterilization.
But the argument that an immoral, oversexed girl would pass on those traits genetically could no longer plausibly be made. So Popenoe offered another rationale, one that had been recently formulated and recommended by the American Eugenics Society.
Heredity, Popenoe told Golden, is "not particularly the issue in this case... [But] I suppose, we should all answer negatively the question whether a young woman such as you describe would be a desirable mother."
ALEXANDRA MINNA STERN, HISTORIAN: In the Thirties, the eugenic rationale for sterilization begins to morph into a kind of more generalized understanding that this person isn’t fit to be a parent.
WENDY KLINE, HISTORIAN: That turns the whole argument about eugenics on its head, because the determining question was not, “Will she spread her genetic defect?” but, “Will she make a desirable mother?”
NARRATOR: When the trial of the two surgeons got underway in San Francisco, Ann's questionable capacity to mother was the centerpiece of the defense. The girl's sexual behavior alone, Golden argued, cast grave doubt on her ability to provide good moral and intellectual training to her offspring.
In the end, the argument had little effect on the judge, who, after six days of listening to testimony, abruptly called a halt to the proceedings and dismissed the case on the grounds that sterilization was legal in California.
But in the public mind, sterilization had been effectively recast––as a preventative measure against inept parenting.
PAUL A. LOMBARDO, HISTORIAN: This not a story that happens in an institution. It’s a story about a socialite. Nevertheless, the same themes of needing to sterilize people for their own good come up. Forget about heredity. These people will be unable to take care of their children so the humane thing to do is not to let them have any.
KEITH WAILOO, HISTORIAN: Eugenics simply becomes part of the machinery of how these state institutions function. Hereditary defect is no longer part of the conversation and it’s simply a question of a state attempting to use all the tools available to limit the number of people who were seen to be a social and economic burden.
NARRATOR: By the close of the 1930's, more than 30,000 Americans had been sterilized nationwide.
ADAM COHEN, WRITER: I think eugenics appeals to some real strong elements in the human psyche. One part of that, the postive part, is that there is a desire among people to perfect things. The negative side is, we're also a species that is very prone to tribalism. We're very prone to believe that our people are the right people, and other people are a threat.
NARRATOR: For a time, the enemies within American society were eclipsed by those without, and the nation's attention diverted by a conflict that consumed much of the world.
Then came the liberation of Buchenwald and Dachau––and the chilling evidence of eugenic policies carried to a monstrous extreme.
DR. SIDDHARTHA MUKHERJEE, WRITER: By the mid-1940s, the full horror of what happens in Nazi Germany becomes apparent––the movement from sterilization to extermination, the killing of several millions based on this kind of idea of the betterment of human race. And it creates a vast embarrassment for the American Eugenics Movement.
DANIEL KEVLES, HISTORIAN: People were repelled and began to turn away from eugenics, and "eugenics” became a dirty word.
KEITH WAILOO, HISTORIAN: The Holocaust, being tied to a wide range of eugenic practices, is a blemish on humans as a species, and it undercuts any notion that eugenics was a positive force in American society.
ARCHIVAL: Surgical sterilization was thought to be too slow and too expensive to be used on a mass scale.
ADAM COHEN, WRITER: After the war when the Allies put the Nazis on trial at Nuremburg, one of the charges was eugenic sterilization and the lawyer for the Nazi who was charged said, you know, “How can you charge my client with the crime of eugenic sterilization when your own US Supreme Court said this was okay?”
NARRATOR: By the end of the 1940's, the eugenics movement had faded from the mainstream of American life.
But the laws that had been passed in the name of eugenics would remain on the books for decades––barring some people from entering the country and others from marriage, and subjecting thousands to forced sterilization at the hands of the state.
By the time such practices finally came to an end, in the 1970s, the total number of sterilized Americans would exceed 60,000.
And no matter the cost or the casualties, the scientific betterment of humanity would remain an irresistible aspiration––tempting generations to come with the promise of perfection.
NATHANIEL COMFORT, HISTORIAN: We believe in science. We want science to solve social problems, okay, and we wanna make ourselves better. I think everybody wants to do that.
THOMAS C. LEONARD, HISTORIAN: There is this idea that remains a kind of hope that if we just get the science right, if only the right people are put in charge, that we can engineer our way to a better world.
NALONDRA NELSON, SOCIOLOGIST: Some of the greatest social changes that have ever been accomplished have occurred because people were really willing to imagine impossible things. But the future that American eugenicists imagined was only a future for some.
DANIEL KEVLES, HISTORIAN: And so now the debate is are we going to use technology to try to fulfill Galton’s dream, if you will, of taking charge of our own evolution? Of course it was a pipe dream, but nevertheless it is a dream that persists. We have reason to be apprehensive about this and the test tube bears watching.
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Additional funding for The Eugenics Crusade provided by |
3461 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3726098/ | Neuroleptic Malignant Syndrome: A Review for Neurohospitalists - PMC
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Neurohospitalist
. 2011 Jan;1(1):41–47. doi: 10.1177/1941875210386491
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Neuroleptic Malignant Syndrome
A Review for Neurohospitalists
Brian D Berman
Brian D Berman, MD, MS
1 Department of Neurology, University of Colorado Denver School of Medicine, Aurora, CO, USA
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1 Department of Neurology, University of Colorado Denver School of Medicine, Aurora, CO, USA
✉
Brian D. Berman, Department of Neurology, University of Colorado Denver School of Medicine, Academic Office 1, Mail Stop B-185, 12631 East 17th Avenue, Aurora, CO 80045, USA Email: brian.berman@ucdenver.edu
Issue date 2011 Jan.
© The Author(s) 2011
PMC Copyright notice
PMCID: PMC3726098 PMID: 23983836
Abstract
Neuroleptic malignant syndrome (NMS) is a life-threatening idiosyncratic reaction to antipsychotic drugs characterized by fever, altered mental status, muscle rigidity, and autonomic dysfunction. It has been associated with virtually all neuroleptics, including newer atypical antipsychotics, as well as a variety of other medications that affect central dopaminergic neurotransmission. Although uncommon, NMS remains a critical consideration in the differential diagnosis of patients presenting with fever and mental status changes because it requires prompt recognition to prevent significant morbidity and death. Treatment includes immediately stopping the offending agent and implementing supportive measures, as well as pharmacological interventions in more severe cases. Maintaining vigilant awareness of the clinical features of NMS to diagnose and treat the disorder early, however, remains the most important strategy by which physicians can keep mortality rates low and improve patient outcomes.
Keywords: neuroleptic malignant syndrome, movement disorders, neurohospitalist
Background
Neuroleptic malignant syndrome (NMS) is a severe disorder caused by an adverse reaction to medications with dopamine receptor-antagonist properties or the rapid withdrawal of dopaminergic medications. The first reported case of NMS appeared in 1956, shortly after the introduction of the antipsychotic drug chlorpromazine (thorazine).1 Additional case reports quickly followed, and in a 1960 study French clinicians gave the syndrome its current name when they reported on the adverse effects of the newly introduced neuroleptic haloperidol and characterized a “syndrome malin des neuroleptiques.”2 Pooled data from 1966 to 1997 suggested the incidence of NMS ranges from 0.2% to 3.2% of psychiatric inpatients receiving neuroleptics3; however, as physicians have become increasingly aware of the syndrome and as newer neuroleptic agents have become available, the incidence has declined more recently to around 0.01% to 0.02%.4 Although NMS occurs only rarely, it remains an unpredictable and potentially life-threatening neurologic condition that hospitalists must be able to recognize, as early identification and proper medical management are essential to ensure improved patient outcomes.
Clinical Presentation
The diagnosis of NMS is based on history and the presence of certain physical examination and laboratory findings.5,6 Patients typically develop NMS within hours or days after exposure to a causative drug, with most exhibiting symptoms within 2 weeks and nearly all within 30 days.7 Although NMS has classically been characterized by the presence of the triad of fever, muscle rigidity, and altered mental status, its presentation can be quite heterogeneous, as reflected in the current Diagnostic and Statistical Manual of Mental Disorders (Fourth Edition [DSM-IV] criteria (see Table 1 ).8 The clinical course typically begins with muscle rigidity followed by a fever within several hours of onset and mental status changes that can range from mild drowsiness, agitation, or confusion to a severe delirium or coma.
Table 1.
Diagnostic and Statistical Manual of Mental Disorders (Fourth Edition [DSM-IV]) Research Criteria for Neuroleptic Malignant Syndrome8
A.Development of severe muscle rigidity and elevated temperature associated with the use of neuroleptic medication.
B.Two (or more) of the following:
(1) Diaphoresis
(2) Dysphagia
(3) Tremor
(4) Incontinence
(5) Changes in level of consciousness ranging from confusion to coma
(6) Mutism
(7) Tachycardia
(8) Elevated or labile blood pressure
(9) Leukocytosis
(10) Laboratory evidence of muscle injury (eg, elevated CPK)
C.The symptoms in criteria A and B are not due to another substance or a neurological or other general medical condition.
D.The symptoms in criteria A and B are not better accounted for by a mental disorder.
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Abbreviation: CPK, creatinine phosphokinase.
Signs of autonomic nervous system instability that frequently accompany NMS include labile blood pressure, tachypnea, tachycardia, sialorrhea, diaphoresis, flushing, skin pallor, and incontinence. Once symptoms appear, progression can be rapid and can reach peak intensity in as little as 3 days. Although muscle rigidity is the most frequently described motor sign, a large number of additional extrapyramidal motor findings have been reported including tremor, chorea, akinesia, and dystonic movements including opisthotonos, trismus, blepharospasm, and oculogyric crisis.3,9,10 Other symptoms that have been associated with NMS include dysphagia, dyspnea, abnormal reflexes, mutism, and seizures.3,11–13
Characteristic laboratory findings seen in NMS include elevated creatinine phosphokinase (CPK) due to rhabdomyolysis and leukocytosis, but these are neither specific for the syndrome nor present in all cases.14 When rhabdomyolysis is present, it can be severe enough to cause renal failure, requiring hemodialysis.13 Additional common laboratory abnormalities include a metabolic acidosis and iron deficiency.15 The cerebrospinal fluid (CSF) and imaging studies are usually normal, but an electroencephalogram (EEG) may show nongeneralized slowing.7
Causative Agents
The primary trigger of NMS is dopamine receptor blockade and the standard causative agent is an antipsychotic. Potent typical neuroleptics such as haloperidol, fluphenazine, chlorpromazine, trifluoperazine, and prochlorperazine have been most frequently associated with NMS and thought to confer the greatest risk. Although atypical neuroleptics appear to have reduced the risk of developing NMS compared to typical neuroleptics,10 a significant number of cases have been reported with most atypical neuroleptics including risperidone,16 clozapine,17 quetiapine,18 olanzapine,19 ariprazole,20 and ziprasidone.21 Neuroleptic malignant syndrome has also been associated with nonneuroleptic agents with antidopaminergic activity such as metoclopramide,22 promethazine,10 tetrabenzine,23 droperidol,24 diatrizoate,25 and amoxapine.26
The abrupt cessation or reduction in dose of dopaminergic medications such as levodopa in Parkinson disease may also precipitate NMS.27 The rapid switching from one type of dopamine receptor agonist to another in such patients has also been associated with NMS,28 and there may be some risk of NMS associated with the abrupt withdrawal of Parkinson medications that are not known to have direct dopaminergic activity such as amantadine29 and tolcapone.30 Neuroleptic malignant syndrome has also been rarely associated with a number of other medications not known to have any central antidopaminergic activity such as lithium,31 desipramine, trimipramine, dosulpin,32 and phenelzine (Table 2 ).33
Table 2.
Neuroleptic and Nonneuroleptic Medications Associated With Neuroleptic Malignant Syndrome
| A. Neuroleptics | B. Nonneuroleptics with antidopaminergic activity |
:--- |
| (1) Typical |
| a. Haloperidol | (1) Metoclopromide |
| b. Fluphenazine | (2) Tetrabenazine |
| c. Chlorpromazine | (3) Reserpine |
| d. Prochlorperazine | (4) Droperidol |
| e. Trifluoperazine | (5) Promethazine |
| f. Thioridazine | (6) Amoxapine |
| g. Thiothixene | (7) Diatrizoate |
| h. Loxapine | C. Dopaminergics (withdrawal) |
| i. Perphenazine |
| j. Bromperidol | (1) Levodopa |
| k. Clopenthixol | (2) Dopamine agonists |
| l. Promazine | (3) Amantadine |
| (2) Atypical | (4) Tolcapone |
| a. Clozapine | D. Others |
| b. Risperidone | (1) Lithium |
| c. Olanzapine | (2) Phenelzine |
| d. Quetiapine | (3) Dosulepin |
| e. Ziprasidone | (4) Desipramine |
| f. Aripiprazole | (5) Trimipramine |
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Abbreviation: NMS, neuroleptic malignant syndrome.
Differential Diagnosis
Many medical conditions can mimic the presentation of NMS, with some of the more common being heat stroke, central nervous system (CNS) infections, toxic encephalopathies, agitated delirium, status epilepticus, and more benign drug-induced extrapyramidal symptoms.3,34 Heat stroke frequently presents with fever and altered level of consciousness, but it can be distinguished by a more abrupt onset and the more common presence of dry skin, hypotension, and limb flaccidity rather than extrapyramidal signs.35 Importantly, neuroleptic medications can predispose patients to hyperthermia, making them prone to heat stroke, especially if contributing factors such as hot weather, dehydration, or excessive exercise or agitation are present.
Central nervous system infection must also be considered early in someone presenting with the clinical features of NMS to avoid any delay in the appropriate treatment. In addition to fever and mental status changes, hallmarks of a CNS infection include a history of prodromal illness, headaches, meningeal signs, focal neurological signs, seizures, and frequently positive CSF and neuroimaging studies. If an infectious etiology is suspected, a lumbar puncture and blood, urine, and CSF cultures are mandatory, and an EEG may be required to rule out seizure activity.
Often complicating the diagnosis of NMS is the large number of drug-induced syndromes that can have motor and cognitive features that resemble the condition. The use of neuroleptic agents has been associated with a variety of adverse motor effects including parkinsonism, acute dystonia, acute akathisia, tremor, and tardive dyskinesia,8 and several other classes of drugs at toxic levels may cause symptoms resembling NMS such as serotinergic agents, anticholinergics, monoamine oxidase inhibitors, tricyclics, lithium, meperidine, and fenfluramine.36 Intoxication syndromes from drugs of abuse such as cocaine, amphetamine, methamphetamine, phencyclidine, and 3,4-Methylenedioxymethamphetamine (MDMA [aka Ecstasy]) can produce hyperthermia, mental status changes, and autonomic dysfunction and can easily be confused with NMS.34 Abrupt withdrawal syndromes from alcohol and benzodiazepine can also be associated with altered mental status and muscle rigidity, and there is at least one report of a case of an NMS-like syndrome resulting from withdrawal of baclofen.37
Serotonin syndrome, which presents with altered mental status, autonomic changes, and motor features related to serotonin excess, shares a number of similarities with NMS.38 Nevertheless, it can typically be distinguished by history, the absence of leukocytosis and elevated CPK, and the presence of gastrointestinal symptoms (eg, nausea, vomiting, and diarrhea) and motor features other than muscle rigidity such as tremor, ataxia, myoclonus, hyperreflexia.39 Malignant hyperthermia, a severe drug-induced reaction linked to defective calcium-related proteins, may also present clinically like NMS. However, because it is triggered by potent inhaled anesthetic agents or depolarizing muscle relaxants, usually history alone is able to discern the two syndromes.40
Lethal catatonia is a life-threatening psychiatric disorder that can present with clinical features of fever, rigidity, akinesia, and altered mental status.41 Although it can be difficult to distinguish it from NMS, the motor features in lethal catatonia are typically preceded by a few weeks of behavioral changes including ambivalence, apathy, withdrawal, automatisms, extreme negativism, and psychotic agitation.42 As lethal catatonia typically requires neuroleptic treatment as opposed to being caused by such treatment, rapid clinical differentiation between these two disorders is extremely important (Table 3 ).
Table 3.
Differential Diagnosis for Neuroleptic Malignant Syndrome
| Differential Diagnosis | Distinguishing Features |
:--- |
| Infectious | |
| 1. Meningitis or encephalitis 2. Brain abscess 3. Sepsis 4. Rabies | History of prodromal viral illness, headaches or meningeal signs Presence of seizures or localizing neurological signs Brain imaging CSF studies |
| Metabolic | |
| 1. Acute renal failure 2. Rhabdomyolysis 3. Thyrotoxicosis 4. Pheochromocytoma | Renal or thyroid function tests Absence of neuroleptic treatment Presence of severe hypertension Significantly elevated catecholamines and metanephrines |
| Environmental | |
| 1. Heat stroke 2. Spider envenomations | History of exertion or exposure to high temperatures Hot dry skin, skin lesion suggestive of spider bite Absence of rigidity Abrupt onset |
| Drug-induced | |
| 1. Malignant hyperthermia 2. Neuroleptic-induced syndromes 1. Parkinsonism 2. Acute dystonia 3. Acute akathisia 4. Tardive dyskinesia 5. Postural tremor 3. Nonneuroleptic-induced syndromes 1. Serotonin syndrome 2. Anticholinergic delirium 3. Monoamine oxidase inhibitor toxicity 4. Lithium toxicity 5. Salicylate poisoning 6. Strychnine poisoning 7. Drugs of abuse (cocaine, amphetamine, methamphetamine, MDMA, phencyclidine) | History of inhalational anesthetics Family history of malignant hyperthermia Presence of hyperkinesias Positive toxicology/drug-level screen Low or normal CPK Presence of nausea, vomiting, diarrhea Presence of anticholinergic signs (dilated pupils, dry mouth, dry skin, urinary retention) Presence of rash, urticaria, or eosinophilia History of drug dependence, abuse, or overdosages |
| Drug-withdrawal syndrome 1. Alcohol 2. Benziodiazepine 3. Baclofen 4. Sedatives 5. Hypnotics | History of drug dependence, abuse, or overdosages Absence of neuroleptic treatment Toxicology screen |
| Neurological or psychiatric disorder 1. Parkinsonism 2. Nonconvulsive status epilepticus 3. Lethal catatonia | Absence of fever or leukocytosis Presence of hyperkinesias, later emergence of rigidity Prior history of catatonic states Absence of neuroleptic treatment EEG |
| Autoimmune 1. Polymyositis | Proximal weakness Abnormal EMG or muscle biopsy Presence of cancer or interstitial lung disease |
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Abbreviations: CPK, creatinine phosphokinase; CSF, cerebrospinal fluid; EEG, electroencephalography; EMG, electromyography; MDMA, 3,4-Methylene dioxymethamphetamine.
Pathophysiology
The underlying pathophysiologic mechanisms of NMS are complex and elements still debated among experts, but most agree that a marked and sudden reduction in central dopaminergic activity resulting from D2 dopamine receptor blockade within the nigrostriatal, hypothalamic, and mesolimbic/cortical pathways helps explain the clinical features of NMS including rigidity, hyperthermia, and altered mental status, respectively.12,34 This theory is supported by the observation that the primary cause of NMS is the use of antipsychotic drugs that specifically block dopamine receptors, and in particular D2 receptors, and that the syndrome can also be induced by abrupt dopamine withdrawal. Additional support comes from a dopamine receptor imaging study of 1 patient with NMS demonstrating a complete lack of D2 receptor binding in the acute phase,43 and another study showing low levels of the dopamine metabolite homovanillic acid in the CSF of patients with acute NMS.44
D2 dopamine receptor antagonism, however, does not explain all the presenting signs and symptoms of NMS, nor does it explain its occurrence with antipsychotic medications with lower D2 activity and medications without known antidopaminergic activity. This has led some to propose that sympathoadrenal hyperactivity, resulting from the removal of tonic inhibition within the sympathetic nervous system, may play a key role in the pathogenesis of NMS.45 Abnormalities in the sympathetic system are supported by the frequent presence of autonomic symptoms in NMS as well as demonstrated changes in the urine and plasma catecholamine levels in patients with NMS. Some have hypothesized that NMS shares pathophysiological similarities with malignant hyperthermia and that a defect in calcium regulatory proteins within sympathetic neurons may be the key factor that brings about the onset of NMS.46
Another system that also appears to play a role in the signs and symptoms of NMS is the peripheral skeletal muscle system. Release of calcium has been shown to be increased from the sarcoplasmic reticulum of muscle cells with antipsychotic usage, possibly leading to increased muscle contractility and rigidity, breakdown of muscle, and hyperthermia.12 To date, however, none of the theories put forth as the underlying cause of NMS have been able to explain why only a small fraction of patients exposed to neuroleptics develop the condition. Furthermore, it remains unknown why patients who develop NMS are usually able to continue being treated with similar medications and, at times, even the same offending agent.
Risk Factors
The main risk factors for developing NMS are the initiation or increase in dose of a neuroleptic medication and the potency and administration form of that drug.47,48 The use of high-dosed, high-potency and long-acting or intramuscular depot forms of neuroleptics, as well as a rapid increase in dosage of neuroleptics, both increase the risk of developing NMS. The concurrent use of multiple neuroleptics, or concomitant taking of predisposing drugs such as lithium, also appear to confer an increased risk.9,13 Although NMS can occur at anytime during neuroleptic treatment and no definite correlation between the duration of exposure to a neuroleptic and risk of developing the condition has been found, it is less likely to occur if a patient has been on a stable dose of their antipsychotics for a long period of time and there are no issues of noncompliance.3,13
A variety of other risk factors have emerged from epidemiological and case studies of NMS which include dehydration, physical exhaustion, exposure to heat, hyponatremia, iron deficiency, malnutrition, trauma, thyrotoxicosis, alcohol, psychoactive substances, and presence of a structural or functional brain disorder such as encephalitis, tumor, delirium, or dementia.47–49 Males under 40 years are often thought to be at greater risk of developing NMS as well, but it remains unclear whether this elevated risk is primarily due to the increased incidence of neuroleptic use in this population. Postpartum women may also be at slightly higher risk of developing NMS.50 Reports of identical twins and a mother and 2 of her daughters all presenting with NMS suggest that a genetic risk factor for NMS may exist,51 and some limited genetic investigations help support existence of genetic component to the condition,46 possibly through a genetically associated reduction in the function of the D2 dopamine receptor.52
Treatment
Neuroleptic malignant syndrome in hospitalized patients is considered a neurologic emergency as a delay in treatment or withholding of therapeutic measures can potentially lead to serious morbidity or death. As such, some consider it prudent to treat for NMS even if there is doubt about the diagnosis.53 Due to its rarity, however, systematic clinical trials in NMS are difficult to perform and so no evidence-based treatment approach exists. Nevertheless, effective general guidelines have been gleaned from case reports and analyses.54
Treatment of NMS is individualized and based on the clinical presentation, but the first step in essentially all cases consists of cessation of the suspected offending neuroleptic pharmacologic agent. If the syndrome has occurred in the setting of an abrupt withdrawal of a dopaminergic medication, then this medication is reinstituted as quickly as possible. The next key step in the management of NMS is the initiation of supportive medical therapy. Aggressive hydration is often required, especially if highly elevated CPK levels threaten to damage the kidneys, and treatment of hyperthermia with cooling blankets or ice packs to the axillae and groin may be needed. Metabolic abnormalities may need to be corrected, and bicarbonate loading should be considered in some cases as it may be beneficial in preventing renal failure.55 Patients with NMS may be at increased risk of morbidity due to renal failure and disseminated intravascular coagulation (DIC) secondary to rhabdomyolysis,34 deep venous thrombosis and pulmonary embolism resulting from dehydration and immobilization, aspiration pneumonia because of difficulty swallowing combined with an altered mental status, as well as other medical complications including cardiopulmonary failure, seizures, arrhythmias, myocardial infarction, and sepsis, and so many cases require intensive care monitoring and support.3,13,56,57
In more severe cases of NMS, empiric pharmacologic therapy is typically tried. The two most frequently used medications are bromocriptine mesylate, a dopamine agonist, and dantrolene sodium, a muscle relaxant that works by inhibiting calcium release from the sarcoplasmic reticulum. Anecdotal reports and meta-analyses suggest these agents may shorten the course of the syndrome and possibly reduce mortality when used alone or in combination.58,59 Bromocriptine is given to reverse the hypodopaminergic state and is administered orally (or via nasogastric tube), starting with 2.5 mg 2 or 3 times daily and increasing doses by 2.5 mg every 24 hours until a response or until reaching a maximum dose of 45 mg/d.13,34,58,59 Dantrolene can be administered intravenously starting with an initial bolus dose of 1 to 2.5 mg/kg followed by 1 mg/kg every 6 hours up to a maximum dose of 10 mg/kg/d.13,34,58,59 Oral dantrolene is used in less severe cases or to taper down from the intravenous form after a few days with doses that range from 50 to 200 mg/d. Due to a risk of hepatoxicity, dantrolene is typically discontinued once symptoms begin to resolve. Bromocriptine, however, is generally maintained for at least 10 days for NMS related to oral neuroleptics and 2 to 3 weeks for depot neuroleptics.
Other dopaminergic agents besides bromocriptine have been used including amantadine hydrochloride,60 levodopa,61 and apomorphine.62 Additional pharmacologic agents that may have some utility in treating NMS are benzodiazepines,63 which can be helpful in controlling agitation but may also ameliorate symptoms and hasten recovery in milder cases, carbamezapine,64 and clonidine.65 In cases that do not respond to standard medical care, electroconvulsive therapy has been reported to improve some of the symptoms of NMS and may be effective.66,67
Recurrences of NMS do occur, especially when a patient is restarted on a neuroleptic with high potency or too quickly after their initial episode.12,68,69 Most patients who require continued antipsychotic treatment, though, are able to have a neuroleptic safely reintroduced with proper precautions including very slow titration and careful monitoring after a waiting period of about 2 weeks for an oral neuroleptic and at least 6 weeks for a depot form.14,70 Although NMS is considered an idiosyncratic reaction, it is generally felt to be prudent to use a different neuroleptic than the one that was originally associated with the development of the syndrome.12,13,71
Prognosis
Initial reports of mortality rates from NMS were over 30%, but increased physician awareness and introduction of newer neuroleptic medications over the last few decades have helped reduce them to closer to 10%.71 When recognized early and treated aggressively, NMS is usually not fatal and a majority of patients will recover completely between 2 and 14 days.3,7 But if diagnosis and treatment are delayed, resolution can require several weeks or longer, and surviving patients may have residual catatonia or parkinsonism, or significant morbidity secondary to renal or cardiopulmonary complications.10,13,34 When death does occur, it is usually attributable to arrhythmias, DIC, or cardiovascular, respiratory, or renal failure. Thus, early recognition and initiation of therapeutic measures by physicians remain paramount to reducing the number of severe cases of NMS and limiting this significant source of morbidity and mortality among patients receiving antipsychotics.
Footnotes
The author(s) declared no potential conflicts of interests with respect to the authorship and/or publication of this article.
The author(s) received no financial support for the research and/or authorship of this article.
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3462 | https://www.cs.purdue.edu/homes/spa/courses/cs182/algorithms-rego.pdf | Algorithms and Growth of Functions 1 Algorithms The growth of functions Complexity of Algorithms Algorithms 2 An algorithm is a finite sequence of precise instructions for performing a computation or for solving a problem. An algorithm is defined on specified inputs and generates an output stops after finitely many instructions are executed. A Recipe is an Algorithm The set of steps to assemble a Piece of Furniture is an Algorithm. Properties of Algorithms 1.
Input 2.
Output 3.
Definiteness 4.
Correctness 5.
Effectiveness 6.
Finiteness 7.
Generality An algorithm has input values from a specified set. Properties of Algorithms 1.
Input 2.
Output 3.
Definiteness 4.
Correctness 5.
Effectiveness 6.
Finiteness 7.
Generality An algorithm has input values from a specified set. From each set of input values, an algorithm produces output values from a specified set. The output values are the solution to the problem. Properties of Algorithms 1.
Input 2.
Output 3.
Definiteness 4.
Correctness 5.
Effectiveness 6.
Finiteness 7.
Generality An algorithm has input values from a specified set. From each set of input values, an algorithm produces output values from a specified set. The output values are the solution to the problem. The steps of an algorithm must be defined precisely. Properties of Algorithms 1.
Input 2.
Output 3.
Definiteness 4.
Correctness 5.
Effectiveness 6.
Finiteness 7.
Generality An algorithm has input values from a specified set. From each set of input values, an algorithm produces output values from a specified set. The output values are the solution to the problem. The steps of an algorithm must be defined precisely. An algorithm should produce the correct output values for each set of input values. Properties of Algorithms 1.
Input 2.
Output 3.
Definiteness 4.
Correctness 5.
Effectiveness 6.
Finiteness 7.
Generality An algorithm has input values from a specified set. From each set of input values, an algorithm produces output values from a specified set. The output values are the solution to the problem. The steps of an algorithm must be defined precisely. An algorithm should produce the correct output values for each set of input values. It must be possible to perform each step of an algorithm exactly and in a finite amount of time. Properties of Algorithms 1. Input 2. Output 3. Definiteness 4. Correctness 5. Effectiveness 6. Finiteness 7. Generality An algorithm has input values from a specified set. From each set of input values, an algorithm produces output values from a specified set. The output values are the solution to the problem. The steps of an algorithm must be defined precisely. An algorithm should produce the correct output values for each set of input values. It must be possible to perform each step of an algorithm exactly and in a finite amount of time. An algorithm should produce the desired output after a finite (but perhaps large) number of steps for any input in the set. Properties of Algorithms 1. Input 2. Output 3. Definiteness 4. Correctness 5. Effectiveness 6. Finiteness 7. Generality An algorithm has input values from a specified set. From each set of input values, an algorithm produces output values from a specified set. The output values are the solution to the problem. The steps of an algorithm must be defined precisely. An algorithm should produce the correct output values for each set of input values. It must be possible to perform each step of an algorithm exactly and in a finite amount of time. An algorithm should produce the desired output after a finite (but perhaps large) number of steps for any input in the set. The procedure should be applicable for all problems of the desired form, not just for a particular set of input values. How to express an Algorithm 12 C code int is_prime(int m) { int i; for (i=2; i<m;i++) if (m % i ==0) return 0; return 1; } Java code class SpecialInt { int m; boolean is_prime() { for (i=2; i large then large := si endif i := i + 1 endwhile return(large) end find_large Search Algorithms Search Find a given element in a list. Return the location of the element in the list (index), or 0 if not found. Linear Search Compare key (element being searched for) with each element in the list until a match is found, or the end of the list is reached. Binary Search Compare key only with elements in certain locations. Split list in half at each comparison. Requires list to be sorted. Linear Search 17 Find the location of an element X in an array of possible unsorted items Linear Search Exercise 19, 1, 17, 2, 11, 13, 7, 9, 10, 5, 15, 6, 14, 20, 16, 12, 4, 18, 3, 8 How many comparisons to find: 17? 21? Binary Search 19 Find the location of an element X in an array of sorted items Binary Search Exercise 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 How many comparisons to find: Find 7 Find 21 Sort Sort Put the elements of a list in ascending order Bubble Sort Compare every element to its neighbor and swap them if they are out of order. Repeat until list is sorted. Insertion Sort For each element of the unsorted portion of the list, insert it in sorted order in the sorted portion of the list. Bubble Sort 22 Bubble Sort Exercise 10, 2, 1, 5, 3 Insertion Sort 24 Insertion Sort Exercise 10, 2, 1, 5, 3 Greedy Algorithms The goal of an optimization problem is to maximize or minimize an objective function. One of the simplest approaches to solving optimization problems is to select the “best” choice at each step. Greedy Change-Making 27 Give an algorithm for making n cents change with quarters, dimes, nickels, and pennies, and using the least total number of coins. Make Change 69 cents: 56 cents: The Halting Problem 29 Is there a procedure that does the following: Takes as input a program and input to that program and determines whether that program will eventually stop when run on that input, for any program and input No, there is no such program. Algorithms and Growth of Functions 1 Algorithms The growth of functions Complexity of Algorithms Algorithms 2 An algorithm is a finite sequence of precise instructions for performing a computation or for solving a problem.
An algorithm is defined on specified inputs and generates an output stops after finitely many instructions are executed.
Search Algorithms Search Find a given element in a list. Return the location of the element in the list (index), or 0 if not found.
Linear Search Compare key (element being searched for) with each element in the list until a match is found, or the end of the list is reached.
Binary Search Compare key only with elements in certain locations. Split list in half at each comparison. Requires list to be sorted.
Sort Sort Put the elements of a list in ascending order Bubble Sort Compare every element to its neighbor and swap them if they are out of order. Repeat until list is sorted.
Insertion Sort For each element of the unsorted portion of the list, insert it in sorted order in the sorted portion of the list. Greedy Algorithms The goal of an optimization problem is to maximize or minimize an objective function.
One of the simplest approaches to solving optimization problems is to select the “best” choice at each step.
Greedy Change-Making 6 Give an algorithm for making n cents change with quarters, dimes, nickels, and pennies, and using the least total number of coins.
Make Change 69 cents: 56 cents: The Halting Problem 8 Is there a procedure that does the following: Takes as input a program and input to that program and determines whether that program will eventually stop when run on that input, for any program and input No, there is no such program.
The Growth of functions 9 The time required to solve a problem using a procedure depends on: Number of operations used Depends on the size of the input Speed of the hardware and software Does not depend on the size of the input Can be accounted for using a constant multiplier The growth of functions refers to the number of operations used by the function to solve the problem.
Complexity of Algorithms 10 The complexity of an algorithm refers to the amount of time and space required to execute the algorithm.
Computing the amount of time and space used without having the actual program requires one to focus on the essential features that affect performance.
Analyzing algorithm find_largest 11 Time of execution depends on the number of iterations of the while loop.
Performance does not generally depend on the values of the elements.
How many iterations are executed? n−1 The time needed is linearly proportional to n.
Example 12 for i := 1 to n do for j:=1 to n do si := si+ sj number of iterations executed: n2 time needed: proportional to n2 Big-O Notation 13 Estimate the growth of a function without worrying about constant multipliers or smaller order terms.
Do not need to worry about hardware or software used Assume that different operations take the same time.
Addition is actually much faster than division, but for the purposes of analysis we assume they take the same time.
Big-O 14 Example 15 X2 + 2x + 1 <= x2 + 2x2 + x2 for x >=1 x2 + 2x2 + x2 = 4x2 Witness C = 4 K = 1 Example 16 Assume n2 is O(n) Then C,k ∀ n>k, n2 <= Cn n <= C But no constant is bigger than all n contradiction Big-𝑂for Polynomials Example 18 Give a big-O estimate for f(x) = 5x2-18x+20 Solution 5𝑥2−18𝑥+20≤5𝑥2+20for 𝑥>0 5𝑥2+20≤5𝑥2+20𝑥2for 𝑥>1 5𝑥2+20𝑥2=25𝑥2≤𝐶𝑔(𝑥) for 𝑥>1 Let 𝑔(𝑥)=𝑥2 𝒇(𝒙)is 𝑶(𝒙𝟐). 𝑪=𝟐𝟓, 𝒌=𝟏 Example 19 Give a big-O estimate for the sum of the first n positive integers Solution 1+2+⋯+𝑛≤𝑛+𝑛+⋯+𝑛=𝑛2 1+2+⋯+𝑛is 𝑂(𝑛2 ), 𝐶=1,𝑘=1 Example 20 Give a big-O estimate for the factorial function f(n)=n!
Give a big-O estimate for the logarithm of the factorial function Solution 𝑛!=1⋅2⋅3⋅⋯⋅𝑛≤𝑛⋅𝑛⋅𝑛⋅⋯⋅𝑛=𝑛𝑛 𝑛! is 𝑂(𝑛𝑛) log(𝑛!)≤log(𝑛𝑛)=𝑛log𝑛 log(𝑛!) is 𝑂(𝑛log𝑛) Basic Growth Functions 21 Constant O(1) Logarithmic O(log n) Linear O(n) Linearithmic O(n log n) Polynomial O(n2) Exponential O(nn) Factorial O(n!) Useful Big-𝑂Estimates The Growth of Combinations of Functions Example Give a big-𝑂estimate for 𝑓𝑛= 3𝑛log 𝑛! + 𝑛2 + 3 log 𝑛 O(n2 log n) Big-Ω Example Show that 8𝑥3 + 5𝑥2 + 7 is Ω 𝑥3 8𝑥3+5𝑥2+7≥8𝑥3 for x > 0 C=8, k =0 Big-Θ 27 Big- Θ (big theta) 𝑓𝑛is 𝑂𝑔𝑛 and Ω 𝑔𝑛 𝑓𝑛is 𝑂𝑔𝑛 and 𝑔𝑛is 𝑂𝑓𝑛 𝑓𝑛is Θ 𝑔𝑛 ↔g 𝑛is Θ 𝑓𝑛 ∃𝐶1, 𝐶2 , 𝑘∀𝑛> 𝑘𝐶1𝑔𝑛≤𝑓𝑛≤𝐶2𝑔𝑛 𝑓𝑛is of order 𝑔𝑛 𝑓𝑛and 𝑔𝑛are of the same order Example 28 Show that 3x2 + 8xlog x is Θ x2 Big-o 3x2 +8x log x <= 11x2 C=11, k =1 Big-omega x2 <= 3x2 + 8x log x Big-Θ for Polynomials 29 Let 𝑓𝑥= 𝑎𝑛𝑥𝑛+ 𝑎𝑛−1𝑥𝑛−1 + ⋯+ 𝑎1𝑥+ 𝑎0.
Then, 𝑓𝑥is of order 𝑥𝑛.
“𝑓𝑥is bounded [above and below] by 𝑔𝑥” Example: 3𝑥8 + 10𝑥7 + 221𝑥2 + 1444 is of order 𝑥8 |
3463 | https://www.gauthmath.com/solution/1820887260626166/f-6-f-3-12-s-Math-games-and-Math-help | Solved: f= 6) f 3=12 s Math games and Math help [Math]
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f= 6) f 3=12 s Math games and Math help
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To find the value of f, solve the equation f x 3 = 12. Divide both sides by 3: f = 12 / 3 = 4.
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3464 | https://www.kristakingmath.com/blog/diagonal-of-a-right-rectangular-prism | Finding the diagonal of a right rectangular prism
Formula for the diagonal in terms of the length, width, and height of the rectangular prism
The diagonal of a right rectangular prism goes from one corner of the prism, across the interior volume, all the way to the opposite corner of the prism.
Hi! I'm krista.
I create online courses to help you rock your math class. Read more.
You can find the length of a diagonal of a right rectangular prism using
???d=\sqrt{{{l}^{2}}+{{w}^{2}}+{{h}^{2}}}???
where ???d??? is the length of the diagonal, and ???l???, ???w???, and ???h??? are the length, width, and height, respectively.
How to find the length of the diagonal
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Want to learn more about Geometry? I have a step-by-step course for that. :)
Length of the diagonal of a right rectangular prism
Example
What is the length of the diagonal of the right rectangular prism?
Not all of the dimensions here are the same. Change ???60\text{ mm}??? to centimeters first.
???60\text{ mm}=6\text{ cm}???
Then the dimensions are
Plugging these into the formula for the diagonal, we get
???d=\sqrt{{{l}^{2}}+{{w}^{2}}+{{h}^{2}}}???
???d=\sqrt{{{7}^{2}}+{{12}^{2}}+{{6}^{2}}}???
???d=\sqrt{49+144+36}???
???d=\sqrt{229}??? cm
Let’s try another one.
The diagonal of a right rectangular prism goes from one corner of the prism, across the interior volume, all the way to the opposite corner of the prism.
Example
Find the width of the right rectangular prism.
We just need to plug the dimensions we’ve been given into the formula for the diagonal.
???d=\sqrt{{{l}^{2}}+{{w}^{2}}+{{h}^{2}}}???
???7\sqrt{2}=\sqrt{{{5}^{2}}+{{w}^{2}}+{{3}^{2}}}???
Manipulate the equation to solve for ???w???.
???(49)(2)={{5}^{2}}+{{w}^{2}}+{{3}^{2}}???
???98=25+{{w}^{2}}+9???
???98=34+{{w}^{2}}???
???64={{w}^{2}}???
???8=w???
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3465 | https://thejas.com.pk/index.php/pjhs/article/view/1226 | Comparative Efficacy of Diode Laser System versus Intense Pulse Light (Ipl) In Management of Unwanted Hair | Pakistan Journal of Health Sciences
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2023: Volume 04 Issue 12 (December Issue)/
Original Article
Comparative Efficacy of Diode Laser System versus Intense Pulse Light (Ipl) In Management of Unwanted Hair
Diode Laser System versus Intense Pulse Light
Authors
Sara Ilyas Combined Military Hospital, Abbottabad, Pakistan
Majid Hussain Combined Military Hospital, Abbottabad, Pakistan
Muhammad Adeel Siddiqui Combined Military Hospital, Abbottabad, Pakistan
Bushra Muzaffar Combined Military Hospital
Kiran Gul Combined Military Hospital, Abbottabad, Pakistan
Daniyal Sajjad Combined Military Hospital, Abbottabad, Pakistan
DOI:
Keywords:
Diode Laser System, Unwanted Hair, Intense Pulse Light System
Abstract
Unwanted hair growth can be caused by several factors, including genetics, systemic illness, and even drug reactions. According to the underlying medical condition, excessive hair growth was labelled as either hirsutism or hypertrichosis. Objective: To compare the efficacy of diode laser system versus intense pulse light in the management of unwanted hair among female patients. Methods: Patients in this randomized controlled study had hirsutism. Group A received powerful pulse light therapy for three sessions, one month apart, while Group B received diode laser treatment. To determine the effectiveness and side effects in both groups, patients were monitored for four months and subsequently assessed clinically. Results: Out of these total 60 patients, intense pulse light system was more efficacious in achieving patient satisfaction (73.3%) at the end of three months as compared to diode laser (26.7%) (p value < 0.001). Conclusions: This randomized controlled trial showed that intense pulse light therapy was more efficacious in management of unwanted hair compared to diode laser therapy.
References
AL-Hamamy HR, Saleh AZ, Rashed ZA. Evaluation of effectiveness of diode laser system (808 nm) versus Intense Pulse Light (IPL) in the management of unwanted hair: A split face comparative study. International Journal of Medical Physics, Clinical Engineering and Radiation Oncology. 2015 Jan; 4(01): 41. doi: 10.4236/ijmpcero.2015.41006. DOI:
Łakuta P, Marcinkiewicz K, Bergler-Czop B, Brzezińska-Wcisło L, Słomian A. Associations between site of skin lesions and depression, social anxiety, body-related emotions and feelings of stigmatization in psoriasis patients. Advances in Dermatology and Allergology/Postępy Dermatologii i Alergologii. 2018 Feb; 35(1): 60-6. doi: 10.5114/pdia.2016.62287. DOI:
Sachdeva S. Hirsutism: evaluation and treatment. Indian Journal of Dermatology. 2010 Jan; 55(1): 3. doi: 10.4103/0019-5154.60342. DOI:
Hafsi W and Badri T. Hirsutism. Treasure Island (FL); StatPearls: 2020.
Agrawal NK. Management of hirsutism. Indian Journal Endocrinology and Metabolism 2013 Oct; 17(1): S77–S82. doi: 10.4103/2230-8210.119511. DOI:
Puri N. Comparative study of diode laser versus neodymium-yttrium aluminum: garnet laser versus intense pulsed light for the treatment of hirsutism. Journal of Cutaneous and Aesthetic Surgery. 2015 Apr; 8(2): 97. doi: 10.4103/0974-2077.158445. DOI:
Jo SJ, Kim JY, Ban J, Lee Y, Kwon O, Koh W. Efficacy and safety of hair removal with a long-pulsed diode laser depending on the spot size: a randomized, evaluators-blinded, left-right study. Annals of Dermatology. 2015 Oct; 27(5): 517-22. doi: 10.5021/ad.2015.27.5.517. DOI:
Szima GZ, Janka EA, Kovacs A, Bortely B, Bodnar E, Sawhney et al. Comparison of hair removal efficacy and side effect of neodymium: Yttrium‐aluminum‐garnet laser and intense pulsed light systems (18‐month follow‐up). Journal of Cosmetic Dermatology. 2017 Jun; 16(2): 193-8. doi: 10.1111/jocd.12312. DOI:
Rizwan M and Hameed A. Treatment of idiopathic facial hirsutism with medroxyprogesterone acetate iontophoresis. Journal of Pakistan Association of Dermatologists. 2009; 19(2): 90-4.
Gupta G. Diode laser: Permanent hair" Reduction" Not" Removal". International Journal of Trichology. 2014; 6(1): 34. doi: 10.4103/0974-7753.136762. DOI:
Hu AC, Chapman LW, Mesinkovska NA. The efficacy and use of finasteride in women: a systematic review. International Journal of Dermatology. 2019 Jul; 58(7): 759-76. doi: 10.1111/ijd.14370. DOI:
Thaysen‐Petersen D, Erlendsson AM, Nash JF, Beerwerth F, Philipsen PA, Wulf HC et al. Side effects from intense pulsed light: Importance of skin pigmentation, fluence level and ultraviolet radiation—A randomized controlled trial. Lasers in Surgery and Medicine. 2017 Jan; 49(1): 88-96. doi: 10.1002/lsm.22566. DOI:
Załęska I and Atta-Motte M. Aspects of diode laser (805 nm) hair removal safety in a mixed-race group of patients. Journal of Lasers in Medical Sciences. 2019; 10(2): 146. doi: 10.15171/jlms.2019.23. DOI:
Behboodi Moghadam Z, Fereidooni B, Saffari M, Montazeri A. Measures of health-related quality of life in PCOS women: a systematic review. International Journal of Women's Health. 2018 Aug: 397-408. doi: 10.2147/IJWH.S165794. DOI:
Alizadeh N, Ayyoubi S, Naghipour M, Hassanzadeh R, Mohtasham-Amiri Z, Zaresharifi S et al. Can laser treatment improve quality of life of hirsute women?. International Journal of Women's Health. 2017 Oct: 777-80. doi: 10.2147/IJWH.S137910. DOI:
Shrimal A, Sardar S, Roychoudhury S, Sarkar S. Long-pulsed Nd: YAG laser and intense pulse light-755 nm for idiopathic facial hirsutism: a comparative study. Journal of Cutaneous and Aesthetic Surgery. 2017 Jan; 10(1): 40. doi: 10.4103/0974-2077.204582. DOI:
Goh CL. Comparative study on a single treatment response to long pulse Nd: YAG lasers and intense pulse light therapy for hair removal on skin type IV to VI–is longer wavelengths lasers preferred over shorter wavelengths lights for assisted hair removal. Journal of Dermatological Treatment. 2003 Dec; 14(4): 243-7. doi: 10.1080/09546630310004171. DOI:
Saeed BT. Comparative Study of Diode Laser Versus Intense Pulsed Light (IPL) for the management of Hirsutism in Sulaimani Government. Kurdistan Journal of Applied Research. 2020 Dec: 40-8. doi: 10.24017/science.2020.ICHMS2020.5. DOI:
Abdul-Hussein AA, Razzaq SA, Shak HH. Effectiveness of diode laser versus intense pulsed light in hirsutism: a prospective and comparative study in Samawa city. International Journal of Psychosocial Rehabilitation. 2020 Apr 1; 24(02). doi: 10.37200/IJPR/V24I2/PR200511. DOI:
N Alhayani N and S Alkubaisi J. The Efficiency of Intense Pulse Light in the Treatment of Hirsute Ladies. Al-Anbar Medical Journal. 2020 Dec; 16(2): 46-9. doi: 10.33091/amj.2020.171027. DOI:
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CITATION
DOI: 10.54393/pjhs.v4i12.1226
Published: 2023-12-31
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How to Cite
Ilyas, S., Hussain, M., Siddiqui, M. A., Muzaffar, B., Gul, K., & Sajjad, D. (2023). Comparative Efficacy of Diode Laser System versus Intense Pulse Light (Ipl) In Management of Unwanted Hair : Diode Laser System versus Intense Pulse Light . Pakistan Journal of Health Sciences, 4(12), 163–167.
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3466 | https://math.stackexchange.com/questions/1829811/changing-y-mxb-equation-into-axby-c | Stack Exchange Network
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Changing $y=mx+b$ equation into $ax+by=c$
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$\begingroup$
I'm stuck on this question and I'm not totally sure how to transform an equation of the form $y=mx+b$ into an equation of the form $ax+by=c$. This is how far I have gotten
$$y=-\frac{1}{3}x+\frac{29}{3}$$
How do I transform this into an $ax+by=c$ equation?
algebra-precalculus
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edited Jun 17, 2016 at 15:02
mvw
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Student909Student909
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Typically when changing from the the $y = mx+b$ form (slope-intercept form) to the $ax+by=c$ form (standard form) you want $a$ and $b$ to be integers. First we can multiply both sides by three (because this is the denominator of the slope $m$), then just move the term with $x$ to the other side of the equation by adding $x$ to both sides: $$\begin{align} y&=-\frac{1}{3}x+\frac{29}{3}\ 3(y)&=3\left(-\frac{1}{3}x+\frac{29}{3}\right)\ 3y&=-x+29\\ x + 3y &= 29 \end{align}$$
More generally, if you wanted $a$,$b$, and $c$ to be integers in the standard form, you would first have to take the denominators of $m$ and $b$ from the slope-intercept form and multiply through by their least common multiple (in our case here, that least common multiple happens to just be $3$).
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edited Jun 17, 2016 at 14:48
answered Jun 17, 2016 at 14:43
Mike PierceMike Pierce
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$\begingroup$ If you're going to invent a requirement of integer coefficients, you should at least discuss how this isn't possible if $m$ and/or $b$ is irrational ... $\endgroup$
hmakholm left over Monica
– hmakholm left over Monica
2016-06-17 15:27:38 +00:00
Commented Jun 17, 2016 at 15:27
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Generally when you write it in the format $ax +by =c$ you want the coefficients to be integers for simplicity. But dividing the coefficients won't affect the properties of the line in anyway. The following equations represent the same line.
$y= \frac{-1}{3} x +\frac{29}{3}$
$ \frac{1}{3} x+y=\frac{29}{3}$
$x+ 3y=29$
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edited Jun 17, 2016 at 14:53
answered Jun 17, 2016 at 14:47
Atulya JainAtulya Jain
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$\begingroup$
You transform it into $$ \frac{1}{3} x + y = \frac{29}{3} $$ by adding $(1/3) x$ to both sides of the equation. So you got $a = 1/3$ and $b = 1$ and $c = 29/3$.
In fact for $\lambda \ne 0$ the set of solutions $(x,y)$ does not change, if we multiply both sides by $\lambda$: $$ \frac{\lambda}{3}x + \lambda y = \frac{29\lambda}{3} $$ Among all those equations one can decide for a nice looking one, in this case $\lambda = 3$ looks a bit less complex (no division): $$ x + 3y = 29 $$
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edited Jun 17, 2016 at 15:03
answered Jun 17, 2016 at 14:40
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$\endgroup$
4
$\begingroup$ dont you have to divide it by 3? $\endgroup$
Student909
– Student909
2016-06-17 14:41:27 +00:00
Commented Jun 17, 2016 at 14:41
$\begingroup$ @Student909 you probably meant multiply? $\endgroup$
windircurse
– windircurse
2016-06-17 14:43:27 +00:00
Commented Jun 17, 2016 at 14:43
$\begingroup$ Thank you so much for the answer !! $\endgroup$
Student909
– Student909
2016-06-17 14:44:53 +00:00
Commented Jun 17, 2016 at 14:44
$\begingroup$ @windircurse yes I meant multiply, thank you. $\endgroup$
Student909
– Student909
2016-06-17 14:48:50 +00:00
Commented Jun 17, 2016 at 14:48
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Sarala Deshpande
Last answered:
04 Oct 2019
Posted on:
04 Oct 2019
0
Confused With R squared
Hi Teacher, At the time I was done with this chapter, R squared was pretty clear to me, however, now that I moved to learning principles of data science, where R squared is defined by 1-SSR/SST. Could you please explain?
1 answers ( 0 marked as helpful)
365 Team
Instructor
Posted on:
04 Oct 2019
0
Hey Sarala,
As far as I understand, the concept is clear to you but in our lecture, we define R-squared as SSR / SST, while according to another source it is 1 - SSR/ SST, correct?
That's a valid question.
In both cases, what is meant is that the R-squared = Variability explained / Total variability.
Now, according to our framework,SST = Sum of Squares Total; SSR = Sum of Squares Regression; SSE = Sum of Squares Error
In that case,R-squared = SSR / SST, or R-squared = 1 - SSE/SST
Unfortuntely, there is adifferent notationin some books that you may come across. Some sources have the abbreviations as:
TSS (SST) = Total Sum of Squares; RSS(SSR)= Residual Sum of Squares; ESS(SSE) = Estimation Sum of Squares
You can see how this is a problem, as residuals (conceptually) meanerror.And estimation (at least in the case of regression analysis)meansregression.
Using this notation, you can state:R-squared = Variability explained / Total variability = ESS / TSS or as you saw it = 1 - RSS / TSS.
Some people even defineSSR = Sum of Squares Regression Error,stating that SSR stands for the sum of errors.This third abbreviation in my opinion is the most misleading of them all. Do you even need to say the word'regression' here?That to me is basically saying:'You can't make me use your notation. I prefer my own.'
Conceptually, the three notations have the same meaning.Unfortunately, their abbreviations are opposite.
I have seen the first notation (the one from our lectures) used much more often than the others. When creating the R-squared lecture, I put the extra effort to research the usage of each one of those, as I anticipated some confusion. Predominantly, sources were using the first notation, so I stuck with it. I like the second one, but only when they put:TSS, RSS, and ESS as abbreviations. That makes it clear which framework the author is using.
In any case, now you know about this ridiculous confusion in statistics. In the material you are using, just assume that SSR and SSE have switched places. Everything else should be the same.
Best,
The 365 Team
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3468 | https://hextobinary.com/unit/magdensity/from/mtesla/to/tesla | Millitesla to Tesla Converter (mT to T)
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Millitesla to Tesla Converter
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Convert → Tesla to Millitesla
1 Millitesla = 0.001 Teslas
How many Teslas are in a Millitesla?
The answer is one Millitesla is equal to 0.001 Teslas and that means we can also write it as 1 Millitesla = 0.001 Teslas. Feel free to use our online unit conversion calculator to convert the unit from Millitesla to Tesla. Just simply enter value 1 in Millitesla and see the result in Tesla.Convert 1 Millitesla to Teslas
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How to Convert Millitesla to Tesla (mT to T)
By using our Millitesla to Tesla conversion tool, you know that one Millitesla is equivalent to 0.001 Tesla. Hence, to convert Millitesla to Tesla, we just need to multiply the number by 0.001. We are going to use very simple Millitesla to Tesla conversion formula for that. Pleas see the calculation example given below.
1 Millitesla=1×0.001=0.001 Teslas
What is Millitesla Unit of Measure?
Millitesla is a unit of measurement for magnetic flux density. Millitesla is a decimal fraction of magnetic flux density unit tesla. One millitesla is equal to 0.001 tesla.
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What is the symbol of Millitesla?
The symbol of Millitesla is mT. This means you can also write one Millitesla as 1 mT.
What is Tesla Unit of Measure?
Tesla is a unit of measurement for magnetic flux density. It is named after Serbian-American inventor, Nikola Tesla. By definition, a particle moving perpendicularly through a magnetic field of one tesla at a speed of one meter per second and carrying a charge of one coulomb, experiences a force with magnitude of one newton.
What is the symbol of Tesla?
The symbol of Tesla is T. This means you can also write one Tesla as 1 T.
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Millitesla to Tesla Conversion Table
| Millitesla [mT] | Tesla [T] |
--- |
| 1 | 0.001 |
| 2 | 0.002 |
| 3 | 0.003 |
| 4 | 0.004 |
| 5 | 0.005 |
| 6 | 0.006 |
| 7 | 0.007 |
| 8 | 0.008 |
| 9 | 0.009 |
| 10 | 0.01 |
| 100 | 0.1 |
| 1000 | 1 |
Millitesla to Other Units Conversion Table
| Millitesla [mT] | Output |
--- |
| 1 millitesla in tesla is equal to | 0.001 |
| 1 millitesla in microtesla is equal to | 1000 |
| 1 millitesla in gauss is equal to | 10 |
| 1 millitesla in gamma is equal to | 1000000 |
| 1 millitesla in weber/square meter is equal to | 0.001 |
| 1 millitesla in weber/square kilometer is equal to | 1000 |
| 1 millitesla in weber/square decimeter is equal to | 0.00001 |
| 1 millitesla in weber/square centimeter is equal to | 1e-7 |
| 1 millitesla in weber/square millimeter is equal to | 1e-9 |
| 1 millitesla in weber/square micrometer is equal to | 1e-15 |
| 1 millitesla in weber/square nanometer is equal to | 1e-21 |
| 1 millitesla in weber/square mile is equal to | 2589.99 |
| 1 millitesla in weber/square yard is equal to | 0.00083612736 |
| 1 millitesla in weber/square foot is equal to | 0.00009290304 |
| 1 millitesla in weber/square inch is equal to | 6.4516e-7 |
| 1 millitesla in milliweber/square meter is equal to | 1 |
| 1 millitesla in milliweber/square kilometer is equal to | 1000000 |
| 1 millitesla in milliweber/square decimeter is equal to | 0.01 |
| 1 millitesla in milliweber/square centimeter is equal to | 0.0001 |
| 1 millitesla in milliweber/square millimeter is equal to | 0.000001 |
| 1 millitesla in milliweber/square micrometer is equal to | 1e-12 |
| 1 millitesla in milliweber/square nanometer is equal to | 1e-18 |
| 1 millitesla in milliweber/square mile is equal to | 2589988.11 |
| 1 millitesla in milliweber/square yard is equal to | 0.83612736 |
| 1 millitesla in milliweber/square foot is equal to | 0.09290304 |
| 1 millitesla in milliweber/square inch is equal to | 0.00064516 |
| 1 millitesla in microweber/square meter is equal to | 1000 |
| 1 millitesla in microweber/square kilometer is equal to | 1000000000 |
| 1 millitesla in microweber/square decimeter is equal to | 10 |
| 1 millitesla in microweber/square centimeter is equal to | 0.1 |
| 1 millitesla in microweber/square millimeter is equal to | 0.001 |
| 1 millitesla in microweber/square micrometer is equal to | 1e-9 |
| 1 millitesla in microweber/square nanometer is equal to | 1e-15 |
| 1 millitesla in microweber/square mile is equal to | 2589988110.34 |
| 1 millitesla in microweber/square yard is equal to | 836.13 |
| 1 millitesla in microweber/square foot is equal to | 92.9 |
| 1 millitesla in microweber/square inch is equal to | 0.64516 |
| 1 millitesla in unit pole/square meter is equal to Engineering calculation tools | 7957.75 |
| 1 millitesla in unit pole/square kilometer is equal to | 7957747154.59 |
| 1 millitesla in unit pole/square decimeter is equal to | 79.58 |
| 1 millitesla in unit pole/square centimeter is equal to | 0.79577471545942 |
| 1 millitesla in unit pole/square millimeter is equal to | 0.0079577471545942 |
| 1 millitesla in unit pole/square micrometer is equal to | 7.9577471545942e-9 |
| 1 millitesla in unit pole/square nanometer is equal to | 7.9577471545942e-15 |
| 1 millitesla in unit pole/square mile is equal to | 20610470515.46 |
| 1 millitesla in unit pole/square yard is equal to | 6653.69 |
| 1 millitesla in unit pole/square foot is equal to Engineering calculation tools | 739.3 |
| 1 millitesla in unit pole/square inch is equal to | 5.13 |
| 1 millitesla in line/square meter is equal to | 100000 |
| 1 millitesla in line/square kilometer is equal to | 100000000000 |
| 1 millitesla in line/square decimeter is equal to | 1000 |
| 1 millitesla in line/square centimeter is equal to | 10 |
| 1 millitesla in line/square millimeter is equal to | 0.1 |
| 1 millitesla in line/square micrometer is equal to | 1e-7 |
| 1 millitesla in line/square nanometer is equal to | 1e-13 |
| 1 millitesla in line/square mile is equal to Engineering calculation tools | 258998811033.6 |
| 1 millitesla in line/square yard is equal to | 83612.74 |
| 1 millitesla in line/square foot is equal to | 9290.3 |
| 1 millitesla in line/square inch is equal to | 64.52 |
| 1 millitesla in maxwell/square meter is equal to | 100000 |
| 1 millitesla in maxwell/square kilometer is equal to | 100000000000 |
| 1 millitesla in maxwell/square decimeter is equal to | 1000 |
| 1 millitesla in maxwell/square centimeter is equal to | 10 |
| 1 millitesla in maxwell/square millimeter is equal to | 0.1 |
| 1 millitesla in maxwell/square micrometer is equal to | 1e-7 |
| 1 millitesla in maxwell/square nanometer is equal to | 1e-13 |
| 1 millitesla in maxwell/square mile is equal to | 258998811033.6 |
| 1 millitesla in maxwell/square yard is equal to | 83612.74 |
| 1 millitesla in maxwell/square foot is equal to | 9290.3 |
| 1 millitesla in maxwell/square inch is equal to | 64.52 |
Convert Millitesla to Other Magnetic Flux Density Units
Millitesla to Unit Pole/Square Yard
Millitesla to Milliweber/Square Decimeter
Millitesla to Weber/Square Millimeter
Millitesla to Unit Pole/Square Centimeter
Millitesla to Weber/Square Decimeter
Millitesla to Line/Square Yard
Millitesla to Weber/Square Foot
Millitesla to Line/Square Kilometer
Millitesla to Maxwell/Square Millimeter
Millitesla to Unit Pole/Square Meter
Millitesla to Microweber/Square Meter
Millitesla to Milliweber/Square Kilometer
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3469 | https://www.youtube.com/watch?v=Sz4gBlO_CYk | L-2(D): Z_n is a Commutative Ring With Unity (By Division Algorithm) (A Series of Five Lectures)
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L-2(D): Z_n is a Commutative Ring With Unity (By Division Algorithm) (A Series of Five Lectures)
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L:1 Field Theory: Introduction to Field, Subfield & Homomorphisms ( M.Sc. Mathematics, CSIR-NET)
L:2 Z_n, Integer modulo n, Construction से Field बनने तक (In Five Lectures)
(L-1: Construction of Z_n )
Field Theory (definition and Examples), M.Sc. Mathematics, Field Theory, Ring Theory, Field Theory, Concept of Field and Subfields, Field Extensions,
Advanced Abstract Algebra (csir net mathematics field theory ring theory)
This Lecture is designed for students doing M.Sc. Mathematics & preparing For CSIR-NET Examination. This is an introductory lecture for Definition of Field, Subfields and Important results in these topics. In upcoming lectures we will learn more about prime fields, field extensions, splitting fields etc.
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Transcript:
Intro and in fact if n is a prime then zn is a field with respect to addition and multiplication on the congruence classes so in this lecture we present another method to prove that zn is a commutative ring and the method we will discuss here is the method of division algorithm so here the set under consideration will be the set of remainders of a positive integer that is if n is a positive integer then what will be the set here the set of remainders of n when any integer is divided by n so we will explain each and every concept to prove that zn is a commutative ring with unity using the method of vision algorithm so without delaying let's start the lecture Set of remainders so as we discussed that the set is the set of remainders of a positive integer so let n be a fixed positive integer okay then the set under consideration is s is the set of remainders set of remainders when any integer is divided by this fixed integer and so what is the set here set of remainders of the positive integer n and everyone is very much familiar that when an integer is divided by a fixed integer n then it may give the remainder 0 1 2 up to n minus 1 so this is the set of remainders of a positive integer now we will show that this s will turns out to be a commutative ring with respect to two binary operation addition and multiplication so first we will define a binary operation on s define a binary operation addition denoted by addition n on s as follows so for this choose two elements a and b belongs to s then what will be a addition modulo b is least known negative remain non-negative integer when the sum a plus b is divided by n and you know that what is the least non-negative remain no negative integer is nothing but the remainder so this is the remainder when a plus b is divided by a okay and you know that the remainder is when any integer is divided by a fixed integer n the remainder may be 0 1 2 up to n minus 1 so clearly this will be an element of the set because this is the remainder and remainder is 0 1 up to n minus 1 is in remember maybe any integer from 0 to n minus 1 and therefore if we apply this operation then it will be an element of s and it will be unique element of s so therefore what does this imply this implies that let me clear it this implies that is a binary operation on s so this is a binary operation of this now we will use the concept of division algorithm here to proceed further so by division algorithm okay remainder okay where q is the quotient quotient and r is the remainder when a plus b is divided by n okay so it means r is the remainder when a plus b is divided by n okay so this implies what is r r is in our operation r is this okay because this is the binary operation on s and it gives the remainder when a plus b is divided by n so r is this okay so you can write this as a plus b minus and q where q is the quotient so we will use this notation in throughout this lecture and this is very useful observation very useful notation to prove that s is a s turns out to be a ring with respect to addition modulo and and multiplication okay now with the help of this fact we will show that this operation is associative Left Side so what is the claim here we have to show that this operation is associated that means this must hold okay now take the left hand side we will show that left hand side is equal to the right hand side this is the easiest way to prove the associativity so this is replace this by r1 where r1 r1 is the remainder when a plus b is divided by n so you can write this as this where k1 is the quotient when a plus b is divided by n and this is nothing but this okay now replace this when you combine these r one and c with respect to the operation defined above then it will be r2 and you can write here as r2 is the remainder when r1 plus c is divided by n so you can write this as where q2 is coset and r1 plus c is divided by n q 2 is quotient and r 2 is the remainder okay so you can write this as r 1 and c okay since r 2 is the remainder when r1 plus c is divided by n so obviously r2 lies between 0 and n minus 1 because r2 is the remainder okay so rewrite the steps again r 2 is equal to r 1 plus c minus n q 2 r1 plus c minus and q2 and what is r1 r1 is a plus b minus n cubed so replace the value of r1 is here a plus b a plus b minus n k 1 this is the value of r 1 plus c minus n equal to so this is a plus b plus c minus n k 1 plus and q2 or you can write this as a plus b plus c minus n q one plus q so what is r 2 r 2 is the remainder when a plus b plus c is divided by n okay calculate the right hand side Right Side so for right hand side we have to compute this one okay so let's call this vector as r3 so this will be r3 where r3 is b modulo c b plus n model of c okay so this is the remainder when b plus c is divided by n so if you divide b plus c by n let q3 be the quotient then r3 is the remainder so you can represent r3 in this way b plus c minus and q3 okay now again combine a with r3 then suppose it will be r4 so what will be r4 here r4 is this one r3 okay so this is the remainder when a plus r3 is divided by n so suppose q4 is the quotient here so r4 is the remainder when a plus r 3 is divided by n so let's read this the steps r 4 is a plus r 3 minus n q q4 now substitute the value of r3 back so what is r3 r3 is b plus c minus nq 3 okay so again simplify the terms a plus b plus c minus n q3 plus q4 so this is you can write here that this is the remainder when a plus b plus c is divided by n okay now since r2 is also the remainder when so you can write here also it is the remainder when a plus b plus c is divided by n so r 2 is also the remainder when a plus b plus c is divided by a and r4 is also the remainder when a plus b plus c is divided by n since the number a plus b plus is same and every time you are dividing by n so you will get same remainder because remainder is always unique so it means this implies r2 must be equal to r4 and this implies a plus b plus c is equal to a plus b plus c so this establish the associativity in s now we move to the existence of identity part so next we move to the next step existence of identity in s with respect to this operation addition n okay for this let a be an arbitrary element of s because s is the set of remainders of n therefore the element a lies between 0 and n minus 1 and of course this is less than n now we expect that 0 will be the identity let us check that 0 will be the identity with respect to this operation so combine a with 0 with respect to this operation then this is the remainder when a plus 0. is divided by n divided by n so this is the remainder remainder when a plus 0 means a remember when a is divided by n since a is less than n so if you divide a by n then a itself is the remainder because a is less than n okay similarly if you combine 0 with a then again you will get a so it means if a is an arbitrary element of s then a with 0 and 0 combined with a give you a and this implies 0 is the identity in s because s has 0 element 0 1 2 up to n minus 1 and 0 is acting as identity and 0 lies in the set s so 0 is the identity element of s with respect to this operation okay now we will show that this is a group with respect to this operation and we will emphasize that we have every element in s is invertible so this is our next task that we have to Invertible show that every element of s is invertible with respect to this operation so again choose an arbitrary element of s then we have two possibility if this element itself is zero then then it is very easy to see that 0 combined with 0 is again 0 because if you add 0 with 0 then you get the remainder 0 when divided by n and what is 0 0 is the identity of s so this implies if a is a 0 element then 0 is the inverse of itself okay now we will discuss the case when a is known 0 so if a is known zero then we will expect that what will be the inverse of a we check that n minus a will be the inverse of okay so what we claim we claim that n minus a is the inverse of a so for this combine a with n minus a so apply the definition this is the remainder when a plus n minus a is divided by n divided by n and if this is equal to the remainder when a gets cancelled with a when n is divided by n and you know that if a integer a number is divided by itself then what is the remainder 0 similarly if you add n minus a to the left side of a again then you get 0 so it means this means every element in fact non-zero element has inverse n minus a with respect to this operation and 0 is the inverse of itself and if a is non-zero then every non-zero element has inverse n minus a Multiplication now we introduce multiplication on the set s so define a multiplication namely modulo n on s as follows so let a and b be any two arbitrary element in s then we define a multiplication operation on s by this so what is this operation this will give you the remainder when a multiplied by b is divided by n so this will be the binary of i i will show that it will be a binary operation on s since a b is divided by n and this operation will give you the remainder okay so you can write this as a b minus n q where q is the quotient when a b is divided by n so this is the remainder and you know that if any number is divided by an integer is divided by n a fixed integer n then remainder belongs to the set 0 1 2 up to n minus 1 and this set is nothing but x so it shows that this is a binary operation on s this implies is a binary operation on so no need to show closer because binary itself is a closer okay now we will show that this operation is associative on s so next target is is associative on s so what we will do we choose any three arbitrary element a b and c from s and what we show we show that you can group these three element in this way or you can group these three element in this way then result will be same okay so let's pick up the left hand side and we show that the left hand side will give you the result as the right hand side gives okay so our claim will be established Left Hand Side so choose the left hand side see so let we assume that this is r1 this means r1 is the remainder when a b is divided by n so let us assume that q is the quotient a 1 is the quotient then r 1 is a b minus n k 1 ok so this is equal to r1 c okay so let's assume r1 when combined with c is r2 because every time you apply this operation you will get a remainder r okay so here we get the remainder r2 and you can simplify r2 s it is a remainder when 1 into c is divided by n and suppose q 2 is the quotient here okay so r 2 is r 1 c minus n cubed now retrace the steps back r 2 is equal to r 1 c minus n k 2 okay now put the value of r 1 r 1 is a b minus n k 1 times c minus and q2 open the bracket maintain the grouping a b c minus n k 1 c minus n cubed so you can rewrite this step again a b c minus taking n common k 1 c plus q ok since r 2 is the remainder so r 2 belongs to the set s s is nothing but 0 1 2 up to n minus 1 mind it that this is a left hand side is a remainder r 2 is a remainder and this is a quantity this is a number lying between 0 to n minus 1 now we will calculate the right hand side this side okay Remainder so consider the right hand side so assume that b combined with c gives you the remainder r3 okay so this implies r3 is the remainder when b into c is divided by n let's assume q3 is the quotient here okay so r3 is bc minus and q3 so you can rewrite this as r3 okay so now assume a combined with r3 is r4 so what you can write here r4 is the remainder when a into r3 is divided by n suppose q4 is the quotient here so what you get r4 is a into r3 minus and q4 okay so this is r4 now rewrite retrace the steps back r4 is a r3 minus n q4 now substitute the value of r3 b into c minus and q3 minus n cube four okay open the bracket and maintain the grouping b into c minus n a q3 plus q4 so what is r4 r4 is again remainder okay so r4 again belongs to the set 0 1 2 up to n minus 1 so in this equation suppose this is star and this is double star both r 2 and r4 these are remainder both are remainder okay when abc abc is divided by n similarly r4 is the remainder when abc is divided by and it means you are dividing the same quantity by n one time you get the remainder r2 and second time you are getting the remainder r4 but you know that if the number is same the remainders are same okay so this implies r 2 must be equal to r 4 and this implies left hand side is equal to right hand side and this establish the associativity of this operation so now we move to the step existence of unity in s it means the identity with respect to the multiplication okay so for this choose any arbitrary element from the set s okay this implies a is an integer lying between 0 and n minus 1 what do you expect that multiplication identity in multiplication generally turns as 1 okay so let's see but by the definition we have this is the remainder when a into 1 is divided by n so what is this this is the remainder render when into one is a is divided by n and you know that a is an integer less than n this a is an integer less than n and if it divide a number smaller than n by n then what is the remainder number itself so a is the remainder itself because a is less than n okay similarly similarly in the other case you will get the same exact so it means if you combine one with any element of s then you will get the same element back so it means one is the one is the multiplicative identity and identity in multiplication is generally termed as unity so we have established the existence of unity now we will establish that this operation is commutative on s so we have to show that this operation is commutative on s Commutative since you know that the remainder when a b is divided is divided by n is same as the remainder same as the remainder when b a is divided by n it means either you divide a b by n or you divide b a by n you will get the same remainder okay so this implies immediately you can say that this is same as okay for all a and b belongs to s so it means this operation is commutative one yes now we will establish distributive laws on s so let us establish distributive laws Distributed Laws so there are two types of distributed laws left left hand distributed loss and right hand distributed loss and these are studied under this that so let's discuss distributed laws there we have to show that this operation is distributive over this addition okay so generally we have to show that let a b and c be three arbitrary element on s then what do we have to show that left hand distributed last whole so what is left distributed law that means multiplication is distributive over addition from the left hand side that means you can rewrite this as plus okay similarly what is the right hand distributed last it means the multiplication defined is distributive over addition from the right hand side that means that is equal to okay so i will prove the left hand distributive law for you and you can retrace the steps to prove right and distributed law let us check the first part so consider the left hand side from the first part and the technique will be same as we have done in associativity part okay so let it be r1 so what is r1 r1 is this so it means r1 is the remainder when b plus c is divided by n so this is b plus c minus n suppose k 1 is the quotient so you can write r 1 as v plus c minus n k 1 ok now let it be the a combined with r 1 b r2 then what is r2 r2 is the mean this means it is the remainder when a into r1 is divided by and suppose q2 is the quotient here okay so retrace the steps r2 is ar1 minus n q2 so put the value of r1 here so a what is r1 b plus c minus n k 1 minus n cube so this is a times b plus c minus n times a q 1 minus n q Multiplication Over Addition so rewrite this as a times b because this is now ordinary addition and multiplication so you know that every uh sorry ordinary multiplication is always distributed over addition so you can separate this a dot b plus a dot c minus n times a k 1 plus q so what you what is r 2 r 2 is the remainder because r 2 is r2 is obtained by the operation modulo n and multiplication model and on s and it means r2 is the remainder okay so r2 belongs to the set s so you have to keep in mind that r2 is remainder and what is the right hand side you can connect r2 with the fact that r2 is the remainder when a b plus ac is divided by n okay so right here that means r2 is remainder remainder when a b plus ac is divided by a okay now we calculate the right hand side so see the right hand side so right hand side is me correct it see okay so again suppose this is this is r3 and this is r so you can write this as r3 plus now what is r3 r3 is the remainder when a into b is divided by n so you can write here a b minus n q 3 where q 3 is the quotient when a b is divided by m similarly represent r4 as what is r4 r4 is this so what is r4 r4 is ac minus and q4 it means r4 is a remainder when ac is divided by n okay now now see here r3 and r4 are combined with respect to this addition so assume that it will be r5 now represent r5 r5 is r3 added with r4 so it means this is the remainder when r3 times r4 is divided by n so let us assume q5 is the quotient here so r5 is r3 plus r4 divided by n it means r5 is the remainder when r3 plus r4 is divided by n okay so now retrace the steps back what is r5 r5 is r3 plus r4 minus nq okay substitute the value of r3 r3 is ab minus nq3 and what is r4 r4 is ac minus and q4 minus n equal 5 so adjust the sides a b plus ac minus n q 3 minus n cube 4 minus n cube phi so it means a b plus ac minus n times q 3 plus q 4 plus q 5 so what is the left hand side left hand side is the remainder because we are applying either addition modulo n or multiplication module it means every time you will get a number from 0 to n minus 1 so therefore r5 is also in the set 0 to 1 to n minus 1 so it means r 5 is the remainder because every member of this set is less than n and therefore it is the remainder itself r5 is the remainder so what is r2 r2 is the remainder when a b plus ac is divided by n and what is r5 r5 is also the remainder r5 is also remainder remainder when a b plus ac is divided by divided by n so it means r5 is also remainder and you know that a b plus ac is a fixed number same number one time you are dividing it by n you are getting remainder r2 and the second time you are dividing you are getting the remainder r5 but you know that remainder is always unique therefore this implies r2 must be equal to r5 and what is r2 r2 is left hand side and r5 is the so r2 is the left hand side Conclusion r2 is and r5 is it means distributive in fact left distribute left distributive law holds in s with respect to this operations similarly you can show that right distributive laws also hold in s it means s have both left distributive laws and right distributed laws it means you can say that the multiplication model line is the distributive over addition model so from this onward we will denote this set as by we will denote this set s by z okay so you can now conclude that so what is the conclusion from all the facts that zn with respect to this addition and with respect to this this multiplication is a commutative ring with unity so this is the conclusion of this lecture okay students that's all from my side in the next lecture we will extend this result to search that what is the condition on n that it turns out to be a field and you know that very well that it will be a ring if and only if n will be a prime integer omni we have proved this result by the method of congruences but in the next lecture we will prove this result that it is a field when n is a prime by assuming that s is the set of remainders okay so stay tuned for upcoming lecture thank you |
3470 | https://calcworkshop.com/volume-surface-area/cone-cylinder/ | Volume of a Cone and Cylinder 9 Step-by-Step Examples!
// Last Updated: - Watch Video //
Did you know that the volume of a cone and cylinder is related?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
It’s true
You’re going to learn how to use this relationship to find volumes and surface areas of both cones and cylinders in today’s geometry lesson.
Also, you’ll learn how to tackle composite figures made up of cones and cylinders.
Let’s jump in!
There are two famous three-dimensional solids: the cylinder and the cone.
A cylinder is made up of two congruent, parallel, circular bases and one lateral face the shape of a rectangle.
To calculate the volume of the cylinder, we must first find the area of the base and then multiply by the height of the cylinder.
To find the surface area of a cylinder, we use our area and circumference formulas. First, we see the area of the two parallel, circular bases, and then we calculate the area of the rectangular lateral face. The trick for finding the area of this polygon is to know that the length is actually the circumference of the circle!
Don’t worry. We walk through several step-by-step examples in the video below.
Formulas for Volume and Surface Area of a Cylinder
Now, a cone is a solid that resembles an ice cream cone and has only one circular base and one lateral face. Its volume is one-third that of a cylinder, and its surface area is the sum of the area of the base and the lateral face, as Math is Fun accurately states.
How to Calculate the Volume and Surface Area of a Cone
And the method we use to find the area of the lateral surface is similar to finding the area of a triangle where the lateral edge is the hypotenuse. Again, this method is depicted in the video that follows.
Next, you’ll learn how to:
Calculate the volume and surface area for all different types of cylinders and cones.
Use the volume addition postulate to find the volume of composite solids.
Video – Lesson & Examples
58 min
Introduction to video: cylinders and cones
00:00:25 – Formulas for finding the volume and surface area of a cylinder and cone
Exclusive Content for Member’s Only
00:07:22 – Find the volume of the solid (Examples #1-4)
00:16:06 – Find the surface area of the solid (Examples #5-8)
00:26:27 – Find the volume and surface of the composite solid (Example #9)
Practice Problems with Step-by-Step Solutions
Chapter Tests with Video Solutions
Get access to all the courses and over 450 HD videos with your subscription
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Jon, C.04-06-25 - MD, United States
CalcWorkshop is a premier resource for any student seeking to understand, practice, and succeed within the vast areas of mathematics. I am impressed with the quality of instruction and practice problems offered by Jenn. As a engineering student, I always utilize Calcworkshop as a the "first stop" to really understand and gain confidence in the subject. I credit my success in Calculus to Jenn and CalcWorkshop and making me "smarter than the average bear."
Sayida S.04-03-25 - PA, United States
I absolutely love this subscription. At first I was iffy about spending the money, but in the long term it is so worth it. I started watching Jenn almost 3/4ths into my calc 2 class, and I a confidently say I understand the depths to each topic. Before, with my professor I felt that I only understood surface level of each topic. But after the CalcWorkshop videos, I have acronyms, tips, and knowledge I walk away with, that I didn't have before. Genuinely cannot wait to keep using this website for my future endeavors, minoring in Mathematics!
Austin08-08-25 - AZ, United States
I was lost in an AP Calculus class and Jenn's explanations of things like limits and dedicates helped me out tremendously! The break downs of every lesson made it seem like I was learning basic algebra again, and allowed me to do so much better in Calculus.
A Reviewer08-07-25 - CT, United States
I have been so super confident ever since I encountered the Calcworksop. This platform was so convenient for me it enabled me to study effectively. Jenn is an absolute guru, I love her so much. She gets it! I do feel I come to tears with the amount of progress I've made with calculus. If anyone ever has any struggle with calculus, Calcworkshop will rescue you. And in a brief amount of time will you start to make progress. This website is a miracle TRUST ME. Calculus 1,2 and 3 can all be courses that you can get great grades out of from this website. And there are more courses you can do well in for mathematics. Again this website saved my life! Jenn Forever!
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Great online videos and super helpful prep for an 8 week college Calculus class. I felt well prepared and did very well in my course overall.
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CalcWorkshop was wonderful! My Honors Calculus IV professor was a wonderful personality to be around but was incredibly smart and only taught about the theory of the math... rather than with example problems. I would end up spending hours upon hours trying to complete three problems and I knew that I needed more examples to show different workflows. With the Jenns help, I ended up pulling through with a high B in this course, which is honestly my most prized grade that I worked so hard for in undergrad studies so far.
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Corynebacterium diphtheriae
In subject area:Pharmacology, Toxicology and Pharmaceutical Science
Corynebacterium diphtheriae is defined as an aerobic or facultatively anaerobic gram-positive bacillus that causes the infectious disease diphtheria in humans.
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Corynebacteria (including diphtheria)
2009, Encyclopedia of Microbiology (Third Edition)K.F. Smith, D.M. Oram
Originally created to describe Corynebacterium diphtheriae and related bacteria, Corynebacterium is a diverse group of Gram-positive, GC-rich organisms. The genus includes nonpathogens such as C. glutamicum, animal pathogens such as C. pseudotuberculosis, as well as opportunistic and strict human pathogens such as C. jeikeium and C. diphtheriae. The most extensively studied species in the genus, C. diphtheriae, causes the severe respiratory disease diphtheria and secretes a potent exotoxin, diphtheria toxin (DT). DT is encoded in the genome of temperate phage in C. diphtheriae lysogens and its expression is regulated by the chromosomally encoded diphtheria toxin repressor (DtxR). The identification of siderophore molecules and more recently of pili suggests that the pathogenesis of C. diphtheriae is a complex process that requires several factors, in addition to DT. The phenotypic diversity of the genus is exemplified by comparing C. diphtheriae to the only other extensively characterized species, C. glutamicum. C. glutamicum is a nonpathogenic soil-dwelling species that is valuable in biotechnology settings as an overproducer of amino acids. Investigations of the other organisms in this understudied genus are identifying links between corynebacteria and infections in immunocompromised hosts. Further research will be required to fully characterize the pathogenic and industrial potential of corynebacteria.
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Reference work2009, Encyclopedia of Microbiology (Third Edition)K.F. Smith, D.M. Oram
Chapter
Aerobic Gram-Positive Bacilli
2017, Infectious Diseases (Fourth Edition)Guy Prod'hom, Jacques Bille
Corynebacterium diphtheriae, C. ulcerans
Nature
Diphtheria is now a rare disease. This communicable infectious and vaccine-preventable disease may result in acute localized respiratory infection with typical ‘croup’ symptomatology due to the development of oropharyngeal adherent pseudomembranes. The mortality is due to airway obstruction or myocarditis caused by toxin production. Corynebacterium diphtheriae is a small, irregular, nonsporulated, gram-positive bacillus with enlarged extremities and a typical ‘club-shaped’ morphology. A ‘palisade’ or ‘V’ arrangement or clusters with a so-called Chinese-letter appearance are observed in liquid media (Figure 178-3, lanes 1A and 1B). Corynebacterium spp. belong to the broad class Actinobacteria. Phylogenetic studies show that Corynebacterium spp. are closely related to acid-fast bacilli such as Mycobacterium.
Corynebacterium ulcerans shares certain common features with C. diphtheriae. C. ulcerans causes a rare oropharyngeal diphtheria-like illness, as well as extrapharyngeal infections due to the presence of a toxin similar to diphtheria toxin. Taxonomic studies show that C. ulcerans is closely related to C. diphtheriae.31-
Epidemiology
Diphtheria remains endemic in numerous countries throughout the world – Eastern Europe, South East Asia, South America and the Indian subcontinent. In higher-income countries, diphtheria occurs sporadically and most cases are imported from areas of endemicity.32
During the 1990s, an important outbreak occurred in Russia and the newly independent states of the former Soviet Union, with more than 157 000 cases and 5000 deaths. Several factors may explain the re-emergence of epidemic diphtheria, notably the introduction of a toxigenic strain, insufficient coverage of vaccination of children and an increasing proportion of adults with waning vaccine-induced immunity. Contrary to the prevaccine era, where diphtheria was essentially a childhood disease, a shift in the age of patients was observed. The epidemic began in towns and in groups with close contacts (e.g. hospitals, military troops), before dissemination to socioeconomically disfavored groups (e.g. alcoholics).33–35
Achieving a high coverage (90–95%) of primary immunization is mandatory for diphtheria elimination and periodic booster doses for adults are necessary in regions at high risk of diphtheria. Control measures allow the control of epidemic diphtheria. In 1997, the World Health Organization (WHO) identified 10 countries with more than 10 cases for a total number of 15 839 cases reported, whereas in 2006 the number of reported cases decreased to 4000.
Humans represent the sole significant reservoir for C. diphtheriae. Acquisition occurs essentially through direct transmission (or via airborne droplet) with contaminated respiratory secretions or skin lesions.
In temperate climates, respiratory infections due to toxigenic strains predominate, while in the tropics, cutaneous diphtheria is more commonly caused by nontoxigenic strains. A peak incidence of respiratory infections is observed during the cold months.
In diphtheria-endemic areas with high prevalence of cutaneous diphtheria, respiratory diphtheria is rare due to the high rate of natural immunity attained through exposure to cutaneous diphtheria. In most cases, transmission of C. diphtheriae to susceptible individuals results in transient pharyngeal carriage rather than in disease. Cutaneous diphtheria appears to be more contagious than the respiratory form of diphtheria and persists longer than C. diphtheriae infections of the tonsils or nose in a carrier state.
C. ulcerans is a commensal of wild and domestic animals. Human C. ulcerans infection is considered a zoonosis, since human transmission occurs via manipulation of infected dairy animals or consumption of contaminated milk; however, these risk factors are absent in half of the cases. Global epidemiologic data on C. ulcerans infection are missing but C. ulcerans strains represent 58% of toxigenic Corynebacterium strains submitted to WHO reference laboratories.35
Pathogenicity
C. diphtheriae contains pili or fimbriae involved in initial adhesion of the bacteria to host-cell receptors; the pili also promote the aggregation of other bacteria to ensure colonization. These structures contain subunits of pilin proteins, a backbone protein component (SpaA; Spa for sortase-mediated pilus assembly) and two ancillary proteins (SpaB, SpaC). The pili are covalently linked to the peptidoglycan structure of the bacterial cell wall. Minor pilins, SpaB and SpaC, are required for pharyngeal cell adhesion, the specific receptor being currently unknown.36 This type of pilus is encoded on a pathogenicity island, probably acquired by recent horizontal transfer. C. diphtheriae is responsible for a localized predominantly pharyngeal infection with the appearance of a so-called ‘pseudomembrane’ made of fibrin, neutrophilic inflammation and abundant colonies of C. diphtheriae. The major virulence factor is a 58 kDa exotoxin responsible for the systemic complications of diphtheria, notably for myocardial and neurologic toxicity.37 This exotoxin is very effective, since one molecule introduced into a cell can kill it.
The exotoxin gene (tox) is present on a family of corynephages hosted by some C. diphtheriae strains. The expression of toxin is reliant on an iron-dependent regulatory element (dtxR). The presence of iron activates dtxR and blocks the transcription of tox. Under iron-limiting conditions, generally observed on the mucosal surface, dtxR is inactivated and diphtheria toxin is produced. The toxin is excreted through the bacterial cell and diffuses both locally and via the circulation to all organs, the myocardium and the peripheral nerves being the most affected. Diphtheria toxin has a complex activity, resulting in cytolysis through cell-protein inhibition.
Three distinct domains of diphtheria toxin have been described:
•
: the carboxyl-terminal R-domain (fragment B), which is responsible for binding to the heparin-binding epidermal-like growth factor, the specific receptor for the toxin;
•
: internalization into endocytic vesicles, which is mediated by the central part of the toxin;
•
: the amino-terminal C-domain (fragment A) which is delivered in the cytosol and is responsible for the inhibition of protein synthesis by inactivating the eukaryotic elongation factor 2 (eEF-2) through adenosine diphosphate (ADP) ribosylation of diphthamide residue38 (Figure 178-4).
Histology of heart tissues shows necrosis associated with active inflammation in the interstitial space. Cardiac conduction tissue may also be affected. Diphtheria toxin may affect peripheral nerves, causing a demyelination localized around the nodes of Ranvier. In severe cases, axonal degeneration can occur.39
C. ulcerans also carries a corynephage coding for the diphtheria toxin, and the C. ulcerans tox gene has 95% homology with C. diphtheriae tox gene. Most differences are located in the B fragment of the toxin in the translocation (T) region and in the receptor-binding (R) domain.40
Diagnostic Microbiology
Due to the declining incidence of diphtheria, the policy for screening throat swabs varies in different countries. In endemic regions, routine screening for C. diphtheriae in throat swabs is recommended since milder infections or atypical infections may be misdiagnosed. When diphtheria is suspected, clinicians should inform the laboratory.
Cotton or polyester-tipped swabs are appropriate for sampling. When membranes are removed, they should be sent to the laboratory for culture and a swab from areas under the membranes should be obtained. Standard Amies transport media ensure adequate maintenance of viability during transport to the laboratory. When molecular testing is available, Dacron polyester swabs are preferred for sampling. Suspected cases should have specimens taken from both the nose and the throat. For wound infections, swabs or aspirates of inflamed lesions are recommended.
For culture, selective media such as cysteine–tellurite blood agar or Tinsdale medium and nonselective sheep blood agar are recommended.41 When selective media are not available, multiple colonies from sheep blood agar are picked and biochemically identified to rule out C. diphtheriae. Commercial biochemical tests and MALDI-TOF mass spectrometry accurately identify C. diphtheriae strains. C. diphtheriae appear as small 0.5 mm up to 2 mm large colonies with a gray to white opaque surface (see Figure 178-3, lanes 1A and 1B). Some strains (biotype intermedius) appear dysgonic on standard media, their colonies being enlarged on lipid-enriched media (lipophilism). Based on colony morphology and biochemical reactions, C. diphtheriae can be divided into four biotypes: gravis, mitis, belfanti and intermedius. However, biotyping is of limited use and is generally carried out only in reference laboratories.
C. ulcerans grows on standard sheep blood agar media, but the use of selective media such as cysteine–tellurite blood agar or Tinsdale medium facilitates the isolation of pathogenic corynebacteria.
After 24 hours, 1 mm colonies are observed with a gray–white appearance (see Figure 178-3, lanes 2A and 2B) and slight hemolysis on sheep blood agar. Presence of urease and a positive reverse CAMP test (inhibition reaction) differentiate C. ulcerans from C. diphtheriae.42
The detection of toxin is the most important test; molecular tests based on the amplification of the tox gene constitute an alternative to the classic Elek immunoprecipitation test. Recently, the direct detection of the tox gene from clinical specimens has been described; the amplification test can differentiate the C. diphtheriae tox gene from toxin-producing C. ulcerans.43
Some toxigenic C. ulcerans strains show atypical results in real-time PCR for toxin detection. Specific amplification of the C. ulcerans tox gene has been described based on the polymorphic region of the tox gene.44
Clinical Manifestations
Respiratory Tract Disease
C. diphtheriae is responsible for localized infection of the upper respiratory tract. The oropharynx and rhinopharynx are most frequently affected. Symptoms generally occur 2–5 days after transmission, the most common symptoms being a moderate fever and sore throat followed by weakness, painful swallowing and headaches. Classic adherent and gray pseudomembranes are not always present and their extent can vary from small patches on infected tonsils to an extensive involvement of the posterior oropharynx and larynx. A serous or serosanguinous discharge can be observed in nasal diphtheria, which is often milder and chronic. Signs of toxicity may be present with edema of the neck, hoarseness and stridor and a marked local lymphadenopathy. In adults, atypical presentations are observed with oral lesions.40
Respiratory infections due to C. ulcerans are similar to infection due to C. diphtheriae.
Cutaneous Disease
Cutaneous diphtheria appears usually as a chronic and nonspecific disease. In fact, C. diphtheriae may colonize any kind of primary cutaneous lesion. Superinfection with C. diphtheriae begins with vesicles or pustules evolving as skin ulcers, with edematous surrounding tissues. Dark pseudomembranes may be observed during the first 1–2 week(s). Their localization is predominantly on the legs and hands. Coinfection with other pathogens such as Staph. aureus and Streptococcus pyogenes is common. Systemic toxic manifestations of cutaneous diphtheria are rarely seen. Chronic cutaneous infections with gray membranes due to toxigenic C. ulcerans have been described.45
Infections due to C. diphtheriae and C. ulcerans are undoubtedly underreported since cutaneous manifestations may be mild and corynebacteria are often considered as contaminants.
Myocardial Disease
Myocardial toxicity occurs in 10–20% of patients with respiratory diphtheria and is associated with a mortality rate of 40%. Its frequency depends on the extension of the oropharyngeal infection and on the delay until antitoxin therapy is started. Cardiac manifestations appear during the second week of the disease with a dilated cardiomyopathy and depressed left ventricular function. Electrocardiography shows anomalies of conduction and dysrhythmia may be observed with atrioventricular block, bundle branch block and hemi block. Markers of poor prognosis and carditis in children are a high myoglobin level and an increased lactate dehydrogenase (LDH) isoenzyme level, with an LDH1/LDH2 ratio >1.46
Neurologic Disease
Neuronal toxicity appears generally in severe respiratory diphtheria. Distinct types of damage are observed due to the regional effect of the toxin, with paralysis of the soft palate, of the posterior pharyngeal wall with dysphonia, dysphagia and defects of ocular accommodation; a delayed complication consisting of a peripheral neuritis affecting the limbs and/or the diaphragm can also occur up to 3 months later. Recovery is slow and only 20% of patients with diphtherial peripheral neurotoxicity were completely healthy after 1 year.47 The incidence of severe paralysis seems reduced when prompt antitoxin therapy is initiated within the first 2–3 days of infection.
Systemic Disease
Deep-seated infections due to C. diphtheriae (e.g. endocarditis, septic arthritis, osteomyelitis and splenic abscesses) have also been documented. Risk factors such as intravenous drug use, homelessness and alcoholism have been identified. The portal of entry is likely to be skin colonization or skin infections with C. diphtheriae. Strains belonging to different biotypes (gravis, mitis, belfanti) have been isolated and are generally nontoxigenic.48 These strains may represent a potential reservoir for the emergence of toxigenic C. diphtheriae strains since they possess the functional regulation machinery represented by the ‘diphtheria toxin repressor (dtxR) genes’ and thus could become toxigenic by acquiring the tox gene.49
Management
The management of respiratory tract diphtheria includes specific and nonspecific measures. Specific measures include the administration of diphtheria antitoxin and antibiotic treatment for C. diphtheriae eradication, whereas nonspecific measures are the surveillance and support of respiratory and cardiac functions. Neutralization of circulating toxin through diphtheria antitoxin represents the mainstay of diphtheria therapy. Since diphtheria antitoxin is prepared with horse hyperimmune serum, the sensitivity of the patient to horse serum should be tested before administration. According to the extent and duration of the illness, a variable dose of antitoxin is administered (Table 178-2).50 Penicillin or erythromycin constitutes the standard antibiotic treatment. In vitro resistance to penicillin has not yet been described but in vitro penicillin tolerance has been documented and could explain some eradication failures.51 Erythromycin is generally highly active in vitro but inducible resistance has been described.52
Antibiotics are used to reduce the carrier state and are combined with antitoxin to treat the disease. Erythromycin 40–50 mg/kg/day intravenously for 7–14 days is the treatment of choice, but oral penicillin (benzylpenicillin G 50 000 IU/kg/day for 5 days followed by penicillin V 50 mg/kg/day for 5 days) has been shown to be superior in a randomized trial.52
Testing for C. diphtheriae eradication is recommended at the end of treatment. Continue treatment for 10 days if culture remains positive.50
Close contacts should receive a single dose of penicillin or a 7-day course of erythromycin.50
Prevention
The following measures to control diphtheria have been proposed by international experts:
•
: primary prevention of disease based on a childhood immunization program;
•
: secondary prevention through rapid investigation of close contacts of index cases to prevent emergence of secondary cases, with investigation of carriage rate, clinical surveillance, penicillin prophylaxis and administration of a booster dose of diphtheria toxoid-containing vaccine;
•
: tertiary prevention based on early diagnosis and proper management to prevent complications of suspected cases.53
Primary prevention with diphtheria toxoid obtained from formaldehyde-treated toxin is highly effective, providing high immunization rates. Recent epidemics have shown that more than 90% of children should be vaccinated to interrupt efficiently the transmission of epidemic clones.54 The primary vaccination is based on a series of three doses started at 6 weeks of age and given at intervals of 4 weeks. Three doses induce the formation of protective toxin-neutralizing antibodies in 95.5% of children. This series is completed by at least one booster dose in the preschool period. The waning of adult immunity should be prevented with booster doses of diphtheria toxoid every 10 years throughout life. Recently, a new vaccine formulation called Tdap (for tetanus toxoid, reduced diphtheria toxoid and acellular pertussis vaccine) has been recommended for adults aged 19–64 years.55 In endemic regions, booster doses are less necessary since natural boosting of immunity may occur through C. diphtheriae colonization.
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Book2017, Infectious Diseases (Fourth Edition)Guy Prod'hom, Jacques Bille
Chapter
Infectious Endophthalmitis
2013, Retina (Fifth Edition)Travis A. Meredith, J. Niklas Ulrich
Corynebacterium diphtheriae
Corynebacterium diphtheriae is a Gram-positive bacillus that is nonsporulating, noncapsulated, nonmotile, and pleomorphic. Humans are the only known reservoir for C. diphtheriae and spread is by airborne respiratory droplets or by direct contact. Occasionally C. diphtheriae resides on skin. The main cause of virulence is a potent exotoxin producing diphtheria and skin infection. The choice for primary treatment of this organism is penicillin or erythromycin.
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Book2013, Retina (Fifth Edition)Travis A. Meredith, J. Niklas Ulrich
Chapter
Aerobic Gram-Positive Bacilli
2017, Infectious Diseases (Fourth Edition)Guy Prod'hom, Jacques Bille
Nature
Diphtheria is now a rare disease. This communicable infectious and vaccine-preventable disease may result in acute localized respiratory infection with typical ‘croup’ symptomatology due to the development of oropharyngeal adherent pseudomembranes. The mortality is due to airway obstruction or myocarditis caused by toxin production. Corynebacterium diphtheriae is a small, irregular, nonsporulated, gram-positive bacillus with enlarged extremities and a typical ‘club-shaped’ morphology. A ‘palisade’ or ‘V’ arrangement or clusters with a so-called Chinese-letter appearance are observed in liquid media (Figure 178-3, lanes 1A and 1B). Corynebacterium spp. belong to the broad class Actinobacteria. Phylogenetic studies show that Corynebacterium spp. are closely related to acid-fast bacilli such as Mycobacterium.
Corynebacterium ulcerans shares certain common features with C. diphtheriae. C. ulcerans causes a rare oropharyngeal diphtheria-like illness, as well as extrapharyngeal infections due to the presence of a toxin similar to diphtheria toxin. Taxonomic studies show that C. ulcerans is closely related to C. diphtheriae.31-
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Book2017, Infectious Diseases (Fourth Edition)Guy Prod'hom, Jacques Bille
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Coryneform bacteria, listeria and erysipelothrix
2012, Medical Microbiology (Eighteenth Edition)J. McLauchlin, P. Riegel
Corynebacterium diphtheriae
The major disease caused by C. diphtheriae is diphtheria, an infection of the local tissue of the upper respiratory tract with the production of a toxin that causes systemic effects, notably in the heart and peripheral nerves. Diphtheria has virtually disappeared in developed countries following mass immunization, but is still endemic in many regions of the world. Skin infections are prevalent in some countries. Non-toxigenic strains have been associated with endocarditis, meningitis, cerebral abscess and osteoarthritis throughout the world.
Description
C. diphtheriae, like other members of the genus, are non-motile, non-spore-forming, straight or slightly curved rods with tapered ends. They are Gram-positive, but easily decolourized, particularly in older cultures. Cells often contain metachromatic granules (polymetaphosphate), which stain bluish-purple with methylene blue. Snapping division produces groups of cells in angular and palisade arrangements that create a ‘Chinese character’ effect. C. diphtheriae is aerobic and facultatively anaerobic, growing best on a blood- or serum-containing medium at 35–37°C with or without carbon dioxide enrichment. On agar medium containing tellurite, colonies of C. diphtheriae are characteristically black or grey after 24–48 h.
Biotypes of C. diphtheriae named gravis, intermedius or mitis are genomically similar variants exhibiting distinct biochemical features and cultural morphology. Bacilli of the gravis biotype are usually short, whereas those of biotype mitis are long and pleomorphic; biotype intermedius ranges from very long to short rods. In broth medium, C. diphtheriae biotype gravis forms a pellicle and a granular deposit, whereas C. diphtheriae biotype mitis produces a diffuse turbidity. The biotype intermedius forms no pellicle, but a fine granular deposit can be observed.
Pathogenesis
To cause disease C. diphtheriae must:
•
: invade, colonize and proliferate in local tissues
•
: be lysogenized by a specific β-phage, enabling it to produce toxin.
In the upper respiratory tract, diphtheria bacilli elicit an inflammatory exudate and cause necrosis of the cells of the faucial mucosa (Fig. 17.1). The diphtheria toxin possibly assists colonization of the throat or skin by killing epithelial cells or neutrophils.
The organisms do not penetrate deeply into the mucosal tissue and bacteraemia does not usually occur. The exotoxin is produced locally and spread by the bloodstream to distant organs, with a special affinity for heart muscle, the peripheral nervous system and the adrenal glands.
C. diphtheriae can colonize the throats of people who have been immunized against diphtheria or who have become immune as a result of natural exposure, but usually no pseudomembrane develops.
The diphtheria toxin is a heat-stable polypeptide, composed of two fragments: A (active) and B (binding). The toxin binds to a specific receptor on susceptible cells and enters by receptor-mediated endocytosis. The A subunit is cleaved and released from the B subunit as it inserts and passes through the lysosomal membrane into the cytoplasm. Fragment A catalyses the transfer of adenosine disphosphate (ADP)-ribose from nicotinamide adenine dinucleotide (NAD) to the eukaryotic elongation factor 2, which inhibits the function of the latter in protein synthesis. Inhibition of protein synthesis is probably responsible for both the necrotic and neurotoxic effects of the toxin. Production of toxin by lysogenized C. diphtheriae is enhanced considerably when the bacteria are grown in low iron conditions. Other factors such as osmolarity, amino acid concentrations and pH have a role.
The Schick test, an intradermal injection of stabilized diphtheria toxin, was formerly used to determine individual susceptibility to the toxin. Absence of a reaction indicates immunity. Tissue culture neutralization tests, enzyme-linked immunosorbent assay (ELISA) and passive haemagglutination assay to measure serum antitoxin levels, are now preferred. For epidemiological purposes the minimum protective level is considered to be 0.01 international units (IU) of diphtheria antitoxin per millilitre in a serum sample. A level of 0.1 IU/mL is desirable for individual protection.
Non-toxigenic strains of C. diphtheriae may cause pharyngitis and cutaneous abscesses. Systemic disease, including endocarditis, septic arthritis and osteomyelitis, has also been reported. C. diphtheriae biotype belfanti could be involved in the processus of a chronic atrophic rhinitis named ozena. The virulence factors of these strains remain unknown. Conversion of a non-toxigenic strain to a toxigenic strain by phage infection can occur in human populations.
Clinical features
The incubation period of diphtheria is 2–5 days, with a range of 1–10 days. At first, patients present with malaise, sore throat and moderate fever. A thick, adherent green pseudomembrane is present on one or both tonsils or adjacent pharynx. In nasopharyngeal infection, the pseudomembrane may involve nasal mucosa, the pharyngeal wall and the soft palate. In this form, oedema involving the cervical lymph glands may occur in the anterior tissues of the neck, a condition known as bullneck diphtheria.
Laryngeal involvement leads to obstruction of the larynx and lower airways. Organisms multiply within the membranes and toxaemia is prominent. The patient is gravely ill, with a weak pulse, restlessness and confusion. Intoxication takes the form of myocarditis and peripheral neuritis, and may be associated with thrombocytopenia. Visual disturbance, difficulty in swallowing, and paralysis of the arms and legs also occur but usually resolve spontaneously. Complete heart block may result from myocarditis. Death is most commonly due to congestive heart failure and cardiac arrhythmias.
Cutaneous diphtheria occurs mostly in tropical countries. The lesion is usually characterized by an ulcer covered by a necrotic pseudomembrane and may involve any area of the skin. Although the organism usually produces toxin, systemic toxic manifestations are uncommon.
Diagnosis
The diagnosis is made on clinical grounds, supported by a history of diphtheria among contacts, lack of prior immunization or travel in countries where diphtheria is endemic.
The role of the laboratory is to confirm the diagnosis by recovery of C. diphtheriae in culture followed by appropriate tests for detection of toxin production (Fig. 17.2). The clinician should inform the laboratory of the presumptive diagnosis of diphtheria because isolation of C. diphtheriae requires special media. Material for cultures should be obtained on a swab from the inflamed areas surrounding the pseudomembranes.
Direct microscopy of a smear is unreliable because C. diphtheriae is morphologically similar to other coryneforms. The recommended media include blood agar and a selective medium containing tellurite and cysteine. Identification is based on carbohydrate fermentation reactions and enzymatic activities. Commercial kits such as the API Coryne strip provide a reliable identification. Matrix-assisted laser desorption/ionisation time-of-flight (MALDI-TOF) is also a reliable tool for rapid diagnosis of potentially toxigenic Corynebacterium species.
Toxigenicity testing is essential. Production of diphtheria toxin is demonstrated by the agar immunoprecipitation test (Elek test; Fig. 17.3) or by the tissue culture cytotoxicity assay, which has replaced the virulence test in guinea-pigs. The toxin gene can be detected by the polymerase chain reaction (PCR). This test shows excellent correlation with guinea-pig virulence, although there is the rare possibility of a false-positive PCR assay if the strain harbouring the tox gene is unable to express it. The detection of the tox gene by PCR directly from clinical specimens is feasible. All biotypes are potentially toxigenic. Multilocus sequence typing provides high-resolution data appropriate for the epidemiological investigation of diphtheria.
Measurement of antibodies to diphtheria toxin in serum collected before administration of antitoxin may support the diagnosis when cultures are negative. An algorithm for the management of suspect cases of diphtheria is shown in Figure 17.2.
Treatment
If diphtheria is strongly suspected on clinical grounds, treatment should not await laboratory confirmation, which may take several days (Fig. 17.2). Diphtheria antitoxin (hyperimmune horse serum) is given, as antibiotics have no effect on preformed toxin which rapidly diffuses from the local lesions and soon becomes irreversibly bound to tissue cells. Because antitoxin neutralizes only circulating toxin, it should be administered promptly.
Treatment with parenteral penicillin or oral erythromycin eradicates the organism and terminates toxin production. C. diphtheriae is universally sensitive to penicillins but some strains are resistant to erythromycin, tetracyclines and rifampicin. Erythromycin may be preferred to penicillin for elimination of the bacilli from the throat, particularly in treatment of persistent carriers. Some strains are tolerant to the bactericidal action of penicillins, and treatment of complicated infections should contain an association with an aminoglycoside.
Patients should be placed in strict isolation, nursed by staff whose immunization history is documented and have daily platelet counts and electrocardiography.
Epidemiology
Diphtheria has virtually disappeared in developed countries following mass immunization in the 1940s, but is still endemic in many regions of the world. About 50 000 cases of diphtheria occurred in the newly independent states of the former Soviet Union during 1990–1996, leading to infection in short-term visitors from western Europe. Other countries that have experienced outbreaks of diphtheria in recent years include China, Ecuador, Algeria, South-East Asia and the eastern Mediterranean. In the USA, only 45 cases were reported during 1980–1995. In 2002, one case of diphtheria was reported in the USA but more toxigenic strains were referred to North American reference laboratories. In 2003, a total of 896 cases were reported from the World Health Organization European Region; 99% were from Eastern Europe. There were 102 cases of infections caused by toxigenic corynebacteria diphtheria in the UK between 1986 and 2008: 42 C. diphtheriae, 59 C. ulcerans and one C. pseudotuberculosis. Five fatalities were reported, all in unvaccinated patients. Non-toxigenic strains capable of causing mild disease continue to circulate throughout the world. In European countries, carriages rates of non-toxigenic strains ranged from 0 in Ireland to 4.0 per 1000 in Turkey.
Infection is confined to man and usually involves contact with a diphtheria case or a carrier. The most important mode of spread is person-to-person transmission by aerosolized droplets when an infected person coughs, sneezes or talks, or by direct contact with skin lesions. Most clinical infections are probably contracted from carriers rather than symptomatic patients. Prolonged close contact with an infected person and intimate contact increases the likelihood of transmission.
Acquired immunity to diphtheria is due primarily to toxin-neutralizing antibody (antitoxin). Passive immunity in utero is acquired transplacentally and can last for 1 or 2 years after birth. Active immunity can probably be produced by a mild or subclinical infection in infants who retain some maternal immunity. Unimmunized children under 15 years old are most likely to contract diphtheria. The disease is also found among adults whose immunization was neglected. The mortality rate is highest among young children and in people aged over 40 years. Skin infections caused by C. diphtheriae may result in early development of natural immunity against the disease.
C. diphtheriae persists longer in skin lesions than in the tonsils or nose, and cutaneous diphtheria appears to be more contagious than respiratory diphtheria. Untreated people who are infected with the diphtheria bacillus can be contagious for up to 2 weeks, but seldom for more than 4 weeks. If treated with appropriate antibiotics, the contagious period can be limited to less than 4 days. C. diphtheriae can survive in the environment in dust and on dry vomits for several months, and transmission via vomits has been documented. Animal-to-man transmission and food-borne transmission by consumption of contaminated foods such as raw milk have been described, but are very rare.
Control
High population immunity achieved through mass immunization (at least 95% coverage in children and at least 90% coverage in adults) is the most effective measure to control epidemic diphtheria. Immunization with diphtheria toxoid was first introduced in 1923. Large-scale immunization programmes introduced in the 1940s reduced the incidence of diphtheria dramatically, although the disease was not eradicated completely. Immunization schedules are discussed in Chapter 70.
Prevention of secondary cases by the rapid investigation of close contacts is essential. These investigations should include ascertainment of the immunization histories of all home and school contacts. Primary courses of immunization or a booster are given if necessary.
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Book2012, Medical Microbiology (Eighteenth Edition)J. McLauchlin, P. Riegel
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Prevention of Fetal and Early Life Infections Through Maternal–Neonatal Immunization
2011, Infectious Diseases of the Fetus and Newborn (Seventh Edition)James E. CroweJr.
Diphtheria
Corynebacterium diphtheriae is an aerobic gram-positive bacterium which secretes a toxin that inactivates human elongation factor eEF-2, thus inhibiting translation during protein synthesis by human cells. The site of infection, generally the throat, becomes sore and swollen. The toxin can cause damage to the myelin sheaths in the central and peripheral nervous system leading to loss of motor control or sensation. Immunization with diphtheria toxoid has been in widespread use since the 1930s; the vaccine is one of the safest in use. The toxoid can be manufactured from diphtheria toxin treated with formalin to inactivate the toxicity but maintain immunogenicity, and is administered as part of the DPT vaccine beginning at about 2 months. Pertussis toxin (PT) and diphtheria toxin (DT) also have been detoxified genetically by introduction of point mutations that cause a loss of enzymatic activity but retention of binding activity. One mutant DT protein that is a toxoid with a single amino acid mutation at the enzymatic active site, designated CRM197, is the protein carrier for a licensed H. influenzae type B vaccine.
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Book2011, Infectious Diseases of the Fetus and Newborn (Seventh Edition)James E. CroweJr.
Chapter
Other Coryneform Bacteria and Rhodococci
2015, Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases (Eighth Edition)Rose Kim, Annette C. Reboli
Coryneform Bacteria Other Than Corynebacterium Diphtheriae
Corynebacterium was proposed as a genus by Lehmann and Neumann in 1896, having derived the name from the Greek koryne, which means “club,” and bacterion, meaning “little rod.”1 The coryneforms are a diverse group of organisms. Corynebacterium diphtheriae serves as the type species, leading to the term diphtheroids to describe other bacteria sharing similar morphology. Also known as coryneform bacteria, bacteria demonstrating morphology similar to that of corynebacteria include the genera Corynebacterium, Arcanobacterium, Brevibacterium, Dermabacter, Microbacterium, Rothia, Turicella, Arthrobacter, Oerskovia, Leifsonia, Helcobacillus, Exiguobacterium, Cellulomonas, Cellulosimicrobium, and Curtobacterium.2,3 The 16S ribosomal RNA (rRNA) sequencing data show that the genera Corynebacterium and Turicella are more related to the partially acid-fast bacteria and to the genus Mycobacterium than to the other coryneforms discussed in this chapter.3
Coryneform bacteria are widely distributed in the environment as normal inhabitants of soil and water. They are commensals colonizing the skin and mucous membranes of humans and other animals.4,5 In the hospital setting, coryneform bacteria may be cultured from the hospital environment, including surfaces and medical equipment.6 Coryneform bacteria other than C. diphtheriae have been isolated frequently in clinical specimens and were commonly considered contaminants without clinical significance. There is an increasing body of evidence of the pathogenicity of the coryneform bacteria, particularly as a cause of nosocomial infection in hospitalized and immunocompromised patients.7,8 Several of the members of the genus Corynebacterium are better known as pathogens in animals and only incidentally cause infection in humans as zoonoses.
The coryneform bacteria are pleomorphic, demonstrating different forms at various stages of the life cycle, irregularly shaped gram-positive rods that are aerobically cultured, not spore forming, and not partially acid-fast.2,3 A history of misidentification of coryneform bacteria has made interpretation of the medical literature difficult. Initial identification is aided by observation of colony size and appearance, and the presence or absence of hemolysis on sheep blood agar. Odor production by colonies assists in identification, particularly of Brevibacterium casei and Corynebacterium urealyticum. Several of the medically relevant coryneform bacteria are lipophilic, demonstrating enhanced growth with the addition of Tween 80 to the culture medium.
True corynebacteria demonstrate club-shaped gram-positive rods on Gram staining, whereas other coryneform bacteria may not appear distinctly club shaped. Cells demonstrate variable sizes and appearance, from coccoid to bacillary forms, depending on the stage of the life cycle, and Gram-stain results may be uneven. Coryneform bacteria typically form arrangements such as “Chinese letters” or picket-fence configurations as a result of “snapping” after the cells divide. Lack of spore formation helps distinguish them from Bacillus species.2
The spectrum of human infections attributed to the coryneform bacteria is broad but can be understood in two general categories: community-acquired infections and nosocomial infections. Community-acquired infections include pharyngitis, skin and soft tissue infections, native valve endocarditis, genitourinary tract infections, acute and chronic prostatitis, and periodontal infections (Table 207-1).9,10 Many case series of nosocomial infections attributed to coryneform bacteria are in the medical literature and include intravascular catheter-associated septicemia, native and prosthetic valve endocarditis, device-related infections, peritonitis in peritoneal dialysis patients, and postoperative surgical site infections.11,12 Common nosocomial pathogens include Corynebacterium jeikeium, C. urealyticum, Corynebacterium amycolatum, and Corynebacterium striatum (Table 207-2).13 Nosocomial infections with the coryneform bacteria will continue to increase, reflecting the increased numbers of severely ill patients with extended stays in intensive care units and multiple antibiotic exposures.
Taxonomy
The taxonomy of the coryneform bacteria has evolved extensively over the past 30 years and continues to be refined. Hollis and Weaver,14 at the Special Bacteriology Laboratory, Centers for Disease Control and Prevention (CDC) in Atlanta, completed the first extensive compilation of coryneform bacteria isolated from clinical specimens. Coryneform bacteria were grouped based on colony and biochemical characteristics. Since then, further work has been done to analyze these groups and define species. Table 207-3 lists the significant coryneform bacteria and the CDC group to which they previously belonged.
To date, there are more than 80 species of Corynebacterium that have been identified; more than 50 species have been associated with disease in humans.15,16 The use of molecular genetics has resulted in continued revision of the taxonomy of the coryneform bacteria and provides useful information on the epidemiology and pathogenicity of the genera. Molecular genetic studies, such as 16S rRNA and rpoB gene sequencing, are used in reference laboratories to confirm identification at the species level; 16S rRNA gene sequencing has become the standard by which new species are identified.2,3,16-18 Matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF) is another molecular test that is being used for identification of Corynebacterium spp. MALDI-TOF uses a mass spectrometer to analyze proteins that are extracted from the bacteria and compared to a database.19
Microbiology
Because the coryneforms are frequently cultured in polymicrobial infections and may be contaminants in cultures collected with poor sterile technique, clinician communication with the microbiology laboratory is essential to determine when species identification is appropriate. The decision to identify the coryneform bacteria to the species level is recommended when the bacteria are cultured from normally sterile sites, such as blood (two or more positive blood cultures, except when recovered from the same set) or cerebrospinal fluid (CSF), if the bacteria appear in adequately collected clinical material as the predominant organism on Gram staining and have a strong inflammatory reaction, and are from urine specimens where the bacterium (e.g. C. urealyticum) is the only organism recovered with a colony count greater than 104/mL or if it is the predominant bacterium cultured and the total bacterial count is greater than 105/mL.3,16
Media used for initial specimen processing are standard blood agar plates for most specimens, thioglycollate broth for wound cultures, and standard blood culturing systems using continuous monitoring for carbon dioxide (CO2) production. Special media used for species identification include sheep blood agar supplemented with Tween 80, to assess lipid-enhanced growth.3
Identification to the species level in the microbiology laboratory is confirmed by biochemical testing. Initial testing includes the catalase test with 3% hydrogen peroxide. Additional tests include nitrate reduction; urea hydrolysis; esculin hydrolysis; and acid production from glucose, maltose, sucrose, mannitol, and xylose. A frequently used system of biochemical testing for medically relevant coryneform bacteria is the API Coryne system (API-bioMérieux, La Balme les Grottes, France), which includes 20 biochemical tests and will identify many of the important corynebacteria and other coryneform bacteria, including Arcanobacterium spp. and Brevibacterium spp., as well as Rhodococcus equi.20 An evaluation of the API Coryne database 2.0 gave correct identification for 90.5% of the coryneforms tested.21 Another identification system, the RapID CB Plus (Remel, Lenexa, KS), correctly identifies 80.9% of strains to the species level and an additional 12.2% to the genus level. It has the advantage of requiring only 4 hours to perform, compared with 24 hours for the API Coryne system.22 In a few cases, the Christie-Atkins-Munch-Petersen (CAMP) test helps to identify the organism to the species level.2
Susceptibility testing has historically been problematic in the coryneforms.23 The Clinical and Laboratory Standards Institute released standards for susceptibility testing of coryneform bacteria in 2010.24 Isolates uniformly show susceptibility to vancomycin, teicoplanin, and linezolid. Susceptibility to daptomycin and tigecycline has also been demonstrated.25,26 Species of Corynebacterium are capable of expressing the ermX methylase gene, which is linked to the resistance phenotype macrolide-lincosamide–streptogramin B (MLSB); this phenotype confers resistance to erythromycin and clindamycin and is associated with cross-resistance to other antimicrobial agents.27 The vanA gene has been identified in Oerskovia turbata and Arcanobacterium haemolyticum, but no documented infections with vancomycin-resistant coryneforms have appeared in the literature.28 When a clinically important isolate is obtained, susceptibility testing is recommended to ensure antimicrobial activity.
For consistency, the coryneform bacteria are reviewed here within groups identified by the presence or absence of lipid-enhanced culture (lipophilic or nonlipophilic) and fermentation activity.
Nonlipophilic, Fermentative Corynebacteria
Corynebacteria have been divided into lipophilic and nonlipophilic, fermentative and nonfermentative. Lipophilic species have enhanced growth in the presence of certain lipids, such as Tween 80. Fermentative strains produce acid from certain sugars. Advances made in the identification of species in the nonlipophilic fermentative group have resulted in a revision of thinking of the pathogenic role of several species, particularly for Corynebacterium xerosis and C. amycolatum.29 Interpretation of the literature that does not include detailed information on laboratory identification is difficult because of historical misidentification of species in the nonlipophilic fermentative group.
Corynebacterium ulcerans and Corynebacterium pseudotuberculosis
C. ulcerans and C. pseudotuberculosis are members of the C. diphtheriae group and are known primarily as animal pathogens, although disease in humans has been reported as zoonotic infections. Both C. ulcerans and C. pseudotuberculosis may elaborate diphtheria toxin.
C. ulcerans is known primarily as a cause of bovine mastitis but has the potential to elaborate diphtheria toxin and cause an exudative pharyngitis in humans indistinguishable from C. diphtheriae.30,31 In the United Kingdom, C. ulcerans exceeded that of C. diphtheriae as the causative agent in diphtheria infection; the European-based Diphtheria Surveillance Network reported an increase in diphtheria cases attributable to C. ulcerans during a 9-year surveillance period.32,33 In the United States, there has been one reported case of C. ulcerans infection since 2005.34 C. ulcerans has been implicated in human infection because of contact with domesticated animals.35 This has made the identification of the causative organism important for epidemiology, and guidelines for laboratory diagnosis of diphtheria cases have been published.36 The spectrum of illness with C. ulcerans is similar to C. diphtheriae.37 Fatalities have been reported, including sudden death from toxin-induced cardiac injury and a case of fatal necrotizing sinusitis.38 Skin infection by C. ulcerans mimics that of C. diphtheriae.39 Infection of the lower respiratory tract may occur, causing pneumonia and pulmonary nodules.40,41 Treatment of pharyngitis caused by C. ulcerans is similar to treatment of diphtheria, including the use of antibiotics such as erythromycin and diphtheria antitoxin when appropriate.
C. pseudotuberculosis is a significant pathogen in animals, particularly sheep, in which it causes caseous lymphadenitis. Human disease is rare, manifesting as granulomatous lymphadenitis, found mainly in farm workers and veterinarians who have had exposure to infected animals.42 It has been reported to cause a diphtheria-like illness and eosinophilic pneumonia and has also been isolated from soft tissue abscesses in a young butcher.32,43,44 Management of C. pseudotuberculosis infection includes excision of affected lymph nodes and treatment with β-lactam antibiotics, macrolides, or tetracyclines.
Corynebacterium xerosis
C. xerosis is a colonizer of the human nasopharynx, conjunctiva, and skin.45 Historically, C. xerosis has been described in the literature as a pathogen causing serious human disease, especially in immunocompromised hosts, including sepsis, endocarditis, pneumonia, peritonitis, ventriculoperitoneal shunt infection, and postoperative sternal wound infection. Subsequent investigations have questioned the reliability of C. xerosis identification in the microbiology laboratory.46,47 In one study, all isolates originally identified as C. xerosis were in actuality C. amycolatum.47 This calls into question preceding case reports attributing disease to C. xerosis because true C. xerosis isolates apparently are quite rare. C. xerosis infections in humans have included blepharitis, a brain abscess, and a case of sepsis in a pediatric patient with sickle cell disease.48-50 True C. xerosis strains are susceptible to most antibiotics, which helps to distinguish them from C. amycolatum, which demonstrates multiple antibiotic resistances.
Corynebacterium striatum
C. striatum has been one of the more commonly isolated coryneform bacteria in the clinical microbiology laboratory.2,3 As with other nonlipophilic fermentative corynebacteria, a high degree of misidentification of C. striatum has occurred in the past in microbiology laboratories, and investigators have found many isolates to be C. amycolatum on detailed retesting.46,51
C. striatum is ubiquitous and colonizes the skin and mucous membranes of normal hosts and hospitalized patients.52,53 Although it is isolated frequently in polymicrobial infections, its degree of pathogenicity has been unclear, and differentiation of colonization from pathogen-causing infection has been difficult.7,8 In a large series of 150 clinical specimens from which coryneform bacteria had been isolated, C. striatum was identified in 11 isolates, only one of which was considered to be related to an infectious process.7 There is evidence for patient-to-patient transmission of C. striatum in hospital settings, which may account for the frequency with which it is isolated in hospitalized patients.54,55
Reports of true infection confirmed by isolation of C. striatum from a sterile site are relatively rare and have been reported mainly for patients with indwelling devices or immunosuppression. Recent case reports in the literature include native and prosthetic valve endocarditis, pacemaker-related endocarditis, meningitis, pulmonary abscess, septic arthritis, and vertebral osteomyelitis.56 A nosocomial outbreak of C. striatum has been reported in patients with chronic obstructive pulmonary disease.57 Nosocomial endocarditis has been reported in a patient with vascular access for dialysis that was successfully treated with vancomycin and rifampicin.58
C. striatum may be resistant to penicillin but is susceptible to other β-lactams and vancomycin. Resistance has been demonstrated to ciprofloxacin, erythromycin, rifampin, and tetracyclines; there is variable susceptibility to aminoglycosides.59 Resistance to daptomycin has been reported in the setting of prior daptomycin therapy.60
Corynebacterium minutissimum
Defined in 1983 by Collins, C. minutissimum is a colonizer of human skin, particularly moist intertriginous areas.61,62 As with other members of this group, C. amycolatum has been misidentified as C. minutissimum in the past.63 Although C. minutissimum historically has been considered the causative agent in erythrasma, that association has been questioned because cultures tend to show polymicrobial infection.2 Erythrasma is a superficial skin infection occurring in intertriginous areas between skin folds, axillae, groin, and fingers and toes.10 It presents as reddened scaling patches that may be accompanied by pruritus. Skin patches glow coral-red under a Wood's lamp. Diagnosis is made by clinical appearance and symptoms and by culture of skin scrapings. Colonies also appear coral-red under ultraviolet light. Treatment includes topical and oral antibiotics. Recurrences are frequent.
Other rare infections attributed to C. minutissimum include septicemia and endocarditis in immunocompromised patients and patients with indwelling central venous catheters, peritonitis in patients undergoing continuous ambulatory peritoneal dialysis (CAPD), and pyelonephritis.64 A case of bacteremia and meningitis has been reported.65
It has been reported to cause cutaneous granulomas and costochondral abscess in patients with acquired immunodeficiency virus (AIDS) and been implicated as a cause of recurrent breast abscesses. Supporting evidence for the microbiologic diagnosis in several case reports is slim, and these infections may actually have been caused by other members of the nonlipophilic fermentative group.
Corynebacterium amycolatum
Defined as a new species in 1988 by Collins, C. amycolatum was first isolated from the skin of healthy humans.66 Noted for its lack of mycolic acids, the species corresponds to the CDC coryneform groups F-2 and I-2. It is the nonlipophilic coryneform bacteria most frequently isolated from clinical specimens.7,8 C. amycolatum forms small dry nonhemolytic colonies of 1- to 2.0-mm in diameter when cultured at 37° C.3 The organisms are pleomorphic and vary from single organisms to an array of Chinese letters. Because of variability in biochemical reactions, C. amycolatum had been misidentified previously as C. minutissimum, C. xerosis, and C. striatum. Currently, the API Coryne system can correctly identify C. amycolatum, but confirmatory tests should be performed.3
Although case reports of infections attributed to C. amycolatum are rare, many previously reported infections by other members of the nonlipophilic fermentative group were most likely caused by C. amycolatum. Reports with reliable information on organism identification include nosocomial endocarditis after intravenous catheter–related infection, septic arthritis, a case of native valve endocarditis with aorta-to–left atrial fistula, and sepsis in pediatric oncology patients.67-70 Susceptibility testing has shown resistance to penicillins, cephalosporins, macrolides, fluoroquinolones, and rifampin and susceptibility to vancomycin, daptomycin, linezolid, and teicoplanin.71 There is variable resistance to aminoglycosides and tetracyclines.23,26 Reports of successful treatment of endovascular infection include the use of vancomycin and daptomycin in combination with rifampin.58,72
Corynebacterium glucuronolyticum
C. glucuronolyticum was defined in 1995, and since 2000, the species has included those isolates previously identified as Corynebacterium seminale that had been defined by Riegel and co-workers73,74 in 1996. Although it has been isolated from the genitourinary tract of animals, in humans, it may be included in the normal flora of the genitourinary tract. It is commonly isolated from males with genitourinary tract infections and is associated with chronic prostatitis.75 C. glucuronolyticum strains are susceptible to β-lactam antibiotics, gentamicin, rifampin, and vancomycin but demonstrate variable resistance to fluoroquinolones, macrolides, and tetracyclines.23
Other Nonlipophilic, Fermentative Corynebacteria
Corynebacterium argentoratense has been isolated from the throats of healthy volunteers and from mucosal biofilms on adenoid tissue from children with chronic or recurrent otitis media. The clinical significance of this finding is unclear.4,76,77 Corynebacterium matruchotii is identified by its characteristic “whip handle” appearance on Gram staining.2,3 It was previously identified as Bacterionema matruchotii until 1983, when it was reclassified as a Corynebacterium species by Collins. Mainly an inhabitant of the oral cavity of humans and animals, C. matruchotii has been rarely associated with human disease.
In 1998, Funke and colleagues78 identified a new species of Corynebacterium isolated from female patients with symptomatic urinary tract infections. Given the name Corynebacterium riegelii, it is nonlipophilic, weakly fermentative, and facultatively anaerobic. Similar to the lipophilic C. urealyticum, it demonstrates strong urease activity. It is susceptible to penicillins, cephalosporins, gentamicin, fluoroquinolones, rifampin, and tetracyclines.
Corynebacterium confusum was defined in 1998 by Funke and colleagues29; it is nonlipophilic and very slowly fermentative.79 C. confusum has been isolated from a blood culture, foot infections, and a breast abscess.79 Additional nonlipophilic fermentative Corynebacterium spp. identified from human clinical specimens include C. simulans, C. sundsvallense, C. thomssenii, C. freneyi, C. aurimucosum, C. tuscaniae, C. coyleae, C. canis, C. falsenii, C. freiburgense, C. massiliense, C. pilbarense, C. stationis, and C. timonense.80-91
Nonlipophilic, Nonfermentative Corynebacteria
The nonlipophilic, nonfermentative corynebacteria do not produce acid from any sugars and were designated as absolute nonfermenters (ANF) by Hollis and Weaver.14 They are colonizers of the human respiratory tract and ear canal and infrequent pathogens.
Corynebacterium afermentans subsp. afermentans
C. afermentans subsp. afermentans was included in the CDC coryneform group ANF-1 until 1993, when Riegel and co-workers92 defined the species as C. afermentans with two subspecies: C. afermentans subsp. afermentans and C. afermentans subsp. lipophilum. C. afermentans subsp. afermentans is a rare human pathogen but has been reported to cause septicemia in immunocompromised patients.93
Corynebacterium auris
As in the case of Turicella otitidis, C. auris was initially isolated from middle ear fluid of pediatric patients with otitis media and was presumed to be among the pathogens causing otitis media. Subsequent studies have cultured C. auris from the external ear canal and cerumen of healthy subjects, both children and adults, and its role as a pathogen has been discounted.5,94 C. auris is resistant to penicillins, clindamycin, and erythromycin and susceptible to fluoroquinolones, gentamicin, rifampin, tetracyclines, and vancomycin.23
Corynebacterium pseudodiphtheriticum
C. pseudodiphtheriticum is included in the normal bacterial flora of the human upper respiratory tract. Lehmann and Neumann1 described the organism in 1896, giving it the name Bacillus pseudodiphtheriticum. Since 1925, it has been known as C. pseudodiphtheriticum. Historically, C. pseudodiphtheriticum was associated with endocarditis of native and prosthetic valves.95 The first cases of infections at other sites attributable to C. pseudodiphtheriticum became known in 1982, and since then, C. pseudodiphtheriticum has been associated primarily with respiratory infections, particularly in immunocompromised hosts.96,97 It has been isolated from patients with pneumonia and advanced AIDS, as well as children with cystic fibrosis and respiratory infections.98,99 Other sites of infections have been the eye, intervertebral disks, joints, lymph nodes, urine, peritoneal fluid, intravenous catheters, and surgical wounds.100 Although C. pseudodiphtheriticum does not elaborate toxins, it has been isolated from three patients with exudative pharyngitis with pseudomembrane formation, not unlike C. diphtheriae.101 Isolates of C. pseudodiphtheriticum have demonstrated resistance to macrolides and lincosamides but have maintained susceptibility to penicillins, cephalosporins, doxycycline, and glycopeptides. One large case series of 113 C. pseudodiphtheriticum strains from a single institution showed moderate levels of resistance to β-lactams, imipenem, tetracycline, erythromycin, ciprofloxacin, aminoglycosides, and clindamycin; all strains were susceptible to vancomycin and teicoplanin.100
Corynebacterium propinquum
Before 1994, C. propinquum was known as CDC coryneform group ANF-3.102 Primarily isolated from the human respiratory tract, its role as a pathogen is yet to be defined. Isolation of C. propinquum from blood specimens has been reported, but the clinical information necessary to interpret the findings, which include one case of endocarditis, is lacking. C. propinquum has also been isolated from a pulmonary pleural effusion and an infected orthopedic device.
Lipophilic Corynebacteria
Lipophilic corynebacteria are fastidious, slow-growing bacteria that form tiny nonhemolytic colonies on standard media but demonstrate enhanced growth with the addition of lipids to the culture medium.2 The group includes the significant human pathogens Corynebacterium jeikeium and C. urealyticum.
Corynebacterium jeikeium
C. jeikeium was initially described in 1976 as a highly resistant coryneform bacteria causing severe sepsis in patients with hematologic malignancies and profound neutropenia and in one patient with a ventricular CSF shunt.103 In 1979, it was designated as CDC group JK, and in 1988, the designation was revised to C. jeikeium.104 C. jeikeium colonizes the skin of hospitalized patients, especially those treated with multiple antibiotics, and can also be isolated from the hospital environment.105 There is some evidence that patient-to-patient transmission occurs in the hospital.6,106 It is the most frequently isolated Corynebacterium in the acute care setting and is the most important pathogen of the lipophilic corynebacteria.7
Microbiology
C. jeikeium is a pleomorphic gram-positive rod that varies in form from coccobacillary to bacillary; some appear club shaped. It is nonhemolytic on standard media and forms small gray-white colonies on routine culture.2 It is lipophilic, forming large colonies on sheep blood agar supplemented with Tween 80. C. jeikeium does not produce urease or reduce nitrate, and it ferments glucose. It has variable fermentation of galactose and maltose.107
Pathogenicity
C. jeikeium is a cause of severe infections in the hospitalized patient.6 Predisposing factors for infection include immunocompromised states, such as malignancy, neutropenia, and AIDS.6,108 Other risk factors include the presence of indwelling medical devices, such as central venous catheters, peritoneal dialysis catheters, prosthetic valves, and CSF shunts. Prolonged hospital stay, treatment with broad-spectrum antibiotics, and impaired skin integrity are well-described risk factors for development of infection with C. jeikeium.
Infectious processes include septicemia from infected intravascular devices, native and prosthetic valve endocarditis, CSF shunt infections, meningitis and transverse myelitis, and prosthetic joint infections.109,110,111 It has been reported to cause postsurgical infections, peritonitis in patients undergoing CAPD, liver abscess, otitis media, and osteomyelitis of the foot. Skin findings with C. jeikeium infection are common: half of neutropenic patients with C. jeikeium septicemia have reported skin findings, including rash and subcutaneous nodules.112 Palpable purpura has been reported in patients with C. jeikeium endocarditis.113
Treatment
C. jeikeium is resistant to many antibiotics, including penicillins, cephalosporins, and aminoglycosides; there is inducible resistance to macrolides.114,115 It remains susceptible to vancomycin, which is the recommended treatment. Although catheter removal has been routinely recommended in the setting of intravascular catheter–related infection, experience has shown a high success rate in catheter salvage with appropriate antimicrobial therapy.116 Successful treatment with daptomycin and tigecycline have been reported; one case of a daptomycin-resistant strain of C. jeikeium in a previously treated neutropenic patient has been reported.117-119
Corynebacterium urealyticum
First described in 1974, this bacterium was designated as CDC group D2 until 1992, when the name C. urealyticum was proposed.120 C. urealyticum colonizes the skin of 25% to 37% of hospitalized patients. Because of its ability to adhere to uroepithelial cells, it is most commonly associated with urinary tract infections, especially in cases of abnormal anatomy, and has been implicated as the cause of encrusted cystitis and encrusted pyelitis.121,122
Microbiology
Colonies of C. urealyticum are slow growing and lipophilic and appear nonhemolytic and pinpoint when cultured on sheep blood agar under CO2 enrichment for 48 hours.2 It is a strict aerobe, with no growth under anaerobic conditions. On Gram staining, organisms are palisading, non–spore-forming coccobacilli. They are catalase positive and oxidase negative, with a rapid production of urease. Laboratories should be made aware of the need for further investigation of diphtheroid bacilli from urinary tract specimens in the proper clinical setting because C. urealyticum may not grow in standard urine culture.2
Pathogenicity
C. urealyticum is primarily a cause of chronic and recurrent urinary tract infections, occurring mainly in elderly people and those with debilitation or immunosuppression. Additional risk factors include prolonged hospitalization, the use of percutaneous and bladder drainage catheters, and urinary tract procedures.123 It has been reported to cause infections in renal transplant recipients.121,124 Clues to diagnosis of C. urealyticum infection include sterile pyuria, alkaline urine, and the presence of white blood cells and struvite crystals.125 C. urealyticum causes encrusted cystitis, which appears as chronic inflammation of the bladder mucosa with crystal deposits on the bladder mucosa, surrounded by erythema. Encrusted pyelitis may occur if there are abnormalities of the upper urinary tract. In rare cases, C. urealyticum has been reported as a causative agent in peritonitis, endocarditis, pneumonia, septicemia, osteomyelitis, soft tissue infections, and superinfection of wounds.126,127
Treatment
In general, C. urealyticum is resistant to β-lactams, aminoglycosides, and trimethoprim-sulfamethoxazole. There is variable susceptibility to fluoroquinolones, macrolides, and tetracycline.7,126 The treatment of choice is vancomycin, to which it remains susceptible. For urinary tract infections, in addition to vancomycin, endoscopic removal of bladder mucosa encrustations or acidification of urine by instilling acid into the bladder in cases of encrusted cystitis may be required, and urologic consultation is recommended. Percutaneous nephrostomy tube placement and irrigation of upper urinary tract with Thomas' acid solution, in cases of upper tract disease, has been described.128
Other Lipophilic Corynebacteria
C. afermentans subsp. lipophilum is a rarely reported human pathogen.92 It has been reported to cause intravascular catheter–related septicemia, prosthetic valve endocarditis, lung abscess, empyema, and brain abscess. Corynebacterium accolens was previously known as CDC coryneform group 6. There were discrepancies in the definition until 1991, when it was defined further by Neubauer and associates129 and given the name Corynebacterium accolens. Known to colonize the human upper respiratory tract, C. accolens is a rarely reported human pathogen but has been reported to cause septicemia, endocarditis, breast abscess, and pelvic osteomyelitis.130-132 Corynebacterium macginleyi (formally CDC coryneform group G-1) was initially isolated solely from the human eye as a cause of conjunctivitis.133 Case reports of C. macginleyi infection include intravascular catheter–associated blood stream infection, urinary tract infection associated with a bladder drainage catheter, and septicemia in immunocompromised patients.134 Other lipophilic corynebacteria, including Corynebacterium tuberculostearicum (formally CDC coryneform group G-2) and Corynebacterium kroppenstedtii, have been cultured from inflammatory breast tissue in cases of granulomatous mastitis; C. tuberculostearicum has also been associated with postsurgical deep wound infections and osteomyelitis.135-137 Corynebacterium bovis is a cause of bovine mastitis but in humans has been described as a cause of endocarditis, chronic otitis media, central nervous system (CNS) infection, line-related septicemia, and joint infection.138-140 CDC coryneform group F-1 may be a cause of urinary tract infection; it is similar to C. urealyticum in its very rapid urease reaction and differs from the latter in its very high susceptibility on antimicrobial testing.141 Corynebacterium lipophiloflavum has been isolated from a patient with bacterial vaginosis. Corynebacterium resistens is a multidrug-resistant coryneform bacteria isolated from blood, bronchial aspirate, and abscess specimens.142 Corynebacterium ureicelerivorans has been implicated in bacteremia and peritonitis.143,144 New species of lipophilic coryneform bacteria found in human specimens continue to be defined; these include Corynebacterium aquatimens, Corynebacterium sputi, and Corynebacterium pyruviciproducens.
Arcanobacteria
Collins defined the genus Arcanobacterium in 1982, from “arcane,” meaning “mysterious or secret” and “bacterium.”145 For many years, Arcanobacterium haemolyticum was the only species in this genus. However, in 1997, further investigation of several Actinomyces spp. resulted in the reclassification of Actinomyces pyogenes and Actinomyces bernardiae as Arcanobacterium spp. and defined two additional new species of arcanobacteria.146
Arcanobacterium haemolyticum
A. haemolyticum was first isolated by MacLean and co-workers147 in 1946 from American soldiers and Pacific Islanders with pharyngeal and skin infections in the South Pacific. The initial classification as Corynebacterium haemolyticum endured until 1982, when the genus Arcanobacterium was defined by Collins.
Microbiology
A. haemolyticum is a catalase-negative, gram-positive to gram-variable rod that does not form spores and is nonmotile.2 It is β-hemolytic, but expression can vary by culture media and conditions, with hemolysis best observed on human blood agar.148 Growth is enhanced in the presence of CO2. It is known for forming dark pits under the colonies. Poor growth on tellurite helps to differentiate it from C. diphtheriae.
Colony morphology has been described as either rough or smooth type.149 Rough-type colonies are most frequently associated with respiratory isolates; smooth biotypes are most frequently associated with wound isolates. A. haemolyticum does not ferment xylose, which differentiates it from Arcanobacterium (Actinomyces) pyogenes. A positive α-mannosidase test identifies A. haemolyticum and differentiates it from A. pyogenes and other coryneform-like bacteria, including Rhodococcus equi and Erysipelothrix rhusiopathiae. Because of the presence of phospholipase D activity similar to C. ulcerans and C. pseudotuberculosis, the reverse CAMP test will be positive, with inhibition of the hemolytic zone of a β-lysin–producing strain of Staphylococcus aureus.3 Other secreted toxins include neuraminidase and a hemolysin.
Infections in Humans
A. haemolyticum is a well-recognized cause of pharyngitis in humans, with a spectrum of illness from mild to diphtheria-like.150-152 It accounts for about 0.5% of pharyngeal infections overall and 2.0% in individuals in the 15- to 25-year-old age range. In studies, A. haemolyticum has not been isolated from healthy control populations but has been isolated from 2.5% of a symptomatic young adult population.152-154 It is indistinguishable from streptococcal pharyngitis in clinical appearance, and about 50% of cases of pharyngitis are exudative. Cervical adenopathy is usually present. A. haemolyticum pharyngitis is accompanied with an exanthem in about 50% of cases. The rash generally appears after the onset of the pharyngitis and has a variable appearance, often described as an erythematous morbilliform or scarlatiniform rash, appearing on the trunk, neck, and extremities (Fig. 207-1). It may also present as an erythematous urticarial rash with an appearance similar to that of erythema multiforme. Complications of A. haemolyticum pharyngitis include peritonsillar and pharyngeal abscesses, with A. hemolyticum the sole pathogen in 50% of cases in adolescents and young adults, and the remaining 50% coinfected with β-hemolytic streptococci.153
A. haemolyticum has been isolated from soft tissue infections, including chronic ulcers, wound infections, cellulitis, and paronychia. It is frequently a component of polymicrobial infection in this setting but has been isolated as the sole pathogen as well.155 Underlying conditions in polymicrobial chronic ulcers include diabetes and peripheral vascular disease. Post-traumatic wound infections have been reported, as has coinfection or superinfection with leprosy ulcers.156
Lemierre disease with Fusobacterium necrophorum and A. haemolyticum has been reported, accompanied by a skin rash typical for A. haemolyticum infection; Lemierre syndrome and septicemia caused by A. haemolyticum has also been reported.157,158 Sepsis syndrome from A. haemolyticum has been described, occurring in all age groups and without predisposing factors.159 Other infections reported include sinusitis, orbital cellulitis, brain abscess, endocarditis, cavitary pneumonia, and vertebral osteomyelitis. A. haemolyticum may be present in subperiosteal abscesses in periodontal disease.156,160
Treatment
Susceptibility information for A. haemolyticum has been reviewed extensively.161 Although in vitro studies show most strains to be penicillin susceptible, treatment failures may occur because of tolerance and poor penetration into the intracellular space. Other β-lactams have shown in vitro activity as well. Susceptibility data showed low minimal inhibitory concentrations to erythromycin and azithromycin.161 Clindamycin and doxycycline are also efficacious, as are ciprofloxacin and vancomycin. Resistance to trimethoprim-sulfamethoxazole and tetracycline is well documented.162 Three vancomycin-resistant strains of A. haemolyticum expressing the vanA gene were recovered in a surveillance study, but no vancomycin-resistant infections have been reported. Surgical management of wound infections and drainage of soft tissue abscesses are recommended.
Arcanobacterium (Actinomyces) pyogenes
Initially described by Glage in 1903, this organism was initially named Bacillus pyogenes. It was known as Corynebacterium pyogenes until 1982, when it was reassigned to the genus Actinomyces. Since 1997, it has been known as Arcanobacterium pyogenes.146 A. pyogenes is primarily an animal pathogen causing pyogenic infections in cattle, including pneumonia, endometritis, endocarditis, wound infections, and mastitis. Abscess formation is aided by neuraminidases, which facilitate adhesion to host epithelial cells.163 Transmission of A. pyogenes by flies has been proposed. A. pyogenes has not been isolated as normal human flora. Most human cases are acquired in rural settings and include outbreaks of leg ulcers in Thai children, septicemia in a patient with colon carcinoma, polymicrobial-infected diabetic foot ulcers, spondylodiskitis and psoas abscess, subcutaneous abscesses, and intra-abdominal infections.164,165 A case of fatal endocarditis in a patient with no animal contact has been reported.166 A. pyogenes is cultured on sheep blood agar under CO2 enrichment. Colonies are weakly hemolytic at 24 hours and become more strongly hemolytic at 48 hours.2 Differentiation from A. haemolyticum is made by observation of the CAMP reaction, by fermentation of xylose, and by the α-mannosidase test. A. pyogenes is susceptible to most antibiotics, including penicillins, cephalosporins, macrolides, tetracyclines, and aminoglycosides.
Arcanobacterium bernardiae
Originally described as CDC coryneform group 2 in 1987, it was assigned the species name Actinomyces bernardiae in 1995. In 1997, it was transferred to the genus Arcanobacterium as Arcanobacterium bernardiae.146 On Gram staining, it appears as short gram-positive rods without branching. It is identified by the ability to ferment maltose more rapidly than glucose, which separates it from other coryneform bacteria. It is distinguished from A. pyogenes by the inability to ferment sucrose, mannitol, and xylose.2 A. bernardiae is a rare human pathogen, with recovery of the organism from the bloodstream, abscesses, urinary tract, joints, the eye, and wounds; it has also been implicated as a cause of necrotizing fasciitis.167-169
Miscellaneous Coryneform Bacteria
Turicella otitidis
Initially isolated from patients with otitis media, Turicella otitidis is believed to be a colonizer of the human auditory canal and not a true pathogen in this setting because it has been isolated in the same frequency from an asymptomatic control population.5,94,170 It has been reported as a cause of mastoiditis and posterior auricular abscess in immunocompetent children and septicemia in a neutropenic child. T. otitidis is resistant to clindamycin and erythromycin but susceptible to penicillins, cephalosporins, tetracyclines, fluoroquinolones, rifampin, and vancomycin.23
Arthrobacter Species
An environmental coryneform found in animal sheds, schools, and daycare centers, Arthrobacter has rarely been isolated from human clinical specimens. Identified species include Arthrobacter cumminsii, Arthrobacter oxydans, Arthrobacter luteolus, and Arthrobacter albus.171,172 There are reports of septicemia in immunocompromised patients and isolation of Arthrobacter from human urine specimens. One unusual case report was of Whipple's syndrome caused by Arthrobacter.
Brevibacterium Species
Brevibacterium spp. are short coryneforms isolated from milk and dairy products and are known colonizers of human skin.8 They have been identified in environmental dust in schools, daycare centers, and animal sheds. Brevibacteria show biphasic morphology on culture, with young colonies demonstrating typical coryneform features. As colonies age, the organisms mature into cocci or a coccobacillary appearance.2 Brevibacteria have been implicated in causing human foot odor when confining footwear results in a moist environment. Only a few species of Brevibacterium have been noted to cause infection.
Brevibacterium casei is the species of this genus that is most frequently isolated from human clinical specimens. On culture, it forms white-gray colonies with a distinctive cheese odor. On Gram staining, it is a short, club-shaped rod that is catalase positive and non–spore-forming.2,173 Human infections with brevibacteria have most frequently been intravascular catheter–related bloodstream infections, particularly in immunocompromised patients and patients with AIDS. There have been additional reports of meningitis, cholangitis, salpingitis, and peritonitis in patients undergoing CAPD. In addition, there is one report each of prosthetic valve endocarditis and osteomyelitis of the sternum in a neonate after an episode of mastitis in the mother.174 Susceptibility testing shows some resistance to β-lactam antibiotics, fluoroquinolones, clindamycin, and macrolides. Vancomycin is the treatment of choice for serious infections.23,175 Other Brevibacterium spp. that have been reported to cause invasive disease include Brevibacterium sanguinis and Brevibacterium epidermidis.176,177
Dermabacter hominis
Dermabacter spp. were previously identified as CDC group 3 and group 5 coryneform bacteria and are skin colonizers of humans.178 They have been a cause of bacteremia in patients with prolonged hospitalizations and peritonitis in immunocompromised persons undergoing CAPD. Dermabacter has been isolated from a cerebral abscess in a renal transplant recipient and from a patient with chronic osteomyelitis with Actinomyces neuii as copathogen.179D. hominis exhibits variable resistance to many antibiotics, including penicillins, fluoroquinolones, macrolides, chloramphenicol, and tetracyclines, and susceptibility to vancomycin and teicoplanin.23
Rothia dentocariosa and Rothia mucilaginosa
Rothia are found as colonizers of the human oral cavity and have been isolated from dental plaque and in cases of periodontal disease.180 R. dentocariosa has the potential for misidentification as a Dermabacter or Actinomyces spp. in the microbiology laboratory.2
Case reports with reliable information on identification of the organisms have found it to be a pathogen in several cases of native and prosthetic valve endocarditis, including presentations with abscesses, mycotic aneurysms, and vertebral osteomyelitis.181,182 It has also been isolated as a cause of bacteremia without endocarditis. It has been found in cases of pneumonia in patients with leukemia and lung cancer and has caused peritonitis in a patient undergoing CAPD.
R. mucilaginosa, formerly Stomatococcus mucilaginosus, is a normal resident of the human mouth and nasopharynx. On culture, it usually appears as gram-positive cocci in clusters—hence, the previous classification as a Stomatococcus. R. mucilaginosa is a rare cause of sepsis from an oral source, but cases of meningitis and spondylodiskitis have also been reported.183 A case of granulomatous dermatitis attributable to R. mucilaginosa bacteremia has been reported.184 Most isolates are susceptible to ampicillin.
Oerskovia Species
Included in CDC group A-1 and A-2, Oerskovia spp. are rare human pathogens but have been reported to cause infection in immunocompromised hosts, patients with implanted devices, and those with indwelling central venous catheters.2 The spectrum of infections has ranged from bacteremia, endocarditis, meningitis associated with CSF shunt infection, soft tissue infection, prosthetic joint infection, and peritonitis in a patient undergoing CAPD.185-187 One report exists in the literature of endophthalmitis after eye injury with a metallic foreign body.
Microbacterium Species
CDC coryneform group A-4 and A-5 bacteria were defined as Microbacterium spp., and since 1998, the genus Aureobacterium has been reclassified and renamed within the genus Microbacterium.188,189 Microbacterium spp. have been found as a cause of bacteremia in patients on an oncology ward and in specimens from patients with endophthalmitis.190 Most commonly, it has been a nosocomial pathogen in debilitated and immunocompromised patients. In a study of 50 human isolates, the most common species recovered were M. oxydans, M. paraoxydans, and M. foliorum.191
Leifsonia aquatica
Corynebacterium aquaticum was reclassified in 2000 as Leifsonia aquatica.192 Because of inconsistencies of identification and confusion with Aureobacterium in previous reports, it has been difficult to determine the pathogenicity of this species. It is expected that future case reports will help to clarify this. Case reports for Leifsonia aquatica are rare, although C. aquaticum had been reported to cause septicemia in immunocompromised hosts, peritonitis in patients on CAPD, and bacteremia in a hemodialysis patient.193 In addition, urinary tract infection in a neonate and meningitis in an infant have been reported.
Other medically relevant coryneform bacteria include the genera Exiguobacterium, Cellulomonas, Helcobacillus, Curtobacterium, and Cellulosimicrobium.3
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Book2015, Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases (Eighth Edition)Rose Kim, Annette C. Reboli
Chapter
History, Science and Methods
2014, Encyclopedia of Food SafetyA.A. Zasada
Occurrence in the Environment and Food
Corynebacterium species are a widely distributed group of bacteria, typically found in the environment in soil and on the skin and mucous membranes of human beings and animals. Corynebacteria have also been documented to survive for long periods of time on objects that have been touched by infected individuals.
Corynebacterium ulcerans is a commensal in animals and has been isolated from a wide host of domestic and wild animals (e.g., dogs, cats, horses, goats, cows, pigs, camels, monkeys, squirrels, and otters). The animals may serve as reservoirs for human infection. Moreover, the bacterium causes mastitis in cattle and goats. In mastitis infection, the bacteria are present in milk. A significant portion of human C. ulcerans infections has been associated with the consumption of raw dairy products. Also handling of infected animals and their products could be a source of infections. Recently, contact with companion animals, such as dogs and cats, has been recognized as a way of infection.
Corynebacterium pseudotuberculosis is a rare zoonosis. Similar to C. ulcerans, raw dairy products are the source of infection as well as handling of infected animals and their products (e.g., butchery). Sheep, followed by goats, are the most commonly affected animal hosts; but horses, cattle, and deer may also be infected. Corynebacterium pseudotuberculosis has also been isolated from milk in mastitis cases.
Corynebacterium diphtheriae is traditionally considered as a nonzoonotic pathogen. However, recent reports of horses as carriers of C. diphtheriae highlight the possible emergence of a new route of human infection. Corynebacterium diphtheriae has also been isolated from a cat. In older publications, isolation of C. diphtheriae from other animals has been reported as well. The species is infrequent cause of bovine mastitis and has been associated with dermatitis with pyrexia in cattle. It is thought to be transferred to cows from infected dairy workers. Moreover, C. diphtheriae has been documented in other domestic animals, including equids and canids. The dairy products obtained from infected animals and handling of infected animals could be a source of infection for humans.
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Pleiomorphism in
2012, Advances in Applied MicrobiologyLeif A. Kirsebom, ... Brännvall M. Fredrik Pettersson
2.2.1Corynebacterium spp.
In 1884, Klebs and Löffler discovered Corynebacterium diphtheriae, the causative agent of diphtheria (Holmes, 2000). Early microscopy studies revealed that cultures of this high GC-content Gram-positive bacteria (which belongs to the order actinomycetales, that is, the same order as Mycobacterium spp. and Streptomyces spp., see below) showed different morphologies. Apparently, C. diphtheriae can grow both as rods and as coccoids as well as form long filaments. The shape and filamentation depend on the growth conditions (Davis & Mudd, 1954; Denny, 1903; Hewitt, 1951). Moreover, early isolates showed that the diphtheria bacillus yielded two types of colonies, rough and smooth ones (Yü, 1930). Interestingly, in an electron microscopy study, Kawata and Inoue (1965) observed remarkable bodies inside aged C. diphtheria cells. As discussed above, MreB or its homologue is a common cytoskeletal element for rod-shaped bacteria. However, corynebacteria seem to lack MreB but nevertheless exist as rods. In other rod-shaped bacterium such as E. coli and B. subtilis, the cell walls are synthesized laterally. In contrast, the new cell walls in Corynebacterium spp. are synthesized at the poles as revealed from studies of C. glutamicum (Letek et al., 2008). Perhaps the lack of MreB homologues confers upon Corynebacterium spp. the capability to change its shape dependent on growth conditions. In this context, we note that overexpression of specific serine/threonine protein kinases (PknA or PknB) in C. glutamicum results in coccoid shape (Fiuza et al., 2008; see below). Hence, these kinases might be part of a pathway involved in regulating the shapes of the bacteria. Interestingly, the gene organization of pknA and pknB in the three actinomycetes Streptomyces coelicolor, M. tuberculosis, and C. glutamicum is well conserved (Molle & Kremer, 2010). Together, these observations might indicate that switching into different shapes and growth depending on the environment might be a general characteristic of the actinomycetes (see below).
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Diphtheria toxin
2015, The Comprehensive Sourcebook of Bacterial Protein Toxins (Fourth Edition)Daniel Gillet, Julien Barbier
Symptoms, treatment, prophylaxis, and epidemiology of diphtheria
Diphtheria is a pharyngal infection caused by strains of the Gram positive bacillus Corynebacterium diphtheriae (rarely Corynebacterium ulcerans or Corynebacterium pseudotuberculosis) carrying a lysogenic bacteriophage, usually corynephage β. This phage carries the tox gene, which encodes the DT [1–3]. The transcription of this gene is activated by lack of iron [2,4]. The toxin is responsible for the major symptoms, morbidity, and mortality of the disease. The infection spreads over the pharynx, larynx, and tonsils. A thick, adherent, whitish or gray-green membrane, the pseudomembrane, develops in the throat and gives the disease its name (Greek: δ ι φ θ ρ α “membrane”, www.cnrtl.fr). The toxin, released locally, reaches the bloodstream and causes toxic myocarditis and neuritis, leading to death. In some cases, death occur from obstruction of upper aerial tracks and suffocation. During the 19th century, the death rate could reach 80% among European children younger than 10 years old . Treatment was based on the removal or piercing of pseudomenbranes with tubes, local disinfection, and tracheotomy. Today, treatment of diphtheria is based on subcutaneous injections of immunoglobulins purified from the serum of horses immunized against DT. The disease has practically vanished from countries where this vaccination is systematic. The very few observed cases are either imported or occur in unvaccinated subjects or aged individuals lacking booster immunization . Unfortunately, epidemics happen in countries with lack of proper vaccination coverage, with death rates as high as 40% among infants [7–10]. Diphtheria is highly contagious and propagated by contact with patients or their belongings. The bacteria survive in the patients’ rooms for months—maybe years. Cutaneous infections also exist.
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Book2015, The Comprehensive Sourcebook of Bacterial Protein Toxins (Fourth Edition)Daniel Gillet, Julien Barbier
Related terms:
Diphtheria Toxin
Amino Acid
Diphtheria
Antiinfective Agent
Bacteriophage
Infection
Receptor
Escherichia coli
Infectious Agent
Disease
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3472 | https://kconrad.math.uconn.edu/blurbs/ugradnumthy/minus1squaremodp.pdf | WHEN IS −1 A SQUARE MODULO PRIMES?
KEITH CONRAD When p is an odd prime, there are two natural ways to pair offnonzero numbers mod p: pair each number with its multiplicative inverse or with its additive inverse. Using both of these ideas will lead to a proof of the following pattern for when −1 mod p is a square: −1 ≡□mod p ⇐ ⇒p = 2 or p ≡1 mod 4.
Theorem 1 (Wilson). For every prime p, (p −1)! ≡−1 mod p.
Proof. In the product (p −1)! = 1 · 2 · 3 · · · (p −2)(p −1) on the right side the factors run through all the nonzero numbers mod p. For each k from 1 to p −1, there’s a k′ from 1 to p −1 such that kk′ ≡1 mod p.
Let’s pair together multiplicative inverses mod p. As long as k′ ̸= k, both k and k′ are in the product for (p −1)! and they cancel each other mod p (each is the other’s multiplicative inverse). The only time there isn’t cancellation of k by k′ in (p −1)! mod p is when k′ = k, which means k2 ≡1 mod p. That is the same as k ≡±1 mod p (since p is prime), so k = 1 and k = p−1.
Therefore (p −1)! ≡1 · (p −1) ≡−1 mod p.
□ Example 2. When p = 7, 6! = 1 · 2 · 3 · 4 · 5 · 6 = 1 · (2 · 4)(3 · 5) · 6, where we collected inverse pairs mod 7: 2 and 4, and 3 and 5. Reducing the product modulo 7, terms cancel and we’re left with 6! ≡1 · (1)(1) · 6 ≡6 ≡−1 mod 7.
The conclusion of Wilson’s theorem is always false if the prime p is replaced by a composite number n: when n is composite we can’t have (n −1)! ≡−1 mod n, because if (n −1)! ≡ −1 mod n then every integer from 1 to n −1 is invertible mod n, which forces n to be prime.1 Thus we have a primality test: (1) n > 1 is prime if and only if (n −1)! ≡−1 mod n.
At first this might seem superior to the Fermat test, which can often detect composites but doesn’t ever prove primality. However, (1) is useless for large n because there is no known fast way of computing (n −1)! mod n, unlike the fast way of computing an−1 mod n for the Fermat test by writing the exponent n −1 in base 2 to reduce the exponentiation mod n to repeated squaring mod n.
We now put Wilson’s theorem to work.
Theorem 3. For every prime p ̸= 2, −1 ≡□mod p if and only if p ≡1 mod 4.
1In fact, if n is composite then (n −1)! ≡0 mod n when n ̸= 4 while (4 −1)! ≡2 mod 4.
1 2 KEITH CONRAD Proof. First suppose −1 ≡□mod p, say −1 ≡x2 mod p. We don’t know much about x, so how can we use x to prove p ≡1 mod 4? Raise both sides to the (p −1)/2-power: (−1)(p−1)/2 ≡(x2)(p−1)/2 ≡xp−1 mod p.
Since x ̸≡0 mod p (why?), we know xp−1 ≡1 mod p by Fermat’s little theorem since p is prime. Thus (2) (−1)(p−1)/2 ≡1 mod p.
The left side is a power of −1, so it is either 1 or −1. Since p > 2 we have −1 ̸≡1 mod p, so (2) tells us that (−1)(p−1)/2 = 1 in Z. Therefore the exponent (p −1)/2 is even, say (p −1)/2 = 2k for some k ∈Z. Thus p = 1 + 4k, so p ≡1 mod 4.
For the converse direction, assume p ≡1 mod 4. To prove −1 ≡□mod p we will use Wilson’s theorem: −1 ≡(p −1)! = 1 · 2 · . . . · (p −2)(p −1) mod p. Let’s pair together additive inverses mod p: write numbers in the second half of the product as negatives of numbers in the first half, modulo p: p −k ≡−k mod p. Then (p −1)!
= 1 · 2 · · · p −1 2 p + 1 2 · · · (p −2)(p −1) ≡ 1 · 2 · · · p −1 2 − p −1 2 · · · (−2)(−1) | {z } (p−1)/2 terms modp ≡ (−1)(p−1)/2 · 1 · 2 · · · p −1 2 p −1 2 · · · (2)(1) mod p ≡ (−1)(p−1)/2 p −1 2 !
2 mod p.
Since (p −1)! ≡−1 mod p by Wilson’s, theorem, this congruence says (3) −1 ≡(−1)(p−1)/2 p −1 2 !
2 mod p.
So far this is valid for all odd primes p. If p ≡1 mod 4 then (p−1)/2 is even, so (−1)(p−1)/2 = 1 and the congruence (3) tells us that −1 ≡x2 mod p where x = ((p −1)/2)!.
□ Example 4. The first three primes that are 1 mod 4 are 5, 13, and 17. We have 5 −1 2 ! = 2! ≡2 mod 5, 13 −1 2 ! = 6! = 720 ≡5 mod 13, and 17 −1 2 ! = 8! = 40320 ≡13 mod 17, and you can verify that these numbers square to −1 mod p in each case.
Remark 5. If p ≡3 mod 4 then (3) implies (((p −1)/2)!)2 ≡1 mod p, so ((p −1)/2)! ≡ ±1 mod p. There is no easy way to say when ((p−1)/2)! mod p is 1 or −1 if p ≡3 mod 4. See
WHEN IS −1 A SQUARE MODULO PRIMES?
3 Theorem 3 is not true for odd composite n > 1: the condition −1 ≡□mod n is not equivalent to the condition n ≡1 mod 4. For example, 21 ≡1 mod 4 but −1 ̸≡□mod 21.
However, one direction is true: if −1 ≡□mod n then n ≡1 mod 4. Indeed, for each prime factor p of n, p ̸= 2 and −1 ≡□mod n = ⇒−1 ≡□mod p Thm 3 = ⇒p ≡1 mod 4.
Since n is the product of its prime factors, we get n ≡1 mod 4.
Appendix A. Comments on Wilson’s Theorem Wilson’s theorem is named after John Wilson, who conjectured the result as a student.
His conjecture was published in the book Meditationes Algebraicae written (in Latin) by his teacher Edward Waring in 1770. The page containing the statement of Wilson’s theorem can be found in [1, Figure 2] (see the paragraph that begins “Sit n numerus primus”).
Lower down on the same page is the following claim by Waring, which shows how limited his perspective was on number theory: “Proofs of propositions of this kind are made more difficult by the fact that one can’t imagine a convenient notation for prime numbers.” Other mathematicians were more imaginative than Waring: the first proof of Wilson’s theorem was given by Lagrange in 1771, using polynomials mod p, and Lagrange also observed that the congruence (n −1)! ≡−1 mod n only holds for prime n. In 1801, Gauss [2, Article 77] proved Wilson’s theorem by the argument we gave that pairs together multiplicative inverses modulo p, and he criticized Waring’s comment about the lack of notation for primes, saying [2, Article 76] “In our opinion, truths of this kind should be drawn from notions rather than from notations.” Gauss also proved a generalization of Wilson’s theorem to all moduli m ≥2 using the product of the units mod m: Y a∈(Z/(m))× a ≡ ( −1 mod m, if x2 ≡1 mod m has at most two solutions, 1 mod m, if x2 ≡1 mod m has more than two solutions.
The first condition occurs not only when m is prime. It also happens when m is 4, an odd prime power, and twice an odd prime power.
While (p −1)! ≡−1 mod p for all primes p, it is rare that (p −1)! ≡−1 mod p2. Such p are called Wilson primes and the only examples up to 1013 are 5, 13, and 563. Probabilistic heuristics suggest there should be infinitely many such primes, and also that they should be extremely rare. Concerning the infinitude of Wilson primes, Harry Vandiver once wrote “This question seems to me to be of such a character that if I should come to life any time after my death and some mathematician were to tell me that it had been definitely settled, I think I would immediately drop dead again.” References G. L. Alexanderson, L. F. Klosinski, “About the cover: Waring’s problems,” Bull. Amer. Math. Soc.
55 (2018), 375–379. URL
C. F. Gauss, Disquisitiones Arithmeticae (English edition), Springer-Verlag, 1986.
J–L. Lagrange, “D´ emonstration d’un th´ eor` eme nouveau concernant les nombres premiers,” Nouveaux M´ emoires de l’Acad´ emie Royale des Sciences et Belles-Lettres, Berlin (1771), 425–438. URL https:// gallica.bnf.fr/ark:/12148/bpt6k229222d/f426.
H. S. Vandiver, “Divisibility Problems in Number Theory,” Scripta Mathematica 21 (1955), 15–19. URL book other versions. |
3473 | https://zhuanlan.zhihu.com/p/98996379 | 直答
切换模式
什么是充分条件、必要条件、充要条件？
727 人赞同了该文章
反过来，有死就一定有生，没有死就必然没有生。所以，死也是生的充要条件。
生与死是互为充要条件。
编辑于 2020-01-05 01:04
写下你的评论...
108 条评论
默认
最新
王某人
充分条件就是逻辑加关系：A1+A2+A3...+An => A ，
必要条件就是逻辑乘关系：A1A2A3...An => A 。
就这么简单的事情，为什么还要讨论那么长时间，搞那么复杂干嘛？
2022-05-20
海风
加法，中间某个因素为0，仍然成立。
乘法，中间某个要素缺失（为0），就绝对不能成立（就是必要条件不存在，则结果不可能发生）
2022-10-07
杨洋
充分就是或，必要就是与，计算机上的与或非
2022-11-07
一缕sunlight
先赞了。另外我真纳闷了，人们在提高中要考数学，那么最基本的是要搞清楚数学问题，为什么知乎乃至全网上数学类讨论很少，却充斥着那么多社科类讨论。
2022-05-04
瑞可可
人一旦讲逻辑 很多事情就不好瞒下去了
2023-01-12
smile-finder
没有门槛呗
2024-05-17
阿狸和云
我觉得手机有电就是开手机的充要条件吧，因为手机没电就开不了，但有电是可以开手机的。并不是说要你强制开手机
2020-06-15
夯实人生
手机有电是手机开的必要条件之一，不是充分条件。手机可能坏了
2020-09-17
无眠空斗
wx59c24634d4bca564
不对 手机有电只是能开启手机的众多条件中的一个
2020-08-01
知乎用户Wg5RqY
2024-03-24
明码标价
这图没解释对
2024-06-21
天明
好图，直观！
2024-04-12
棒棒鸡
看了更不懂
2022-08-17
宁情深
😡
2020-07-06
且听风吟
浦口人是南京人的充分条件。不是南京人就一定不是浦口人，南京人是浦口人的必要条件
2022-03-25
兽医院主任医师
浦口人p→南京人q
p是q的充分条件，p不是q的必要条件
q也是p的必要条件，因为如果p成立，那q也成立（q→p）
2023-11-12
且走且行
A是B的必要条件，是不是不仅无A无B，而且有B必有A
2021-02-03
遇水架桥
这两个是逆否的关系，所以是等价的关系
2021-06-02
nationworld
是的，a是b的必要条件说明在b发生的情况下a一定发生，也就是b->a,这个的逆否就是
-a->-b
02-25
Reda
终于理清了！简言之如果是充分条件那么在逻辑上就一定会导致该结果发生，而必要条件只是某结果的促成条件之一，比如勤劳是致富的必要不充分条件，因为致富的条件有很多而勤劳只是促成致富的条件之一，并不是全部。
2022-12-11
周一
勤劳和致富是既不充分也不必要的关系
2023-08-31
qwer
你这里隐藏了一个前提，就是致富必须要勤劳
2023-03-26
一只追风少年
第二个例子是错的。
2022-06-27
Z先生
逆否命题等价原命题，你再试试
2023-12-05
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3474 | https://www.youtube.com/watch?v=TeSf9NyPwFc | Writing a Parametric Equation Given 2 Points
MATH OMG
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Posted: 8 Feb 2016
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In this video you will learn how to take two points and write a parametric equation for those two points using the MATH OMG method. If you have any questions, please make sure to leave them in the comments!Easier equations thanks to Hope!
If you like what you see make sure to comment, like, subscribe & share!
x= x1+ t(x2 - x1),
y= y1 + t(y2 - y1)
x1 and y1 come from coordinate A, x2 and y2 come from coordinate B.
26 comments
Transcript:
Intro [Music] hey guys it's Cassie here with math OMG today we're going to be learning how to find parametric equations for a line given two points so if we look at this problem the first thing that we noticed is we're given an A and A B coordinate a is -23 and B is 36 our goal is going to be to take these two points and make a parametric equation which means put each equation in respect to T with Y and X so the first thing we need to look at is what's our initial and what's our terminal point so our initial point in this case is going to be the first given value which is a and our terminal point is B and then we are going to create two What is a parametric equation equations so a parametric equation means we have x equals something with respect to T and Y equals something with respect to T now here are kind of the way that I like to think about it this will not be found in a textbook or at least not any of the textbooks that I looked at so don't try to go and like find these equations somewhere else this was just the way that it seemed most logical to me so this is kind of how I set it up and then how I teach my students how to do it so the first thing we have to do is find an equation with respect to x and t so it's going to be in this format which looks very similar to when you are finding slope intercept form of a linear function you have your variable in this case x equals your initial amount plus some we can call it slope times T and then you also have an equation with respect to Y and T and then we need to figure out okay so when T is zero my X needs to be -2 my y needs to be three so that means my X for this equation Y for this equation and then when t equal 1 my X needs to be three and my y needs to be six and these numbers right here the zero and one will always be the case in every textbook I've looked at that's always the case is we're trying to figure out between those two parameters of zero and one we're trying to figure out that line and then our game plan for doing this is just going to be to use Writing a parametric equation these formulas so as we go through and write this the first thing I'm going to do is separate this so I need to make an equation for an x equals and for a y equals now this A1 for x and I'm going to write this down so I'm just going to write down all the information I need so that I'm not confused going back and forth I need to figure out in my equation which is a1+ WT I need to make an equation so I need to find A1 and I need to find W and I need to know these facts when T is Zer my X is supposed to be -2 and when T is is one my X is supposed to be three and I got those from these equations so this part right here represents your X and then the second coordinate represents your y just like in a normal XY point so my A1 is pretty easy it's always my initial point x value so the initial was a and the initial x value of a of point a is -2 so a is -2 and then I don't know but I do know that I have a t value of one when X is three so if I know those three unknowns then all I have to do is solve for w so when X is three A1 is always -2 so that's going to stay -2 plus W time or t not time T which is one and then I just solve for w so 3 = -2 + w w 1 is W then I add two to both sides and I get w = 5 so the equation for x is going to be X = my initial x value which was -2 plus W which I solved for to be 5 T so that is my parametric equation for x then I do the same thing for y so Y is equal to B1 plus ZT and again I have similar restrictions so my B1 that's my first initial X or yalue sorry my yvalue so I'm going to say that is three and then I have a w or sorry Z that I don't know so I'm going to write down Z I'm trying to find that I don't know that and then I also know that when T is equal to 1 my y value needs to equal six so again I plug all of that information in and then I solve for Z so Y is 6 B1 is three Z is unknown and T is 1 so then it's 6 = 3 + z I subtract three from both sides 3 is equal to Z so this equation for y is y equals our initial y value which is 3 plus Z which is 3 T so here are my two parametric equations right here and right here those are my parametric equations now if it asked me to do this for a line then I would say that these are my two equations and that my T is going to be from negative Infinity to Infinity which should make sense you want your line to go on for forever so your T values need to go on forever that's how you create a line to create the line segment we just want the line segment between those two points so then the line segment between those two points is going to between be between these two t values that we created the line with so for the line segment we would say t needs to be in between zero and one and that's it we've created parametric equations of a line given two points thank you so much for watching if you guys have questions please make sure to leave them in the comments and I will do my best to respond and let you know or clarify any ideas and again if you want more information on parametric equations make sure you subscribe to my channel and check out the playlist on parametric equations so that you know everything there is to know with them thank you guys so much for watching my videos If you want to get better at math subscribe to my videos here if you want more information on math click on my website link here [Music] |
3475 | https://www.kenhub.com/en/videos/humerus-bone | Video: Humerus
You are watching a preview. Go Premium to access the full video:
Anatomy, bony landmarks and function of the humerus.
Highlights
| |
| 0:10 Overview |
| 0:41 Bony landmarks of the proximal end |
| 1:53 Body |
| 2:05 Bony landmarks of the distal end |
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Transcript
Hey, everyone. It’s Matt from Kenhub! And in this tutorial, we will discuss the anatomy, definition, and function of the humerus.
The humerus is a long bone that connects the elbow to the shoulder ...
Hey, everyone. It’s Matt from Kenhub! And in this tutorial, we will discuss the anatomy, definition, and function of the humerus.
The humerus is a long bone that connects the elbow to the shoulder blade. It provides a base of support for the muscles of the shoulder, the upper arm, and the lower arm.
You’ve probably heard people say that they hit their funny bone, and you’ve probably also heard people say, “Why is it called that?” and, “It hurts really bad, and it isn’t funny at all.” Well, it’s called that because what they hit is actually the humerus. So it’s a play on the word “humerous.”
Let’s discuss some of the anatomical features of the humerus. Here is the head of the humerus. The humeral head is projected medially and superiorly and articulates with the glenoid cavity of the scapula to form the glenohumeral joint (also known as the shoulder joint).
The next structure that we find on the humerus is the anatomical neck that you see now highlighted in green.
The anatomical neck is an area between the head and the greater and lesser tubercles of the humerus.
On these two images, you see two prominences known as the greater tubercle and lesser tubercle of the humerus. These areas serve as attachment points for some of the major muscles of your shoulder.
Now, we’re looking at another neck: the surgical neck of the humerus. This is a narrow area found distal to the tubercles that is a common site for fractures.
The surgical neck is in close contact with two other important structures: the axillary nerve and the posterior humeral circumflex artery.
You’re now seeing another structure highlighted known as the body of the humerus (often referred to as the shaft).
The picture here shows the lateral epicondyle of the humerus. This is a protuberance found lateral to the capitulum that gives attachment to the extensor muscles of the forearm.
And here, you now see the medial epicondyle of the humerus. This medial protuberance serves as an origin point for the flexor muscles of the forearm.
Switching from the posterior view to the anterior view, we see the capitulum of the humerus. This is a rounded projection found at the distal end of the humerus and articulates with the radius.
And last but not least, the trochlea of the humerus from the anterior view as well. The trochlea is an articular cylinder that connects with another bone known as the ulna.
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3476 | https://www.merckmanuals.com/professional/pediatrics/congenital-renal-transport-abnormalities/bartter-syndrome-and-gitelman-syndrome | honeypot link
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Merck ManualProfessional Version
Bartter Syndrome and Gitelman Syndrome
(Bartter's Syndrome; Gitelman's Syndrome)
ByChristopher J. LaRosa, MD, Perelman School of Medicine at The University of Pennsylvania
Reviewed ByMichael SD Agus, MD, Harvard Medical School
Reviewed/Revised Modified Oct 2024
v1098157
View Patient Education
Bartter syndrome and Gitelman syndrome are autosomal recessive kidney disorders characterized by fluid, electrolyte, urinary, and hormonal abnormalities, including renal potassium, sodium, chloride, and hydrogen wasting; hypokalemia; hyperreninemia and hyperaldosteronism without hypertension; and metabolic alkalosis. Findings include electrolyte, growth, and sometimes neuromuscular abnormalities. Diagnosis is assisted by urine electrolyte measurements and hormone assays but is typically a diagnosis of exclusion. Treatment consists of nonsteroidal anti-inflammatory drugs (for Bartter syndrome) and electrolyte replacement.
Pathophysiology|
Etiology|
Symptoms and Signs|
Diagnosis|
Treatment|
Key Points|
Bartter and Gitelman syndromes are rare disorders, and there are few data to determine their prevalence. The annual incidence of Bartter syndrome was 1.2/million in a Swedish database study (1). The prevalence of Gitelman syndrome is estimated to be approximately 1 in 40,000 in the United States (2) and approximately 3% in Japan and China (3).
General references
Rudin A. Bartter's syndrome. A review of 28 patients followed for 10 years. Acta Med Scand. 1988;224(2):165-171.
Ji W, Foo JN, O'Roak BJ, et al. Rare independent mutations in renal salt handling genes contribute to blood pressure variation. Nat Genet. 2008;40(5):592-599. doi:10.1038/ng.118
Hsu YJ, Yang SS, Chu NF, Sytwu HK, Cheng CJ, Lin SH. Heterozygous mutations of the sodium chloride cotransporter in Chinese children: prevalence and association with blood pressure. . Heterozygous mutations of the sodium chloride cotransporter in Chinese children: prevalence and association with blood pressure.Nephrol Dial Transplant. 2009;24(4):1170-1175. doi:10.1093/ndt/gfn619
Pathophysiology of Bartter Syndrome and Gitelman Syndrome
Bartter syndrome and Gitelman syndrome result from
Defective sodium chloride reabsorption
In Bartter syndrome, the defect is in the ascending thick limb of the loop of Henle. In Gitelman syndrome, the defect is in the distal tubule.
In both syndromes, the impairment of sodium chloride reabsorption causes mild volume depletion, which leads to increases in renin and aldosterone release, resulting in potassium and hydrogen losses. Sodium wasting contributes to a chronic mild plasma volume contraction reflected by a normal to low blood pressure despite high renin and angiotensin levels.
In Bartter syndrome, there is increased prostaglandin secretion as well as a urinary concentrating defect due to impaired generation of the medullary concentration gradient.
In Gitelman syndrome, hypomagnesemia and a low urinary calcium excretion are common.
The features at clinical presentation vary (see table Some Differences Between Bartter Syndrome and Gitelman Syndrome).
Table
Some Differences Between Bartter Syndrome and Gitelman Syndrome
Table
Some Differences Between Bartter Syndrome and Gitelman Syndrome
Some Differences Between Bartter Syndrome and Gitelman Syndrome
| Feature | Bartter Syndrome | Gitelman Syndrome |
| Location of kidney defect | Ascending loop of Henle (mimics effects of loop diuretics) | Distal tubule (mimics effects of thiazides) |
| Urinary calcium excretion | Normal or increased, commonly with nephrocalcinosis | Decreased |
| Serum magnesium level | Normal or decreased | Decreased, sometimes greatly |
| Renal prostaglandin E2 production | Increased | Normal |
| Usual age at presentation | Before birth to early childhood, often with intellectual disability and growth disturbance | Late childhood to adulthood |
| Neuromuscular symptoms (eg, muscle spasms, weakness) | Uncommon or mild | Common |
Etiology of Bartter Syndrome and Gitelman Syndrome
Both syndromes are usually autosomal recessive, although sporadic cases and other types of familial patterns can occur. Of note, there is an X-linked mutation in the MAGED2 gene, which can cause severe antenatal Bartter syndrome that is transient and resolves by 1 to 2 years of life.
There are several genotypes of both syndromes (see table Subtypes of Bartter Syndrome); different genotypes can have different manifestations (1).
Table
Subtypes of Bartter Syndrome
Table
Subtypes of Bartter Syndrome
Subtypes of Bartter Syndrome
| Subtype | Gene (Protein)† | Age of Onset | Clinical Features |
| I | SLC12A1 (NKCC2)† | Antenatal/neonatal | Polyhydramnios, prematurity, hypokalemia/alkalosis, polyuria, hypercalciuria, nephrocalcinosis |
| II | KCNJ1 (ROMK1)† | Antenatal/neonatal | Similar to type I |
| III | CLCNKB (ClC-Kb)† | Later onset (childhood) | Similar to type I, may be less severe Some children present with Gitelman phenotype as ClC-Kb found in the distal convoluted tubule and in the connecting tubule |
| IVa | BSND (Barttin)† | Antenatal/neonatal | Similar to type I, nephrocalcinosis less common Associated with sensorineural hearing loss |
| IVb | CLCNKA (ClC-Ka)† and CLCNKB (ClC-Kb)† | Antenatal/neonatal | Similar to type IV Associated with sensorineural hearing loss |
| V | CASR (CaSR) | Later onset | Bartter phenotype with low/normal intact parathyroid hormone, hypocalcemia, hypercalciuria, and nephrocalcinosisBartter phenotype with low/normal intact parathyroid hormone, hypocalcemia, hypercalciuria, and nephrocalcinosis Due to CaSR gain of function, which may reduce ROMK and NKCC2 activity |
| Protein abbreviations: Barttin = beta subunit of ClC-Ka and ClC-Kb; CaSR = calcium-sensing receptor; ClC-Kb = basolateral chloride channel kidney B; ClC-Ka = basolateral chloride channel kidney A; NKCC2 = Na-K-2Cl channel; ROMK = luminal potassium channel. |
| |
| Protein abbreviations: Barttin = beta subunit of ClC-Ka and ClC-Kb; CaSR = calcium-sensing receptor; ClC-Kb = basolateral chloride channel kidney B; ClC-Ka = basolateral chloride channel kidney A; NKCC2 = Na-K-2Cl channel; ROMK = luminal potassium channel. |
Etiology reference
Fulchiero R, Seo-Mayer P: Bartter syndrome and Gitelman syndrome. Pediatr Clin North Am 66(1):121–134, 2019. doi: 10.1016/j.pcl.2018.08.010
Symptoms and Signs of Bartter Syndrome and Gitelman Syndrome
Patients with Bartter syndrome tend to present prenatally or during infancy or early childhood. Patients with Gitelman syndrome tend to present during late childhood to adulthood.
Of note, some patients, especially those with Gitelman syndrome, are asymptomatic and diagnosed incidentally after blood tests are done.
Bartter syndrome can manifest prenatally with intrauterine growth restriction and polyhydramnios. Different forms of Bartter syndrome can have specific manifestations, including hearing loss, hypocalcemia, and nephrocalcinosis, depending on the underlying genetic defect. Children with Bartter syndrome, more so than those with Gitelman syndrome, may be born prematurely and may have poor growth and development postnatally, and some children have intellectual disability.
Inability to retain potassium, calcium, or magnesium can lead to muscle weakness, cramping, spasms, tetany, or fatigue. This is especially apparent in Gitelman syndrome. Polydipsia, polyuria, salt cravings, and vomiting may be present in both syndromes.
Most patients with Bartter syndrome or Gitelman syndrome have low or low-normal blood pressure and may have signs of volume depletion.
In general, neither Bartter syndrome nor Gitelman syndrome typically leads to chronic kidney disease.
Diagnosis of Bartter Syndrome and Gitelman Syndrome
Serum and urine electrolyte levels
Exclusion of similar disorders
Genetic testing
Bartter syndrome and Gitelman syndrome should be suspected in children with characteristic symptoms or incidentally noted laboratory abnormalities, such as metabolic alkalosis and hypokalemia. Measurement of urine electrolytes shows high levels of sodium, potassium, and chloride that are inappropriate for the euvolemic or hypovolemic state of the patient.
Diagnosis is by exclusion of other disorders:
Primary and secondary aldosteronism can often be distinguished by the presence of hypertension and normal or low plasma levels of renin (see table Distinguishing Primary and Secondary Aldosteronism).
Surreptitious vomiting or laxative abuse can often be distinguished by low levels of urinary chloride (usually < 20 mmol/L).
Surreptitious diuretic abuse can often be distinguished by low levels of urinary chloride and by a urine assay for diuretics.
A 24-hour measurement of urinary calcium or the urine calcium/creatinine ratio may help distinguish the 2 syndromes; the levels are typically normal to increased in Bartter syndrome and low in Gitelman syndrome.
Definitive diagnosis, including identification of disease subtypes, is through genetic testing, which is now becoming more widely available.
Children of carriers have a 25% chance of being affected by a recessive form, so asymptomatic siblings should be screened for electrolyte derangements, primarily hypokalemia and metabolic alkalosis, as well as hypomagnesemia. Parents of an affected child can consider consulting a genetic counselor regarding prenatal and preimplantation genetic screening for subsequent pregnancies.
Treatment of Bartter Syndrome and Gitelman Syndrome
For Bartter syndrome, nonsteroidal anti-inflammatory drugs (NSAIDs)
Sodium, potassium, and magnesium supplements
Because renal prostaglandin E2 secretion contributes to the pathogenesis in Bartter syndrome, NSAIDs (eg, oral indomethacin or ibuprofen) may be used (Because renal prostaglandin E2 secretion contributes to the pathogenesis in Bartter syndrome, NSAIDs (eg, oral indomethacin or ibuprofen) may be used (1, 2). Selective cyclooxygenase (COX)-2 inhibitors (eg, celecoxib) also may be used in patients with Bartter syndrome. If COX-2 inhibitors are used, patients may be given medications to suppress gastric acid (). Selective cyclooxygenase (COX)-2 inhibitors (eg, celecoxib) also may be used in patients with Bartter syndrome. If COX-2 inhibitors are used, patients may be given medications to suppress gastric acid (1, 2).
Electrolyte supplementation is the mainstay of management. Regimens usually include sodium chloride, usually 5 to 10 mEq/kg/day. Additionally, potassium chloride supplementation should be given, initially about 1 to 3 mEq/kg/day. Patients with magnesium wasting should be given magnesium salts, but this therapy may be limited by the development of diarrhea. Some Electrolyte supplementation is the mainstay of management. Regimens usually include sodium chloride, usually 5 to 10 mEq/kg/day. Additionally, potassium chloride supplementation should be given, initially about 1 to 3 mEq/kg/day. Patients with magnesium wasting should be given magnesium salts, but this therapy may be limited by the development of diarrhea. Somemagnesium salts, such as aspartate, citrate, or lactate, have better bioavailability. A high solute load, such as that resulting from sodium supplementation, should be avoided in patients who have a urinary concentrating defect, secondary nephrogenic diabetes insipidus, or both because it exacerbates polyuria and polydipsia resulting from obligate water loss and could precipitate significant hypernatremia. In general, electrolyte supplementation should try to maintain adequate serum levels with minimal fluctuation, thus, dosing should be spread out as long as it does not significantly increase the risk of nonadherence.
Although potassium-sparing diuretics, angiotensin-converting enzyme inhibitors, and angiotensin receptor blockers have been used in some patients, current consensus is that these therapies are largely unproved (1).
Thiazide diuretics for management of hypercalciuria are generally not recommended. They may complicate sodium supplementation, which can worsen the risk of nephrolithiasis and nephrocalcinosis.
Nutritional optimization is important, especially for infants and young children.
Exogenous growth hormone may be considered to treat short stature.
Treatment reference
Konrad M, Nijenhuis T, Ariceta G, et al: Diagnosis and management of Bartter syndrome: Executive summary of the consensus and recommendations from the European Rare Kidney Disease Reference Network Working Group for Tubular Disorders. Kidney Int 99(2):324–335, 2021. doi: 10.1016/j.kint.2020.10.035
Key Points
Both Bartter and Gitelman syndromes have impaired sodium chloride reabsorption, which causes mild volume depletion, leading to increases in renin and aldosterone release, resulting in urinary potassium and hydrogen losses.
Manifestations vary depending on genotype, but growth and development may be affected and electrolyte abnormalities may cause muscle weakness, cramping, spasms, tetany, or fatigue.
Diagnosis involves serum and urinary electrolyte measurement; genetic testing is becoming more available for confirmation and identification of the Bartter subtypes.
Treatment involves electrolyte replacement; for Bartter syndrome, NSAIDs also are given.
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3477 | https://www.investopedia.com/terms/a/amortization.asp | Skip to content
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Table of Contents
Table of Contents
What Is an Amortization Schedule?
How a Loan Amortization Schedule Works
How to Calculate Loan Amortization
Loan Amortization Schedule vs. Loan Term
Benefits of a Loan Amortization Schedule
How Amortization Schedules for Intangible Assets Work
Example of an Intangible Asset Amortization Schedule
Example
The Bottom Line
What Is an Amortization Schedule? How to Calculate With Formula
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What Is an Amortization Schedule?
An amortization schedule is a chart that tracks the falling book value of a loan or an intangible asset over time. For loans, it details each payment’s breakdown between principal and interest. For intangible assets, it outlines the systematic allocation of the asset’s cost over its useful life.
Key Takeaways
Amortization schedules outline the payments needed to pay off a loan and how the portion allocated to principal versus interest changes over time.
Early in the loan term, the interest portion is larger due to the higher loan balance.
Over time, the interest portion shrinks, and a larger portion of each payment goes toward reducing the principal balance.
Businesses use a different kind of amortization schedule to expense intangible assets over their useful life.
How a Loan Amortization Schedule Works
Most people use “amortization schedule” in the context of loans, where it outlines how a loan is paid down over time. It details the total number of payments and the proportion of each that goes toward principal versus interest. Principal is the unpaid loan balance, excluding any interest or fees, while interest is the cost of borrowing charged by lenders.
At the start of the loan term, when the loan balance is highest, a higher percentage of each payment goes toward interest. Over time, as the loan balance decreases, the interest portion shrinks, and more of each payment goes toward the principal.
Fast Fact
Accountants use amortization to spread out the costs of an asset over the useful lifetime of that asset.
How to Calculate Loan Amortization
The formula to calculate the monthly principal due on an amortized loan is as follows:
Principal Payment=TMP−(OLB×12 MonthsInterest Rate)where:TMP=Total monthly paymentOLB=Outstanding loan balance
The total monthly payment is typically specified when you take out a loan. However, you may need to calculate the monthly payment if you are attempting to estimate or compare monthly payments based on a given set of factors, such as loan amount and interest rate. If you need to calculate the total monthly payment for any reason, the formula is as follows:
Total Payment=Loan Amount×[(1+i)n−1i×(1+i)n]where:i=Monthly interest paymentn=Number of payments
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You’ll need to divide your annual interest rate by 12. For example, if your annual interest rate is 3%, your monthly interest rate will be 0.25% (0.03 annual interest rate ÷ 12 months). You'll also multiply the years in your loan term by 12. For example, a four-year car loan would have 48 payments (four years × 12 months).
Loan Amortization Schedule vs. Loan Term
Though related, loan amortization schedule and loan term are not the same. Loan amortization refers to the schedule over which payments are calculated, while loan term is the period before the loan is due. For example, a loan may be amortized over 30 years but have a 10-year term. In this case, payments are based on a 30-year schedule, but at the end of the 10-year term, the remaining balance (a balloon payment) must be paid off or refinanced.
Benefits of a Loan Amortization Schedule
Though it may look daunting, a loan amortization schedule is a powerful tool. Consider these benefits:
Budgeting: Knowing exactly how much you’ll owe every month can help you budget.
Transparency: Seeing the total interest cost can help you understand the full cost of the loan, so you can compare it against other loan offers.
Tax deductions: Some types of interest (such as home mortgage interest) may be tax-deductible, making it important to partition principal from interest contributions.1
Early repayment: With an amortization schedule, you can see how reducing the loan balance with early payments can cut your total interest costs and shorten the loan term (but beware of prepayment penalties).
How Amortization Schedules for Intangible Assets Work
Businesses also use amortization schedules to account for the declining value of intangible assets like patents, trademarks, and goodwill. They do this to understand their earnings better, comply with accounting standards like GAAP, and sometimes reduce their taxable income.2
The process is similar to how tangible assets are depreciated. Typically, businesses use the straight line method to allocate the cost of an intangible asset evenly over its expected useful life. For example, a $10,000 patent with a 10-year useful life would be amortized at $1,000 per year ($10,000 /10). Unlike loan amortizations, no principal or interest is involved, making the calculation more straightforward. Divide the asset’s cost evenly over its useful life.
Example of an Intangible Asset Amortization Schedule
| Year | Beginning Book Value | Amortization Expense | Ending Book Value |
| 1 | $10,000 | $1,000 | $9,000 |
| 2 | $9,000 | $1,000 | $8,000 |
| 3 | $8,000 | $1,000 | $7,000 |
| 4 | $7,000 | $1,000 | $6,000 |
| 5 | $6,000 | $1,000 | $5,000 |
| 6 | $5,000 | $1,000 | $4,000 |
| 7 | $4,000 | $1,000 | $3,000 |
| 8 | $3,000 | $1,000 | $2,000 |
| 9 | $2,000 | $1,000 | $1,000 |
| 10 | $1,000 | $1,000 | $0 |
Important
The IRS has schedules that dictate the total number of years in which tangible and intangible assets are expensed for tax purposes.
Example of Amortization
For purposes of illustration, consider a $30,000 car loan at 3% interest with a term of 4 years. The monthly payment is going to be $664.03. That is arrived at as follows:
$30,000×(0.0025×1.002548)−10.0025×1.002548
In the first month, $75 of the $664.03 monthly payment goes to interest.
$30,000 loan balance×3% interest rate÷12 months
The remaining $589.03 goes toward the principal.
$664.03 total monthly payment−$75 interest payment
The total payment remains constant over each of the 48 months of the loan while the amount going to the principal increases and the portion going to interest decreases. In the final month, only $1.66 is paid in interest because the outstanding loan balance is minimal compared with the starting loan balance.
| Loan Amortization Schedule | | | | |
| Period | Total Payment Due | Computed Interest Due | Principal Due | Principal Balance |
| | | | | $30,000 |
| 1 | $664.03 | $75 | $589.03 | $29,410.97 |
| 2 | $664.03 | $73.53 | $590.50 | $28,820.47 |
| 3 | $664.03 | $72.05 | $591.98 | $28,228.49 |
| 4 | $664.03 | $70.57 | $593.46 | $27,635.03 |
| 5 | $664.03 | $69.09 | $594.94 | $27,040.09 |
| 6 | $664.03 | $67.60 | $596.43 | $26,443.66 |
| 7 | $664.03 | $66.11 | $597.92 | $25,845.74 |
| 8 | $664.03 | $64.61 | $599.42 | $25,246.32 |
| 9 | $664.03 | $63.12 | $600.91 | $24,645.41 |
| 10 | $664.03 | $61.61 | $602.42 | $24,042.99 |
| 11 | $664.03 | $60.11 | $603.92 | $23,439.07 |
| 12 | $664.03 | $58.60 | $605.43 | $22,833.64 |
| 13 | $664.03 | $57.08 | $606.95 | $22,226.69 |
| 14 | $664.03 | $55.57 | $608.46 | $21,618.23 |
| 15 | $664.03 | $54.05 | $609.98 | $21,008.24 |
| 16 | $664.03 | $52.52 | $611.51 | $20,396.73 |
| 17 | $664.03 | $50.99 | $613.04 | $19,783.69 |
| 18 | $664.03 | $49.46 | $614.57 | $19,169.12 |
| 19 | $664.03 | $47.92 | $616.11 | $18,553.02 |
| 20 | $664.03 | $46.38 | $617.65 | $17,935.37 |
| 21 | $664.03 | $44.84 | $619.19 | $17,316.18 |
| 22 | $664.03 | $43.29 | $620.74 | $16,695.44 |
| 23 | $664.03 | $41.74 | $622.29 | $16,073.15 |
| 24 | $664.03 | $40.18 | $623.85 | $15,449.30 |
| 25 | $664.03 | $38.62 | $625.41 | $14,823.89 |
| 26 | $664.03 | $37.06 | $626.97 | $14,196.92 |
| 27 | $664.03 | $35.49 | $628.54 | $13,568.38 |
| 28 | $664.03 | $33.92 | $630.11 | $12,938.28 |
| 29 | $664.03 | $32.35 | $631.68 | $12,306.59 |
| 30 | $664.03 | $30.77 | $633.26 | $11,673.33 |
| 31 | $664.03 | $29.18 | $634.85 | $11,038.48 |
| 32 | $664.03 | $27.60 | $636.43 | $10,402.05 |
| 33 | $664.03 | $26.01 | $638.02 | $9,764.02 |
| 34 | $664.03 | $24.41 | $639.62 | $9,124.40 |
| 35 | $664.03 | $22.81 | $641.22 | $8,483.18 |
| 36 | $664.03 | $21.21 | $642.82 | $7,840.36 |
| 37 | $664.03 | $19.60 | $644.43 | $7,195.93 |
| 38 | $664.03 | $17.99 | $646.04 | $6,549.89 |
| 39 | $664.03 | $16.37 | $647.66 | $5,902.24 |
| 40 | $664.03 | $14.76 | $649.27 | $5,252.96 |
| 41 | $664.03 | $13.13 | $650.90 | $4,602.06 |
| 42 | $664.03 | $11.51 | $652.52 | $3,949.54 |
| 43 | $664.03 | $9.87 | $654.16 | $3,295.38 |
| 44 | $664.03 | $8.24 | $655.79 | $2,639.59 |
| 45 | $664.03 | $6.60 | $657.43 | $1,982.16 |
| 46 | $664.03 | $4.96 | $659.07 | $1,323.09 |
| 47 | $664.03 | $3.31 | $660.72 | $662.36 |
| 48 | $664.03 | $1.66 | $662.36 | $0.00 |
The Bottom Line
Reading an amortization schedule is one thing, but knowing how to create one is another. Use this newfound skill to analyze and compare loan offers and business earnings. The more you know, the better financial decisions you can make.
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3478 | https://math.stackexchange.com/questions/1498398/controllability-of-a-system-with-point-matrices | linear algebra - Controllability of a system with point matrices - Mathematics Stack Exchange
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Controllability of a system with point matrices
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Consider the scalar nonlinear system
d d t x=sin x+uf(x,u)d d t x=sinx+u⏟f(x,u)
with equilibrium point (x∗,u∗)=(0,0)(x∗,u∗)=(0,0)
We have ∂f∂x=cos x∂f∂x=cosx and ∂f∂u=1∂f∂u=1.
Thus ∂f∂x(x∗,u∗)=cos(0)=1∂f∂x(x∗,u∗)=cos(0)=1 and ∂f∂u(x∗,u∗)=1∂f∂u(x∗,u∗)=1
So linearisation around the equilibrium yields
d d t Δ x=A Δ x+B Δ u d d t Δ x=A Δ x+B Δ u
with A=∂f∂x(x∗,u∗)A=∂f∂x(x∗,u∗) and B=∂f∂u(x∗,u∗)B=∂f∂u(x∗,u∗). Hence
d d t Δ x=Δ x+Δ u d d t Δ x=Δ x+Δ u
I want to determine if (A,B)(A,B), i.e. (1,1)(1,1) is controllable, but can this be determined for singular matrices such as these? For instance, I don't believe the controllability matrices, C=[B A B A 2 B⋯A n−1 B]C=[B A B A 2 B⋯A n−1 B] can be computed.
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edited Oct 26, 2015 at 13:14
Jason BornJason Born
asked Oct 26, 2015 at 13:05
Jason BornJason Born
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In this case n=1 n=1, so controllability matrix is 1×1 1×1 matrix which is equal to B B. Since B B is nonsingular, i.e. ≠0≠0 when scalar, the system is controllable.
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answered Oct 26, 2015 at 15:32
obareeyobareey
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If A A and B B were both zero then the system wouldn't be controllable, since B B would be singular?Jason Born –Jason Born 2015-10-29 17:56:30 +00:00 Commented Oct 29, 2015 at 17:56
@user3482534 That's right. Actually A A doesn't need to be zero, only B=0 B=0 is enough for uncontrollability in this case.obareey –obareey 2015-10-30 06:03:48 +00:00 Commented Oct 30, 2015 at 6:03
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3479 | https://www.gauthmath.com/solution/1835046882274369/-t-_0frac-4sin-4x-cos-4xdx-square-Simplify-your-answer-Type-an-exact-answer-usin | Solved: ∈t _0^((frac π)4)(sin (4x)+cos (4x))dx=□ (Simplify your answer. Type an exact answer, usin [Calculus]
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∈t _0^((frac π)4)(sin (4x)+cos (4x))dx=□ (Simplify your answer. Type an exact answer, using radicals as needed.)
Show transcript
Gauth AI Solution
100%(4 rated)
Answer
The answer is 1/2
Explanation
Indefinite Integration: We begin by computing the indefinite integral of the given expression. Applying the linearity of integration and standard trigonometric integral formulas, we have:
∫(sin(4x) + cos(4x))dx = (-1/4)cos(4x) + (1/4)sin(4x) + C, where C is the constant of integration.
Definite Integration: We now evaluate the definite integral over the specified interval [0, π/4]. Using the Fundamental Theorem of Calculus, we substitute the limits of integration into the antiderivative obtained in Step 1:
[(-1/4)cos(4x) + (1/4)sin(4x)] evaluated from 0 to π/4.
Evaluating at Limits: Substituting the upper and lower limits of integration, we get:
[(-1/4)cos(π) + (1/4)sin(π)] - [(-1/4)cos(0) + (1/4)sin(0)]
Simplification: Using the known values of trigonometric functions at 0 and π (cos(π) = -1, sin(π) = 0, cos(0) = 1, sin(0) = 0), we simplify the expression:
[(-1/4)(-1) + (1/4)(0)] - [(-1/4)(1) + (1/4)(0)] = (1/4) - (-1/4) = 1/2
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3480 | https://www.youtube.com/watch?v=eMUjcJ203Ao | Escasez | Khan Academy en Español
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Posted: 16 Jan 2018
Transcript:
y todo en la economía está basado en la noción de la escasez de recursos que significa escasez en el contexto cotidiano significa que no existe cantidad suficiente o ilimitada de algo que las personas necesitan es decir cantidad limitada de recursos los ejemplos más comunes especialmente en las clases de economía de los recursos escasos son la tierra y con esto me refiero a recursos naturales la energía incluso podría ser el agua la madera y también los metales obviamente también se refiere a terrenos podríamos representarlo en esta imagen claramente vemos un terreno de cultivo incluso podemos decir que la energía solar que hace que funcione todo esto entra en esta categoría de recursos naturales otro tipo de recurso escaso del que se habla mucho en las clases de economía es el trabajo y el trabajo lo asociamos inmediatamente con las personas estamos hablando de la fuerza de trabajo es decir hablamos de los trabajadores y las horas de trabajo que realizan no sólo físicamente también intelectualmente aquí vemos una foto de personas trabajando otro recurso del que se habla mucho en el contexto de la escasez es el capital el concepto de capital puede ser un poco confuso al principio pero se refiere a las cosas que tienen las personas para fabricar otras cosas para comprender mejor esto pausa en el vídeo observen estas imágenes y traten de encontrar las cosas que se refieren al capital en estas imágenes podemos encontrar varias cosas que funcionan como capital hay cosas obvias como estas herramientas las cuales son un claro ejemplo de lo que es capital y también estos edificios construidos para albergar animales y cosechas son cosas que alguien hizo para ayudar a crear otras cosas también encontramos cosas poco obvias como estas líneas de corriente eléctrica alguien las fabricó para ayudarnos a distribuir energía eléctrica que a su vez se usa para crear otras cosas este es otro ejemplo de capital un cuarto recurso del cual escucharán hablar en economía es el saber cómo combinar los otros tres tipos de recursos y a este le llamaremos organización algunas personas también hablarán de la tecnología como recurso escaso pero organizar se refiere a cómo combinar los demás recursos para obtener algo mejor alguien decidió que este era el mejor lugar para colocar estas construcciones alguien tuvo que decidir el mejor lugar para poner los postes de electricidad alguien decidió qué cosas cultivar aquí y cómo sembrar las en esta parte alguien decidió qué tipo de trabajo se necesitaba aquí qué tipo de terreno y qué tipo de capital era necesario para construir esta fábrica el equipo necesario para fabricar lo que sea que fabriquen aquí a esto nos referimos cuando hablamos de organización aquí lo importante es que vean a su alrededor no hay una cantidad ilimitada de terreno para un propósito específico no hay una cantidad ilimitada de recursos naturales energía madera o ciertos metales tampoco hay una cantidad ilimitada de trabajo solo hay ciertas personas que pueden trabajar y no existe una cantidad ilimitada de capital es decir no tenemos una cantidad infinita de edificios o de equipo y tampoco hay una cantidad infinita de organización hay cierta cantidad de energía y tiempo que podemos usar para pensar en cómo combinar todas estas cosas conforme avancemos con los vídeos de economía veremos la escasez de estos cuatro recursos que aparecen una y otra vez nos vemos en otro vídeo |
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部首: 木 部首笔画: 4画 总笔画: 6画
康熙字典笔画: (襍:18;雜:18)
汉程字典详细解释返回分类
汉字五行：水
五笔编码：vsu
仓颉编码：KND
四角号码：40900
UniCode：U+6742
繁体字集：襍雜
异体字集：雑
姓名用字：在线起名
汉程字典解释 杂
（雜）
zá
多种多样的，不单纯的：杂乱。杂沓。杂感。杂志。杂货。杂居。杂务。杂品。错综复杂。私心杂念。
混合：夹杂。混杂。杂交。
纯
笔画数：6；
部首：木；
笔顺编号：351234
索引参考 [故训彙纂]：2076|2154.11
[康熙字典]：页510第18(点击查看原图)
[ 汉语字典 ]：卷2页1155第07
详细解释 杂
雜、襍
zá
【动】
(形声。从衣,集声。本义:五彩相合)
同本义〖multicoloured〗
画绘之事,杂五色。——《周礼·考工记》
混合;搀杂〖mix;mingle〗
故先王以土与金木水火杂,以成百物。——《国语·郑语》
彼此错杂。——清·徐珂《清稗类钞·战事类》
又如:杂烩菜(将各种剩菜合并在一起的菜淆);夹杂(搀杂);杂就(参杂而成);混杂(混合搀杂)
聚会;聚集〖gettogether;assemble〗
杂,聚也。——《广雅》
四方来杂,远乡皆至。——《吕氏春秋》
又如:杂物(聚集事物)
杂
雜
zá
【形】
驳杂不纯〖heterogeneous;mixed〗
夹岸数百步,中无杂树。——晋·陶潜《桃花源记》
又如:杂面(杂合面粉作的面条)
杂彩三百匹。——《玉台新咏·古诗为焦仲卿妻作》
杂花生树。——南朝梁·丘迟《与陈伯之书》
昧没而杂。——唐·柳宗元《柳河东集》
杂然而前陈。——宋·欧阳修《醉翁亭记》
少杂树。——清·姚鼐《登泰山记》
又如:杂裳(前黑后黄的下衣)
紊乱〖disorderly;chaotic〗。如:杂错(交错混杂);嘈杂(声音杂乱;喧闹);庞杂(多而杂乱)
繁琐;细碎〖(ofwritingsandspeech)manyandmiscellaneouswithtrifles〗。如:杂冗(零杂事务;公务繁忙的谦辞);杂言(不主一家而杂采众家的言论);杂说(博采众家的学说)
众多〖ofpopulationmultitudinous;numerous〗
杂然相许。——《列子·汤问》
又如:杂沓(众多而纷杂的样子);杂袭(众多的样子);杂玩(各种玩物)
各种〖various〗。如:杂趁(多种非正式职业;做零活);杂服(各种服制);杂帛(各种细绢的通称)
正项以外〖extra;otherthan〗。如:杂学旁收(指不去攻读八股文,而喜好诗词、曲赋、小说等其他门类的学问)
交错,交会〖crisscross〗
嘈嘈切切错杂弹。——唐·白居易《琵琶行(并序)》诗
杂植兰桂竹木。——明·归有光《项脊轩志》
杂
雜
zá
【副】
都;共同〖together;jointly;common〗
杂曰:“投诸渤海之尾,隐土之北。”——《列子·汤问》
杂
雜
zá
【名】
旧时等外的小官为杂职,如清代九品未入流之类或统称为佐杂〖minorofficial〗
听我将、弁、参、杂管领。——清·严如熤《苗防备览要略》
传统戏曲角色名。元杂剧、明清传奇以至京剧里的“杂”,一般扮演杂差、百姓等人物〖apartoftraditionalopera〗
杂绯衣扮秦国引院子梅香各乘车行上。——《长生殿》
〖量〗通“匝”(zā)。圈〖circle〗
守宫三杂,外环,隅为之楼。——《墨子》
并行而杂,是礼之中流也。——《荀子·礼论》
以数杂之寿。——《淮南子·诠言》
杂拌儿
zábànr
〖mixedpreservedfruits〗∶多种类相杂混合的食品
杂拌儿糖
〖mixture;hotchpotch〗∶比喻各种杂事的集萃
杂拌儿趣闻
杂草
zácǎo
〖weed;rankgrass〗指各种野草
这块土地上的杂草必须清除掉
杂凑
zácòu
〖ragbag;knocktogether〗把不同类别的事或人凑合于一处
这是一支杂凑的队伍,不堪一击
杂感
zágǎn
〖randomthoughts〗∶对现实生活和社会现象的各种零散感受
〖atypeofliteraturerecordingthoughts〗∶抒发零散感受的文字
杂工
zágōng
〖choreman〗干多种杂活的工人
杂环
záhuán
〖heterocycle〗一个杂环的环系统或一个杂环的化合物
杂烩
záhuì
〖mixedstew〗∶各种菜合在一起烩成的菜
素杂烩
〖stew〗∶比喻各种事物的集萃
杂婚
záhūn
〖mixedmarriage;promitivepromiscuity〗不同种族或宗教信仰的人们之间的婚姻
杂货
záhuò
〖sundrygoods〗各种各样的生活日用品
杂货店
杂货店
záhuòdiàn
〖emporiumorgrocery〗出售杂货的商店
杂记
zájì
〖jottings;notes〗∶记载杂项的笔记;零碎的笔记
〖miscellanies(asatypeofliterature)〗∶写风景、琐事、感想等的一种文体
杂技
zájì
〖acrobatics〗指车技、口技、顶碗、走钢丝、变戏法等技艺
在杂技场表演杂技
杂家
zájiā
〖theEclectics,aschoolofthoughtsflourishinginancientChina;miscellaneousscholars〗∶先秦时期的一个思想流派,以吕不韦为代表,融会各家学说,内容比较庞杂
〖sage〗∶指知识广博,什么都懂一些的人
杂件，杂件儿
zájiàn,zájiànr
〖sundrygoods〗杂货;各种小用品
杂件什物
杂交
zájiāo
〖hybrid〗一品系与另一品系之间的交合繁殖
杂交水稻
杂居
zájū
〖twoormorenationalitieslivetogether〗指若干民族在一个地区居住
满汉杂居
杂剧
zájù
〖poeticdramasettomusic,flourishingintheYuanDynasty〗宋代的一种以滑稽调笑为特点的表演。元代发展成戏曲,每本多为四折,每折由同宫调同韵的北曲套曲和宾白组成。明清两代的杂剧每本不限四折
杂粮
záliáng
〖foodgrainsotherthanwheatandrice;coarsecereals;miscellaneousgraincrops〗稻谷、小麦以外的各种粮食
五谷杂粮
杂流
záliú
〖craftman〗旧时对手艺工人的蔑称
工艺杂流。——清·薛福成《观巴黎油画记》
亦作“杂沓
骈罗列布,鳞从杂沓兮。——《汉书·扬雄传》
杂录
zálù
〖varia〗∶文学杂文集
〖potpourri〗∶杂集
杂乱
záluàn
〖wilderness;bemixedanddisorderly;inajumble〗繁杂而凌乱
一头杂乱的白发
杂乱无章
záluàn-wúzhāng
〖mess;rambling;bedisorderlyandunsytematic〗∶无条理、无规律
一篇杂乱无章的长篇大论
杂牌，杂牌儿
zápái,zápáir
〖inferiorbrand〗非正规的;非正牌或名牌的
这辆自行车是杂牌的
杂七杂八
záqī-zábā
〖miscellaneous;assortmentofbitofeverything〗形容十分混杂
他有处理杂七杂八事情的天才
杂糅
záróu
〖blend;mingle;mix〗交错混杂,浑然一体
中西杂糅,珠联璧合
杂色
zásè
〖varicolored〗∶具有各种颜色的
杂色的大理石
〖motley;motle〗∶混杂不纯的颜色
穿着专为他们设计的杂色的奇异服装
杂食
záshí
〖omnivorous〗兼食动植物性食物
杂食有利于健康
杂耍
záshuǎ
〖sideshow;vandeville〗旧时对曲艺、杂技等技艺的合称。亦称“什样杂耍”
杂税
záshuì
〖miscellaneouslevies;sundrytax〗旧指正税以外的各种税
苛捐杂税
杂说
záshuō
〖variousopinions〗∶各种说法
杂说不一
〖scatteredessays〗∶零碎的论说文章
杂碎
zásuì
〖choppedcookedentails(ofsheeporoxen)〗牛、羊内脏做成的熟食;繁杂琐碎;比喻心肠
我看他没安好杂碎
杂遝
zátà
〖benumerousanddisorderly〗众多杂乱的样子
人马杂遝。——清·薛福成《观巴黎油画记》
亦作“杂沓”
骈罗列布,鳞从杂沓兮。——《汉书·扬雄传》
杂谈
zátán
〖tittle-tattle〗各种命题、不拘一格的论述
杂务
záwù
〖oddjobs;sundryduties〗正事以外的琐碎事务
还有些杂务要处理
杂物
záwù
〖truck〗∶无价值的小零碎物品
〖mess〗∶各种各样杂乱的东西
杂项
záxiàng
〖sundries〗各种名目的
国家的杂项税收
杂役
záyì
〖mendoingodd-jobs〗旧指受雇做杂事的人
杂音
záyīn
〖noise〗心、肺、机器、收音机等发出的不正常声音
杂志
zázhì
〖journal;magazine;periodical〗∶期刊活的定期出版物
《高等教育杂志》第10期
〖jottings〗∶杂记
杂质
zázhì
〖foreignsubstance〗一种物质中所夹杂的不纯成分
杂种
zázhǒng
〖hybrid〗∶杂交产生的子代种系
无取杂种。——南朝梁·丘迟《与陈伯之书》
〖bastard;sonofabitch〗∶粗鲁的骂人话
杂字
zázì
〖collectionofwords〗把各种常用字缀集成韵,以便于记诵的字册
六言杂字
字形结构 [ 首尾分解查字 ]：九木 (jiumu)
[ 笔顺编号 ]：351234
[ 笔顺读写 ]：撇折横竖撇捺
汉语字典解释[①]［zá］
［《廣韻》徂合切，入合，從。］
亦作“襍1”。“杂1”的繁体字。“籴2”的被通假字。亦作“雑1”。
(1)组合；配合。
(2)混杂；参杂。
(3)驳杂；不精纯。
(4)紊乱；使紊乱。
(5)多；繁多。
(6)装饰。
(7)兼及。
(8)共同，一起。
(9)一种诗体。参见“雜詩”。
(10)古代杂剧、传奇中生、旦、净、丑以外的一种行当，一般扮演各种临时上场、无关重要的人物。
(11)通“集”。集合；聚集。
(12)通“匝”。圆周。
康熙字典解释 【戌集中】【隹字部】雜【廣韻】徂合切【集韻】【韻會】昨合切，𠀤 音 䕹。【說文】五彩相合也。【玉篇】糅也。【易·坤卦】夫 𤣥 黃者，天地之雜也。【周禮·冬官考工記】畫繢之事雜五色。 又【禮·玉藻·雜帶註】雜，猶飾也，卽上之韠也。 又【玉篇】同也。【廣韻】集也。【易·繫辭】雜物撰德。【疏】言雜聚天下之物。 又【揚子·方言】碎也。【易·繫辭】其稱名也，雜而不越。【疏】辭理雜碎，各有倫序，而不相乖越。 又【玉篇】厠也。 又最也。 又【廣韻】帀也。 又穿也。 又鳥名。【爾雅·釋鳥】爰居，雜縣。【疏】爰居，海鳥也，一名雜縣。 又【集韻】七盍切【韻會】【正韻】七合切，𠀤 音囃。【公羊傳·成十五年】諸大夫皆雜然曰：仲氏也，其然乎。【釋文】雜，七合反，又如字。
音韵参考 [ 平水韵 ]：入声十五合
[ 国 语 ]：zá
[ 粤 语 ]：zaap6
[ 闽南语 ]：chap8
说文解字详解【卷八】【衣部】 编号：5307
雜,徂合切 ,五彩相會。从衣集聲。
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3482 | https://help.datylon.com/tutorials/formatting-date-and-time-labels-charts | How to format date and time labels in charts
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How to format date and time labels in charts
Learn how to format date and time labels and axes for Datylon Line, Area, and Scatter charts
This article covers date & time formatting settings only. For Text and Number settings please refer to the following article.
Date & time formatting is only supported for Bar, Line, Area and Scatter charts.
There are two steps in formatting: to define the format of the input data, and to define the format of labels. Follow the steps below to visualize date & time labels and axes correctly:
Formatting input data in the Formatting tab
Formatting the visualization in the Styles tab
1. Formatting input data
To ensure correct date & time labels and axes, the input data has to be formatted correctly first.
The result of data update using swap workbook might differ from direct copy/paste, as the data updated using swap workbook can contain the local format setting. To make the result consistent between different import options use the same forced formatting.
Standard strings
The following standard strings are supported by default, meaning no specific formatting in the Formatting tab is required. So formatting can be kept as None to visualize them correctly as a Date & Time label when below strings are used.
US short time format strings, separated by a dash (-), or a slash (/):
2019
2019-09
2019-09-30
2019-09-30T14:30:00
US long time format as in
time formats with or without time and year
Unix Time Stamp
A number counting the seconds as of January 1st, 1970 at UTC
Formatting tab
In all other cases, the input data shall first be formatted in the Formatting tab.
The data in the datasheets can have many different formats regarding date & time. Sometimes as year only, up to full year/month/day with hours, minutes and seconds.
In the Formatting tab, you can specify the format of the incoming data after selecting the relevant column or row and clicking the Date & Time button.
In the Date Format list, you have several default options next to Custom, which allows you to define a custom date & time format string by using any of the specifiers below:
%a- abbreviated weekday name.
%A- full weekday name.
%b- abbreviated month name.
%B- full month name.
%c- the locale’s date and time, such as%x, %X.
%d- zero-padded day of the month as a decimal number [01,31].
%e- space-padded day of the month as a decimal number [ 1,31]; equivalent to%_d.
%f- microseconds as a decimal number [000000, 999999].
%H- hour (24-hour clock) as a decimal number [00,23].
%I- hour (12-hour clock) as a decimal number [01,12].
%j- day of the year as a decimal number [001,366].
%L- milliseconds as a decimal number [000, 999].
%m - month as a decimal number [01,12].
%M - minute as a decimal number [00,59].
%p - either AM or PM.
%q - quarter as a decimal number [1,4]
%Q- milliseconds since UNIX epoch.
%s- seconds since UNIX epoch.
%S- second as a decimal number [00,61].
%u- Monday-based (ISO 8601) weekday as a decimal number [1,7].
%U- Sunday-based week of the year as a decimal number [00,53].
%V- ISO 8601 week of the year as a decimal number [01, 53].
%w- Sunday-based weekday as a decimal number [0,6].
%W- Monday-based week of the year as a decimal number [00,53].
%x- the locale’s date, such as%-m/%-d/%Y.
%X- the locale’s time, such as%-I:%M:%S %p.
%y- year without century as a decimal number [00,99].
%Y- year with century as a decimal number.
%Z- time zone offset, such as-0700,-07:00,-07, or Z.
%%- a literal percent sign (%).
Directives marked with an asterisk () may be affected by the locale definition.
For example:
%Y refers to the year, %m the month and %d the day in numbers. These specifiers can be combined with other characters: e.g. "2022-07-04", as in the 4th of July 2022, should be defined as %Y-%m-%d.
2. Formatting thevisualization
If the standard strings are used, or once the format of the input data is set correctly, check the DateTime button in Styles > X or Y-axis.
In the DateTime Format field you can then define how the axis labels are visualized by using the same specifiers as above.
The Locale option allows you to select one of 31 languages for the full and abbreviated month and weekday names.
Example 1
There is a dataset with various formats in "Date" column. In the given case all the formats are automatically recognized by Datylon so there is no need to change anything in the Formatting tab. Each value will be transformed into a unified format:
"2022" will be interpreted as "2022-01-01 00:00:00".
"2022 Feb" will be interpreted as "2022-02-01 00:00:00".
"2022 Apr 02" will be interpreted as "2022-04-02 00:00:00" etc.
If a month or date is missing it will be recorded as "01".
If hours, minutes or seconds are missing it will be recorded as "00".
If a year is missing it will be recorded as "2001".
The next step is to set the X-Axis. Currently it is set to "Categorical" type.
This means that all the data from Date column will be perceived as text. To change that select the Datetimetype.
Now data points are dispersed according to data. If we want to place ticks at the start of every month we should set Min Date to "2022-01-01 00:00:00", turn on Major ticks and grid and Custom ticks amount, set Ticks Interval to "1 month" and set Starting Tick to "2022-01-01 00:00:00". To apply full name of the month for X-Axis labels set Time Format to "%B" (the list of available options can be seen above in the current article).
Example 2
If we use a dataset with date & time values that couldn't be recognized automatically by Datylon, changes have to be made in the Formatting:
Formatting>Date (column with date & time values) >Date & Time>Date Format>Custom>%H - %M %d %b %y (for the current example)
After that data would be in a format that is recognisable by Datylon. One can proceed to styling as shown above.
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3483 | https://artofproblemsolving.com/wiki/index.php/1952_AHSME_Problems/Problem_8?srsltid=AfmBOopPasEZNrT5SK8ggRN0na6k6iRX_b5_w-r1cKlbsWQsA9ZpYkrO | Art of Problem Solving
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1952 AHSME Problems/Problem 8
Problem
Two equal circles in the same plane cannot have the following number of common tangents.
Solution
Two congruent coplanar circles will either be tangent to one another (resulting in common tangents), intersect one another (resulting in common tangents), or be separate from one another (resulting in common tangents). Having only common tangent is impossible, unless the circles are non-congruent and internally tangent.
See also
1952 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 7Followed by
Problem 9
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25•26•27•28•29•30•31•32•33•34•35•36•37•38•39•40•41•42•43•44•45•46•47•48•49•50
All AHSME Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
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3484 | https://flexbooks.ck12.org/cbook/ck-12-basic-geometry-concepts/section/3.4/primary/lesson/alternate-interior-angles-bsc-geom/ | Skip to content
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3.4
Alternate Interior Angles
Written by:Dan Greenberg | Lori Jordan |
Fact-checked by:The CK-12 Editorial Team
Last Modified: Aug 01, 2025
Alternate Interior Angles
Alternate interior angles are two angles that are on the interior of @$\begin{align}l\end{align}@$ and @$\begin{align}m\end{align}@$, but on opposite sides of the transversal.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
If @$\begin{align}l || m\end{align}@$, then @$\begin{align}\angle 1 \cong \angle 2\end{align}@$
Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
If then @$\begin{align}l || m\end{align}@$.
What if you were presented with two angles that are on the interior of two parallel lines cut by a transversal but on opposite sides of the transversal? How would you describe these angles and what could you conclude about their measures?
Examples
For Examples 1 and 2, use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.
Example 1
@$\begin{align}\angle EAF \cong \angle FJI\end{align}@$
None
Example 2
@$\begin{align}\angle EFJ \cong \angle FJK\end{align}@$
@$\begin{align}\overleftrightarrow{CG} || \overleftrightarrow{HK}\end{align}@$
Example 3
Find the value of @$\begin{align}x\end{align}@$.
The two given angles are alternate interior angles and equal.
@$$\begin{align}(4x-10)^\circ & = 58^\circ\
4x & = 68\
x & = 17\end{align}@$$
Example 4
True or false: alternate interior angles are always congruent.
This statement is false, but is a common misconception. Remember that alternate interior angles are only congruent when the lines are parallel.
Example 5
What does @$\begin{align}x\end{align}@$ have to be to make @$\begin{align}a || b\end{align}@$?
The angles are alternate interior angles, and must be equal for @$\begin{align}a || b\end{align}@$. Set the expressions equal to each other and solve.
@$$\begin{align}3x+16^\circ & = 5x-54^\circ\
70 = 2x\
35 = x\end{align}@$$
To make @$\begin{align}a || b, \ x = 35\end{align}@$.
Review
Is the angle pair @$\begin{align}\angle 6\end{align}@$ and @$\begin{align}\angle 3\end{align}@$ congruent, supplementary or neither?
Give two examples of alternate interior angles in the diagram:
For 3-4, find the values of @$\begin{align}x\end{align}@$.
For question 5, use the picture below. Find the value of @$\begin{align}x\end{align}@$.
@$\begin{align}m\angle 4 = (5x - 33)^\circ, \ m\angle 5 = (2x + 60)^\circ\end{align}@$
Are lines @$\begin{align}l\end{align}@$ and @$\begin{align}m\end{align}@$ parallel? If yes, how do you know?
For 7-10, what does the value of @$\begin{align}x\end{align}@$ have to be to make the lines parallel?
@$\begin{align}m\angle 4 = (3x-7)^\circ\end{align}@$ and @$\begin{align}m\angle 5 = (5x-21)^\circ\end{align}@$
@$\begin{align}m\angle 3 = (2x-1)^\circ\end{align}@$ and @$\begin{align}m\angle 6 = (4x-11)^\circ\end{align}@$
@$\begin{align}m\angle 3 = (5x-2)^\circ\end{align}@$ and @$\begin{align}m\angle 6 = (3x)^\circ\end{align}@$
@$\begin{align}m\angle 4 = (x-7)^\circ\end{align}@$ and @$\begin{align}m\angle 5 = (5x-31)^\circ\end{align}@$
Review (Answers)
Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
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Trevor M.
asked • 04/21/23
Standard form equation for a hyperbola
Find the standard form equation for a hyperbola with vertices at (5,0) and (-5,0) that passes through the point (6,11).
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As the vertices are only affected by the x-values, the standard form for a hyperbola that opens sideways would be (x-h)2/a2 - (y-k)2/b2 = 1 where (h,k) is the center of the hyperbola, "a" is half the length of the major axis, and "b" is half the length of the minor axis.
Our center in this case is (h,k)=(0,0) because of our vertices (5,0) and (-5,0) in which (0,0) is directly in-between.
Using either one of the vertices, we can use it to determine the values of a2 and b2:
(x-h)2/a2 - (y-k)2/b2 = 1
(5-0)2/a2 - (0-0)2/b2 = 1
52/a2 = 1
25 = a2
x2/25 - y2/b2 = 1
Because we are given a point of (6,11), we can determine the value of b2:
x2/25 - y2/b2 = 1
62/25 - 112/b2 = 1
36/25 - 121/b2 = 1
-121/b2 = -11/25
121/b2 = 11/25
3025 = 11b2
275 = b2
Thus, the equation for the hyperbola with vertices (±5,0) and passes through (6,11) is x2/25 - y2/275 = 1
Hope this helped!
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3486 | http://calculuscourse.maa.org/sample/Chapter5/Section5-3/Chapter5-3-4M.html | Section 5-3-4
Chapter 5
Modeling with Differential Equations
5.3 Periodic Motion
5.3.4 Period and Frequency
A function that repeats over and over, as do sine and cosine, is called periodic, and the horizontal length of a pattern that repeats is called a period. The shortest repetition length is called the fundamental period. We can also think of the fundamental period as being the time required to complete a single cycle (if time is the independent variable). Thus, sine and cosine are periodic functions with fundamental period 2 π 2 π. These functions are also periodic with period 4 π 4 π, with period 6 π 6 π, with period 8 π,...8 π,.... If there is no danger of confusion, "fundamental period" is often shortened to just "period." We adopt that convention in the rest of this section.
A concept closely related to period is frequency, which is the rate at which periods are being completed. Thus, if the unit of time is seconds, and a periodic function has period 5 5 seconds, then its frequency is 1/5 1/5 cycles per second. In general, frequency and period are reciprocals of each other. The unit for frequency is cycles per unit of time, and the unit for period is units of time per cycle.
Note 1 – Constant functions
Activity 4
For each of the following repeating phenomena, determine the period and frequency (in compatible time units) of whatever function describes the phenomenon.
Normal heart beat
Rotation of the Earth on its axis
Earth orbiting the Sun
Standard alternating current
Second hand on a clock
Minute hand on a clock
Hour hand on a clock
Comment on Activity 4
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Contents for Chapter 5 |
3487 | https://math.stackexchange.com/questions/1300284/polynomial-of-degree-2-has-at-most-2-roots | real analysis - Polynomial of degree $2$ has at most $2$ roots - Mathematics Stack Exchange
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Polynomial of degree $2$ has at most $2$ roots
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Asked 10 years, 4 months ago
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$\begingroup$
Suppose that $P: \mathbb{R} \rightarrow \mathbb{R}$ is a real polynomial of degree exactly $2$. Prove $P$ has at most two roots.
Let $P(x)=a_2 x^2 +a_1 x +a_0$ for all $x \in \mathbb{R}$. I tried to assume there are more than $2$ roots and contradict it using Rolle's theorem but it wasn't working out. I am stuck.
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asked May 26, 2015 at 23:25
snowmansnowman
3,817 8 8 gold badges 44 44 silver badges 79 79 bronze badges
$\endgroup$
5
1 $\begingroup$Suppose by contradiction that it has three roots. Then...$\endgroup$abiessu –abiessu 2015-05-26 23:29:04 +00:00 Commented May 26, 2015 at 23:29
$\begingroup$Or you can always break out the hammer and use the fundamental theorem of algebra, but I seriously doubt your teacher intends you to do that :)$\endgroup$Alan –Alan 2015-05-26 23:33:42 +00:00 Commented May 26, 2015 at 23:33
$\begingroup$...the derivative $P'$ which is a polynomial of degree $1$ has two distinct roots.$\endgroup$user226387 –user226387 2015-05-26 23:34:53 +00:00 Commented May 26, 2015 at 23:34
$\begingroup$@math what does that mean though? I don't see how P' shows anything about the roots...$\endgroup$snowman –snowman 2015-05-26 23:37:24 +00:00 Commented May 26, 2015 at 23:37
$\begingroup$My comment completes that of abiessu.$\endgroup$user226387 –user226387 2015-05-26 23:39:12 +00:00 Commented May 26, 2015 at 23:39
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I was going to go with the systems of equations approach, but got beaten to the chase. So here is a more "calculus" type answer.
Let the polynomial be $P(x)=ax^2+bx+c$. Then we can write the derivative as $P'(x)=2ax+b$.
Without loss of generality suppose that $a>0$. If it isn't then just multiply the polynomial by $-1$, which will not change its roots.
Note that $P'(x) > 0 $ when $x>-b/2a$ and $P'(x)<0$ when $x<-b/2a$. This means that $P$ is strictly increasing on $(-\infty,-b/2a)$ and then strictly decreasing on $(-b/2a,\infty)$. From this we should be able to conclude that it can only cross the $x-axis$ at most twice, but that is not yet a proof.
With the above in mind suppose that there are three roots $x_1,x_2$, and $x_3$. Then Rolle's theorem tells us that the derivative of $P$ is zero in the intervals $[x_1,x_2]$ and $[x_2,x_3]$. This is impossible because the derivative is linear and only has one root. Therefore there cannot be three roots.
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edited May 26, 2015 at 23:46
answered May 26, 2015 at 23:37
SpencerSpencer
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$\begingroup$I think this is the intended approach in a calculus class. +1. (Except you should have $2ax+b$, of course.)$\endgroup$Ian –Ian 2015-05-26 23:38:12 +00:00 Commented May 26, 2015 at 23:38
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Suppose there were three (distinct) roots were $a,b,c$. Then
$$\begin{pmatrix} a^2 & a & 1 \ b^2 & b & 1 \ c^2 & c & 1 \end{pmatrix}\begin{pmatrix} a_2 \ a_1 \ a_0 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}$$
As $\begin{pmatrix} a_2 \ a_1 \ a_0 \end{pmatrix}$ is a non-zero vector, that must mean the the matrix $$M = \begin{pmatrix} a^2 & a & 1 \ b^2 & b & 1 \ c^2 & c & 1 \end{pmatrix}$$ has non-trivial kernel or equivalently it has zero determinant.
However $\det M = -(a-b)(b-c)(c-a) \neq 0$.
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edited May 27, 2015 at 0:33
answered May 26, 2015 at 23:30
Simon SSimon S
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$\begingroup$I find it amazing that this was one a real analysis past paper...$\endgroup$snowman –snowman 2015-05-26 23:32:00 +00:00 Commented May 26, 2015 at 23:32
1 $\begingroup$Lots of ways to skin this cat.$\endgroup$Simon S –Simon S 2015-05-26 23:32:47 +00:00 Commented May 26, 2015 at 23:32
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Hint:
This is purely algebraic:
$\alpha$ is a root of $p(x)$ if and only if $p(x)$ is divisible by $x-\alpha$. Then remember $\deg(p(x)q(x))=\deg p(x)+\deg q(x)$.
Edit: This result can be generalised further: a polynomial over any field (or integral domain) of degree $d$ has at most $d$ roots.
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edited May 26, 2015 at 23:44
answered May 26, 2015 at 23:30
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3488 | https://www.youtube.com/watch?v=3Xt9XzLu-S0 | RMS Molecular Speed and Graham's Law of Effusion
Groce Chemistry
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Posted: 20 Jan 2021
This video discusses the calculation for rms molecular speed and relates it to the movement of gases through porous containers (effusion). The difference between diffusion and effusion is defined and an example is worked for finding the ratio of effusion rates for two gases. This video is meant to supplement General Chemistry II (CHEM 162) at Big Bend Community College.
Transcript:
RMS Molecular Speed i want to talk today about how quickly gas particles move because i've said in past videos and what we understand about matter is that gas particles are pretty free to move around we kind of get that from kinetic molecular theory that the more free particles are to move around then the more we can define the phase of matter based on that freedom and flexibility of motion so how well can they move past each other how quickly do they move how much space is between them all of that factors in to kind of the way that we think about the different phases of matter and gases are pretty unique because they move around so quickly there's such large spaces between them and the particles themselves are fairly teeny tiny so there's all these kind of interesting things like gas laws but we can also use them to study what is called molecular speed and the effects of kind of the movement of those gases from the containers that they're in so we're going to talk about graham's law of effusion in this video as well in order to get to graham's law of effusion though we need to talk about rms molecular speed rms stands for root mean square root mean square molecular speed and what that really means is that we're taking the average molecular speed given an average kinetic energy which sounds like we're making a lot of assumptions there and we kind of are i mean we can't have a little stopwatch and a meter stick and a little atomic nanometer stick to uh to look at the rate at which individual molecules are going to be moving so we have to think about them in terms of an average kinetic energy and we know that average kinetic energy is something that we can measure with a temperature scale right the average kinetic energy we can compare to a scale that scale will give us an idea of kind of hotness and coldness of a substance and if we want to think about this in terms of individual molecules because we're chemists so we think about things on the particle level then the average molecular speed given that average kinetic energy is this rms molecular speed and this is something that we can actually kind of put some numbers to and some numbers that give us a general sense of how quickly these things are actually moving so a couple things the rms molecular speed is abbreviated with a u which i always say is kind of a catch-all variable for when we've run out of variables then this is the one that's left over u is also used for things like internal energy um which is probably why it's used here if i was to guess but i don't know 100 sure on that so it's you um that's equivalent to the square root of three r t t meaning the absolute temperature so we have to think about things in terms of kelvin so let me write this out absolute temp so the absolute temperature of course is in our si unit of kelvin m m here is your molar mass and r is the same gas constant that we've talked about in past videos with our various gas laws like boyle's law and the ideal gas law that r is the same but there's a useful version of r that is 8.31 which is the same as it is in kilopascals but it's equal to joules per mole kelvin 8.31 joules per mole kelvin or another way to write that if we unpack the joules here is 8.31 kilogram meters squared per second squared and then mole kelvin so recall that a joule is a kilogram meter squared per second squared so if i kind of plug that in then that'll actually make these units make sense because we have our kelvin here the molar mass here has to be in because of the joule it has to be in kilograms per mole so that's kind of a weirdness because we don't usually have to convert molar mass we usually just take it from the periodic table and that's in grams per mole but we have to in order to use rms molecular speeds equation here we have to use kilograms per mole and you can see that there's no there's no pressure term in here so this is independent of pressure um so this is just the average molecular speed based on the movement and the size essentially of these things so um so we can use that equation to solve problems that look like this if you have an identity of the gas let's take diatomic nitrogen for example and a temperature then we can figure out the rms molecular speed so our equation is this three r t over the molar mass so our temperature is given in celsius 455 degrees c it's an absolute temperature so we need to convert to kelvin so when we add our 273.15 then we end up with 728 kelvin and then for dinitrogen my diatomic nitrogen my elemental nitrogen then the molar mass is 28.02 grams per mole so one nitrogen 14.01 i multiplied by 2 because it's diatomic and now the problem is it's in grams per mole i need it in kilograms because of these joules because of this form of r that i'm using so if i convert this to kilograms then i can just kind of move the decimal around and if i just push the decimal around a little i end up with this kilograms per mole or you could write it you know 2.802 times 10 to the negative second kilograms per mole however you want to think of it and now i just plug these values in so i take the square root of 3 times the r kilograms meter squared per second squared mole kelvin quite the unit times the temperature 728 kelvin divided by the molar mass in kilograms times 10 to the negative two kilograms per mole see this and when i plug and chug through this kind of see how this thing divides out my kelvin will divide out my moles divide out my kilograms divide out and then i take the square root of meters squared per second squared which gives me a velocity it just leaves me with meters per second so when you plug this in you get a 805 meters per second which is a speed or velocity how quickly these things are moving that's fast i hate 805 meters per second for one of these teeny tiny little molecules it's pretty impressive so it kind of gives you a sense of scale for these things at really warm temperatures you can play around with temperature and see how that impacts the speed you can play around with the different molecules and see how the mass impacts the speed and that's kind of how we use these types of things now the reason that i mention rms molecular speed in terms of gases Effusion is to get into effusion so i want to define a couple terms first there's diffusion which you've probably heard of before because diffusion can be used for a number of different phases but for gases it's defined as one gas spreading out evenly through another gas so if i had a compartment like this and i have one gas on one side and one gas on the other side and i take away that partition then my gases are going to spread out and they're going to spread out evenly throughout the system and they're going to spread out kind of through each other they're going to diffuse through each other so there's no barrier or anything to those guys mixing together and that's what they're going to do is they're going to spread out that's the behavior of gases it's the behavior of matter to naturally spread out now if fusion effusion is when gas flows out through a tiny pore or a tiny hole in a container so let's say that i have a balloon that is full of gas here and if i look at my kind of let's just take blow up this little segment of my balloon let's say that i have here's the balloon surface and let's make it a helium balloon because that's nice and festive looks like it's laughing at you so we have the helium balloon here and the helium and you've probably noticed this if you've ever had a balloon over time even if you don't undo the knot or even if you don't poke a hole in it the balloon still gets smaller the gas still leaves the material that they're made of has enough space in between the particles that these teeny tiny gas particles are going to effuse through those tiny little holes in the container so it's not like there's like a portal or something that's allowing these things not like transport protein through a cell membrane or something like that we're talking about these teeny tiny little pores these things can naturally seep through because they're so small and they're moving so quickly so this is effusion so teeny tiny and we have a higher concentration of helium on the inside lower on the outside so it's going to move that direction naturally now graham's law of effusion says if we do this at a constant temperature and pressure so if i'm at atmospheric pressure and i keep the temperature of my system relatively constant then that rate of effusion so how quickly this process happens is inversely proportional to the square root of the molar mass so square root of molar mass sounds like that rms molecular speed we just talked about inversely proportional so as one goes up the other goes down so the rate right the rate is going to increase as the mass decreases which kind of makes sense right if i have smaller mass smaller particles smaller amounts of stuff right mass is a measure of how much matter there is then that's going to increase how quickly these things are going to leave the container and we can actually measure how quickly this happens or compare rates of how quickly this happens using graham's law this inverse proportionality here so let's look at an example of this Example we're going to calculate the ratio of effusion rates of molecules of hydrogen so h2 which is elemental hydrogen to helium from the same container at a constant temperature and pressure so a couple things the rate of hydrogen's effusion is equal to or proportional to one over the molecular or molar mass of hydrogen right so it's inversely proportional to the square root of the mass that's graham's law so there's this kind of general proportionality here probably more correct to have the proportional symbol and the rate of helium's effusion is proportional to 1 over the molar mass of helium so if we want to calculate the ratio of effusion rates then i would say well let's look at the rate of h2 compared to the rate of helium and since these guys are proportional and we're talking about ratios here then the proportionalities are also going to be equivalent so i can say well then 1 over the square root of the molar mass of hydrogen divided by 1 over the square root of the molar mass of helium will give me this ratio right so that will give me by how much more hydrogen i'm assuming is going to move faster is going to effuse through that surface than helium because we're in the same container how much more hydrogen am i going to lose by how much how what speed by what rate okay so if i do some math magic then i can kind of combine these guys together and i'm going to because i'm taking the inverse inverse flip these around and do this and then i'm going to do this and actually plug these guys in so i go to the periodic table and i find that the molar mass of helium is 4.003 grams per mole and my hydrogen is 2 times my 1.08 okay so when i plug that in to my handy dandy calculator i get 1.409 and what does that mean well what that means then is that hydrogen fuses 1.409 times faster than my helium i set up my ratio to see kind of by how much larger is this guy than this guy you could have flipped it the other direction and this would have been a less than num one number and that would have been fine too you would have said that's how much faster helium is uh knowing that it's slower so it doesn't matter which way you set this kind of problem up but the interpretation of this result then is that h2o fuses the 1.4 times faster than helium because of its smaller mass okay so that's one way to use graham's lava fusion it wraps up and uses that rms molecular speed and kind of gives us a sense of how quickly these things are moving around as always if you have any questions on this don't hesitate to reach out otherwise i'll talk to you again soon |
3489 | http://physics.bu.edu/~redner/542/book/rw.pdf | Chapter 2 RANDOM WALK/DIFFUSION Because the random walk and its continuum diffusion limit underlie so many fundamental processes in non-equilibrium statistical physics, we give a brief introduction to this central topic.
There are several complementary ways to describe random walks and diffusion, each with their own advantages.
2.1 Langevin Equation We begin with the phenomenological Langevin equation that represents a minimalist description for the stochastic motion of a random walk. We mostly restrict ourselves to one dimension, but the generalization to higher dimensions is straightforward.
Random walk motion arises, for example, when a microscopic bacterium is placed in a fluid.
The bacterium is constantly buffeted on a very short time scale by the random collisions with fluid molecules.
In the Langevin approach the effect of these rapid collisions is represented by an effective, but stochastic, external force η(t). On the other hand, if the bacterium had a non-zero velocity in the fluid, there would be a systematic frictional force proportional to the velocity that would bring the bacterium to rest. Under the influence of these two forces, Newton’s second law for the motion of the bacterium leads to the Langevin equation mdv dt = −γv + η(t).
(2.1) This equation is very different from the deterministic equation of motion that one normally encounters in mechanics. Because the stochastic force is so rapidly changing with time, the actual trajectory of the particle contains too much information. The velocity changes every time there is a collision between the bacterium and a fluid molecule; for a particle of linear dimension 1µm, there are of the order of 1020 collisions per second and it is pointless to follow the motion on such a short time scale. For this reason, it is more meaningful physically to study the trajectory that is averaged over longer times. To this end, we need to specify the statistical properties of the random force. Because the force is a result of molecular collisions, it is natural to assume that the force η(t) is a random function of time with zero mean, ⟨η(t)⟩= 0. Here the angle brackets denote the time average. Because of the rapidly fluctuating nature of the force, we also assume that there is no correlation between the force at two different times, so that ⟨η(t)η(t′)⟩= 2Dγ2δ(t −t′). As a result, the product of the forces at two different times has a mean value of zero. However, the mean-square force at any time has the value D. This statement merely states that the average magnitude of the force is well-defined.
In the limit where the mass of the bacterium is sufficiently small that it may be neglected, we obtain an even simpler equation for the position of the bacterium: dx dt = 1 γ η(t) ≡ξ(t).
(2.2) In this limit of no inertia (m = 0) the instantaneous velocity equals the force. In spite of this strange feature, Eq. (2.2) has a simple interpretation—the change in position is a randomly fluctuating variable.
This corresponds to a naive view of what a random walk actually does; at each step the position changes by a random amount.
15 16 CHAPTER 2. RANDOM WALK/DIFFUSION One of the advantages of the Langevin equation description is that average values of the moments of the position can be obtained quite simply. Thus formally integrating Eq. (2.1), we obtain x(t) = Z t 0 ξ(t′) dt′.
(2.3) Because ⟨ξ(t)⟩= 0, then ⟨x(t)⟩= 0. However, the mean-square displacement is non-trivial. Formally, ⟨x(t)2⟩= Z t 0 Z t 0 ⟨ξ(t′)ξ(t′′)⟩dt′ dt′′.
(2.4) Using ⟨ξ(t)ξ(t′)⟩= 2Dδ(t−t′), it immediately follows that ⟨x(t)2⟩= 2Dt. Thus we recover the classical result that the mean-square displacement grows linearly in time. Furthermore, we can identify D as the diffusion coefficient. The dependence of the mean-square displacement can also be obtained by dimensional analysis of the Langevin equation. Because the delta function δ(t) has units of 1/t (since the integral R δ(t) dt = 1), the statement ⟨ξ(t)ξ(t′)⟩= 2Dδ(t −t′) means that ξ has the units p D/t. Thus from Eq. (2.3), x(t) must have units of √ Dt.
The Langevin equation has the great advantage of simplicity. With a bit more work, it is possible to determine higher moments of the position. Furthermore there is a standard prescription to determine the underlying and more fundamental probability distribution of positions. This prescription involves writing a continuum Fokker-Planck equation for the evolution of this probability distribution. The Fokker-Planck equation is in the form of a convection-diffusion equation, namely, the diffusion equation augmented by a term that accounts for a global bias in the stochastic motion. The coefficients in this Fokker-Planck equation are directly related to the parameters in the original Langevin equation. The Fokker-Planck equation can be naturally viewed as the continuum limit of the master equation, which represents perhaps the most fundamental way to describe a stochastic process. We will not pursue this conventional approach because we are generally more interested in developing direct approaches to write the master equation.
2.2 Master Equation for the Probability Distribution Discrete space and time Consider a random walker on a one-dimensional lattice that hops to the right with probability p or to the left with probability q = 1 −p in a single step. Let P(x, N) be the probability that the particle is at site x at the N th time step. Then evolution of this occupation probability is described by the master equation P(x, N + 1) = p P(x −1, N) + q P(x + 1, N).
(2.5) Because of translational invariance in both space and time, it is expedient to solve this equation by transform techniques. One strategy is to Fourier transform in space and write the generating function (sometimes called the z-transform). Thus multiplying the master equation by zN+1 eikx and summing over all N and x gives ∞ X N=0 ∞ X x=−∞ zN+1eikx [P(x, N + 1) = p P(x −1, N) + q P(x + 1, N)] .
(2.6) We now define the joint transform—the Fourier transform of the generating function P(k, z) = ∞ X N=0 zN ∞ X x=−∞ eikx P(x, N).
In what follows, either the arguments of a function or the context (when obvious) will be used to distinguish transforms from the function itself. The left-hand side of (2.6) is just the joint transform P(k, z), except that the term P(x, N = 0) is missing. Similarly, on the right-hand side the two factors are just the generating 2.2. MASTER EQUATION FOR THE PROBABILITY DISTRIBUTION 17 function at x −1 and at x + 1 times an extra factor of z. The Fourier transform then converts these shifts of ±1 in the spatial argument to the phase factors e±ik, respectively. Thus P(k, z) − ∞ X x=−∞ P(x, N = 0)eikx = zu(k)P(k, z), (2.7) where u(k) = p eik + q e−ik is the Fourier transform of the single-step hopping probability. For the initial condition of a particle initially at the origin, P(x, N = 0) = δx,0, the joint transform becomes P(k, z) = 1 1 −zu(k).
(2.8) We now invert the transform to reconstruct the probability distribution. Expanding P(k, z) in a Taylor series, the Fourier transform of the generating function is simply P(k, N) = u(k)N. Then the inverse Fourier transform is P(x, N) = 1 2π Z π −π e−ikx u(k)N dk, (2.9) To evaluate the integral, we write u(k)N = (p eik + q e−ik)N in a binomial series. This gives P(x, N) = 1 2π Z π −π e−ikx N X m=0 N m pm eikm qN−m e−ik(N−m) dk.
(2.10) The only non-zero term is the one with m = (N + x)/2 in which all the phase factors cancel. This leads to the classical binomial probability distribution of a discrete random walk P(x, N) = N!
( N+x 2 )!( N−x 2 )! p N+x 2 q N−x 2 .
(2.11) Finally, using Stirling’s approximation, the binomial approaches the Gaussian probability distribution in the long-time limit, P(x, N) → 1 √2πNpqe−[x−N(p−q)]2/2Npq.
(2.12) This result is a particular realization of the central-limit theorem—namely, that the asymptotic probability distribution of an N-step random walk is independent of the form of the single step distribution, as long as the mean displacement ⟨x⟩and the mean-square displacement ⟨x2⟩in a single step are finite; we will present the central limit theorem in Sec. 2.3.
Continuous time Alternatively, we can treat the random walk in continuous time by replacing N by continuous time t, the increment N →N + 1 with t →t + δt, and finally Taylor expanding the master equation (2.5) to first order in δt. These steps give ∂P(x, t) ∂t = w+P(x −1, t) + w−P(x + 1, t) −w0P(x, t) (2.13) where w+ = p/δt and w−= q/δt are the hopping rates to the right and to the left, respectively, and w0 = 1/δt is the total hopping rate from each site. This hopping process satisfies detailed balance, as the total hopping rates to a site equal the total hopping rate from the same site.
Again, the simple structure of Eq. (2.13) calls out for applying the Fourier transform. After doing so, the master equation becomes dP(k, t) dt = (w+eik + w−e−ik −w0) P(k, t) ≡w(k) P(k, t).
(2.14) For the initial condition P(x, t = 0) = δx,0, the corresponding Fourier transform is P(k, t = 0) = 1, and the solution to Eq. (2.14) is P(k, t) = ew(k)t. To invert this Fourier transform, let’s consider the symmetric case 18 CHAPTER 2. RANDOM WALK/DIFFUSION where w± = 1/2 and w0 = 1. Then w(k) = w0(cos k −1), and we use the generating function representation for the modified Bessel function of the first kind of order x, ez cos k = P∞ x=−∞eikxIx(z) (10), to give P(k, t) = e−t ∞ X x=−∞ eikxIx(t), (2.15) from which we immediately obtain P(x, t) = e−tIx(t).
(2.16) To determine the probability distribution in the scaling limit where x and t both diverge but x2/t remains finite, it is more useful to Laplace transform the master equation (2.13) to give sP(x, s) −P(x, t = 0) = 1 2P(x + 1, s) + 1 2P(x −1, s) −P(x, s).
(2.17) For x ̸= 0, we solve the resulting difference equation, P(x, s) = a[P(x+1, s)+P(x−1, s)], with a = 1/2(s+1), by assuming the exponential solution P(x, s) = Aλx for x > 0; by symmetry P(x, s) = Aλ−x for x < 0.
Substituting P(x, s) = Aλ−x into the recursion for P(x, s) gives a quadratic characteristic equation for λ whose solution is λ± = (1 ± √ 1 −4a2)/2a. For all s > 0, λ± are both real and positive, with λ+ > 1 and λ−< 1. We reject the solution that grows exponentially with x, thus giving Px = Aλx −. Finally, we obtain the constant A from the x = 0 boundary master equation sP(0, s) −1 = 1 2P(1, s) + 1 2P(−1, s) −P(0, s) = P(1, s) −P(0, s).
(2.18) The −1 on the left-hand side arises from the initial condition, and the second equality follows by spatial symmetry. Substituting P(n, s) = Aλx −into Eq. (2.18) gives A, from which we finally obtain P(x, s) = 1 s + 1 −λ− λx −.
(2.19) This Laplace transform diverges at s = 0; consequently, we may easily obtain the interesting asymptotic behavior by considering the limiting form of P(x, s) as s →0. Since λ−≈1 − √ 2s as s →0, we find P(x, s) ≈(1 − √ 2s)x √ 2s + s ∼e−x √ 2s √ 2s .
(2.20) We now invert the Laplace transform P(x, t) = R s0+i∞ s0−i∞P(x, s) est ds by using the integration variable u = √s.
This immediately leads to the Gaussian probability distribution quoted in Eq. (2.26) for the case ⟨x⟩= 0 and ⟨x2⟩= 1.
Continuous space and time When both space and time are continuous, we expand the master equation (2.5) in a Taylor series to lowest non-vanishing order—second order in space x and first order in time t—we obtain the fundamental convection-diffusion equation, ∂P(x, t) ∂t + v ∂P(x, t) ∂x = D ∂2P(x, t) ∂x2 , (2.21) for the concentration P(x, t). Here v = (p −q)δx/δt is the bias velocity and D = δx2/2δt is the diffusion coefficient. Notice that the factor v/D diverges as 1/δx in the continuum limit. Therefore the convective term ∂P ∂x invariably dominates over the diffusion term ∂2P ∂x2 . To construct a non-pathological continuum limit, the bias p−q must be proportional to δx as δx →0 so that both the first- and second-order spatial derivative terms are simultaneously finite. For the diffusion equation, we obtain a non-singular continuum limit merely by ensuring that the ratio δx2/δt remains finite as both δx and δt approach zero.
To solve the convection-diffusion equation, we introduce the Fourier transform P(k, t) = R P(x, t) eikx dx to simplify the convection-diffusion equation to ˙ P(k, t) = (ikv −Dk2)P(k, t), with solution P(k, t) = P(k, 0)e(ikv−Dk2)t = e(ikv−Dk2)t, (2.22) 2.3. CENTRAL LIMIT THEOREM 19 for the initial condition P(x, t = 0) = δ(x). We then obtain the probability distribution by inverting the Fourier transform to give, by completing the square in the exponential, P(x, t) = 1 √ 4πDt e−(x−vt)2/4Dt.
(2.23) Alternatively, we may first Laplace transform in the time domain. For the convection-diffusion equation, this yields the ordinary differential equation sP(x, s) −δ(x) + vP(x, s) = DP ′′(x, s), (2.24) where the delta function reflects the initial condition. This equation may be solved separately in the half-spaces x > 0 and x < 0.
In each subdomain Eq. (2.24) reduces to a homogeneous constant-coefficient equation that has exponential solutions. The corresponding solution for the entire line has the form c+(x, s) = A+e−α−x for x > 0 and c−(x, s) = A−eα+x for x < 0, where α± = v ± √ v2 + 4Ds /2D are the roots of the characteristic polynomial. We join these two solutions at the origin by applying the joining conditions of continuity of P(x, s) at x = 0, and a discontinuity in ∂c ∂x at x = 0 whose magnitude is determined by integrating Eq. (2.24) over an infinitesimal domain which includes the origin. The continuity condition trivially gives A+ = A−≡A, and the condition for the discontinuity in P(x, s) is D P ′ +|x=0 −P ′ −|x=0 = −1.
This gives A = 1/ √ v2 + 4Ds. Thus the Laplace transform of the probability distribution is c±(x, s) = 1 √ v2 + 4Ds e−α∓|x|.
(2.25) For zero bias, this coincides with Eq. (2.20) and thus recovers the Gaussian probability distribution.
2.3 Central Limit Theorem The central limit theorem states that the asymptotic N →∞probability distribution of an N-step random walk is the universal Gaussian function P(x, N) → 1 √ 2πNσ2 e−(x−⟨x⟩)2/2Nσ2, (2.26) where ⟨x⟩and ⟨x2⟩are respectively the mean and the mean-square displacement for a single step of the walk, and σ2 = ⟨x2⟩−⟨x⟩2. A necessary condition for the central limit theorem to hold is that each step of the walk is an independent identically distributed random variable that is drawn from a distribution p(x) such that ⟨x⟩and ⟨x2⟩are both finite. We now give a simple derivation of this fundamental result. For simplicity we give the derivation for a one-dimensional system, but this derivation can immediately be extended to any dimension.
When the steps of the random walk are independent, the probability distribution after N steps is related to the probability after N −1 steps by the recursion (also known as the Chapman-Kolmogorov equation) PN(x) = Z PN−1(x′)p(x′ →x) dx′.
(2.27) This equation merely states that to reach x in N steps, the walk first reaches an arbitrary point x′ in N −1 steps and then makes a transition from x′ to x with probability p(x′ →x). It is now useful to introduce the Fourier transforms f(k) = Z ∞ −∞ f(x)eikx dx f(x) = 1 2π Z ∞ −∞ f(k)e−ikx dk to transform Eq. (2.27) to the algebraic equation PN(k) = PN−1(k)p(k) that we iterate to give PN(k) = P0(k)p(k)N. At this stage, there is another mild condition for the central limit theorem to hold—the initial condition cannot be too long range in space. The natural condition is for the random walk to start at the origin, P0(x) = δx, 0 for which the Fourier transform of the initial probability distribution is simply P0(k) = 1. Then the Fourier transform of the probability distribution is simply PN(k) = p(k)N, (2.28) 20 CHAPTER 2. RANDOM WALK/DIFFUSION so that PN(x) = 1 2π Z ∞ −∞ p(k)N e−ikx dk.
(2.29) To invert the Fourier transform, we now use the fact that the first two moments of p(x) are finite to write the Fourier transform p(k) as p(k) = Z ∞ −∞ p(x) eikx dx = Z ∞ −∞ p(x) 1 + ikx −1 2k2x2 + . . .
dx = 1 + ik⟨x⟩−1 2k2⟨x2⟩+ . . .
Now the probability distribution is PN(x) ∼1 2π Z ∞ −∞ [1 + ik⟨x⟩−1 2k2⟨x2⟩]N e−ikx dx ∼1 2π Z ∞ −∞ eN ln[1+ik⟨x⟩−1 2 k2⟨x2⟩] e−ikx dx ∼1 2π Z ∞ −∞ eN[1+ik⟨x⟩−k2 2 (⟨x2⟩−⟨x⟩2)] e−ikx dx (2.30) We now complete the square in the exponent and perform the resulting Gaussian integral to arrive at the fundamental result PN(x) ∼ 1 √ 2πNσ2 e−(x−N⟨x⟩)2/2Nσ2.
(2.31) 2.4 Connection to First-Passage Properties An intriguing property of random walks is the transition between recurrence and transience as a function of the spatial dimension d. Recurrence means that a random walk is certain to return to its starting point; this occurs for d ≤2. Conversely, d > 2 the random walk is transient in that there is positive probability for a random walk to never return to its starting point. It is striking that the spatial dimension—and not any other features of a random walk—is the only parameter that determines this transition.
The qualitative explanation for this transition is quite simple. Consider the trajectory of a typical random walk. After a time t a random walk explores a roughly spherical domain of radius √ Dt while the total number of sites visited during this walk equals to t. Therefore the density of visited sites within an exploration sphere is ρ ∝t/td/2 ∝t1−d/2 in d dimensions. For d < 2 this density grows with time; thus a random walk visits each site within the sphere infinitely often and is certain to return to its starting point. On the other hand, for d > 2, the density decreases with time and so some points within the exploration sphere never get visited.
The case d = 2 is more delicate but turns out to be barely recurrent.
+ t-t 0 r,t 0 = 0,r r,t Figure 2.1: Diagrammatic relation between the occupation probability of a random walk (propagation is represented by a wavy line) and the first-passage probability (straight line).
2.4. CONNECTION TO FIRST-PASSAGE PROPERTIES 21 We now present a simple-minded approach to understand this transition between recurrence and tran-sience. Let P(r, t) be probability that a random walk is at r at time t when it starts at the origin. Similarly, let F(r, t) be the first-passage probability, namely, the probability that the random walk visits r for the first time at time t with the same initial condition.
For a random walk to be at r at time t, the walk must first reach r at some earlier time step t′ and then return to r after t −t′ (Fig. 2.1). This connection between F(r, t) and P(r, t) may therefore be expressed as the convolution P(r, t) = δr,0 δt,0 + Z t 0 F(r, t′) P(0, t −t′) dt′.
(2.32) The delta function term accounts for the initial condition. The second term accounts for the ways that a walk can be at r at time t. To reach r at time t, the walk must first reach r at some time t′ ≤t. Once a first passage has occurred, the walk must return to r exactly at time t (and the walk can also return to r at earlier times, so long as the walk is also at r at time t). Because of the possibility of multiple visits to r between time t′ and t, the return factor involves P rather than F. This convolution equation is most conveniently solved in terms of the Laplace transform to give P(r, s) = δr,0 + F(r, s)P(0, s). Thus we obtain the fundamental connection F(r, s) = P(r, s) P(0, s), r ̸= 0 1 − 1 P(0, s), r = 0, (2.33) in which the Laplace transform of the first-passage probability is determined by the corresponding transform of the probability distribution of diffusion P(r, t).
We now use the techniques of Section A.2 to determine the time dependence of the first-passage probability in terms of the Laplace transform for the occupation probability. For isotropic diffusion, P(r = 0, t) = (4πDt)−d/2 in d dimensions and the Laplace transform is P(0, s) = R ∞ 0 P(0, t) e−st dt.
As discussed in Section A.2, this integral has two fundamentally different behaviors, depending on whether R ∞P(0, t) dt diverges or converges. In the former case, we apply the last step in Eq. (A.6) to obtain P(0, s) ∝ Z t∗=1/s (4πDt)−d/2 dt ∼ ( Ad(t∗)1−d/2 = Adsd/2−1, d < 2 A2 ln t∗= −A2 ln s, d = 2, (2.34) where the dimension-dependent prefactor Ad is of the order of 1 and does not play any role in the asymptotic behavior.
For d > 2, the integral R ∞P(0, t) dt converges and one has to be more careful to extract the asymptotic behavior by studying P(0, 1) −P(0, s). By such an approach, it is possible to show that P(0, s) has the asymptotic behavior P(0, s) ∼(1 −R)−1 + Bdsd/2−1 + . . . , d > 2, (2.35) where R is the eventual return probability, namely, the probability that a diffusing particle random walk ultimately reaches the origin, and Bd is another dimension-dependent constant of the order of 1. Using these results in Eq. (2.33), we infer that the Laplace transform for the first-passage probability has the asymptotic behaviors F(0, s) ∼ 1 −Ads1−d/2, d < 2 1 + A2(ln s)−1, d = 2 R + Bd(1 −R)2sd/2−1, d > 2, (2.36) From this Laplace transform, we determine the time dependence of the survival probability by approxi-mation (A.9); that is, F(0, s = 1 −1/t∗) ∼ Z t∗ 0 F(0, t) dt ≡T (t∗), (2.37) where T (t) is the probability that the particle gets trapped (reaches the origin) by time t. For what follows, we also define the survival probability S(t) = 1 −T (t), which is simply the probability that the particle has 22 CHAPTER 2. RANDOM WALK/DIFFUSION not reached the origin by time t. Here the trick of replacing an exponential cutoffby a sharp cutoffprovides an extremely easy way to invert the Laplace transform. From Eqs. (2.36) and (2.37) we thus find S(t) ∼ Adtd/2−1, d < 2 A2(ln t)−1, d = 2 (1 −R) + Cd (1 −R)2 t1−d/2, d > 2.
(2.38) where Cd is another d-dependent constant of the order of 1. Finally, the time dependence of the first-passage probability may be obtained from the basic relation 1 −S(t) ∼ R t F(0, t) dt to give F(0, t) = −∂S(t) ∂t ∝ td/2−2, d < 2 t−1(ln t)−2, d = 2 t−d/2, d > 2.
(2.39) It is worth emphasizing several important physical ramifications of the above first-passage properties.
First, the asymptotic behavior is determined by the spatial dimension only and that there is a dramatic change in behavior when d = 2. For d ≤2, the survival probability S(t) ultimately decays to zero. This means that a random walk is recurrent and is certain to eventually return to its starting point, and indeed visit any site of an infinite lattice. Finally, because a random walk has no memory, it is “renewed” every time a specific lattice site is reached. Thus recurrence also implies that every lattice site is visited infinitely often.
We can give is a simple physical explanation for this efficient visitation of sites. After a time t a random walk explores a roughly spherical domain of radius √ Dt.
The total number of sites visited during this exploration is also proportional to t. Consequently in d dimensions, the density of visited sites within this exploration sphere is ρ ∝t/td/2 ∝t1−d/2. For d < 2, ρ diverges as t →∞and a random walk visits each site within the sphere infinitely often. This feature is termed compact exploration. Paradoxically, although every site is visited with certainty, these visitations take forever because the mean time to return to the origin, ⟨t⟩= R t F(0, t) dt, diverges for all d ≤2.
Finally, we outline a useful technique to compute where on a boundary is a diffusing particle absorbed and when does this absorption occur. This method will provide helpful in understanding finite-size effect in reaction kinetics. For simplicity, consider a symmetric nearest-neighbor random walk in the finite interval [0, 1]. Let E+(x) be the probability that a particle, which starts at x, eventually hits x = 1 without hitting x = 0. This eventual hitting probability E+(x) is obtained by summing the probabilities for all paths that start at x and reach 1 without touching 0. Thus E+(x) = X p Pp(x), (2.40) where Pp(x) denotes the probability of a path from x to 1 that does not touch 0. The sum over all such paths can be decomposed into the outcome after one step (the factors of 1/2 below) and the sum over all path remainders from the location after one step to 1. This gives E+(x) = X p 1 2 Pp(x + δx) + 1 2 Pp(x −δx) = 1 2[E+(x + δx) + E+(x −δx)].
(2.41) By a simple rearrangement, this equation is equivalent to ∆(2)E+(x) = 0, (2.42) where ∆(2) is the second-difference operator. Notice the opposite sense of this recursion formula compared to the master equation Eq. (2.5) for the probability distribution. Here E+(x) is expressed in terms of output from x, while in the master equation, the occupation probability at x is expressed in terms of input to x.
For this reason, Eq. (2.41) is sometimes referred to as a backward master equation. This backward equation is just the Laplace equation and gives a hint of the deep relation between first-passage properties, such as the exit probability, and electrostatics. Equation (2.42) is subject to the boundary conditions E+(0) = 0 2.4. CONNECTION TO FIRST-PASSAGE PROPERTIES 23 and E+(1) = 1; namely if the walk starts at 1 it surely exits at 1 and if the walk starts at 0 it has no chance to exit at 1. In the continuum limit, Eq. (2.42) becomes the Laplace equation E′′ = 0, subject to appropriate boundary conditions. We can now transcribe well-known results from electrostatics to solve the exit probability. For the one dimensional interval, the result is remarkably simple: E+(x) = x!
This exit probability also represents the solution to the classic “gambler’s ruin” problem: let x represent your wealth that changes by a small amount dx with equal probability in a single bet with a Casino. You continue to bet as long as you have money. You lose if your wealth hits zero, while you break the Casino if your wealth reaches 1. The exit probability to x = 1 is the same as the probability that you break the Casino.
Let’s now determine the mean time for a random walk to exit a domain. We focus on the unconditional exit time, namely, the time for a particle to reach any point on the absorbing boundary of this domain. For the symmetric random walk, let the time increment between successive steps be δt, and let t(x) denote the average exit time from the interval [0, 1] when a particle starts at x. The exit time is simply the time for each exit path times the probability of the path, averaged over all trajectories, and leads to the analog of Eq. (2.40) t(x) = X p Pp(x) tp(x), (2.43) where tp(x) is the exit time of a specific path to the boundary that starts at x.
In analogy with Eq. (2.41), this mean exit time obeys the recursion t(x) = 1 2 [(t(x + δx) + δt) + (t(x −δx) + δt)] , (2.44) This recursion expresses the mean exit time starting at x in terms of the outcome one step in the future, for which the initial walk can be viewed as restarting at either x+δx or x−δx, each with probability 1/2, but also with the time incremented by δt. This equation is subject to the boundary conditions t(0) = t(1) = 0; the exit time equals zero if the particle starts at the boundary. In the continuum limit, this recursion formula reduces to the Poisson equation Dt′′(x) = −1. For diffusion in a d-dimensional domain with absorption on a boundary B, the corresponding Poisson equation for the exit time is D∇2t(r) = −1, subject to the boundary condition t(r) = 0 for r ∈B. Thus the determination of the mean exit time has been recast as a time-independent electrostatic problem! For the example of the unit interval, the solution to the Laplace equation is just a second-order polynomial in x. Imposing the boundary conditions immediately leads to the classic result t(x) = 1 2D x(1 −x).
(2.45) First passage probability and the gambler’s ruin problem Consider a random walk in a finite interval of length N. The two boundary sites are absorbing, i.e., the random walker immediately disappears upon reaching these sites. Suppose that the starting position of the random walk is n, with 0 ≤n ≤N. What is Fn, the probability that the walker first reaches the boundary at site N? We can write a simple recursion formula for the first-passage probability. With probability 1/2, the walk steps to site n−1, at which point the exit probability to site N is Fn−1. Similarly, the walk steps to site n+1 with probability 1/2, where the exit probability is Fn+1. Thus the first passage probability satisfies the discrete Poisson equation Fn = 1 2(Fn−1 + Fn+1), (2.46) with the boundary conditions F0 = 0 and FN = 1. The solution is simple: Fn = n N .
(2.47) This first passage probability also solves a neat probability theory problem.
In a fair coin-toss game, the probability that a gambler ruins a Casino equals the wealth of the gambler divided by the combined wealth of the gambler and casino. Gambling is most definitely a bad idea...
24 CHAPTER 2. RANDOM WALK/DIFFUSION 2.5 The Reaction Rate Suppose that you wanted to hit the side of a barn using an ensemble of blind riflemen that fire bullets in random directions as your incident beam. What is the rate at which the barn is hit? Theorists that we are, let’s model the barn as a sphere of radius R. A patently obvious fact is that if the radius of the barn is increased, the number of bullets that hit our theoretical barn increases as its cross-sectional area. In d spatial dimensions, the cross section therefore scales as Rd−1. Now suppose that we take away the rifles from our blind marksmen and give them the task of hitting the barn simply by wandering around. Surprisingly, the rate at which the blind riflemen diffuse to the barn is proportional to Rd−2 for d > 2. Thus in the physical case of 3 dimensions, the absorption rate is proportional to the sphere radius rather than to its cross section! Even more striking— for d ≤2 the absorption rate no longer depends on the radius of the absorbing sphere. The rate at which diffusing particles hit an absorbing sphere is the underlying mechanism of diffusion-controlled reactions. Because of the centrality of this topic to reaction kinetics and because it represents a nice application of first-passage ideas, we now determine this reaction rate.
As in the original Smoluchowski theory for the reaction rate, we fix a spherical absorbing particle of mass mi radius Ri at the origin, while a gas of non-interacting particles each of mass mj and radii Rj freely diffuses outside the sphere. The separation between the absorbing sphere and a background particle diffuses with diffusion coefficient Di + Dj, where Di is the diffusion coefficient of a droplet of radius Ri. When the separation first reaches a = Ri + Rj, reaction occurs. The reaction rate is then identified as the flux to an absorbing sphere of radius a by an effective particle with diffusivity D = Di + Dj.
The concentration of background particles around the absorbing sphere thus obeys the diffusion equation ∂c(⃗ r, t) ∂t = D∇2c(⃗ r, t), (2.48) subject to the initial condition c(⃗ r, t = 0) = 1 for r > a and the boundary conditions c(r = a, t) = 0 and c(r →∞, t) = 1. The reaction rate is then identified with the integral of the flux over the sphere surface K(t) = −D Z S ∂c(⃗ r, t) ∂r r=a dΩ.
(2.49) There are two regimes of behavior as a function of the spatial dimension. For d > 2, the loss of reactants at the absorbing sphere is sufficiently slow that it is replenished by the re-supply from larger distances. A steady state is thus reached and the reaction rate K is finite. In this case, the reaction rate can be determined more simply by solving the time-independent Laplace equation, rather than the diffusion equation (2.48).
The solution to the Laplace equation with the above initial and boundary conditions is c(r) = 1 − a r d−2 .
The flux is then −D ∂c ∂r|r=a = D(d −2)/a and the total current is the integral of this flux over the surface of the sphere K = (d −2)Ωd Dad−2, where Ωd = 2πd/2/Γ(d/2) is the area of a unit sphere in d dimensions. We translate this flux into the reaction kernel for aggregation by expressing a and D in terms of the parameters of the constituent reactants to give Kij = (d −2)Ωd (Di + Dj)(Ri + Rj)d−2.
(2.50) We can express this result as a function of reactant masses only for the physical case of three dimension by using Ri ∝i1/3, while for the diffusion coefficient, we use the Einstein-Stokes relation Di = kT/(6πηRi) ∝ i−1/3, where kT is the thermal energy and η is the viscosity coefficient to obtain Kij ∝2kT 3η (R−1 i + R−1 j )(Ri + Rj).
(2.51) What happens for d < 2? We could solve the diffusion equation with the absorbing boundary condition and the unit initial condition, from which the time-dependent flux and thereby a time-dependent reaction rate 2.5. THE REACTION RATE 25 Dt r c(r,t) a Figure 2.2: Sketch of the concentration about an absorbing sphere according to the quasi-static approxima-tion. The near- and far-zone concentrations match at r = √ Dt.
can be deduced. However, it is simpler and more revealing to apply the general quasi-static approximation.
Because of its simplicity and general utility, we now present the quasi-static calculation of the reaction rate.
The basis of the quasi-static approximation is that the region exterior to the absorbing sphere naturally divides into “near” and “far” zones. In the near zone, which extends to a distance √ Dt from the sphere, diffusing particles have ample time to explore this near zone thoroughly and the concentration is nearly time independent. In the complementary far zone there is negligible depletion because diffusing particles that are more distant than √ Dt typically will not hit the sphere in a time t. Thus in the far zone the concentration c(r) = 1 for r > √ Dt.
Based on this picture, we merely solve the Laplace equation in the near zone a < r < √ Dt with the time-dependent boundary conditions c(r = √ Dt) = 1, to match to the static far-zone solution, and c(a) = 0.
The general solution is c(r) = A + Br2−d, and matching to the boundary conditions gives c(r, t) = 1 −(a/r)d−2 1 −(a/ √ Dt)d−2 → √ Dt r !d−2 t →∞ for d = 1.
(2.52a) For d = 2, we can still apply the same quasi-static approach because diffusion is still recurrent, so that a qualitatively similar depletion layer builds up around the absorbing sphere. Now, however, the general solution to the Laplace equation is c(r) = A + B ln r. Apply the boundary conditions at r = a and r = √ Dt leads to c(r, t) = ln(r/a) ln( √ Dt/a) →ln r ln t t →∞ for d = 2.
(2.52b) Finally, we substitute the above expressions for the concentration into the definition of the time-dependent reaction rate from Eq. (2.49) to obtain the reaction rate.
K(t) ∝ D × (Dt)(d−2)/2 d < 2; 4πD ln Dt/a2 d = 2; Dad−2 d > 2.
(2.53) Notice that the rate does not depend on the cluster radius for d ≤2. This surprising fact arises because of the recurrence of diffusion in d ≤2 so that two diffusing particles are guaranteed to eventually meet independent of their radii.
26 CHAPTER 2. RANDOM WALK/DIFFUSION Problems Section 2.2 1. Find the generating function for the Fibonacci sequence, Fn = Fn−1 + Fn−2, with the initial condition F0 = F1 = 1; that is, determine F(z) = P∞ 0 Fnzn. Invert the generating function to find a closed form expression for Fn.
2. Consider a random walk in one dimension in which a step to the right of length 2 occurs with probability 1/3 and a step to the left of length 1 occurs with probability 2/3. Investigate the corrections to the isotropic Gaussian that characterizes the probability distribution in the long-time limit. Hint: Consider the behavior of moments beyond second order, ⟨xk⟩with k > 2.
3. Solve the gambler’s ruin problem when the probability of winning in a single bet is p. The betting game is repeated until either you are broke or the casino is broken. Take the total amount of capital to be $N and you start with $n. What is the probability that you will break the casino? Also determine the mean time until the betting is over (either you are broke or the Casino is broken). More advanced: Determine the mean time until betting is over with the condition that: (i) you are broke, and (ii) you break the Casino. Solve this problem both for fair betting and biased betting.
4. Consider the gambler’s ruin problem under the assumptions that you win each bet with probability p ̸= 1/2, but that the casino has an infinite reserve of money. What is the probability that you break the casino as a function of p? For those values of p where you break the casino, what is the average time for this event to occur?
Section 2.4 5. For r ̸= 0 and t > 0, explicitly verify Eq. (2.32) in one dimension.
Solution.
Notes The field of random walks, diffusion, and first-passage processes are classic areas of applied probability theory and there is a corresponding large literature. For the more probabilistic aspects of random walks and probability theory in general, we recommend 15; 3; 11; 9. For the theory of random walks and diffusion from a physicist’s perspective, we recommend 12; 13; 14. For first-passage properties, please consult 8; 9. |
3490 | https://www.ictr.edu.cn/Uploads/File/2022/07/19/%E5%AE%9E%E9%AA%8C%E5%8C%BA%E5%B7%A5%E4%BD%9C%E5%86%85%E9%83%A8%E4%BA%A4%E6%B5%81%E6%9D%90%E6%96%992022%E5%B9%B42%E6%9C%88%E5%88%8A.20220719103349.pdf | 基础教育课程改革实验区 工作内部交流材料 总第29期 〔2022〕 第 2期 课程教材研究所 2022年3月15日 总目录 实验区经验展示 ..................................................................................... 1 基地校经验展示 ................................................................................. 197 目 录 重庆南岸实验区 探索学科作业体系 发展学生核心素养 ...... 胡 平 王永强 王 冰 1 ——重庆市南岸区小学语文学科落实“双减”的实践案例 “三增”提质量 “三减”消总量 “三变”添能量 ....... 刘平兴 8 ——重庆市南坪中学校“双减”工作的实践探索 广州南沙实验区 悉心谋划“5+2”，课后服务质量优 .......................... 14 ——“双减”背景下区域课后服务供给的南沙做法 将“减负提质增优”放在心上 ......................... 李明秋 20 ——广州市南沙区金隆小学落实“双减”政策侧记 成都锦江实验区 共建有温度的家庭教育生态圈 ............... 成都市锦江区教育局 25 ——家庭教育锦江模式探索 “五优五引”建设高质量作业体系 ........... 成都市锦江区教育局 30 ——成都市锦江区实施“优作业管理行动” ⅰ 重庆南川实验区 作业改革小切口 撬动教学大变革 ... 重庆市南川区教育科学研究所 36 — — 重 庆 市 南 川 区 “ 12345” 作 业 管 理 体 系 的 构 建 与 实 践 江苏常州实验区 “蓬勃生长”的书院儿童 ................. 常州市武进区实验小学 40 ——“课后服务”工作的探索与实践 着眼于关键能力提升的作业管理 ........... 常州市教科院附属中学 46 山东临沂实验区 强化课程育人 助推“双减”落地 ................ 临沂市教育局 52 江苏徐州实验区 “四位一体”推进义务教育阶段作业减负增效 . 徐州市云龙区教育局 58 “减”与“加”的智慧调和之道 ............. 徐州市大马路小学校 65 ——徐州市大马路小学校“双减”背景下课堂提质增效的校本实践 二七一教育实验基地 “双减”之下，变革学习方式是课堂育人的根本之策 ..... 赵丰平 69 ——271 教育大单元整体学习研究的实践范式 “双减”背景下小学数学深度学习研究与实践 ........... 孙瑞华 75 郑州高新实验区 家校社共育，培育时代新人 .......... 郑州高新区管委会社会事业局 81 聚焦单元整体，优化作业设计 ........ 郑州高新区管委会社会事业局 87 ⅱ 王府教育实验基地 王府学校落实“双减”政策主要做法 ... 北京王府外国语学校小学部 98 开设“家长学校”：“双减”背景下“校家社”协同育人的新举措 ............................................. 蔡启彬 黄 潇 104 成都都江堰实验区 全面落实“双减”政策 营造绿色教育生态 .... 成都都江堰实验区 109 ——都江堰市“双减”工作推进情况汇报 关于分层作业与家校协同育人的几点思考 ............... 孙 洁 115 广东珠海实验区 提质减负，守正推新 .......................... 珠海市香山学校 119 学在乐中 乐在“考”中 ............................. 黄美健 125 ——“双减”下改进考试测评方式的实践探索 重庆江津实验区 重庆江津：“双减”政策落地生根，美好教育“减负提质” .... 131 作业巧设计 孩子多快乐 ............................. 凌 荣 135 深圳龙华实验区 “双减”背景下课堂提质增效的实践探索 .... 深圳市龙华区教育局 140 1+1+1>3——“双减”政策下的玉龙样态 ..................... 145 鄂尔多斯康巴什实验区 “双减”之下开新局，精细管理谱新篇 ......................... ................................. 鄂尔多斯市康巴什区第五小学 157 “有效作业研究”牵引“减负提质”列车 . 鄂尔多斯市康巴什实验区 151 ⅲ 广州黄埔实验区 党建引领“双减” 助推学生全面个性发展 ... 广州市黄埔区教育局 163 平台+服务+课程 完善校内课后托管 ..... 广州黄埔区东荟花园小学 170 昆明市盘龙区教育体育局 将“双减”落实到位 让教育回归本真 ...... 昆明市盘龙区明通小学 173 ——明通小学“双减”工作案例 以基于学情的分层作业设计促教学提质 ..... 昆明市盘龙区桃源小学 179 ——盘龙区桃源小学落实“双减”工作案例 井冈山市教育局 打造高效课堂 减轻学生课业负担 ................... 宁冈中学 189 用好“加法”运算，破解“双减”难题 ............. 宁冈山小学 192 ——井冈山小学落实“双减”工作主要做法 ⅳ 基地校 “双减”背景下幼儿园育人质量提升的实践探索 ................. ................................... 北京市西城区三教寺幼儿园 197 自主共享 个性飞扬 ......................... 成都师范附属小学 204 ——“1+X”课后服务模式助力“双减”政策落地 促进深度学习的实践性作业优化设计 ............... 刁善玉思佳 211 ——以“制作数字故事”为例 北京光明小学课后服务工作的研究性实践........... 北京光明小学 217 ——学生兴趣·爱好·特长拓展课程 增强作业管理透明度， 助力“双减”落地见实效 .. 庞涌洪 蒋明权 223 将课后服务课程融入学校整体课程体系之中 .... 成都市盐道街小学 229 ——成都市盐道街小学的新探索 三“点”赋能，融合育人 ................. 赵玉如 余 悦 侯 毅 238 发展核心素养，培育时代新人 .............. 山东省青岛第二中学 245 ——青岛二中“十个一”项目助力“双减”落地 基于教学改进视角下的作业减负行动 ................... 杨国军 251 “双减”背景下以“优雅生活者”为导向的趣味英语作业布置 ..... ................................................... 李一丹 257 优化教学 迎双减政策落地 ................ 孙宽正 曹文霞 夏 宁 262 五育融合视野下落实双减的育才行动 . 吴明平 臧玲 陈开文 詹滢 叶德元269 做好课后服务，实现“五育并举” ....... 北京二十一世纪国际学校 283 “双减”之下作业设计提质增效的实践探索 ............. 李素娟 289 减负与提质并举 有效与优质共存 ....................... 徐琳 296 历史学科作业优化设计 ........................ 天津市五十中学304 ⅴ 【重庆市南岸实验区】 探索学科作业体系 发展学生核心素养 ——重庆市南岸区小学语文学科落实“双减”的 实践案例 重庆市南岸区教师进修学院 胡 平 王永强 重庆市天台岗雅居乐小学 王 冰 “双减”政策对学科课堂教学高质量发展提出了前所未 有的挑战，重庆市南岸区把小学语文作业的提质减负作为教 育质量提升的重要抓手。针对作业数量过多、质量不高、学 科坚持以学生为本，遵循教育规律，区校协同共建共享核心 素养导向的单元整体作业资源，从源头发力，减轻学生过重 作业负担，提升育人水平，发展学生核心素养。 一、科学建构学科单元整体作业设计体系 在探索落实核心素养导向的课堂教学改进的基础上，南 岸区探索建构了核心素养导向的单元整体作业设计流程。 （一）立高质量作业设计之关键前提 高质量作业设计的基础是提高课堂实效，夯实“双减” 主阵地。以“课堂教学”为抓手，南岸区从2016 年开始，整 体布局，全学段探索落实核心素养导向的课堂教学改进。区 域先后制定了推进核心素养导向的课堂教学改进的指导意 见、行动方案，以基于课程标准的单元整体教学设计与实施 为抓手，强化核心素养导向的课堂教学改进行动实效。继区 1 域小学语文“作业研究项目”“小学语文教师作业设计及命 题能力提升”项目，2021 年，由区教师进修学院小学语文教 研员领衔、小学语文中心组协同、学科教师联动，组成了区 校研修团队， 共同推行 “核心素养导向的单元整体作业设计” 改革实践。 （二）探高质量作业设计之关键举措 基于专家的指导和实践经验，研修团队建构了素养导向 的单元整体作业设计流程（图1），通过课标、教材和学情 的多维研读，整体规划单元作业目标，设计“积累与发展” “阅读与表达”“跨学科综合”三类单元作业，同时开发了 学生单元学习质量评价表及教师作业设计质量评价表，跟踪 评价学生表现发展变化的轨迹，及时调整教学策略，优化作 业设计。 图1 小学语文素养导向的单元整体作业设计流程 2 这一流程为设计高质量作业提供了可操作的路径，提升 了研修团队对语文学科课程整体的把握能力和设计能力。 二、优化实施学科单元整体作业设计行动 通过选、 改、 创等方式， 区校研修团队于2021 年秋季学 期共设计了96 个单元整体作业任务，覆盖统编小学语文1-6 年级教材上册所有课程内容，形成了区校协同共建共享的 作业资源库。 （一）作业内容紧扣目标，实现结构化和体系化 在单元作业设计中，始终坚持以单元作业目标为导向。 依据语文学科性质、课程标准要求、教材编排体系和学生认 知规律制定单元作业目标。基于此，将作业内容结构化为三 种类型：“积累与梳理”“阅读与表达”“跨学科综合”。 “积累与梳理” 指向识字、 写字、 朗读、 字词句 （段） 运用、 日积月累等学段基础目标的巩固与运用，重在学习规范地运 用国家通用语言文字；“阅读与表达”指向单元语文要素的 专项迁移运用，选择匹配单元文本类型的新文本，参照教材 助学系统的题型， 重在面向全体， 发展阅读和表达关键能力； “跨学科综合”指向综合运用语文及其他学科的知识尝试解 决现实生活中的问题，发展问题解决、团队合作、实践创新 等综合素养。设计时可以语文课程学习为基础，链接“展示 台”“快乐读书吧”“综合性学习”等设计作业任务，也可 围绕其他学科、社会生活中有意义的话题，开展探究活动。 三种类型的作业设计，既关注了学科内不同年级必备知 识、关键能力的纵向衔接，跨学科作业还关注了同一年级不 同学科之间知识、能力的横向关联，并且发展了学生尊重他 3 人、认真做事、得体交流、团结合作等必备品格，真正将作 业的过程变成了发展学生核心素养的育人过程。 （二）作业分层紧扣学情，满足差异化和个性化 “双减” 政策强调 “鼓励布置分层、 弹性和个性化作业” 。 作业设计应根据学生的先前经验、认知能力、学习风格、能 力差异等，设计不同层次的有针对性的作业，同时辅以分层 评价与辅导，使不同层次的学生获得最大限度发展。分层作 业包括：A 类基础性作业，重在学会的巩固；B 类挑战性作 业，侧重理解与运用；C 类综合性作业，重在开放性和综合 运用知识解决问题的能力。作业弹性分层、动态管理，在教 师的指导下，学生可以根据自己学习和理解的实际情况自由 选择，调动每一类学生的主观能动性。 台岗集团校设计的 《将相和》作业 （图2） ， 根据学生的 理解、接受能力的不同，围绕同一目标，搭建支架，把复杂 的作业进行分解，满足不同学力学生差异化发展。《俗世奇 图2 《将相和》分层作业 4 人》整本书阅读作业（图3），让学生自己选择喜欢的或者 能够证明自己学习结果的方式方法，满足了不同学习风格的 学生个性化需求。 图3 冯骥才《俗世奇人》“菜单式”作业 （三）作业情境紧扣素养，追求生活化和真实化 作业设计应以促进学生语文课程核心素养发展为出发 点和落脚点，紧密结合课堂所学，关注学生生活中的热点问 题，创设真实的学习情境，设计动手操作、主题观察、人际 交流、跨媒介创意表达等跨学科作业，培养发展学生自主、 综合学习的能力。如南坪实验集团一年级上册第四单元作业 设计，紧扣培养学生主动识字学段目标和借助同学姓名识字 的单元目标，设计“我和汉字交朋友”跨学科综合活动，分 成了借助同学姓名识字、借助班牌、小区路牌、车站站牌、 街道广告牌识字等多个生活情境，既解决了一年级小朋友不 会认同学名字的现实问题，还发展了学生自主识字的兴趣， 更重要的是促进了人与人之间交流与交往的能力。 依托作业资源库，区域小学语文教师可以较为轻松地为 学生提供个性化服务，满足学生的多样化需求，进一步形成 5 备、教、学、评一体化的教学格局，促进区域教育高质量发 展。 三、科学评价学科单元整体作业设计实效 在评价作业完成情况时，南岸区更注重学生的学习过程、 取得的进步，培养自主学习能力，保护学生的学习兴趣。 （一）立足生长性，满足学生自主选择的多样评价方式 作业实效既着眼于班级整体，又关注班级中不同层次学 生的能力水平和发展程度。学生可以自主选择组合，生成属 于自己的个性化作业单，在达标的基础上，满足差异需求。 如珊瑚中铁小学三年级下册第四单元“手可摘星辰”作业设 计，改变传统的作业布置“一刀切”的模式，将作业设计为 游戏集星的形式，制定游戏规则，规定闯关总星级，由学生 自主选择不同星级难度的作业。在完成任务目标的基础上， 减少了学生的作业题量，关注了学生在兴趣、能力和学习基 础等方面的个体差异，激发了学生学习的自主性和积极性。 （二）立足过程性，跟踪记录学生表现和反思改进教学 每个单元学习结束后，组织学生借助单元学习质量评价 表，回顾反思作业过程，教师跟踪统计分析学生作业完成情 况，真实全面收集学情，对本单元学习效果作出比较精准的 判断。教师借助作业质量评价表，结合学生学习情况，进一 步反思作业设计质量，为优化设计、调整教学提供了实证依 据。 （三）立足激励性，调动学生主观能动的评价机制 聚焦“双减”，注重作业评价主体的多元协同，采取教 师评价、学生自我评价、小组合作评价和家长参与评价。立 6 足激励性评价，教师对优、中、差作业的评价可以不按同一 标准。如对学优生坚持高标准严要求，激励不断超越自己； 对中等生以激励性评价为主，促进积极向上；对学困生采用 鼓励评价， 发现闪光点。 同时， 如果学困生订正后还是不会， 就需要进行补充性练习，在练习中可以进行鼓励性加分。让 每个学生都有“跳一跳够得着”的成就感。 作业的变革影响着课堂教与学方式的改变。课程育人意 识与科学作业理念的增强，引领教师研读课标，转变教学方 式， 从规模化、 经验化作业布置转向核心素养导向的选择性、 发展性、进阶性作业设计，支撑“双减”有效落实，促进学 生全面发展、健康成长。 7 “三增”提质量 “三减”消总量 “三变”添能量 ——重庆市南坪中学校“双减”工作的实践探索 重庆市南坪中学校党委书记、校长 刘平兴 “双减”工作是党中央坚持以人民为中心的发展思想， 立足实现第二个百年奋斗目标战略全局，着眼解决教学质量、 教育公平、教育生态存在的突出问题作出的重大决策部署。 重庆市南岸区南坪中学牢记“国之大者”，将“双减”工作 置于基础性、先导性、战略性位置，注重“加”与“减”的 辩证法，把握“破”与“立”的方法论，探索实践出“三增” “三减”实践路径，初步取得了“三变”的可喜成绩。 一、“三增”提质量 减轻学生负担，根本之策在于强化学校教育的主阵地作 用，全面提高学校教学质量。南坪中学坚持破立结合、先立 后破，以平台建设、评价改革和实践活动为抓手，以校内优 质的教育供给填补学生教育个性需求“真空”。 （一）增优质平台，让优质教育触手可及 一是增设网络远程平台。充分利用互联网和教学资源， 引入30 余名知名教师进驻网络平台， 开发微课视频， 推送教 学资源， 把线上答疑作为常态化课后服务， 建成南坪中学 “空 中课堂”，及时解决学生学习中遇到的问题，减轻家长辅导 压力，打通学生课后服务“最后一公里”。二是增设特色课 程平台。充分发挥南坪中学艺术特色学校优势，着眼于中学 生个性发展需求，在拓展课程基础上增设“六大”扬长特色 8 课程，建立起“辅导+拓展+扬长”的课后服务三阶课程。比 如，与美国华盛顿州雪兰社区学院、加拿大康乃狄克州肯特 学校缔结为友好学校，增设典范英语阅读课、圣三一口语课 等英语特色国际课程，提供个性化学习机会。三是增设师资 培训平台。搭建教师校内竞赛平台，开展党员示范课、青年 教师创新课、同一学科“同课异构”等活动，提升教师课堂 教学效能。积极推进名师、名工作室建设，组建南中教育智 库，引领教师职业成长。 （二）增评价措施，让作业负担科学可控 一是常态化考评。 对标对表 “双减” 政策要求， 制定 《南 坪中学作业管理办法》，明确教师发展中心对各学科作业布 置、完成、批改情况，每月开展 1 次抽查，每学期开展 2 次 全面检查，及时了解作业布置工作中亮点和不足。二是学生 间互评。开展基于证据的“三五”学生互相评价方式：五个 维度（自我评价、组内互评、小组长评价、学科长评价、学 习助理评价） 、 五个模块 （身心素养、 美德素养、 人文素养、 科学素养、艺术素养）、五个阶段（每日、每周、每月、每 学期、每学年），让评价在时间、空间上实现延展，关注人 的持续发展。三是反向式测评。修改完善《南坪中学教师考 核评价实施办法》 ， 把作业设计、 布置、 批改、 分析、 反馈、 辅导等环节纳入对教师专业素养培养和教学实绩考核评价 手段。采取“背靠背”式的座谈会、学生和家长开放性问卷 调查等多种方式开展评教，促进对教师作业管理的定性和定 量测评。 9 （三）增课后实践，让素质教育成就精彩 一是增加校内社团实践。学生社团是书本理论知识与生 活实践有机结合的重要载体。仅今年，学校就新增爱音社、 书法社、 动漫社、 Wonderland乐言英语社团等 12 个由学生自 主管理的实践社团，丰富学生课后校园文化生活，培养学生 艺术情趣， 促进学生心理健康发展。 二是增加校外研学实践。 校外集体研学实践是学校教育和校外教育衔接的创新形式， 是综合实践育人的有效途径。地理教研组和语文教研组分别 组织了《南岸地名文化》和《南岸景点科考》等集体研学实 践，既让学生亲近自然，又开拓视野。三是增加校内公益服 务实践。公益服务能有效促进学生智力、能力和品德的全面 发展。建立《南坪中学校中学生志愿服务记录档案》，增加 开学报道、公益讲座、防电诈宣传、抗击疫情宣传等志愿服 务活动，在公益活动中培养学生的社会担当、实践能力，促 进学生德智体美劳全面发展。 二、“三减”消总量 习近平总书记指出，义务教育最突出的问题之一是中小 学生负担太重， 短视化、 功利化问题没有根本解决。 “双减” 的落地见效离不开“减”，“减”的应是学生的学业负担、 家长的焦虑恐慌。南坪中学始终紧抓学生和家长这两大主体， 着力在“减”字上做文章，确保减到位、减彻底。 （一）为作业减厚度 作业管理是落实“双减”政策的重中之重。学校从优化 作业时间入手，探索实施“四时作业法”，全面减轻学生课 后作业负担。一是“计时”。实施教师试做制度，推算各层 10 次学生完成作业所需时间，在清单上进行标明，报年级学科 组备案，促进作业时长管理精准化。二是“限时”。由班主 任每天整合各科作业，汇总计算，统一发布，全面保障书面 作业平均完成时间不超过90 分钟， 避免 “九龙治水， 各管一 段”的现象发生。三是“错时”。针对部分学科因为内容需 要（如语文学科作文），花费时间可能会超过预先分配的时 间，由班主任与其他学科教师协商，减少其它学科书面作业 时间，适当增加实践性作业。四是“及时”。要求教师既要 做到“及时批”，对作业给予评价或等级，又要做到“及时 改”，引导学生订正，强化作业批改与反馈的育人功能。 （二）为学习减难度 一是强化教案审核。全面推行集体备课制度，由年级学 科组长对教案进行把关，审查难易程度和配置梯度，确保教 师不随意增减课时、提高难度、加快进度。二是优化校本课 程。结合南岸区三期课改，持续深入实施“美德品行、身心 健康、人文品位、艺术雅趣、科学智慧”为核心幸福素养课 程体系，建成南中特色课程群，激发学生学习兴趣。三是深 化课堂改革。充分利用南中特色“幸福e课堂”，开展混合式 教学，提高学生课前、课中、课后的效率，全面提升课堂教 学成效，打造高效课堂。 （三）为家长减热度 一是突出宣传引导。 学校通过微信公众号加大对 “双减” 政策的宣传， 转发中央和市区相关要求20 余条， 确保家长熟 悉相关政策要求。 二是突出沟通疏导。 开学前， 设立 “双减” 工作咨询电话， 方便家长反映问题和意见， 及时改进 “双减” 11 工作不足。三是突出教育指导。第一时间召开学校、年级、 班级三级家长会，讲清“双减”工作在解决学生“体质弱、 心理脆、负担重”等问题上的作用，引导家长树立正确的教 育观念。 三、“三变”添能量 检验一切工作的成效，最终都要看人民是否真正得到了 实惠。努力办好人民满意的教育是“双减”工作的最终落脚 点,是教育改革的重要价值和归宿。 只有在学生素养、 家长观 念、学校生态上实现综合性转变，才能真正实现“双减”目 标。 （一）学生素养变优。通过学校管理方式和课程体系的 重新建构，学生学习方式实现了从内容到形式、从时间到空 间的转变，学生有更多独立思考的时间，有利于促进学生自 主质疑、自主探究，促进创新能力的培养，推动学生在人文 底蕴、科学精神、学会学习、健康生活、责任担当、实践创 新等方面实现了素养优化提升。 （二）家长观念变新。通过家、校、社联动，家长改变 依赖的思想，重视对孩子的陪伴，亲子关系得到很好改善； 改变包办思想， 尊重孩子的心声， 让孩子的课余生活更精彩； 逐步去除“唯分数论”，重视对孩子学习习惯的培养，成就 孩子终身发展需要；去除“教育焦虑”，促使孩子形成学习 的内驱力，让孩子身心健康发展。 （三）学校生态变好。通过“三增”“三减”，学校健 全教学管理规程，深入实施新课程新教材，科学制订教学计 划，严格执行均衡编班的法律规定，学校在精品课程培育、 12 规范招生行为、尊重个体差异、均衡配置师资等方面实现了 良性发展。 虽然南坪中学 “双减” 工作卓有成效， 积累了一些经验， 但还需在深度上挖掘、广度上拓展、力度上强化，方能谱写 南中教育高质量发展新篇章，助力南岸教育优质均衡发展。 13 【广州南沙实验区】 悉心谋划“5+2”，课后服务质量优 ——“双减”背景下区域课后服务供给的南沙做法 2021 年7 月24 日，中共中央办公厅、国务院办公厅印 发《关于进一步减轻义务教育阶段学生作业负担和校外培训 负担的意见》，对“双减”工作进行全面部署，提出要提升 学校课后服务水平，满足学生多样化需求，并从保证课后服 务时间、提高课后服务质量、拓展课后服务渠道、做强做优 免费线上学习服务等方面明确了具体要求。教育部也做出部 署，确保秋季开学后，课后服务实现学校全覆盖。每周5 个 工作日， 保证每天都要开展， 每天课后服务不少于2 个小时。 南沙区作为双减全国试点城市之一广州市的一个行政 区，全区上下积极学习、认真解读“双减”政策，深刻领会 “双减”工作的重要意义，把全面落实“双减”政策作为头 号工程，全区73 所义务教育学校100%开展课后服务，73 所 学校课后服务时间100%达标。全区73 所学校充分利用每天 的课外服务时间，从“规范、阅读、强体、增艺、勤劳”五 个方面综合施策，通过悉心谋划，优化课程、活动供给，实 现减负提质、减压育人。 一、规范：调研充分具体，流程透明可见 一是充分调研，确保课后服务高效有质量。全区各校根 据相关文件精神，制定了《2021 学年学生校内课后服务实施 方案》，课后服务工作手册、安全应急预案等一系列方案措 14 施。 并通过教师会议， 家长会议收集征求了教师和家长意见， 进行广泛的调研、 讨论， 明确工作思路,确定服务内容， 完善 工作流程，课后服务工作得到了有效开展。 二是成立领导小组，确保落实各类方案。为统筹、协调 校内课后服务工作，除区教育局成立“双减”工作专班外， 全区各校成立了“双减”工作领导小组。由校长任组长，小 组办公室设在教导处，工作小组负责统筹、协调校内课后服 务工作，工作小组办公室负责日常工作，建立工作小组联络 员机制。家长是学生校内课后服务的重要责任方和参与方， 学校邀请家委会进行共同监督，对校内课后服务工作质量定 期进行评价，总结经验，共同进步。 三是明确课后服务申报流程。 全区各校本着 “家长自愿， 校内实施，有效监管”的原则，遴选第三方服务机构、明确 课后服务申报流程、召开课后服务课程遴选会议，对学生和 家长进行宣传。在这过程中，家委会代表全程参与。 二、阅读：文字浸润人生，阅读看护心灵 重视阅读对学生精神成长的重要作用。在减少学生课外 作业的同时，利用课后服务时间引导学生多读书、读好书。 根据《中小学生课外读物进校园管理办法的通知》要求，南 沙区中小学已100%完成了学校课外读物推荐的自查自纠， 100%制定了 《课外读物进校园审核制度》 ， 2021 年底将100% 完成2021 年课外读物推荐书目的报备工作。 通过自查自纠， 有效地防止了问题读物进入校园。同时，持续加强阅读资源 库建设，开展学校阅读资源建设及应用的评比活动，积极推 动学校开展阅读资源建设，利用好学校阅读资源开展有意义 15 的阅读活动，丰富师生的精神世界，助力“双减”政策更好 落地。 营造深厚的阅读氛围。各校校园文化建设中融入“书香” 元素，学校走廊、楼梯间、架空层都挂有名人名言、名人故 事、成语故事、推荐阅读等，各班教室每学期会定期出与阅 读相关的黑板报；高标准建设图书馆和阅览室，有条件的学 校还设有开放式读书吧、开放式阅览室；各校建设有书籍种 类较多的班级图书角，还有些学校配备了电子图书和有声图 书，有专门的电子书包室。课后时间，学生静静地挑上一本 书，默默享受书籍带来的快乐。 多形式开展读书活动。如读书分享会，图书介绍会，汇 编读书札记、绘制读书小报、荟聚读书分享会、诵读中华经 典、展演名著片段、聚思专题论文等。 三、强“体”：运动强身固本，精神养志培元 动起来。全区各校积极探索落实“每天运动一小时，健 康生活一辈子”的理念，开展趣味性、多形式的体育健康活 动，强壮学生身体，培养积极锻炼的好习惯。全区各校积极 落实每生必须有一项体育技能的要求，按课程方案要求开足 开齐，上好体育课；多元多彩，开展体育活动。课后，学生 的体育活动多了起来，学校操场热闹了起来，有的打篮球、 打网球、打乒乓球，有的踢足球、踢键子，有的跑步，有的 游泳，有的做健身操，还有的练习单双杠、跳高、跳远、跳 街舞等。学生动起来了，身体素质不断强起来。 睡得好。为保障学生充足的睡眠时间，提高学生午睡质 量，区内不少学校纷纷开设午睡课程，尤其难得的是学校想 16 尽办法，只为让午休学生能100%“平躺睡”。部分学校在教 室安装推拉床，如广州大学附属中学南沙实验学校；部分学 校购买午睡垫，如实验小学、滨海实验学校等；部分学校在 午休室安装固定床铺。为了能平躺睡，各校在分配午休教室 时， 切实考虑到了小学生和初中生不同的特点和需求。 小学， 一个午休教室里面的学生均为本行政班级的学生，避免了走 班或各班级混住，提高了管理的效率。初中，结合初中生的 生理和心理特征，划分男女生午休教室。学校和午托机构建 立联动机制，午休前，午托机构教师负责清点人数，午休期 间，由午休教师全程看班，并由学校行政值班人员负责每日 巡查，保障学生在校睡得好。 四、增艺：鸣琴回雪悦耳目，赤橙黄绿怡性情 学校根据《关于全面加强和改进新时代学校美育工作的 意见》，精心打造美育教育生态，用心营造美育教学氛围， 创新开发美育学科能力，丰富学生的精神世界，要求每名学 生至少掌握一项艺术技能。 音乐项目丰富多彩， 合唱、 合奏、 集体舞，从一个旋律，到一份乐谱；从一个节拍，到一场和 声； 从一个动作， 到一台视听盛宴， 南沙的美育既面向人人， 建立常态化普及型美育机制；又尊重差异，鼓励个性化创造 性艺术发展机制。 课后美术课程异彩纷呈。书法、绘画、陶艺，活动多， 样式新，学生兴趣足，参与率高。如，有些学校新开设了岩 彩画社团， 在笔墨间留存传统艺术魅力， 丰富时代审美价值。 从一块天然矿物质岩石，到一幅粗犷写意画；从历史遗存的 祖辈艺术，到眼前鲜活的绘画乐趣。坚守中华文化立场，坚 17 定文化自信。 五、勤“劳”：返察自然知本源，辛勉劳作全意志 在课后服务中，南沙区不断强化劳动教育主阵地作用， 在全面提升校内教育教学质量的同时，积极探索劳动教育与 “双减”政策有效结合的新路径，开展形式多样的活动，将 劳动教育融入学生的日常学习与生活，探索具有南沙特色的 劳动教育实施模式，提升中小学生劳动能力，赋能学生全面 发展。目前全区共有市劳动教育试点学校5 所，城乡结对学 校5 所，市劳动教材试验学校20 所，市校园小农田建设14 所。 各校充分利用学校现有条件，开展多形式的劳动教育。 尤其是结合道德与法治教材中的相关主题，开展各类主题活 动，在活动中引导学生劳动，培育学生的劳动技能。如联系 生活中有尊老孝亲意义的节日，比如母亲节、父亲节、重阳 节等，为家长创设情境，比如：“我送妈妈一日闲”，“爷 爷奶奶请坐下”，“这周我当家”等，让家长鼓励孩子主动 地多做家务劳动。组织学生参加力所能及的公益劳动，比如 主动擦拭楼梯扶手，协助保洁人员清扫电梯间和楼道，帮助 小区花工栽种绿植、 修建枝叶等等。 组织学生开展劳动竞赛， 如榄核中学利用学校植物园，引导各班种植中药材比赛；金 隆小学利用校园小农场，各班开展种菜竞赛等。 各校还积极与校外劳动基地或是周边村社联动，让学生 走进社会，在劳动中体验，在劳动中在成长，不断丰富“劳 动教育”的新形式。 18 六、辅导：一生一案促进步，合作学习共提升 南沙区非常重视学生作业的课后辅导，由各学科教研员 带头，各学科开展优化作业设计研究，要求作业量要减少， 质要提升。对于课后学生作业辅导，针对不同学生、不同学 科的特点，做到一生一案。作业辅导中，重视学生主动性的 发挥，引导学生组建合作学习小组，小组内同学相互帮助， 共同提高。 19 将“减负提质增优”放在心上 ——广州市南沙区金隆小学落实“双减”政策侧记 广州市南沙区金隆小学 李明秋 2021年7月， 中央办公厅、 国务院办公厅印发了 《关于进 一步减轻义务教育阶段学生作业负担和校外培训负担的意 见》（以下称《意见》）后，金隆小学通过组织干部教师认 真解读“双减”政策，深刻领会“双减”工作的重要意义， 决定把全面落实“双减”政策、确保减负提质作为学校2021 学年办学工作的头号工程，并立即相继采取了一系列举措， 现已取得阶段性成效。 一、全面而深入做好上级“双减”政策解读 “双减”启动以后，很多家长坚决拥护。之所以坚决拥 护，是因为自己的孩子可以在家不做作业、少做作业，周末 可以不参加培训班、少参加培训班；但同时家长也有一些担 忧， 是因为这些家长马上会想到， 中考、 高考不还是要考吗？ 既然要考，没有大量的作业训练如何保证学业成绩？没有校 外培训，如何让自己的孩子脱颖而出？面对上述情况的出现， 金隆小学将全面、深入地解读《意见》等各级“双减”类文 件放到了首要的工作位置。 一是在行政会议对《意见》进行解读与讨论；二是在全 体教师会对《意见》进行解读，然后分学科组进行讨论；三 是在家长委员会常务会议对《意见》进行解读与讨论；四是 通过线上方式向全体家长对《意见》进行解读。通过解读与 20 讨论，让全体干部教师与全体家长对“双减”政策有了更深 入、更全面、更准确的认识，也更加明晰了“双减”政策对 贯彻立德树人这一教育根本任务所具有的重大意义，统一了 思想，坚定了人人积极参与“双减”政策落地工作的决心。 二、细致而周到做好学校“双减”举措说明 上级“双减”政策如何在学校落地，就近而言，具体表 现为“五项管理”。要确保“五项管理”的实施到位，没有 家长的理解与配合是不可能达到管理工作的预期目标的。为 此，金隆小学在落实“五项管理”的过程中，尤其重视了对 学校“五项管理”相关工作制度出台前的起草、决策与实施 前的说明工作。一是关于“五项管理”相关工作制度的起草 工作均由相关部门完成，确保工作的专业性；二是关于“五 项管理”相关工作制度的讨论与修改，均征求了家长委员会 成员意见，均通过了学校行政会议审核；三是关于“五项管 理”相关工作制度的实施前，均向面向全体教师、全体家长 进行公示，并以班为单位向家长进行说明；四是利用好家校 顺畅的沟通渠道，及时收集实施过程中产生的新问题，并据 实对相关制度加上“人文关怀”方面的修正；五是校主要领 导与专家通过线上进行专业辅导，让家长对家庭教育、心理 健康教育有了更科学的认识与知识储备。 三、快速而优质推进校内课后服务工作 “双减”政策其中的一个目标就是保障学生体质健康， 而校内课后服务正是保障学生身体健康的一个重要抓手，学 校在推进校内课后服务工作上，坚决践行本校慧心教育理念， 坚持 “将孩子的健康成长放在心上” ， 努力做到了快速落实、 21 优质推进。 一是把家长困难放在心上， 让课后服务贴心、 暖心。 9月 1日， 学校一校三区的午间与课后服务就全面铺开， 只要家长 有服务需求，就供给，并一律采取先服务再收费的工作推进 准则。家长接送孩子时间不同，素质拓展需求不同，学校就 在服务中按不同批次送孩子到校门口。针对学校有不少家长 都要在18：00才下班这情况，第三段课后服务不仅延迟到了 18:30后， 而且以随到随接、 全免费的方式安排。 校内托管服 务通过家长承担、社会资助、财政适当补贴等方式多渠道解 决经费来源，多渠道合理分担成本，素质类课程收费不高于 南沙区少年宫同类型课程收费标准。对于家庭困难学生，金 隆小学更是送上暖心行动，家长可向班主任提交书面申请书、 家庭特困相关证明材料，经校内课后领导小组审核同意后， 可酌情减免部分或全部服务费。 二是把营养午餐放在心上，率先全区之先启动食堂自主 经营。 通过上级、 学校与家长三方的鼎力合作， 2021年1月， 学校食堂对食材供应、人力资源服务顺利地实现了分开招标。 2021年2月，金隆小学成为了南沙公办小学中第一所推进自 主经营本校食堂的学校。 2021年8月， 学校克服困难对金沙路 校区厨房进行了设备完善， 将 “教师专用厨房” 升级为了 “师 生厨房”。现在，金隆小学一校三区的师生们不仅都吃上了 在本校厨房加工的热饭热菜，而且还吃上了“全新鲜食材、 优质花生油、 优质丝苗米与每周都配置了牛肉、 牛奶、 水果” 的营养午餐。 三是把健康休息放在心上，据实推进优质午睡方式。现 22 在， 金隆小学本部参加午托人数2120人， 金沙路校区866人， 海滨路校区326人。 面对在学生午托人数众多、 教室面积较小 的情况，家校紧密协同，制定了安全、具体、可行的“一班 一案”午托平躺睡工作方案，目前，三个校区3000多名学生 已在校内全部实现了平躺睡。 四是把全面发展放在心上，创新课后服务实施。学校制 定了《2021学年学生校内课后服务实施方案》，在全区率先 提出并实施“5＋2”校内课后服务，目前，学校素质拓展服 务共开展体育类、 艺术类、 科普类等240多个社团， 不仅极大 地丰富了学生校园生活，更为促进学生全面而有个性的发展 奠定了扎实的基础。 四、积极而主动地开展校内教学的“减负提质”工作 家长对 “双减” 工作焦虑或顾虑， 究其根本就是担心 “双 减”政策会以降低教育质量为代价。因此，学校果断提出了 “着眼课堂与作业改革两大阵地”“坚持减负与提质同行” 的工作策略，以实际行动保障教学优质、消除家长担忧。 一是继续坚决践行“爱与活”的“慧心课堂文化”，努 力提升课堂教学质量。“双减”工作全面实施后，学校课堂 对保障教学质量的重要性将更加突出。为充分发挥每堂课在 学科教学中的作用，学校加大了教师教学技能培训力度，加 强了对教师精心备课的指导强度，更加努力地维护学生的课 堂主体功能，引导、指导学生喜欢学习、我要学习、我会学 习，让“互爱、爱学、学活、活用”的“慧心课堂文化”引 领师生共建有质量的课堂、高质量课堂。 二是全面推进“作业改革的研究与实践”，服务“减负 23 提质增优”。如果说课堂教学是师生互动、生生互动环境下 的一种学生学习活动，那么作业就是一种学生自主性很强的 学习活动。为了让作业这一学习活动充分实现“减负提质增 优”的目标，学校在四个方面进行了探索与尝试。第一，以 适合的教育理念为指导，以自然单元为整体，以学科育人为 主要目标，努力设计与布置“适合每个学生学业水平提升的 作业”；第二，根据学科属性、内容属性、学生认知特点与 学生行为习惯特点，设计、组织与推进“适合作业特征的作 业指导工作”；第三，根据政策要求、人文关怀、方法指导 等三个方面的要求，组织与落实“适合每个学生的作业批阅 工作”；第四，根据反映全面、突出问题、关注个体、讲求 实效的要求， 组织与落实 “适合每次作业的作业讲评工作” 。 金隆小学对落实“双减”政策的一系列实践与尝试，始 终围绕 “减负提质” 展开。 现在， 教师与家长在参与推进 “双 减”工作方面的目标明确、态度积极，学生精神状态好、学 习积极性高、身心也更健康。 学校在落实“双减”政策的有关工作中形成了一定的特 色、亮点。 一是“减负、提质齐迈步，师生、家长少顾虑”； 二是“手机、手表不入课，慧心课堂更优质”； 三是“自主供餐精选材，健康饮食育未来”； 四是“学生午休均平躺，健康睡眠高质量”； 五是“课题研究来推动，作业提质正加速”； 六是“课后服务全覆盖，家长放心学生爱”； 七是“社会资源都调动，家校合作有深度”。 24 【成都锦江实验区】 共建有温度的家庭教育生态圈 ——家庭教育锦江模式探索 成都市锦江区教育局 习近平总书记指出： “家庭是人生的第一所学校， 家长是 孩子的第一任老师， 要给孩子讲好 ‘人生第一课’ ， 帮助扣好 人生第一粒扣子，迈好人生的第一个台阶” 。 锦江区现有户籍人口45万，但教育服务人口超过100万， 学生家庭来源地区不同，结构层次不同。在课程教材研究所 的指导和支持下，锦江区委区政府将家庭教育纳入公共服务 体系，结合时代特点，积极寻求突破，努力提升家庭教育质 量。现从以下五个方面予以总结。 一、顶层设计，系统推进 一是明确思路。 以 “全纳、 公平、 优质、 均衡” 为目标， 以 “政府主导、 部门协作、 学校主抓、 家长主体、 社会支持” 为基本策略，以“平台搭建、课程建设、队伍培养、服务家 庭”为工作重点，全面推进家庭教育工作。 二是组建机构。2016年，依托锦江区社区教育学院，成 立了 “锦江区中小学 （幼儿园） 家庭教育研究与指导中心” ， 负责课题研究、队伍建设培训、活动策划和家庭教育专家顾 问团管理。 三是完善机制。成立由区政府分管领导牵头，相关部门 25 负责人参与的家庭教育工作领导小组。建立社治委、财政、 教育、民政、文广新、司法、法院、检察、团委、妇联、关 工委、街办、社区等单位联动的工作机制。将家庭教育工作 作为公众督导团日常挂牌督导的重要内容，纳入学校考核。 二、多维并进，筑牢阵地 一是建设数字化平台。充分发挥线上教育资源可选择、 易参与、可重复等优势，搭建“锦江全时空”学习交流互动 平台。联合成都广播电视大学，率先在全省试点开设“爸妈 成长团” 微信矩阵， 开展家庭教育专题课程、 个案咨询指导、 答疑解惑，目前累计授课2000余次。在“锦江教育云平台” 开设 “锦江心育” 模块， 通过微信公众号开设 “濯锦讲堂· 家 庭教育云课堂” ， 推送家庭教育微课程、 微视频课程100余节， 自制《智慧家长小课堂》微课程20节，开展《 “双减”后，家 庭教育怎样做？》 《以<家庭教育促进法>开拓家校社协同育 人新格局》 《家庭教育容易出现的偏差》 等直播课程30余次， 单个课程同时在线量超过2万人次。 二是优建学校阵地。加强对家长学校的规范管理，全区 中小学 （幼儿园） 做到 “有牌子、 有班子、 有经费、 有教师、 有教材、有计划、有活动、有总结” 。目前，全区所有中小学 （幼儿园）家长学校覆盖面达100%，成都市家庭教育示范校 6 所，区级家庭教育品牌活动36 个。各校（园）在开展好传 统家校共育课程的基础上， 创新特色课程， 如 “云竹会客厅” “爸爸妈妈故事会” 等， 家长参与学校家庭教育活动率达95% 以上。同时各学校持续开展“四季家庭公约”活动，运用线 上线下相结合的方式，追踪辅导特殊家庭，有效解决家庭教 26 育实际问题。 三是内化主题活动。 围绕 “传承好家训 弘扬好家风” 主 题， 开展 “寻找最美家庭” “好家风 好家训 好家规传承行动” “孝心家庭”等评选活动，树典型、立榜样。围绕家庭和谐 发展主题，开展“每周七点半”亲子阅读、父亲节专题、青 春期话题等主题分析、交流活动。社区学院联合教科院，根 据不同年龄段孩子和家长特点及需求，分学段开展“智慧家 长·父母成长营”主题沙龙活动。各学校结合实际，开展了 丰富多彩的教育活动，如： 《让幸福看得见》 《关于沟通哪些 事》 《新生适应指导》 《一套家谱•诉说家族故事》等。 四是助力社会组织。借助锦江区作为全国首批“和谐社 区建设示范城区” “全国社区管理和服务创新实验区” 发展优 势，通过购买服务等形式，引进社会力量，协助学校家庭教 育。 联合区检察院在内的29家单位， 在全区所有学校设置 “心 雨梦工厂”工作站，共同关护未成年心理健康，此项工作受 到了最高人民检察院的充分肯定。依托街道社区教育学校、 社区教育工作站和院落学习室，建成“区—街道—社区—院 落” 四级家庭教育平台， 每年开展送教活动100余场， 暑期 “社 区雏鹰”公益活动200余场；建立“四点半学校”21个，解决 部分未参加课后延时服务学生家长的后顾之忧。 三、注重科学，强化指导 一是队伍专业指导。 组织 “德育宣讲团” “专家讲师团” “家庭教育指导师” ， 每年面向学校、 社区、 街道开展专业送 教活动。 组织 “千名党员教师进社区” 活动， 开展问卷调查、 发放家教宣传资料和家庭教育读本，进行现场答惑解疑等。 27 二是专著系统引导。为破解家庭教育凭经验，缺乏可依 据、可参照标准和内容的难题，锦江区研发并出版了涵盖幼 儿园、 小学、 初中和高中四个学段的 《智慧家长指导丛书》 ， 通过丰富的案例、全面的问题诊断和系统的方法指导，提高 家长家庭教育能力。各学校还基于自身特色，开发了《溯源 明道——述说家族故事》 《我的育子故事》 《孝心孩子、孝心 家庭》等家庭教育读本 20 余本。 三是建档个性指导。为筛查发现心理行为异常的学生建 立个人追踪档案，对于一般心理健康问题的学生，学校专职 心理健康老师通过家庭访谈、学生沟通等形式，指导父母改 进家庭教育方式，调整学生心理行为；对于有心理健康问题 需医学治疗的学生， 追踪学生诊疗过程， 并向家庭提供支援和 辅导，帮助学生更好地改善心理状况。 四、夯实基础，建设队伍 一是凝聚专家团队。选聘业内知名专家和社会贤达组建 “锦江区家庭教育智囊团” ，为区域家庭教育改革和发展出 谋划策。组建讲师团。采取人才引进和本土培育相结合的方 式，组建“锦江区家庭教育讲师团” ，每年面向学校、社区、 街道开展公益送教活动。组建专业志愿者服务队。招募有一 定教育心理学知识、热衷服务的家长，组成志愿服务队，协 助学校，为有需求的家长提供有针对性的家庭教育指导服务。 二是培养名优导师。 在全市率先开展 “家庭教育指导师” 培训，每年在全区学校中遴选名优德育干部、班主任、科任 教师100余名， 定时定点定主题， 采用 “集中培训+分散自学” “线下培训+线上学习”和“理论学习+实际操作” “论坛+比 28 赛”等方式开展区域家庭教育指导师培训，五年共培训“家 庭教育指导师” 657名， 2020年遴选优秀学员培训 “家庭教育 讲师”60名。通过不断调整和完善，已构建起区级家庭教育 指导师及讲师的课程，实现了全区各中小学、幼儿园家庭教 育指导师全覆盖，大大提升了我区教师家庭教育专业水平。 五、创新思路，扎实研究 一是聚焦家庭教育重点、难点、热点问题等开展专题研 究。在研项目有《基于“家文化”主题活动课程的家校教育 共同体构建研究》 等50余项。 通过成果报告会、 经验分享会、 案例讲演会等形式实现区域共享，极大地提高了区域家庭教 育专业性和针对性。 二是把家庭教育情况纳入锦江教育质量综合监测。从家 庭教育环境、家庭教育方式、家校合作、亲子关系等方面， 开展周期性监测，为家庭教育提供科学指导。 随着工作推进， 锦江家庭教育逐渐构建起了 “三有” （有 课程、 有师资、 有阵地） 、 “三全” （全天候、 全方位、 全覆盖） 家庭教育服务品牌，形成了学校、家庭、社会“三位一体” 的家庭教育新格局。监测显示，锦江区学生心理健康状态发 展良好，幸福感等积极情绪指标持续上升，孤独、焦虑等消 极情绪指标持续降低，在沟通合作、承受挫折、心理复原、 意志品质等方面的心理指标均居全市前列。 锦江区将时刻牢记总书记对家庭教育的要求，在家庭教 育发展评价，家庭教育社会支持网络建设等方面着力，让家 庭教育成为学校文化、家庭生活的重要组成部分，营造有温 度的家庭教育生态圈。 29 “五优五引”建设高质量作业体系 ——成都市锦江区实施“优作业管理行动” 成都市锦江区教育局 作为课程教材研究所“五育并举人才培养实验示范区” ， 成都市锦江区一直在推进区域高质量教育体系建设上下功 夫，特别是“双减”政策实施以来，区域充分发挥学校教育 主阵地作用，以重点任务推进为主要抓手，在课程教材研究 所的指导下， 提出了校内减负 “六大工程、 二十五项行动” ， 其中 “锦江优作业管理行动” ， 抓住高质量作业管理中减负增 效的关键节点，以“五优五引”推动学校建设高质量作业体 系，全面强化作业育人功能，构建起良好教育生态，切实减 轻了学生过重学习负担，实现了小学生作业不出校门，有效 缓解了家长焦虑情绪，促进了小学生全面发展、健康成长。 一、优制度规引 锦江区教育局加强作业管理工作，印发了《关于加强学 生作业管理的指导意见》 ， 全面压减作业总量和时长， 减轻学 生过重作业负担。区教科院加快出台《锦江区中小学作业设 计与实施指导意见》 ，加强研究并指导学校建设校本化作业 体系，努力实现教练考一致。 各学校按照“一校一案”落实作业管理的主体责任，健 全了以校长为第一责任人的作业管理机制，全面重构学校的 作业管理制度、重构学校课堂教学结构、重构课后服务管理 30 制度，加强作业的全过程管理，提升作业育人功能，提升学 校教育教学质量。各学校加强作业管理闭环，完善了作业管 理细则，建立了作业校内公示制度、作业面批面改及时反馈 制度，实行作业总量审核监管和质量定期评价制度，把作业 设计、批改、指导和反馈情况纳入对教师专业素养和教学实 绩的综合考核评价。 二、优教研导引 区教科院以“好作业促进好课堂”为主题，将作业设计 与实施纳入教研体系， 加强对作业设计与实施的培训、 研究、 指导， 推动区域教研从学科课堂教学逐步转向 “课程、 教学、 评价”整体性教研。一是健全教师作业设计实施能力的培训 制度。将作业设计理念、自主设计作业能力纳入教师上岗培 训和岗位培训内容，纳入教师业务技能考核评比范围，切实 提升教师作业设计实施水平。全区先后组建语文、数学等学 科骨干教师培训班，开展单元作业设计培训与研究；定期召 开学校优作业管理主题分享交流，引领学校重建作业系统的 结构，重建课堂结构，提升干部教师作业管理与研究的专业 能力。二是强化作业设计与实施的教研制度。区教科院围绕 作业设计与实施履行好研究、指导职责，切实加强高质量作 业设计教研工作， 各学科教研员每学期做到 “五个一” ， 即一 份推进“好作业促进好课堂”改革计划，至少开展一次作业 实证调研，至少组织一次课程视域下“基于深度学习的单元 作业设计研究”的主题教研活动，至少完成一份作业设计的 评析和一份作业研究的相关总结。三是研发学科作业设计与 实施示例。区教科院组织各学科教研员和骨干教师，分学科 31 研究作业示例，包括学科作业主要类型、作业示例、实施建 议等，供教师优化作业设计时参照。四是建立优秀作业评选 与展示交流制度。区教科院积极支持鼓励学校教师系统化选 编、改编、创编符合学生学习规律、体现素质和能力培养导 向的基础性作业，定期开展优秀作业评选，建立优秀作业资 源库，加强优质作业资源共建共享。 各学校将作业设计与实施、批改与反馈作为校本教研的 重点内容，在教研组、备课组深入开展作业有效性研究，以 发展学生相关学科核心素养为目的，以开阔学生学科视野、 掌握学科方法、拓展学生思维深度和广度为质量标准，紧扣 课标教材和学生学情，立足单元整体，同步研究教材解读与 作业设计，课堂教学与课时作业，课后服务与巩固作业，项 目学习与综合实践作业，学习评价与作业实施，形成在课程 目标引领下的备、教、学、评一体化的新教学格局。学校每 周进行一次教学集中备课，其中一项内容就是学生作业，备 课组要研究确定作业数量和时间，制定出符合要求的作业计 划，然后由各学科老师根据实际情况微调。虽然学生的作业 量在减少，但是诊断、巩固和学情分析功能在增加，用优化 作业的“处方” ，助力精准教学，让“好作业”与“好课堂” 同频共振，让“轻负担、高质量”与学生相伴。 三、优科研牵引 一是课题研究让学生作业更优效。锦江区积极采取“总 课题+子课题+教师小专题”的方式，于2020年11月全面启动 了学校作业设计系列课题研究。首先，确立了区域总课题研 究，锦江区作为国家级优秀教学成果奖《提升中小学作业设 32 计质量的实践研究》的推广应用示范区，结合历经十年的深 度学习研究成果，确立了《基于深度学习的中小学作业设计 与管理研究》的课题，在作业设计、实施、评价和管理上进 行探索。 其次， 由区教科院教研员率领课题项目组开展研究。 围绕 “系统观视阈下学科单元作业设计与实施研究” “学科实 践性作业设计与实施研究” “大概念统整的学科单元作业设 计与实施研究” 等组织40余所学校展开研究。 第三， 设立 “作 业设计、实施、评价、管理”教师小专题研究专项。区教科 院设立教师小专题作业专项，研制发布作业小专题选题指南， 全区共立项93项小专题，500余名教师围绕学科实践性作业 设计与实施、中小学实践性作业设计与实施等展开研究。二 是技术赋能让学生作业更精准。探索将信息技术融入作业管 理，成师附小万科分校等学校，积极探索运用新技术手段， 把学生作业进行集中扫描，分析学生作业结果、作业目标达 成情况，关注不同学生的结果差异，深层次探寻产生差异的 原因。定期整理学生作业结果，反思教学中存在的偏差，及 时优化和改进教学。对错误较为集中的题目在课堂集中讲解， 错误不集中的则根据错题生成的学生名单，利用自习或课后 进行解疑答惑。 四、优联动推引 一是成立作业研究工作坊，教研员领衔各学科作业设计 研究。教研员组织全区优秀教学干部、骨干教师，分学科主 题化开展作业系列研究和培训，给予学校教师思路指引、方 法指导、操作建议。二是名师工作室率先探索，形成单元作 业案例的编制经验。从合理设定作业目标、科学提供作业内 33 容、整体把控作业难度、合理预估作业时间、灵活选择作业 类型、丰富作业设计形式等方面明确作业编制的核心任务， 把握作业案例编制要点，提高教师自主设计作业能力。三是 建立作业研究先行校团队，集群推动开展作业研究的学校共 研。由区教科院统筹指导，定期开展课堂教学与作业研究， 将作业设计、作业布置纳入教研科研体系，扭转一些学校作 业数量过多、质量不高、功能异化等突出问题，减轻学生过 重作业负担，发挥好作业的诊断、巩固、学情分析等功能。 目前已在川师附属实验学校、成师附小万科分校、沙河堡小 学等学校开展了 “基于UBD教学理念下的作业设计” 、 “培育核 心素养的作业整体优化探索”等研究成果分享交流。 五、优评价指引 一是健全作业管理督导评价机制。区教育局督导室把作 业管理情况列为学校办学水平评估、规范办学行为检查、责 任督学日常监管的重要内容，定期开展专项督导，并形成督 导报告，纳入年度目标考核。二是建立作业抽查评估工作机 制。将作业设计与实施作为常规调研视导的重点内容之一， 定时到校开展集体视导、课后服务专题巡视，将学校作业公 示、课堂作业情况、课后服务作业时长及完成情况、随机抽 查作业设计和批改反馈情况、随机访谈师生家长等情况，纳 入学校常规管理评估考核。三是建立作业实施情况评价机制。 将作业设计、实施与命题能力以及家校共育指导能力作为评 价教师教育教学能力的内容之一，将作业设计、批改和反馈 情况纳入对教师专业素养和教学实绩的考核评价。四是建立 学校作业监测机制。推进作业监测数据智能采集水平提升， 34 建构作业的绿色指标多元评价体系，将学生作业负担纳入教 育质量监测范围，逐步向社会公布作业管理监测情况。 锦江区加强作业管理， 深化作业研究， 发挥学校主动性， 提升作业效能，在作业变革的路上，砥砺前行，智慧成长。 35 【重庆南川实验区】 作业改革小切口 撬动教学大变革 ——重庆市南川区“12345”作业管理体系的构建与实践 重庆市南川区教育科学研究所 为落实“双减”政策，南川实验区立足区域实际，以作 业设计与实施为抓手，构建了“12345”作业管理体系，积极 探索减负提质增效的南川路径，通过作业改革小切口，撬动 区域教学大变革。 一、聚焦一个主题 全区作业改革围绕“减负增效提质”的主题，坚持顶层 设计和基层学校实践相结合，上下联动，齐抓共管，通过精 细化管理、科学化实施，守好育人主阵地，打好“双减”主 动战。 二、抓好两大行动 一是开展“问课行动”。紧扣“基于标准的教学”课堂 改革，制定问课要点“四步十问”，落实问课指导专家，建 立蹲点联系制度， 重点关注作业环节， 强调 “目标先于内容” “评价先于任务”的理念，将作业设计前置，注重教学评的 一致性，开展入校跟踪问课指导900余次。 二是开展“能力提升行动”。围绕作业设计与实施，区 级教研部门组织开展教师培训与教研活动50余次。遴选、推 送优秀作业设计和管理案例100余份， 供全区教师学习， 拓宽 36 教师作业设计能力提升渠道。学科教研员深入学校、深入课 堂，开展作业设计与实施的调查、评估与指导。 三、出台三个文件 一是聚焦课堂改革。出台《南川区“基于标准的教学” 课堂教学改革行动指南》，理清工作思路，落实推进机制， 明确基本要求，采取九大措施，着力提升课堂教学效率，全 面提升教育教学质量。 二是强化常规管理。出台《重庆市南川区中小学教学常 规基本要求》，明确课堂、作业、教研等教学常规的底线要 求，要求作业“四布置”“两不布置”，统筹作业时间和数 量，杜绝将学生作业变成家长作业。 三是加强作业指导。出台《南川区“基于标准的教学” 课堂改革作业设计与实施要点》，指导学科作业的有效设计 和实施，发挥作业的检测、巩固、反馈功能，重视学生综合 能力的培养，强调学科作业的育人功能。 四、落实四方责任 一是行政推动。基教科、督导室等行政科室结合区域现 状做好顶层设计，出台“1+5”配套文件促进“双减”政策的 全面落实。为加强督导考核，区教委成立专门的作业管理工 作小组，加大对作业管理和实施情况的督查和监管，将作业 改革效果纳入学校办学质量评价和综合目标考核。 二是教研引导。区级教研机构通过“送教送培”“网络 教研”“教学视导”等形式，发挥学科研究中心、名师工作 室等骨干研究力量，指导学校、教师有效开展作业设计专题 研究，通过优秀作业设计评比、先进作业管理经验交流、作 37 业设计课题研究、作业设计与实施学术研讨等，搭建交流平 台，培育推广典型案例。 三是学校主责。各中小学校发挥主体作用，自主设计符 合校情和学情的作业管理方案。建立学科统筹协调机制，规 范作业布置；建立教师试做机制，团队集体设计作业，优化 作业质量；建立作业反馈机制，设置底线标准，提高作业实 效。 四是家庭配合。通过家长会、家委会、公众号等渠道， 加大“双减”政策的宣传与解读，明确家庭教育责任，有效 落实家校合作。让作业管理透明化，有效建立家长监督机制 和作业反馈渠道，同时避免家庭盲目加码，形成家校育人合 力。 五、做细五个环节 一是严把设计关。 作业设计要关注课标要求、 教学目标、 学生学情，紧扣教材内容，落实教学评一致；要注重巩固基 础、结合新知、趣味拓展，推行差异化作业，分层实施；要 注重整体设计、联系生活、丰富内涵，凸显素养导向，培养 学生关键能力。 二是严把呈现关。按照课时作业、周末作业、假期作业 等不同类型，控量提质，拓展创新作业样式。做实做好基础 性作业，确保教学内容的有效检测；强化实践性、探究性、 综合性作业，强调学以致用；探索跨学科、项目式、长周期 作业以及 “互联网+” 作业， 促进线下作业与线上作业有机融 合，优势互补。 三是严把使用关。学科作业必须经备课组集体研讨、学 38 校相关负责人审核后方可实施，并根据实际情况及时进行调 整、修正。利用钉钉家长群、微信家长群同步发布作业，建 立学生、家长、教管中心等监督渠道，对超量作业和滥用教 辅等现象，学校必须严肃处理。 四是严把反馈关。 作业要做到有布置、 有批改、 有评讲、 有反馈，减少使用等级、数字等判断性反馈，重点开展评语 式反馈、个性化讲评和及时持续反馈。教师要利用现代信息 技术，建立学生日常作业档案，跟踪分析，根据作业反馈出 的问题开展自我诊断、反思和改进教学。 五是严把查评关。学校要加大作业常规管理力度，通过 定期查阅学生作业，及时发现作业设计和实施中的问题，与 教师进行反馈、沟通、分析，并持续追踪整改效果。把作业 设计、批改和反馈情况纳入教师专业素养和教学实绩的考核 中，切实落实作业管理的各项要求。 39 【江苏常州实验区】 “蓬勃生长”的书院儿童 ——“课后服务”工作的探索与实践 常州市武进区实验小学 积极心理学创始人塞利格曼教授，曾用“蓬勃人生”一 词表述具有持久幸福的含义，而最美的教育样态无疑也是 “蓬勃生长”的。 自“双减”政策出台以来，中小学课后服务倍受社会关 注。而面对“课后服务”这一作为解决家长急难愁盼问题的 民生工程，武实小有着自己朴素的认知：课后服务2 小时不 是时间和技能的简单堆砌，不是放学后的“托管”，而是坚 持“实践育人”的“第二课堂”。衡量课后服务工作的核心 指标应该是：家长满意、学生喜欢、教师认可并乐于参与其 中。让“三点半”后的校园生活依旧呈现“蓬勃生长”的幸 福教育样态。 一、蓬勃生长：有同频的思想 课后服务是一个创新举措，特别需要教师、学生、家长 达成共识。武实小第一时间通过问卷调查、师生访谈、家委 会等多种方式了解各方对于课后服务的真实需求与困惑。针 对家长普遍关心的几个问题，如：课后服务为什么做、做什 么、怎么做、谁来做、如何监督过程、如何评价效果等进行 多维、深度的沟通，实现同频共振。 40 基于沟通，抓牢课后服务的四个“破局点”。找准出发 点：安全第一、作业优先、兴趣为本、活动育人，精准定位 服务宗旨；把握切入点：坚持自愿原则，尊重学生、家长、 教师意愿， 科学设计服务环节； 明确着力点： 作业管理精准， 六艺课程创优，高效筑好服务主阵地；确定支撑点：扁平化 管理，优化提升服务质量。 目标明确后，还要有清晰的路径指引和强大的行政支持。 于是，学校管理团队进一步明晰了“十步走”流程指引：做 足功课，研读政策，制定实施方案；解读文件，思想引领， 做好宣传动员；统筹“家校社”各方资源力量；项目遴选， 确定课后服务项目； 提供课程菜单， 供学生、 家长自主选择； 班级汇总、年级统筹、责任部门全面协调；绘制课后服务学 校总表、年级总表、班级总表；课后服务全面运行，扁平化 管理；期末成果展，开展项目评价与反馈；撰写课后服务工 作总结报告。 在这样具体的流程指引下，学校课后服务循序渐进，稳 步运行。教师们从思想认同到行为再造，主动投身与适应课 后服务。 二、蓬勃生长：有活力和朝气 武实小“三点半”后的校园生活依旧呈现“蓬勃生长” 的教育样态，课后服务内容设计如下： 1.作业管理，夯实基本盘 作业是课后服务的重要内容，是评价服务质量的第一印 象。贯彻教育部办公厅《关于加强义务教育学校作业管理的 通知》的文件精神，结合“五项管理”的规定要求，我们在 41 课后服务的第一时段中重点做好以下工作：从总量协调，确 保作业上限和学科协调的“天花板”；从时间管理，推动作 业时速运营“效率板”；从管理制度，规范作业批改和主课 教师看班陪伴、答疑辅导的“规范板”。 如语文组侧重作业与大单元备课研究，形成《武实小 语文作业指南》，以星级难度分层作业，以能力指向归类 设计，布置弹性作业、阅读与生活创意作业及周末“无作业 日”等内容；数学组侧重从课堂出发，提质增效，用足用 好每节课，完成了《武实小数学课堂范式》；综合学科侧 重学科融合与实践作业研究等。各学科打好作业管理“组 合拳”，保障武实小学生在校高效率完成高质量作业。 2.“六艺时刻”，彰显多样性 课后延时服务的实质， 不仅是一种服务， 更是一种产品。 在实际操作中，针对课后服务的第二时段“六艺时刻”，学 校也经历了这样的三个阶段：第一阶段：“教师可操作，学 生兴趣弱”。为了让更多孩子接受优质的课后服务，针对每 个老师的特长和爱好进行了统计，开设丰富多彩的课程。第 二阶段：“学生兴趣浓，教师压力大”。对社团安排、学校 作息时间、教师课务等方面进行协调完善。第三阶段：“从 需求出发，统筹家校社资源”。孩子年段不同，家长的期待 也不同，一个月后再一次进行了课后服务改进。经过这样的 三步进阶，明确了本校课后服务的宗旨：家长满意、学生喜 欢、教师认可，真正实现让服务贴心，让产品优质。 目前，学校的“六艺时刻”的内容基于书院课程的五大 主题：文化传承、益智创新、艺美实践、运动健康、生命安 42 全，设计了百余节课程选项。既有校级层面的统一主题，如 周一“上善家长讲堂”、周二“书院大秀场”、周三“专题 教育” 。 每周一的 “上善家长讲堂” ， 邀请家长做一日讲师， 参与的家长根据自身职业特点、专业特长精心备课。在活动 中还会请他的孩子为其颁发奖状，小朋友都倍感光荣。“上 善家长讲堂”的设置也解决了学校没有时间开教师会议的这 一难题。 也有分年级、 分层次的 “一班一品” 自主类 “项目菜单” ， 即一年级一策，一班一案，最大限度满足学生多样化的学习 需求。 低年级： “游戏化课程” 。 遵循学生好奇、 好动的特点， 以 “游戏化课程” 为抓手， 助推幼小科学衔接。 一年级以 “人” 聚集游戏玩伴，打破学校围墙，利用青少年活动中心、文化 馆等资源聘请专业老师，实现“家、校、社”协同、“双导 师”联盟，解决了师资的难点；二年级以“物”开发游戏项 目，依托游戏器材篮球架、绳毽、轮滑等，通过混龄混班实 现破圈式学习，让孩子们不仅增体能，还学会交朋友。 中高年级：“36度课程”。小学高年级学生正处于情感 转折期， 自主意识逐渐增强。 于是， 在高年级延展学习空间， 开放物型场馆营造沉浸式学习场境，打造“36度课程”。有 “一日校长”、学科吉尼斯、书院梦工厂、上善文学社等自 治式生长活动，让学生有充分的学习自主权。 把课后服务嵌入儿童成长的关键节点，贯穿学生的校园 生活。9月，每周三课后服务的专题教育统一关注“食育”课 程和劳动课程。如课后服务期间补充能量的小零食吃什么， 43 怎么吃，吃完如何打扫，生活中的用餐礼仪又有哪些，都成 为了课程进行研究。10月，随着疫情突发，专题教育则是围 绕抗疫模范、防疫知识、心理疏导、如何健康居家等进行指 导，让课后服务切实指向儿童生活的真实情境与问题。 三、蓬勃生长：有质量的提升 在课后服务的运行中，定位儿童的学习与蓬勃生长的道 路上，学校需要一个调整速度的节拍器。只有精细化管理才 能让其推动最优化，效果最大化。学校把它视作提升学校管 理水平的重要机遇。 强化管理以保证质量：全面完善课后服务扁平化管理制 度。一位校长蹲点一个级部，所有中层以网格管理员深入班 级、年级、条线进行课后服务管理，即蹲点班级日查、值日 当天年级巡查、门口放学值岗，而校长室的课后服务一日简 报，更是让所有教师了解当天课后服务所有情况。这样的扁 平化管理，确保了家校的无缝对接与课后服务的秩序与安全。 用“共”的理念推动每一个人参与管理，彼此赋能。 评价导向以提升质量：在整体架构课后服务之初，学校 构思，要有显性的、家长切实看得见的评价，才能赢得家长 的肯定，增强课后服务的质量。为此，学校设计了“上善美 德卡” ， 关注学生性格的养成和习惯的提升， 从礼仪、 劳动、 阅读、好学和健美五个方面，引导学生做好每一天；并把每 日记录汇成“行动每一周”积累表，让学生与家长“看见” 收获，反思得失。期末还有“书院成果展”暨课后服务汇报 展，班班展示，生生参与。既有动态呈现，也有静态作品展 示，全面展现课后服务阶段学生的学科综合素养。 44 课后服务两个月后，学校又一次进行了学生、家长、教 师问卷调查。在反馈中，学校真实感受到学生和家长从中得 到的安全感、获得感、幸福感。 “双减”之下学校面临很多的变革与抉择，武实小因时 而动，顺势而为。循着“一切为了孩子”的质朴理念去做学 校课后服务管理中的微革新， 在试误中完善， 在完善中优化， 力求实现教育、学校、孩子的“蓬勃生长”。 45 着眼于关键能力提升的作业管理 常州市教科院附属中学 2019年6月中共中央 国务院印发的《关于深化教育教学 改革全面提高义务教育质量的意见》 、 2021年4月教育部办公 厅印发的《关于加强义务教育学校作业管理的通知》对作业 管理进行了规范要求。2021年7月，中共中央办公厅 国务院 办公厅印发了《关于进一步减轻义务教育阶段学生作业负担 和校外培训负担的意见》，直接提出“全面压减作业总量和 时长，减轻学生过重作业负担”的要求。作业管理既是“双 减” 的内容， 也是 “五项管理” 的重要内容。 作为基层学校， 只有做好作业管理，才能真正将“双减”和“五项管理”落 到实处；也只有做好作业管理，才能解决好作业数量过多、 质量不高、功能异化等问题，才能将关键能力的培养落实落 地。常州市教科院附中围绕作业管理做了一些思考与实践， 尝试破解中小学作业管理松、总量多、质量差、指导缺、反 馈慢等问题。 一、真实掌握作业的实际需求 树立一个理念。常州市教科院附中在作业设计中一直坚 持“作业设计先于教学设计”的理念。教学管理上，在课程 表编排时就考虑每天作业总量，做到作业总量设计先于课表 编排， 均衡考试科目与考察科目， 有效防止某一天作业量 “井 喷”，将“每天书面作业不超过90分钟”落到实处。教学设 计上，从单元和课时两个方面，把作业设计、学生作业完成 46 情况作为重要方面。这既是教-学-评一体化的基本要求，也 是教学逆向设计的基本操作。 梳理两份清单。附中教师以教材为本，逐章逐节分析知 识点、能力要求，梳理出知识点清单和能力要求清单，梳理 清楚它们之间的逻辑关系。教师研究清楚这两份清单，有助 于从整个学段角度认识本学科的知识与能力要求，不仅做到 课程标准在心中，还对标准达成的路径做到心中有数，这更 是平时设计作业内容、形式等方面的基础。 掌握三个实情。一是教学活动之后期望达到的水平层次， 二是学生在课堂教学过程中真实达到的水平层次，三是回顾 与本部分教学相关章节作业的达成度。这是控制作业难度、 提升作业有效性的基础，同时与作业总量的控制也息息相关。 作业难度过大， 往往就是在这个维度出了问题， 并导致教-学 -评的一致性出现严重偏差。 二、整体搭建作业的设计模型 明晰作业功能。作业作为课堂教学的延伸，其根本任务 是巩固核心知识，发展关键能力，应该具有“诊断、巩固、 学情分析等功能”。 搭建设计模型。 附中搭建的作 业设计模型包括三个核心要素： 明 确教学目标、 梳理教学内容、 确定 能力水平。 目标需要兼顾学科课程 标准要求和学生发展实际两个方 面； 内容既需要在面上做到教学内 容全覆盖，又需要在点上勾连相关的已有经验、已有认知； 图1 作业设计模型 47 水平应该对不同层次进行清晰界定。应对这样的要求，必须 既要注重夯实学科基础知识和基本技能，又要注重弥补关键 能力的不足。根据《义务教育化学课程标准（2011年版）》 和沪教版九年级《化学》第6章第2节教材，结合学生整体水 平，我校化学学科梳理、确定本部分的教学内容、目标及其 能力水平，如表1。 表1 教学内容、目标与能力水平 教学内容、目标 能力水平 A. 知道溶液是由溶质 和溶剂组成的。 水平1：a能说出生活中常见溶液的溶质、溶剂。 水平2：b能识记溶质、溶剂的概念；能根据溶质、溶剂给常见溶液进行命名。 水平3：c能根据概念分析常见的溶质、溶剂；能根据溶液名称判断溶质、溶剂。 水平4：d能初步判断化学反应后溶液中的溶质、溶剂。 B. （略） （略） C. （略） （略） D. （略） （略） E. （略） （略） 把控作业难度。 要求新授课作业的正确率应达到90%。 作 业反馈后，如果出现学生正确率过低，不是作业设计水平层 次出了问题，就是课堂教学出了问题。这么做，既夯实了学 科基础知识和基本技能，又弥补了关键能力的不足。 三、科学设计作业的结构蓝图 设计结构蓝图。设计一份高质 量的作业，必须强调结构。作业结 构就相当于人身体的骨架，没有良 好结构的作业是很难在各部分之间 增强内在的联系。结构设计必须紧 扣目标，从总量控制、题型结构、 内容结构、能力结构，以及难度结构等方面进行整体设计， 充分发挥每一道作业的综合育人功能。如，地理学科学习水 图2 作业结构蓝图 48 资源的时候，布置“了解自家近一年每月的用水量、主要用 途，分析家庭节水的具体措施”作业；数学学科轴对称、中 心对称的时候，要求学生制作剪纸作品，感受对称的魅 力…… 鲜活作业情境。英国教育家怀特海认为：“教育只有一 个主题，那就是五彩缤纷的生活。”教师创设与学生真实体 验相切合的作业情境，有助于构建真实生活情境与学科核心 知识、关键能力的内在联系。各学科的课程标准和教材都给 教师和学生提供了许多可供选择的学习情景素材，同时，在 学生日常生活中，也会接触到许许多多相关的情境，这些都 可以成为作业设计的情境资源。只有在作业设计中不断更新 情境，才能促进所学知识、能力在不同情境之间实现迁移， 使习得的核心知识、掌握的关键能力逐步内化为核心素养。 丰富作业类型。做到口头作业与笔头作业相结合，长作 业与短作业相结合，统一作业和个别作业相结合。实践性作 业也是附中作业的重要“题型”，比如在学习溶液以后，化 学老师布置学生充分利用家中常见用品配制一杯20%的蔗糖 溶液，这样的实践性作业，创设了“做中学”的真实情境， 学生经历搜集实践材料-动手设计方案-进行实验操作-总结 反思的实践过程，这既是对课内实验知识的拓展，也可以是 与化学学科相关的课外实践活动，能够与课程标准中对实践 性知识的要求相匹配，也是对所学核心知识的有效应用和关 键能力的有力提升；再比如在学习近代史的时候，历史老师 布置学生走进常州的青果巷，走访盛宣怀、周有光等诸多名 人故居，这不仅光关乎知识的习得，更是爱国爱家乡情怀的 49 激发， 这样的作业， 是对立德树人教育根本任务的直接呼应。 四、切实加强作业的过程指导 完成作业的过程，首先是对核心知识和关键能力进行运 用、内化的过程，也是一个对原有认知不断进行同化、顺应 的动态过程，同化和顺应是内化的两个方面。而知识的内化 是很少，也很难一次完成的，并不是完全的“立刻同化”和 “立刻顺应”，而是“正确概念”和“前概念”之间经过反 复多次碰撞，逐步形成新的认知结构的过程。这个过程往往 由于核心知识掌握不牢、关键能力运用不活等问题，就可能 使学生完成作业的过程不够顺利。因此，学生完成作业，有 时需要教师给予适当指导。 形成作业规范。良好的作业规范是有效学习的关键。附 中教师积极引导学生养成书写整洁、独立思考、认真完成的 作业习惯，还重视培养学生自主检查，按时上缴，自主订正 等规范。 注重学习指导。 指导从两方面进行， 一是作业难易方面， 要充分分析学生的课堂表现，对关键点进行重点提醒、适当 点拨；二是学生个体方面，可能由于知识掌握不到位，或者 学习能力不到位，给予针对性的、个别化的指导，同时对完 成作业的水平层次、上缴时间等给予差异化要求，有力保证 不同学生都得到公平的发展。 加强作业辅导。“双减”以来，大部分学生都在学校上 晚自习，合理编排各班的值班教师，学生晚自习过程中做到 “作业问题不出教室”，所有作业在当天晚自习结束时直接 交掉， 学生回家不再有笔头作业， 提升晚自习的效率和效益。 50 五、有效促进作业的评价反馈 作业从某种意义上说，是学生学习的作品。学校在作业 后期的“批”“评”“改”三个环节有效落实外，还丰富了 评价形式和反馈机制。定期举办学生作业展，让优秀引领优 秀；让学生设计作业，变被动作业为主动作业；让有特殊情 况的孩子免做作业、缓交作业，减少作业的刚性，增加作业 的柔性；通过作业耗时的教师预估、学生记录制度，对作业 总量进行反馈控制；通过学生作业目标卡制度，引导学生合 理规划。同时，学校定期开展以解题、讲题、命题为主题的 “三题”教师培训，帮助教师不断更新作业理念，提升业务 能力。 作业管理是学校教学的常规工作，也是永恒话题。作业 管理没有最好，只有更好。 51 【山东临沂实验区】 强化课程育人 助推“双减”落地 临沂市教育局 临沂市义务教育阶段在校生接近160 万，“双减”工作 关系千家万户。作为课程教材研究所的改革实验区，临沂市 始终坚持“走在前列”目标，根据省教育厅统一部署，结合 全市课改工作，主要做了两件事情：一是课后服务减负担， 二是课程拓展提质量。一方面积极探索开展课后服务，让有 需求的学生能够得到百分之百的保障；另一方面，积极开展 课程拓展，依托临沂革命老区丰富的红色教育资源，开发大 量的研学路线和实践课程，保障每一个沂蒙老区的孩子都能 接受红色教育，实现了“校校有项目，班班有特色，师生齐 参与”的课程育人格局，丰富了学生的学习生活，让孩子们 真正体会到课内学习的快乐，“减负”在润物细无声中顺利 进行。 一、行政推动，硬核“减负” 中央部署“双减”工作之前，临沂就已经开始落实校内 学生课业负担减负。2021 年1 月，市教育局发布了“年度教 育十件实事”，其中就包含“规范学生作业管理和提升基础 教育教学质量水平”。 中央部署“双减”工作之后，市教育局对以往“减负” 工作进行梳理，对“双减”工作重新谋划。在校外培训机构 管控上，严把审批关、严控排查项、严查办学行为；在校内 52 减负上， 积极落实课后延时服务 “5+2” 、 创新组织 “1 小时” 服务、开发多种拓展活动供学生自主选择，切实提升课后延 时服务实效，实现义务教育学校课后延时服务全覆盖、有需 求的学生全覆盖。 为满足家长不同接送需求，真正解群众之所困，指导县 区和学校根据家长需求设置2～3 个不同的延时服务结束时 间点，最早结束时间不得早于5：30，对于结束后仍有需求 的学生学校应继续提供延时托管服务，为家长下班后接送学 生提供便利。对于寄宿制初中学校，划清底线、明确纪律， 严禁将晚自习作为课后服务并收取费用，切实减轻家长负担， 严防课后服务“变质”。 二、“加减”协调，减负提质 “加减”协调就是校内“减负提质”，既要减轻校内学 生课业负担，又要提高教育教学质量。“减”的标准是小学 一、二年级不布置家庭书面作业，三至六年级书面作业完成 时间不超过60 分钟， 初中书面作业完成时间不超过90 分钟。 为落实这一标准，临沂市制定了《临沂市教育局关于开展中 小学作业管理专项整治行动的通知》《临沂市中小学规范办 学行为“十严十禁”》《临沂市教育局关于进一步规范办学 行为减轻中小学生过重课业负担工作实施方案》等系列减负 文件，通过加强监督检查，确保课业负担降下来。在校内课 业负担减负方面，兰山区建立“五级纵横联动”作业管控机 制。纵向上：一级，区教研室制定作业减负方案，并指导学 校落实；二级，学校教务处负责做好两个清单，明确各科作 业该布置什么、不该布置什么；三级，年级负责调度好作业 53 数量和形式；四级，备课组负责作业的质量；五级，由班主 任统筹作业分层。横向上主要是各个学校、各个年级、各个 学科、各个班级的各自创新作业管理机制。 “加减”协调的“加”即提高课后服务水平、提高校内 教育教学质量。在课后服务的提质增效方面，根据《山东省 义务教育学校课后服务工作规范（试行）》，采取“作业辅 导+素质拓展” 的服务形式， 将课后服务分为两个时段， 第一 时段是课业辅导，第二时段根据学校总体课程规划，聚焦育 人目标， 落实素质拓展活动， 促进学生综合素质的全面提升； 在提高教育教学质量方面。大力实施强课提质、强师提质行 动。围绕打造高效课堂，积极开展“六课”活动，通过名师 引领课、骨干教师示范课、每组一节优质课、青年教师研讨 课、新任教师达标课、每人一节展示课等，引领教师关注课 堂， 向45 分钟要质量。 围绕教师素质提升， 启动教师读写工 程、青蓝工程等，借助名师引领作用，带动广大教师更新教 育观念，实现自身专业化发展，提高教育教学质量。 三、疏堵结合，教培“降温” “疏”主要是向校外培训机构，宣传国家、省市关于落 实“双减”的政策，帮助他们认清形势、熟知政策、规范办 学、科学运营、良性发展。在整治校外培训工作期间，教育 局发布了 《致广大家长的一封信》 《致广大培训机构办学者的 一封信》 ，让广大学生家长及校外培训机构举办者熟悉有关 政策规定。通过建立黑白名单公示制度，在主要媒体平台定 时更新全市校外培训机构白、 黑名单。 综合运用网站、 微博、 微信等媒体，进行广泛宣传，引导机构主动积极办证。 54 “堵”也是必要的手段。教育部副部长郑富芝说“教育 改革发展的重点难点在哪里，教育督导就要跟进到哪里”。 临沂市教育局出台了《临沂市民办学校年度检查实施办法 （试行）》，将校外培训机构年度检查纳入年检督导范围， 这也是加强校外培训机构管理、减轻学生校外培训负担的措 施之一。临沂市按照凡新申请三类机构一律“停止审批”、 凡营业机构一律“拉网式排查”、凡违规培训行为一律“专 项整改”的要求，全面落实校外“双减”工作。7 月份，市教 育局联合市行政审批服务局在全市范围内停止审批三类培 训机构。 8 月份， 市教育局联合市民政局、 市场监督管理局、 行政审批服务局开展为期一个月的“证照不全”专项整治， 查处非法办学、违规培训行为，同时对校外培训机构进行安 全隐患大排查大整治。 四、课程育人，家校协同 减负是一项系统工程，解“表”不治本。临沂市充分借 助实验区优势，发挥课程育人功能，一校一策、一生一案、 分类指导、精准服务，从根本上解决学生课业负担过重的问 题。在市域层面，联合市委宣传部，按照“小学讲故事、初 中讲历史、高中讲精神”的思路一体化设计地方课程《沂蒙 精神》。在县域层面，依托突出地域特色开发研学课程，建 设国家级营地和基地4 个， 省级基地6 个， 市级基地23 个， 共组织社会实践活动75 期； 在学校层面， 坚持五育并举， 开 发丰富多彩的校本课程。如临沂朴园小学推行“书包不回家 作业不出校”措施，抓实1 小时学科作业辅导，实现小学生 零作业回家基础上， “搭桥”学校课程创新 “1 小时” 特色活 55 动，开发学科拓展性课程25 门，涉及语文学科的演讲、写 作、趣味语文，科学学科的STEAM 课程、趣味实验、生活大 咖秀等， 开发文艺、 体育、 劳动等7 大系列学生发展性课程。 同时，学校还积极拓宽课后服务渠道，借力校外优质资源， 开发朴雅社团课程35 门， 最大程度满足学生的不同需求， 供 学生自主选择，促进学生德智体美劳全面发展，切实提升课 后延时服务实效。 在解决场地问题上，鼓励学校统筹考虑校内资源，避免 扎堆搞活动，错时错峰地开展特色教学。对于孩子们而言， 每个下午的课后服务都是值得期待的，真正的实现寓教于乐。 至于孩子活动不同步的问题，主要遵从“因材施教”原则， 充分尊重和考虑每一个孩子的学习情况，灵活地开展辅导， 最大程度地来均衡作业时间，保证每一个孩子第二小时的充 分活动。 面对高考、就业竞争的压力，家长在“双减”面前不可 避免地存在着矛盾心理。一方面，对“双减”持支持态度， 另一方面，面对升学的压力，家长又陷入了深深的担忧。对 于家长的这种矛盾心理，临沂市按照“家校沟通畅渠道、多 种方式解疑难”的思路，在家校沟通下足功夫。组织学校通 过家长开学第一课、召开家长会、邀请家长在课后服务时间 进校园、举办家长沙龙等形式，把“双减”政策讲细讲透讲 好，让家长切实感受减负给学生成长带来的好处，从而让家 长成为“双减”政策的宣讲人和代言人，成效显著。 “双减”工作推进以来，受到了社会各界的广泛关注。 8 月20 日，省教育厅微信公众号以“扎实推进‘双减’工 56 作 强化学校育人主阵地作用”为题对临沂市“双减”做法 进行报道。9 月29 日，山东教育电视台《“双减”大家谈》 栏目对临沂市的“双减”经验作专题报道。11 月19 日，山 东教育电视台《聚焦·基础教育》系列节目报道临沂市双减 推进情况。 11 月25 日-26 日， 课程中心 “校家社协同创新育 人项目推进会” 在临沂市召开。 《山东教育报》 《临沂日报》 等也先后对有关经验做法进行专题报道。 57 【江苏徐州实验区】 “四位一体”推进义务教育阶段作业减负增效 习近平总书记强调，减轻学生负担，根本之策在于全面 提高学校教学质量，做到应教尽教，强化学校教育的主阵地 作用。徐州市云龙区教育局坚决扛起责任担当，以推进义务 教育阶段学生作业管理改革为抓手，进一步优化作业管理， 推动素质教育内涵发展，努力探索作业减负增效、提升学生 核心素养的创新之路，让学生学习更好回归校园。 一、建立一个机制，把稳改革之舵，避免作业布置随意 性 1.树牢“四精四必”导向。严格对标中央深化教育教学 改革的部署和教育部加强义务教育学校作业管理的要求，研 究制定《云龙区中小学“作业布置及批改”实施意见》，大 力倡导作业精选、精练、精讲、精批，做到有发必收、有收 必改、有错必纠、有练必评，用严格控制作业总量倒逼老师 改变教学方式、提高教学效率，努力破解作业数量多、质量 不高、功能异化等突出问题。 2.明确“四项考核”重点。区属所有中小学建立作业考 核管理细则和作业公示制度，占比100%。将作业设计能力、 作业量控制、作业有效反馈和作业改革研究纳入教学考核， 通过问卷调查、学生座谈会和月现场会定期调研作业情况， 徐州市云龙区教育局 58 发现典型经验进行推广。 3.强化“四大属性”设计。注重作业布置的计划性、科 学性、针对性和典型性，集体研究不同学段学生特点，设计 形式多样的全科课外阅读类、拓展类、实践类作业，避免布 置非必要的机械重复和大量抄写作业，小学一二年级不布置 书面家庭作业，小学其他年级每天书面作业完成时间平均不 超过60分钟、初中不超过90分钟，适当布置有思维含量的综 合性、创新性题目，坚持学生作业教师必先做，以实践换实 效。全区各校根据教师发展中心出台的文件精神，将作业设 计作为校本教研重点。通过区域集体培训，学校校本培训， 提高教师自主设计作业能力，定期开展优秀作业评选与展示 交流活动，加强优质作业资源共建共享。 二、打造一个平台，探索改革之先，提高作业管理灵活 性 以建设“基于教学改革、融合信息技术的新型教与学模 式”实验区为契机，区域独立研发了云龙区作业管理平台， 对辖区所有学校实行作业专项大数据监管。平台预警设置从 源头上严控作业时间，平台公示作业内容，可有效监控作业 布置的数量和质量。 精心构建符合本地化学情的语文、 数学、 英语、科学四学科区本题库资源引导学科老师根据实际需要， 从中精心选编、改编习题组成课时书面作业单，精准推送符 合学生个性发展需要的靶向作业。通过平台，也可以清晰地 掌握广大一线教师布置的非书面作业情况。如劳动与技术、 体育学科的周作业清单，科学实验、艺术欣赏、研学实践的 59 长周期作业，及时察看是否充分体现“五育并举”。通过平 台， 亦可以掌握广大一线教师布置的语文、 英语的课外阅读， 数学绘本、数学故事的阅读，艺体类、科学类故事的阅读， 察看是否在充分落实区域全学科大阅读的要求。通过各个学 科教师上传作业平台的作业内容，可清晰采集到学校作业布 置的学科覆盖面，特别是非书面作业内容设计、非书面作业 布置频率，平台的音、体、美、劳、STEAM跨学科作业的数据 的采集，可清晰地引导广大教师强化综合性的思维训练和动 手实践类作业设计。 1.用好有效时段提效率。根据作答时段和准确率分析学 生作业习惯，结合最高效率作业时段和作答特征，预测学生 课后学习效果与困难，提前进行干预、帮助、指导，夯实作 业减负增效之基。 2.依据用时分布纠偏差。严格落实管理平台运行机制， 作业内容、时长、发布全过程在平台运行，系统自动核查作 业总时长，对超时的进行预警、阻止发布。区教育局定期对 各校作业时长进行统计分析，把牢作业减负增长方向。 60 3.精准分析个体优设计。结合学生作业时间系数与作业 成绩两项指标，建成学生作答特征四象限模型，辅助老师进 行差异化教学，优化作业布置策略。 4.双向比对作业促均衡。坚持横向对比各学科作业平均 用时、纵向对比学生作业用时差异，及时指导老师调整作业 量及作业难度，推进作业分层设计，并创新实施区域教研员 跨学段作业量衔接管理。 61 三、站稳一个阵地，改变课堂之生态，凸显精准施教有 效性 站稳课堂教学主阵地，促课堂的转变。云龙区“四学” 课堂 （结伴互助学、 精准有效学、 开放自主学、 持续深度学） 是对“学讲”方式的深度推进。通过精准教研全区形成课堂 教学基于课程标准和学生核心素养，结合学科特点，用“大 单元”统整学科知识，全年段覆盖，全学科实施。改变单课 设计教学为单元整体设计教学。通过综合性学习活动，延展 课堂的时空边界，变课堂为“学堂”，变机械传授、重复的 作业练习为师生共学共研。探索在课程视域下以科学设计作 业为着力点助推课堂教学。 云兴小学结合省前瞻性项目 《 “物 联网+植物工厂” ： 指向儿童科学素养的校本实践与探索》 ， 根据教材的进度，将科学课课堂教学搬到了植物工厂，同时 每个班级成立了植物工厂科学研究小组，变革学习方式，进 行项目式学习、主题式学习、常规式学习，形成了课堂、课 外、社团、网络四级立体学习空间。 “四学课堂”带来的是学生学习方式的转变。“四学” 62 课堂中采用的探索融合型作业、分层型作业、游戏型作业、 实践型作业，改变了学生的学习方式。让学生在“思”中完 成作业，在“说”中完成作业，在“玩”中完成作业，在“实 践”中完成作业，在“动手”中完成作业，学生从作业中领 悟到书本中所学不到的生活真谛，培养了应用知识能力、学 科整合能力、实践创新能力和合作交往能力，学生的意志得 以锻炼，潜能得以挖掘，孩子们逐步养成良好的学习习惯， 激发出不断向上的活力，有效地提升了学生核心素养的提升。 云龙“四学课堂”带来的是我区学生在学习动力进步指 数、学业负担进步指数、师生关系进步指数、教师教学方式 指数等影响学业质量的一些关键因素趋向向好。在校长课程 领导力、教师教学方式、师生关系、学校统一补课指数、作 业指数、睡眠指数、学习压力指数、内部学习动力指数、学 习自信心指数、对学校满意度指数、高层次思维能力指数均 等于或高于全市平均指数。成绩的取得靠的就是教育教学改 革引发作业变革， 靠的就是作业改革切实做到了 “减负增效” ， 全方位提升学生核心素养。 四、落实一个保障，凝聚改革之力，监督作业减负稳定 性 坚持把监督检查作为学生作业减负增效的关键一环，成 立局、校两级工作专班，压紧压实各方责任。 1.压实领导责任。把作业减负增效作为一项重要政治任 63 务，纳入全区教育工作和全局党建工作要点、各校年度重点 工作目标，学校班子年度考核一体推进，构建了“一把手” 负总责、分管领导具体抓、全员抓落实的工作格局。 2.压实督学责任。区教育督学包挂到校，定期随机进行 《学生“减负”问卷调查》，全面了解学生平均每天睡眠时 间、完成家庭作业时间、参加学校音、体、美、科技活动情 况、 教辅资料及课外补习等情况， 与学生家长进行电话访谈， 形成调研报告反馈各学校。压实班主任责任。明确班主任是 作业总量控制的第一责任人，通过学生人手一册的《作业调 控本》对每天作业实施监督，定期向中等学业水平的学生了 解作业完成情况、睡眠情况、锻炼游戏情况。 3.实行专员监督。教研部门飞行检查教师备课和学生作 业，实地教研和跟踪听课，帮助老师提升课堂教学效果。聘 请有责任心的家长担任“社会监督员”，从学生完成作业时 间、 睡眠时间等方面进行监督， 发挥好家校沟通反馈的作用。 育人使命， 重任在肩。 云龙教育人坚决落实中央 “双减” 精神， 努力为学生的作业减负增效贡献云龙方案、 云龙实践、 云龙智慧！肩负起新时代教书育人的神圣使命！ 64 “减”与“加”的智慧调和之道 ——徐州市大马路小学校“双减”背景下课堂提质 增效的校本实践 徐州市大马路小学校 “双减”政策的出台，旨在还原教育原本的模样，让教 育回归本真。“减”需要符合规律，需要适应儿童，需要促 进发展。徐州市大马路小学校立足课堂，积极开展课堂提质 增效的校本实践， 不断探寻 “减” 与 “加” 的智慧调和之道。 一、教学目标：“减”散点设计，“加”系统思维 大马路小学坚持十多年的整体把握教材活动现已经过 三次升级迭代，特别强调在学期初即形成对本学期核心目标 的精准定位。教师们坚持以“系统化思维”确立目标，通过 对学科课程标准的解读、学期核心任务的分解以及单元学习 目标的梳理形成具有整体性、连贯性的目标体系。 例如，义务教育阶段语文课程标准对低中高三个学段提 出的 “朗读” 能力的要求分别指向三个层级： “学习用” “用” “能用” 普通话正确、 流利、 有感情地朗读课文,呈现出了由 低到高的学段要求的变化。在一年级第一学期，教师们根据 课标的要求以及统编教材的教学内容，设定了学期朗读教学 核心任务：1.能够准确认读生字，读好陈述句和疑问句；2.
认识自然段，读出自然段之间的停顿；3.学会分角色朗读课 文；4.能够根据不同的语言环境，读好“一”“不”等字的 变调。 65 在学期核心任务的基础上，教师们又结合具体的课文单 元，设计了单元朗读目标。经历这个过程，每一位教师不仅 深度了解将要执教的“这一册”“这个单元”的目标定位， 还在倾听其他教研组的交流中把握学科知识的关联、学科逻 辑的联系，形成对学科核心素养的整体认识与把握。 二、教学内容：“减”课时主义，“加”单元重构 课堂是教育教学的主阵地，大马路小学校提出以“学科 教学主张的再优化”为核心任务的“学科育人”校本行动方 案，着力于立足学科课程，依据课程标准对教材进行校本的 二度开发和深度整合， 基于学科知识的文化意义、 社会功能、 精神价值进行融会贯通，利用整合思维，实现学科内容与社 会生活的关联，强调单元思想和单元视角，以联系、统整的 思维架构学科课程模块和跨学科的内容整合单元。语文学科 的“大单元、大情境、大任务”设计，数学学科基于数学实 验的单元统整以及英语大主题引领下的单元教学日趋成熟。 学校于2021年成功申报的江苏省基础教育前瞻性教学改革 实验项目“学科教学中儿童想象力的培养”更加突显儿童的 认知特征和想象的价值，将引领学校以创新素养培育为目标 开展“指向未来教育”的深度课堂变革。 三、教学实施：“减”单向传授，“加”任务驱动 推进学校课程改革，切实提升中小学课堂教学效率，应 着重培养学生解决现实问题的能力，而传统的教师单向传授 知识点的教学难以达成这样的目标。在单元统整的视角下， 学校积极实践，尝试设计与真实生活紧密联系的，具有挑战 性、综合性的驱动任务，让学生在真实情境中体验“做事” 66 的过程，实现素养的提升。 以统编小学语文三年级下册第六单元为例，在“剃头大 师是怎样炼成的”这个学习活动中，教师们设计了“请你来 劝说”和“试着做解释”两个任务。在这个过程中，教师根 据情境与学生共同思考同一个话题，参与同一个活动，引入 新的故事情节。而学生则在扮演各种角色，建构各种关系， 理解各种现象的过程中生成了新的意义理解。 四、教学评价：“减”随意盲目，“加”精准分析 在大马路小学，聚焦主题，全员卷入的观课、评课已成 为常态。在“双减”背景下，学校又借助智慧课堂的分析系 统，在对课堂教学实时录播的同时，全面采集学生的举手、 读写、应答、互动等数据，借助对数据的精准分析，帮助教 师全面评析学生学习情况，破解教学重点、难点，进一步提 升教学效率。 以一位教师执教《总也倒不了的老屋》为例，初次执教 的活动数据显示学生自主预测时的参与率不高，仅有36.54%， 互动效果不够理想。为了解决这个问题，教师创造性地设计 了绘本故事《小猪变形记》的拓展阅读环节，引导学生将习 得的预测方法运用到陌生的文本中，学生不仅成为阅读的参 与者， 还成为阅读的创造者， 使 “预测” 真正发生。 调整后， 学生的行为占有率提高至73.2%，课堂类型属于理想的对话 型课堂。数据准确反映了各个教学环节中的真实问题，有效 促进了“教-学-评”一致性的达成度。 五、教研保障：“减”低效形式，“加”主题反思 扎实有效的教研活动是提升课堂教学质量的有力保障。 67 学校对集中教研进行了改革，将各个学科的教研时间进行了 分散安排，引导教师围绕“情境设计”“任务规划”“氛围 营造”“作业管理”等主题开展研讨活动。学期末，学校举 行 “做学科育人的明师” 系列活动， 开展 “我的教学微改变” 主题交流。全校教师从课堂上的一个个微改变做起，推陈出 新，并将教育教学实践梳理、提升为专业智慧，形成专业表 达。 学校每学年还开展优秀学科教研组评选活动，各组教师 通过自我审视与发现，凝聚学科教研组的核心价值追求，创 新活动形式，实现活动的序列化、规范化和制度化。形成了 期初 “确立主题、 开展研究” ， 期末 “汇报交流、 团队展示” 的闭环式研修管理方式，提升了学科团队深度、有效的专业 进阶研修能力。 68 “双减”之下，变革学习方式是课堂育人的根本之策 ——271教育大单元整体学习研究的实践范式 山东二七一教育集团 赵丰平 2021年5月21日召开的中央全面深化改革委员会第十九 次会议审议通过《关于进一步减轻义务教育阶段学生作业负 担和校外培训负担的意见》，并配套出台了一系列的措施， 对校内学生作业负担和校外培训机构进行规范治理。从高分 到高素质，“双减”是在为教育大计的稳步发展做减法，是 为下一代的成长成才做加法，减掉功利心，减掉焦虑感，摒 弃应试教育的弊端，培养出完善的素质教育体系，让真正的 教育回归本源。 “双减” 之下， 271教育集团全面落实立德树人根本任务， 全面实施德智体美劳五育并举，聚焦师生生命成长，大力推 进课程课堂改革，彻底改变学习方式，全面实施大单元整体 学习， 减负不减质， 探索出了一条适合的课堂学习变革之路。 一、厘清概念，大单元整体学习是什么 大单元整体学习是从核心价值出发，紧紧围绕课程大概 念，通过整体感知、探究建构、应用迁移、重构拓展四个学 习阶段， 让学生从整体上认知学科知识与逻辑、 结构与本质、 应用与生成、价值与意义，培养学生思维能力的一种循环往 复的自学、对话、建构、生成的整体认知过程。 学科大概念是一个学术问题，围绕抽象核心观念和概念， 把相关知识、能力和结构逻辑建构成一个相对独立、整体的 69 概念体系。学科大概念的特质是恒定不变的学科思想、逻辑 和体系。 课程大概念是基于学科大概念，为育人价值追求整合学 科间知识、社会生活实际、教师学生学习生活经验，符合学 生认知特点，直指学生核心素养养成的整体学习内容，提供 了有利于学生围绕大概念整体思考、创造建构的学习内容。 课程大概念的特质是整体性的、综合性的、生活性的、开源 性的、开放性的、生成性的。 二、溯本求源，大单元整体学习为什么 “双减”之下，学校教育重构力求改变：学生是主体、 学习过程自主发生、整体认知建构课程大概念、学习式自主 对话与探究生成、学习组织相互启发、启动自我系统等等， 关注的是学生未来。所以，推进大单元整体学习既是国家要 求，又是学生生命成长需要，也是271教育的价值追求。 1.大单元整体学习符合认知规律，认知过程给学生。一 是学生整体认知是一个螺旋上升的过程。二是教师应该给学 生问题、场景、任务，不给答案，多策略推动学生思考。三 是知识是学生基于经验的主观建构。四是学习应该在对话中 进行并生成新知。 2.大单元整体学习克服学习碎片化，整体认知不动摇。 学习过程真正成为独立、创造、生成的完整过程，真正解决 了四个碎片化。即学习知识碎片化、学习时间碎片化、探究 过程碎片化、思维建构碎片化。 3.大单元整体学习确保学生主体性，育人目标不游离。 一是教师机械讲解、 训练变成育人， 扭转学生被动学习局面。 70 二是学生按照自己认知来感知世界，释放思维活力，打开多 元思维闸门。三是学生全面、自主、合作、探究，真正成为 学习的主人， 高阶思维成为可能。 四是教师全程引领、 陪伴、 激发。 4.指向高阶思维，学习价值求突破。一是大单元学习目 标指向人的完整发展而非学科知识。二是大单元整体学习基 于核心价值、通向学科素养。三是通过整体认知建构，学生 接触高阶思维。四是知识的形成和任务的完成都在学习过程 中自然而然实现。 三、科学实施，大单元整体学习怎么做 1.大单元整体学习按照四个学习阶段来整体设计实施。 第一学习阶段： 整体感知。 学生自主建构单元知识结构、 能力结构，整体感知、链接、生成。 第二学习阶段：探究建构。探究清楚知识与已有知识、 能力、生活的必然联系，独立建构链接。 第三学习阶段：应用迁移。结合探究建构成果，创新应 用，解决问题，并迁移到新情境中，解决真实问题、生成新 知。 第四学习阶段： 重构拓展。 生成并完善自己的知识结构、 能力结构、逻辑结构和价值意义结构，完成清晰大概念的最 终建构，拓展创新。 71 2.学习目标引领学习全过程。 首先聚焦到真实目标，先确定真实学习目标，真实目标 设计要基于 “一核四层四翼” 、 基于核心价值的要求去设计， 坚决避免就知识而知识。每一个学习阶段都要围绕整体学习 目标由里到外设计学习任务、设计学习过程，并且符合新高 考“一核四层四翼”的要求。 3.设计驱动性学习任务。 基于核心价值， 创造学习任务， 直指真实学习目标追求， 指向学科素养，最终通过各种学习任务的完成过程，全力推 动学生全过程、全天候动脑思考，感知建构。任务设计必须 是逆向设计：核心价值→学科素养→关键能力→必备知识； 而非：必备知识→关键能力→学科素养→核心价值。 4.重构大单元课程学习内容。 一是紧紧围绕课程大概念，整合所有资源，突出学习内 容的整体性、过程性、增值性、挑战性。二是将知识理解、 应用、创造和学科本质、教育本质、教育价值意义逻辑到一 起进行重构。 三是大开大合。 大开， 一定要打破原来的思维， 突破原有知识束缚，开放到位。大合，按照事物本质和学生 72 学习实际重新整合，引领学生结构化思维到位。四是跟学生 的学习经验、已有认知、生活场景、螺旋上升的认知过程紧 密链接在一起进行重构。 四、大单元整体学习课堂实践 271教育课堂学习以马克思主义实践观为理论指导，从 人和世界的相互作用、相互创造入手，把学生的学习过程作 为“一切将成”的生成性思维方式，既有社会总体实践活动 经验， 又有个体生活实践经验。 271教育课堂注重生成， 强调 过程即目的； 注重思维逻辑， 强调学生为中心的个性化学习， 强化学习过程设计与实施。遵从“认识—实践—认识—再实 践—再认识”的马克思主义实践模型，通过认知内化—实践 生成—迁移提升，在这一过程中达成学习目标，落实学科核 心素养，培养德智体美劳全面发展的人。 1.271教育课堂五个特点。 一是271教育课堂学习设计是大单元整体学习设计，特 别重视体系建构。二是271教育课堂特别重视学习目标的制 定。三是271教育课堂学习过程是靠任务驱动来完成。四是 271教育课堂特别重视思维训练。五是271教育课堂的学习方 式是学生自主探究，小组合作学习的方式。 2.271教育课堂的“十六字”要求。 任务驱动：变要我学为我要学； 情境体验：变关注知识为关注人； 真实探究：变关注学习结果为关注学习过程； 迁移提升：变关注实践应用为关注素养达成。 73 （1） 呈现真实情境， 让学生体验、 激发学生探究学习的 欲望。 （2） 明确目标任务， 驱动学生去学内容——认知内化： 学生通过自主学习、合作学习，梳理本课时内容的知识、能 力、方法，并加以内化。 （3） 用所学的知识、 能力、 方法去做任务——实践生成： 通过学的内容来做任务，解决问题，在做任务的过程中加以 总结梳理，内化所学内容，上升到学科理解，习得、重构、 生成新的知识、能力、方法、思维、价值意义。 （4）相互交流新生成的知识、能力、方法、思维、价值 意义，到新的情境中、到社会生活中去解决新的问题，从而 生成对学科核心概念更深刻的理解，这就是迁移提升。 271教育课堂是在真实情境中， 运用知识、 技能、 方法解 决实际问题，在解决实际问题过程中体验过程方法，生成价 值意义，并迁移到社会生活中，激发兴趣、创造意义。既是 一种高阶思维，又是一种真实学习。 “双减” 为学校教育提出新的方向和新的挑战， 271教育 坚定不移的走在党和政府领导下教育正道上，承担“回归教 育本真，创新适性发展，坚守民族梦想，贡献中国教育”的 271教育使命，用实际行动践行“做真教育，真做教育”的铮 铮誓言！ 3.271教育课堂基本的思路。 74 【二七一教育实验基地】 “双减”背景下小学数学深度学习研究与实践 潍坊峡山二七一双语小学 孙瑞华 “双减”减的到底是什么？在《人民教育》杂志中提到 “要减的是给孩子不必要的、重复的学业负担，双减的本质 不是减，而是给孩子更大的发展空间。”为了提升小学数学 育人品质，潍坊峡山二七一双语小学进行了基于核心素养的 小学数学深度学习研究与实践，并提出了“i 数学”的育人 理念。 “i 数学”就是“我的数学”，就是为“这一个”孩子 设计的数学课程。i 的显性含义：有趣的、个性化的、有创 造力的、互联网+；隐形含义：爱。“i 数学”旨在通过整合 课程、改进课堂及关注个体的变革实现数学的深度学习。 一、整合课程，让数学课程鲜活有趣 著名数学大师陈省身先生曾提出“数学好玩”的观念， 即只有让学生体会到数学的好玩，培养学习数学的兴趣，孩 子才能真正深入学习 数学。但在应试教育 的背景下， “考” “练” 抹杀了儿童的数学学 习兴趣。小学教学的 对象是儿童，如果把 75 一串枯燥的数学知识堆砌到一起，儿童肯定是不喜欢的。要 想将课程整合效果发挥到极致，必须要走进儿童的内心，将 数学用儿童的喜欢方式呈现出来。因此学校将数学史、数学 故事、 数学游戏、 数学写绘、 数学日记、 手抄报、 思维导图、 数学调查、数学家的故事纳入到i 数学课程中，让数学呈现 出丰富多彩的样态。同时，为了满足学生的个性化需求，开 发了利于数学思维发展的校本课程，主要包括魔方、数独、 环板扣、趣味数学、象棋等，实现i 数学课程的生活化、情 境化、实践化、体系化。 二、大单元整体学习设计，让数学学习走向深度 作为一线教师，核心工作就是认真备好课，让孩子享受 课堂，但到学生实际应用知识时，效果却不佳。在学生的脑 海中知识碎片化，知识之间没有建立起联系，学生难以形成 深层次的理解。为此，进行了大单元整体学习设计，摒弃单 一的课时设计，改为以单元为单位的整体学习设计，帮助学 生建立起立体且多元化的认知。 在《学历案与深度学习》一书中提到，“教师在专业教 学过程中， 需要了解和解决三个问题： 一是学生应该学到哪； 二是学生已经学到哪里了；三是学生怎么去那里。”这里将 76 其总结为四个方面：在哪里、去哪里、怎样去、到了吗。学 校建立了“单元前测——核心素养——课标、教材--单元大 概念——单元目标——单元任务——课时任务与活动—— 单元后测”的研究路径进行大单元整体学习设计与实践。 （一） 在哪里。 每个学生都有自己的学习起点， 在开始教 学前帮助学生找到自己的学习起点，理清起点与终点的距离， 从而让学生科学规划自己的学习路径，就如同医生在给病人 看病时， 先让病人去做化验 （了解病人的健康起点） 一样,因 此每个单元学习前，先进行预习任务的设计，学生完成前测 后，再对每个孩子的前测结果进行详细分析，了解每个孩子 的学习起点。 （二）去哪里。首先教师要深度研究课程标准，既要研 究本学段课标，也要研究更高学段的课标，还要研读不同版 本的教材。通过研究课标教材，分析挖掘出单元大概念，制 定出明确的单元目标。 （三）怎么去。明确了学生学习起点及目标后，设计单 元任务。任务设计突出数学与生活的双向融合，将生活实际 问题放在数学课程的中心，激发学生自主学习的欲望，发展 学生学数学、 用数学的良好素养， 实现 “以用驱学” 的目的。 （四）到了吗。单元学习结束后，设计重构拓展课，帮 助学生建立起单元知识结构，提升学生综合应用能力。最后 设计指向核心素养的单元后测，通过单元后测形成每个孩子 的数据报告，让教师做明明白白的教学者，让学生做明明白 白的学习者。 通过以上大单元整体学习的设计，以核心素养为起点和 77 终点，让数学学习走向思维深处。 三、关注个体，让学习指向“这一个” 为满足不同孩子的学习需求，研发支撑学生个性化成长 的数学学习文本，文本从目标、评价、情境、任务、问题、 探究、拓展、收获八部分进行了指向“这一个”的设计，旨 在满足“这一个”孩子不同学习阶段的需求。 （一） 目标设计。 教师在每节课之前都先制定学习目标， 目标面向班级整体学生。如何将教师制定的共性目标转化成 学生的个性目标？课前让学生制定自己的目标，每个孩子根 据自己的学习基础和认知水平制定出适合自己的学习目标。 （二）评价设计。明确学习目标之后，先设计出具体清 晰可量化的评价标准，让学生在学习前就清楚知道“我要去 哪里”。每节课都设计了评价接口，学生在学习过程中可随 时检测个人目标达成情况。 （三）情境体验。教材中有很多资源离学生的实际生活 较远，学生不感兴趣或者很难到现场去操作。将教材中的生 活转化成学生身边的生活，让学生可以到现场去操作，真正 感受到数学就在身边。 （四）任务驱动。课堂学习以任务为驱动，将学习内容 包裹在任务中，将任务整体放给学生，让学生探究、发现。 学生在完成任务的过程中学会知识，真正让学习有趣，有挑 战。 （五）问题导向。在不同学习时段设计问题接口。学习 开始的问题接口：课堂之初让孩子根据数学信息提出自己的 问题， 带着问题开启本节课的学习。 课堂过程中的问题接口： 78 课堂推进过程中鼓励孩子随时提问，做好记录，以问题为载 体让学生深入思考。课堂结尾的问题接口：一节课的结束不 是没有问题而是产生新的问题，因此在每节课末尾留出时间， 鼓励孩子提出新的问题，生成新的思考。 （六）深度探究。课堂以“发现问题——提出问题—— 分析问题——解决问题”为主线，鼓励学生用自己喜欢的方 法去解决，小组四人合作，互相分享自己的方法，真正让探 究成为孩子自己的。 （七） 个性拓展。 给 孩子们提供丰富的拓展 资源， 满足不同孩子的不 同需求， 例如： 发散思维 的问题、 实践操作、 分层 问题设计、阅读资源等，孩子们可以根据实际情况选择适合 自己的拓展资源。 （八）收获成长。创设生态化的学习样态，不同的孩子 有自己不同的数学思考。在学习过程中,孩子们尽情的交流、 表达、决策、提问，可以用数学故事、手抄报、数学写绘、 数学日记等自己喜欢的方式创造世界，实现了“每一个”学 生在数学上的整体发展，真正由关注知识到关注人的完整成 长。 通过以上研究与实践，实现了小学数学从课程到课堂再 到指向不同个体学生的差异化资源设计的三维立体的改革， 通过这样的改革实现了小学数学的深度学习。“双减”减的 是负担，提的是能力，“双减”的到来意味着教师必须提升 79 自己的专业素养来改变课程课堂的样态。在“双减”政策的 指向下不断学习、持续探索，让数学教育回归本真，实现从 追求高分的教学到追求素养的深度学习的转变。 80 【郑州高新实验区】 家校社共育，培育时代新人 郑州高新区管委会社会事业局 为深入贯彻习近平总书记在全国教育大会上讲话的精 神，落实立德树人根本任务，郑州高新实验区认真贯彻教育 部《关于加强家庭教育工作的指导意见》文件精神，完善工 作机制，搭建工作平台，创新工作方法，务实推进家庭教育 工作，为形成家庭教育、学校教育、社会教育相结合的全方 位、立体化教育格局克难攻坚，努力推动区域教育事业新发 展。 一、立足“家校社”深度融合，构建教育新生态 郑州高新区社会事业局高度重视“家校社”共育工程建 设工作。一是始终坚持把“家校社”共育作为第一工作，把 “家校社”协同育人工作列入我局重点推进工程和学校工作 目标考核重要内容。严格落实“一门两校”（一所学校既是 学生就读学校又是家长学校）、“一长两职”（一位校长既 是学校校长又是家长学校校长）、“一师双型”（一位老师 既是学科教师又是家长学校教师）、“一教三落实”（家长 学校教学计划落实、教学课时落实、教师队伍落实）的工作 要求。二是始终把家庭教育专业指导队伍建设作为第一工程。 一方面加强家庭教育专业指导师队伍建设， 提升 “软实力” 。 每年组织骨干教师参加高级家庭教育指导师、高级心理健康 指导师、高级学习能力指导师培训，为教师提供系统化、持 81 续性的家庭教育培训，将家庭教育培训作为促进教师专业化 发展的重要内容。另一方面增强教师家庭教育课程研发的能 力，推动高新区教师从“经验型”向“科研型”转变，提升 教师家庭教育科研品质。三是始终坚持把家校合作课程作为 第一抓手。充分利用家长资源，引入专业团队，共同开发研 究家庭教育课程，并在学校内开设家长课堂，为学生提供更 开阔的生涯视野。 近年来，高新区逐步建立健全了“家校社”工作机制， 通过“家庭教育进万家”“学校教师强培训”“社会实践促 发展”等方式，深耕“家校社”融合，建立起家庭、学校、 社会沟通的桥梁，目前“家校社”共育的教育新生态已基本 呈现。 二、探索“家校社”协同机制，构建共育新格局 郑州高新区社会事业局在推进“家校社”协同育人工作 的过程中，积极构建上下贯通、多方联动、各司其职、齐抓 共管的“家校社”协同育人工作格局。 家庭教育层面：一是依托郑州市社区大学“家庭教育进 万家”活动，认真组织开展家庭教育活动；二是依托“康健 家庭教育技能工作室”，在辖区内社区、学校、企业等地广 泛开展 “社区学习共同体 家庭教育真落地” “日行一善” “学 习型家庭”“传承好家风·弘扬正能量”等活动；三是郑州 高新区社区学院邀请家庭教育领域专家面向全区开设“家庭 教育”课程，并与专业机构合作出版《亲子困惑答疑录》系 列书籍。目前，辖区内有1 所社区学院、1 处市级家庭教育 82 技能工作室、28 所社区学校、38 所中小学校和52 所幼儿园 基本能够做到开设家庭教育课程。 学校教育层面：一是以郑州市“百校万家·家校社共育 工程”为载体，不断引导骨干力量和全员共同参与，不断发 现典型、树立典型，发挥主导作用和榜样作用，坚持试点单 位的示范引领； 二是以数字平台为载体， 各学校充分利用 “幼 教三六五”亲子共育平台和中小学校家校共育数字平台，为 教师和家长提供专家讲座、家校共育视频案例等家长教育资 源和家庭教育资源， 有效引导教师、 家长开展针对性的学习； 三是着力提高家长素质， 为推进 “双减” 政策实施， 落实 《家 庭教育促进法》，提升家校协同育人质量，引领家庭教育回 归本质。高新区通过《“双减”政策解读》《“双减”后家 庭教育面临的挑战与对策》《“双减”后家校社共育的新任 务与要求》 等专题讲座为家长解疑答惑； 同时通过专项调研、 给家长一封信等方式，及时掌握家长教育焦虑状况，以便有 针对性地开展活动。 社会教育层面：一是高新区各学校与其所在社区建立 “共建”关系，加强了学校和社区的联系，方便学校组织学 生参与社区内的实践活动，丰富了学生课余生活，增加了学 生社会阅历；二是以传统节日为契机，参与社会志愿服务活 动，开展主题教育实践活动；三是协同政法、司法、宣传、 共青团等职能部门和群团组织发挥职能优势，相互配合，建 立以党政主导、 教育协调、 部门参与、 学校实施 “四位一体” 的工作格局。 83 三、建立“家校社”合作体系，构建共育新发展 “家校社”共育工程是寻求家庭、学校、社会深度合作 的一项长期工程，高新区坚持合作促发展的管理理念，近年 来，在“家校社”共育工作中，逐步探索出了合作发展的经 验。 与法制教育工作结合方面。高新区坚持把青少年普法教 育纳入重要议事日程。 在校内有效开展 “法官进校园 课后普 法助双减”青少年普法教育活动、 “防控疫情 与法同行”主 题征文比赛等丰富多样的法制教育活动。在校外组织学生走 进社区发放宣传材料;走进法院、 检察院等单位参观， “零距 离” 认识法律相关职业， 感受法律威严等。 通过一系列举措， 高新区形成了家校社互联互通的青少年法治教育体系，助力 学生树立法治信仰，践行法治理念，用法律为青少年的健康 成长保驾护航。 与社会实践活动结合方面。高新区管委会高度重视中 小学生社会实践活动，区财政每年拨付不低于200 万元的 社会实践专项经费予以保障学校社会实践活动的开展。一 是积极开展研学实践活动。高新区积极探索包括优秀传统 文化、革命传统教育、国情教育、国防科工和自然生态等 方面的研学线路，以及涵盖生存体验、素质拓展、科学实 践和专题教育等方面的综合实践课程，组织学生开展研究 性学习和研学旅行活动。二是做深、做实、做活劳动教 育。郑州高新区是省内劳动教育高地。2020 年4 月，郑州 高新区开始着力构建12 年中小学一体化劳动教育体系。 84 2021 年4 月，高新区社会事业局获得“郑州市劳动教 育区域推进典型单位”。郑州中学和郑州高新区艾瑞德国 际学校入选首批河南省中小学劳动教育特色学校；郑州大 学实验小学入选第二批河南省中小学劳动教育特色学校。 劳动教育为“双减”赋能，行知合一的教育成长模式，逐 渐成为高新区的教育特色。三是利用区域资源，开放校外 教育场地。高新区社会事业局充分利用区域资源，遴选了 一批对全区中小学生集体参观实行免费开放的校外教育场 所，包括网络安全科技馆、国家超级计算郑州中心展区、 河南省地质调查院、郑州高新区消防大队、郑州美术馆、 郑州太古可口可乐饮料有限公司等，与课后服务完美接 轨，既扩展了学生课后服务空间，又满足了学生多样化需 求。 与爱国主题教育活动结合方面。习近平总书记指出： “要把加强青少年的爱国主义教育摆在更加突出的位置， 把爱我中华的种子埋入每个孩子的心灵深处。”高新区紧 密结合时代特征，丰富教育内容，拓展教育途径，把爱国 主义教育贯穿于青少年成长全过程和各环节。在线下开展 “红领巾心向党”少先队活动案例交流、“离队入团，我 们的青春成长仪式”高新区离队入团仪式、“请党放心， 强国有我”主题队会等爱国主题活动；在线上通过开展 “红领巾讲解员”“习爷爷教导记心中”“红领巾爱学 习”等系列活动，实现课堂内外、线上线下同频共振，使 爱国主义精神深深扎根在青少年心中。 85 “大鹏之动，非一羽之轻也，骐骥之速，非一足之力 也”。家庭、学校、社会三方联动，同向、同力，目标一 致、互相补充，方能最大限度地提升育人效果。下一步， 郑州高新区将以“滚石上山”的信念勇气，深耕“家校 社”共育，为培养担当民族复兴大任的时代新人凝心聚 力，为高新区教育新局面做出更大的贡献。 86 聚焦单元整体，优化作业设计 郑州高新区管委会社会事业局 “双减”政策下，校内减负的一个关键点在作业的高质 量设计。郑州高新实验区立足单元意识，从单元视角整体规 划，针对学习内容的目标及学生学习特点，进行作业设计与 实施，形成完整的作业体系，促进单元目标和学段目标的达 成。关注知识体系、单元目标、课时内容之间的关联性及递 进性，各课时的作业内容要有针对性和延展性，既能帮助学 生巩固复习所学的知识，又能促进学生综合能力的持续发展。 一、细化学科核心素养，设定单元学习目标 细化解读学科课程标准，深入理解学科核心素养，正确 认识学科本质，明确本学科的正确价值观念、必备品格和关 键能力，并细化对应至所在单元，提炼概括单元大概念，科 学设定单元学习目标。 以人教版三年级上册Unit3 Look at me单元作业设计为 例，单元学习目标与学科核心素养细化结构图如下： 87 二、架构单元大概念，呈现思维进阶序列 依据所提炼的单元大概念，架构起单元大概念统领下的 单元知识结构体系、思维能力进阶序列和学科素养培育路径， 设计单元学习目标、课堂教学、学生学习和评价反馈一体化 系统，并提供给学生必要的学习支架和学习资源。 以初中数学北师大版九年级上册《反比例函数》为例。 （一）单元大概念统领下的单元知识结构体系 函数概念是数学学科的一个大观念。函数是初中数与代 数课程领域学习的主线，具有广泛的适应性和解释性。其知 识结构架构如下图： 反比例函数作为一个具体函数，它的教学设计以“函数 观念”统领、体现“总—分—总”的认知策略，即一方面要 在将反比例函数作为函数的下位概念，利用“同化”的学习 方式研究反比例函数相关性质。另一方面，学习反比例函数 88 之后，将反比例函数纳入函数的知识结构之中，形成新的、 相对完善的认知结构。 （二）思维能力进阶序列 实际教学遵循学生的思维发展规律， 构建了五个进阶 序列： 一是引导学生基于现实生活发现和提出问题， 二是 分析实际问题， 建立数学模型； 三是运用反比例函数的知 识，求解模型；四是根据反比例函数的性质，进一步检验 结果和完善模型； 五是举一反三， 解决实际问题 （如下图） 。 （三）目标—教学—学习—评价一体化系统 在各个环节上，明确重点关注内容，形成“教—学— 评”一体化体系（如下表）。 大概念 主题内容 核心本质 课时 数感 数形结合 几何直观 模型思想 应用意识 学科素养 3课时 2课时 1课时 用函数的观 点解决问题 函数的不同 呈现方式 变量间的关系 主题三：反比例函数的 应用 主题二：反比例函数的 图象与性质 反比例函数的 概念 主题一： 函数 89 三、整合重构教材内容，透视整体教学流程 基于课程标准、教材内容和学生认知特点，科学设计本 单元教学课时安排、教学流程、教学活动及教学资源（或对 教材内容整合、重构）等，以单元大概念贯穿单元教学整个 过程，并体现学生习得和思维进阶的具体过程。必要时可用 教学流程图或结构图呈现。 以人教版三年级上册Unit3 Look at me 单元作业设计 为例：《英语课程标准》明确指出，三年级的学生能在图片 的提示下听懂简单的对话并做出适当的反应；能根据录音模 仿说话，并相互交流简单的信息；能在教师的指导下用英语 做游戏并进行简单的交际和角色扮演。具体教学流程透视图 如下图所示： 90 四、设计作业具体内容，阐释意图准确到位 依据单元学习目标，并参照评价标准，整体设计单元作 业具体内容，要求以真实或拟真、具体的情境为载体，设计 典型的问题、任务或活动，体现作业的基础性、综合性、应 用性和创新性特征，促进学生解决真实问题能力的培养；同 时，作业内容设计的具体意图，包括每项作业内容所蕴含的 学科正确价值观念、必备品格和关键能力等。 以人教版Go for it九年级 Unit5 What are the shirts made of?为例： 注：作业内容体现分层，Option A 为提高型作业，难度 较大；Option B 和Option C为基础性作业，难度稍低。学生 根据个人程度自主选择。 91 课时背景1： 从潍坊国际风筝节回来之后， 唐宫小姐姐发 了微博，征集广大网友的对她的发现之旅的建议。 Period 1（第一课时） 作业内容 （Content） Option A从潍坊国际风筝节回来后，唐宫 小姐姐想要继续她的发现之旅。请在她的 微博下留言，为她推荐合适的旅行地点并 列出购物清单建议。 （20分钟） Option B 读一篇题为Colorful kites sail through China sky的文章，完成文 后的习题，并就文章内容作一个简单的总 结。 （20分钟） 作业要求 （Requirements） 1. 在描述你所推荐的物品时，请聚焦在物 品的产地、材料和它们的用途；在进行拓 展阅读的总结时，同样总结出文中物品的 产地、材料和它们的用途。 2. 请尽可能多的使用目标语言：be made in, be made of, be made for等。 作业评价 （Evaluation） 1. 学生在课堂上分享为唐宫小姐姐物品推 介的想法，大家共同选出“必买好物 92 课时背景2：唐宫小姐姐根据同学们的建议来到了福建 安溪——孔明灯的故乡、甘肃庆阳——剪纸的故乡和天津 ——泥人张的故乡。她们在那里领略了传统艺术之美。 Period 2（第二课时） 作业内容 （Content） Option A 选择一种自己感兴趣的传统 艺术形式，可以是课文中的或是其他方 式了解的，收集这种艺术形式的图片， 编辑后发一个plog到自己的朋友圈或其 他社交媒体,让更多的人了解这种艺术形 式。 （30分钟） Option B 为今天所学的课文 Beauty in Common Things 绘制思维导图。 （20分 Top5” ； 2. 教师检查学生是否正确使用了目标语 言。 设计意图 （Design Intent） 本单元的语言重难点为被动语态的使用， 作为本单元的第一课时，是一节听说课， 承载着语言感知、理解，进而运用的功 能。本课时作业结合本节课的语言输入的 内容，让学生结合自己的生活体验，主动 拓宽学习的渠道，运用英语达到沟通与交 流的目的。 93 钟） Option C 完成拓展阅读Ancient embroidery in danger ，完成文后习 题。 （10分钟） 作业要求 （Requirements） 1. 发plog选择至少5张有代表性的图 片，并配上恰当的英文说明（不少于60 词） ；不能使用手机的同学可以将作业内 容以PPT的形式呈现。 2. 思维导图需包含传统艺术形式的材 料、功能、寓意等。 3. 请任意选择两个Options 来完成。 作业评价 （Evaluation） 1. 学生将图片和文字发至自己的社交媒 体，看能够获得多少点赞或评论，获得 点赞或评论最多的三名同学成为今日的 “MVP（the most valuable plogger） ” ；做PPT的同学在班级内进行 展示，由同学们投票选出最佳作品前三 名Most Popular Works Top 3。 2. 教师对思维导图的内容、内在逻辑、 语言和版面设计几个方面进行评价。 设计意图 （Design Intent） 第二课时为阅读课，作为课堂阅读活动 的延伸，我们的作业设计突出了英语学 94 习活动观对于实践应用和迁移创新能力 的培养。通过图文并茂的plog 为我们的 传统文化艺术形式打call。学生在完成 这个作业的过程中，对我们的传统文化 有了更深的认同感和自豪感，他们的文 化品格得到了塑造，学习能力得到了提 升。 课时背景3： 在唐宫小姐姐的发现之旅行进的过程中， 她 们收到了一位在美国读书的中国网友康建的e-mail,姐姐们 将他的经历和困惑写了出来， 供大家探讨。 同学们分析了 “中 国制造”背后的原因，讲述了“中国制造”的现状，树立了 为中华崛起而读书的坚定信念。 课时背景4： 随着时间的推移， 唐宫小姐姐的发现之旅也 进入了尾声。课堂上同学们为唐宫小姐姐即将参加的联合国 文化交流活动写了发言稿，课下他们需要合作完成最后一项 任务，回归我们“Made in China, loved by the world — —让世界爱上中国造”的单元大主题。 五、评估作业质量效果，反思改进全部流程 根据作业实施过程中的实际情况，简要撰写作业设计与 实施评估报告，包括作业设计的能力指向、认知水平、难易 程度、作业时长等要素在作业实施过程中的实践效果及存在 问题，可用数据统计分析辅助说明，同时撰写出作业设计质 量和实施效果的反思。 以部编人教版四年级下册语文第二单元作业设计为例。 95 针对作业链，设计的系统的评价量表作为评价学生是否 保质保量完成作业的参考标准，最大限度的实现“教——学 ——评”的一致性。 （一）针对预习单的评价量表 （二）针对思维导图的量表 96 (三)课文朗诵音频作业评价量表 （四）习作评价心动量表 97 【王府教育实验基地】 王府学校落实“双减”政策的主要做法 北京王府外国语学校小学部 为了更好地落实“双减”政策，北京王府外国语学校小 学部围绕提质增效教学质量、优化作业设计、改进学生评价 方式、升级课后服务、完善家校共育等五个方面采取了一系 列的举措，以保证政策的有效实施。 一、提质增效教学质量 （一）对教师进行全方位培训，指导教师打造“以学生 为中心”的“灵动”课堂 学校根据教育部、区教委的指示精神，对全体教师进行 了关于 “双减” 工作方案以及打造优质课堂的系列集中培训。 指导教师通过研读各学科课程标准，吃透教材与课标关系， 准确设定学期、单元、课时教学目标，课堂教学重点关注教 学目标的落实；指导教师继续规范教学流程，突出课堂教学 严谨度与完整性，优化课堂教学设计，针对不同学科、不同 学段开展互动式、探究式教学，在“让课堂活起来，让学生 动起来”的教学宗旨下，打造灵动课堂。学部还积极与法政 国际教育集团层面学术支持团队沟通，寻求专家团队的引领， 为教师搭建培训平台。如吴正宪老师走进学校，为教师们带 来《“双减”背景下的教学策略》的主题培训。教学专家的 引领，助力教师课堂教学的提质增效。 98 （二）加强学科教研，借助集体智慧协力提升整体教学 质量 学校各学科组充分领会“双减”精神，加强学科教研力 度。各学科组在学科负责人的带领下，深度挖掘课标和教材 的联系，针对不同年级学生学情，精心设计各年级整体教学 计划、学科活动计划、检测评估计划等。学科组集体教研重 点为教师常态课、公开课的评价与定向指导。在学科组集体 教研的基础上，各年级集体备课常态化。集体备课是将课标 落实到日常教学实践的关键，是提高课堂教学实效的前提。 加强集体备课，可以促进优质资源共享，提高教学质量。 （三）完善学校公开课制度，加强听评课管理，支持教 师提升教学质量 学校进一步明晰教师开展公开课的目的，完善学校公开 课制度，“以课代训”，在不增加教师教学负担的前提下， 鼓励并指导教师上好公开课，同组教师积极参与听评课，并 鼓励跨学科听课。学校教学部门指导听课教师从教学准备和 组织的情况、教学的多样性和节奏、教学风格、师生关系、 课堂管理、学习评估等方面观察上课教师的教学情况。在各 学科组教研例会中本着“公开、诚恳”的态度开展评课，助 力教师拓宽教学思路，提升教学落实程度，以每位教师的成 长带动整个学科组团队的共同进步，以此达成全校教学质量 整体提升的教学工作目标。 （四）组织丰富多彩的学科活动，外化课堂教学成果 学科活动是学科课堂教学成果的外化表现。在“让课堂 活起来” 的同时， 学生通过丰富多彩的学科活动， 充分地 “动” 99 起来，将课堂学习的成果充分表现出来。学校在多年教学实 践中，创造性地开展了语文学科“百词斩”活动、“诗词大 会”、查字典比赛；数学学科“数独”大赛、“七巧板大比 拼”、数学“密室逃脱”；英语学科英文故事大赛、英文歌 曲大赛、配音比赛；以及学科融合的STEAM Week等丰富多彩 的活动，提升了学生学习兴趣，为学生提供了展示课堂学习 成果的舞台。 (五) 精准教学管理，对课堂教学进行有效监管 课堂提质增效离不开专业、精准的教学管理。学校为了 确保教学准备工作充分到位、教学实施环节规范合理、教学 评价环节严谨有效，每月由教务处安排组织开展各学科教学 常规检查，对教师教学计划的实施情况，教案的书写、落实 及反思情况，作业的批改情况等进行全面的检查与评价。通 过教学常规检查，教学管理部门全面、系统地了解教师对教 学常规的落实情况， 是指导教学教研、 开展教学改革的前提， 同时也是教学管理部门加强教学管理的依据。 二、优化作业设计 （一）统一认识和要求，不断完善教师作业观念 学期初通过召开学部全体教师会议，以“双减”为关键 词进行全面解读，再次明确作业总量，要求教师们严格执行 “一二年级不布置书面家庭作业，巩固性练习在校内完成； 三至六年级书面作业平均完成时间不超过60分钟”的要求。 同时通过学习和培训，完善教师对于作业的认识：作业除了 能够巩固课堂学习的知识与技能，对于培养学生的学习习惯、 责任心和意志力、学习兴趣，以及对于提升学生解决问题、 100 创新实践和自主管理时间等能力都有重要的意义。统一的认 识和要求，完善的作业观念为教师的作业设计提供了明确的 方向和标准。 （二）开展作业设计研讨 各教研组将作业设计研讨融入日常的教研中，通过集体 备课，认真研读教学目标，进行单元整合，设计符合学生认 知特点、丰富多样的作业。比如：数学组注重作业设计有层 次、有梯度、有思考价值，将数学问题与实际生活更紧密的 结合起来，在高年级设计让学生自主设计数学周刊的跨学科 融合作业，提升学生的综合素养；英语组设计思维导图、单 元海报、创意写作、故事表演、演讲等多种完成方式的作业 提升英文素养；语文组将古诗积累和主题阅读融入日常作业， 扩充学生的积累，帮助学生养成阅读的习惯。 （三）精准教学调研和管理，对作业进行有效监管 学期中由教务处牵头对学生进行随机采访，了解学生对 于各科作业的真实想法，包括完成作业的时间，存在的问题 与困难等，并及时与教师沟通，进行有效的调整。 为了更好的统筹作业，各班建立作业公示制度，班主任 为第一统筹人，及时协调各学科的作业量，并进行作业管理 检查。 为了让家长更好的了解学习内容，各学科每周通过年级 教研讨论并由学科主任审核之后，发送统一的周学习清单， 包含本周学习目标、学习重点和难度、学习内容、作业等内 容，周学习清单对于沟通作业、巩固复习以及家校沟通起到 了很好的促进作用。 101 三、改进学生评价方式 在学生的评价方面，坚持过程评价与结果评价相结合， 对学生进行多元测评。将学生的课堂表现、作业情况、参与 学科活动及成果、单元巩固练习等方面的情况纳入期末综合 成绩，给学生以全面持续评价。过程性评价包括小组活动、 海报展示、演讲等。期末一二年级不组织纸笔考试，以闯关 的形式进行乐考。 四、升级课后服务 学校主张在夯实学科基础这个“一体”的前提下，为学 生提供丰富优质的课后服务，以培育学生拥有艺术和体育的 强壮“两翼”，助力学生未来人生飞得更高、更远。在“双 减”政策下，家长对于孩子素质培养有了新的变化和要求。 艺体课程作为课后服务“菜单式”素质课程的重要组成 部分，不仅仅是让学生掌握一种艺体技能，而是希望学生在 成长过程中有艺术、 体育素养的滋养， 形成丰盈的精神世界、 强健的体魄和对美的领悟力。 2021-2022学年重点进行 “三团” （合唱团、舞蹈团、民乐团）、 “四队” （足球队、篮球队、 羽毛球队、游泳队）的打造，因材施教，为一部分有艺体特 长的孩子提供展示自我的平台。 除了艺体、科技等素质教育课外，还为学生提供了“小 小外交官Young Diplomats”课、“全球小学者 Global Scholars”项目制学习项目、“小作家”班等培优课等，发 展学有余力学生的潜力。与此同时，对急需教师帮扶的学生 开设了学业帮扶指导课。 学校对所有参与课后服务的教师明确岗位要求，明确上 102 课流程、考勤制度、学生安全管理、授课规范、疫情突发情 况上报流程等各方面的管理规定，确保各门课后服务课程有 序、高质量开展。 五、完善家校共育 为了贯彻落实家校社协同育人， 加强家校沟通， 促进 “双 减”高效落地，学部德育处着手开展家委会工作，搭建家校 共育平台，倾听家长的声音，帮助家长树立规则意识，当好 孩子成长的榜样，同时让家长及时了解学校教育教学工作。 对于学部中需要帮助的学生，及时与家长取得联系，让家长 了解孩子的实际情况，协助教师做好教育工作。对于一年级 的小同学，疫情期间家长虽然不能来到学校观看展示课，但 是小学部组织开展了双满月线上直播活动，全方位地向家长 展示孩子们在校学习的成果，让孩子们积极表达自己内心所 想，做到让家长放心安心，活动得到了家长的一致好评。 103 开设“家长学校”：“双减”背景下“校家社”协 同育人的新举措 成都王府外国语学校 蔡启彬 黄 潇 2015年10月，教育部印发的《教育部关于加强家庭教育 工作的指导意见》中明确提出要充分发挥学校在家庭教育中 的重要作用。2021年中共中央办公厅、国务院办公厅印发的 《关于进一步减轻义务教育阶段学生作业负担和校外培训 负担的意见》将“社会”因素加入其中，强调“完善家校社 协同机制”。在以上政策背景下，成都王府外国语学校基于 办学理念和家长现实需求，充分利用社会资源，在学校中开 设家长学校，提升家长的教育教学观念，实现校家社协同育 人的整体目标。 成都王府家长学校自2021年10月18日成立以来，在短短 的一个多月，所开设的演讲、普通话、合唱、舞蹈、美术、 书法、 儿童心理、 英语口语等课程及家教艺术大课堂的分享， 赢得了广泛好评。 一、家长学校的缘起 家长学校的成立，源于每到放学时段，总能看到校门口 焦急等待的家长们，为了避免家长在寒风酷暑的天气中站在 校门口接孩子放学，同时帮助家长建立良好的亲子关系，并 能以科学的家庭教育观形成家校共育合力，成都王府“家长 学校”应运而生。 104 学校一直以来倡导家长热爱教师，教师尊重家长，学校 热爱学生，学生热爱学校的家校关系。家长学校成立后，全 校家长均可以在每周一至周四规定教学授课时间段内走进 学校参与学习，与孩子一起成长。 二、家长学校的具体举措 学校将特色艺体课程作为家长学校的选修课程，面向全 体家长进行推广。在前期主要培养家长对艺体的兴趣，在后 期主要进行专项提高，让家长成为文化、艺体双修人才，持 续培养家长内在与外显的综合素养。学校根据资源的优势和 艺体教师的特长以及学校的场地器材的实际情况，选择最合 适的资源加以开发利用。 学校拥有现代化的卫星录播接收教室、计算机教室、国 家标准物理、 生物、 化学实验室和剑桥国际考试委员会 （CIE） 授权考试中心；多媒体报告厅（500个席位）、阶梯教室、室 外体育场、足球场、室内标准地板篮球场、羽毛球场、乒乓 球场、跆拳道场、形体房、钢琴房、声乐练声房等；全校视 频监控系统，所有的教室都配备了空调，为家长们提供良好 的学习环境。 基于对学校资源以及家长现实需求的综合分析，成都王 府为家长开设了《普通话》《英语口语》《儿童心理教育》 《书法》 《篮球》 《羽毛球》 《唯美古典舞》 《拉丁舞》 《瑜 伽》《合唱》《走进毕加索—丙烯画》《遇见大师莫奈—池 塘里的形与色》 《走进梵高的向日葵—丙烯画》 《家长讲坛》 《电影大课堂》等十余门课程。家长可以在周一至周四下午 18:20—19:20的一个小时中根据课程表 （见表1） 自由选择。 105 表1 家长课堂的课程安排表 日期 时间 周一 周二 周三 周四 18:20—19:20 1.普通话 2.英语口语 3.儿童心理教育 4.书法 1.体育 2.舞蹈 3.合唱 4.美术 家长讲坛 电影 大课堂 课程开设结合了家长在家庭教育方面的现实需求。教师 精心选择授课内容，结合学生及家长的具体特征，通过互动 性强、生动精彩的讲授方式，使广大家长自觉自愿地参与其 中。以《儿童心理教育》为例，成都王府外国语学校的心理 教师们，根据孩子们在学校的表现，孩子们在心理课上的情 况，与孩子进行一对一心理辅导的真实案例，归纳出不同年 龄段孩子的心理特点，根据不同年龄段孩子的心理特点，与 家长沟通交流。让家长知晓在孩子不同的年龄段，作为父母 怎么做更有利于孩子的心理健康发展。通过课程，有助于缓 解孩子和父母的紧张关系，解决孩子与父母之间的各种冲突， 营造和谐的家庭氛围，促进孩子和父母的共同成长。 三、家长学校的实施保障 为了真正促进家长在文化、艺体、育儿等方面成长，成 都王府对师资队伍、内容、家长参与度三个方面予以严格把 关，旨在让更多的家长能够自愿参与到高质量的家长学校学 习中来。 首先，建设认真负责且素质较强的教师队伍。家长来自 各条战线，文化程度高低不一，认识水准、兴趣爱好不尽相 106 同，参不参加全凭自愿。家长学校作为一所业余学校，师资 全是聘请，如何以一堂堂应家长所需、急家长所急的家教课 来向家长有效地普及家教知识，是一个需要面对的问题。对 此，学校对外请家委会负责人、部分家校委成员共商教学计 划；对内对艺体教师进行整合，动用一切资源，进行统筹安 排。教师的聘请采取内聘(即自己的音体美舞书法老师)与外 聘相结合的方式，以保证家长学校的教学质量。 其次，加强家长学校的内容与社会“热点”的联系。 学 校认为:要有效地传授家教的理论与方法，必须抓住家长最 关注、最急需解决的“热点”问题。只有这样家长学校才能 产生强大的吸引力， 才能调动家长的积极性。 因此， 抓住 “双 减如何减”“疫情背景下家长孩子的情绪管控”“家庭教育 的重要性”“家庭教育的原则和方法”等几个时代热点，通 过线上（云课堂）、线下（家教知识大讲堂）跟家长们分享 《新形势下如何做一个合格家长》《优良的环境有利于子女 成材》 《少年儿童的劳动责任感培养》 《正确指导子女学习》 《学龄初期少儿心理特点及教育》《家长如何管控自己的情 绪》等。通过家教系列大讲堂的分享，家长收获很大，反响 也很强烈。 最后，利用资源优势提升家长的参与度与认可度。家长 学校开办以来，学校根据资源优势，开设了普通话、英语口 语、儿童心理教育、书法、体育、舞蹈、合唱、美术、家长 讲坛和电影大课堂等丰富多彩的课程，每门课程都由专业的 教师进行授课。家长依据学校提供的课程表及时间安排，根 据自己的实际情况，各取所需选报自己喜欢的课程，参与度 107 和认同度都很高。喜欢运动的家长和体育教师在运动场上挥 汗如雨地打乒乓球、篮球、羽毛球；喜欢健美的家长和舞蹈 老师认真练习基本功；喜欢普通话的、音乐的、美术的、心 理的家长也都有相应的选择。 四、家长学校的成效 在参与家长学校前，部分家长表示“虽然自己的阅历较 深，工作经验丰富，但教育子女的方法毕竟欠缺，很有必要 加强学习。 ” 有位家长说： “像这样坐着听课， 确实不舒服， 然而为了孩子， 我却是很想听的。 ” 积极的探索和实践证明， 创办家长学校，不但促进了社会各界对教育的了解和关心， 而且对于提高家长素质，改进家教方法起到了很好的作用。 家长对孩子的影响是深远的，从孩子身上能看到父母的 影子，也能从父母的身上看到孩子的未来。对于父母来讲， 教养孩子就是一场修行，最重要的是一颗积极向上的心，与 他一起进步，共同成长。 家长学校的举办必须立足在常抓不懈和务求实效上，同 时需要采取一些切实可行的措施。 只有这样， 才能办出特色、 办出水平，才能切实有效地帮助家长提高教育水平。 成都王府“家长学校”的设立将拓宽家校共育的路径， 帮助家长们发现自己的闪光点，在学习的过程中去发现美并 创造美，让家长们等待孩子放学的时间更加充实和快乐。 108 【成都都江堰实验区】 全面落实“双减”政策 营造绿色教育生态 ——都江堰市“双减”工作推进情况汇报 成都都江堰实验区 为全面贯彻落实立德树人根本任务，立足校内提质增效、 校外规范管理两方面，都江堰市将“双减”工作作为撬动全 市教育高质量发展的新起点、新支点。在强化组织领导、创 新监测体系、优化作业设计、丰富课后服务、治理培训机构 五个方面做足功课、下足功夫，全力构建学校、家庭、社会 协同育人新格局，切实减轻学生课业负担，努力办好人民满 意的教育。 一、加大治理力度，进一步规范校外培训行为 （一）快速启动“双减”工作。为全面贯彻党的教育方 针，准确把握“双减”要义，加快推动“双减”工作落地见 效，市委市政府召开专题会议，统筹部署，成立由市委分管 负责同志任组长的规范校外培训风险防控专项工作组，制定 《都江堰市规范校外培训风险防控专项工作方案》；做好宣 传发动、规范清理、隐患排查、风险管控等系列工作；落实 职责任务、细化工作措施，确保“双减”工作稳妥审慎、快 速推进。 （二） 加强正面宣传引导。 牢牢把握 “双减” 舆论导向， 向全市6万余名义务教育段学生印发《致全市参加校外培训 的中小学生及家长的一封信》，通过都江堰电视台、 “i都江 109 堰教育”、家长微信群等广泛宣传“双减”政策，精准推送 “双减”讯息20余次，积极引导家长树立正确教育观念，营 造良好“双减”社会氛围。 （三）抓实培训机构治理。坚持校内保障与校外治理两 手抓、两手硬，依法依规、多措并举、疏堵结合，抓好校外 培训机构清理整顿。 2021年7月以来， 组织对全市学科类77家、 综合类50家、艺体类59家共计186家校外培训机构进行全面 排查，准确掌握其营业状态、法人情况等信息，精准梳排风 险点，建立监管数据库、隐患问题台账和“黑白名单”。对 发现的隐患苗头问题，组织联合执法、联合惩戒，坚决取缔 无证培训机构，杜绝违规培训行为发生。进一步探索完善校 外培训机构资金监管体系，统筹做好学科类培训机构“营转 非”和变更培训类别工作。截至11月，收到“营转非”机构 申请23家，剥离义务教育段培训转型转向机构申请65家，注 销机构申请18家，义务教育段学科类培训机构压减率为 42.74%，排名成都市义务段学科类培训机构“压减”情况第 3名。 （四）化解多元矛盾纠纷。全市建立17个部门联动的多 元矛盾纠纷化解调处工作机制，成立专项工作组，构建形成 市级、镇（街道）、社区（村）三级联动综合监管机制，全 面排查，精准研判，坚持校内校外双向发力，主动化解隐患 问题。收到涉及爱贝斯英语教育培训学校信访70余件，均已 有效回复。 二、规范办学行为，全面强化学校主阵地作用 （一）强化制度建设，健全规范管理长效机制 110 加强 “五项管理” 工作， 相继制定了作业、 睡眠、 手机、 读物、体质健康管理的实施方案，以及《都江堰市中小学校 进一步做好规范教师布置和批改作业实施办法》《都江堰市 “双减”工作实施方案（征求意见稿）》等协同治理文件。 强化学校主阵地作用，坚持源头化治理、清单化管理，做到 “五项管理”与减负工作整体设计、同步实施、一体推进。 如团结小学全体行政班子，率先“个性化”解读“双减”政 策的目的意义，在思想认知上高度重视，形成了基于团小实 情的“双减”校体化认识主张，做到“线”要合、“量”要 减、“质”要加。 （二）强化校本研修，进一步提高课堂教学质量 都江堰市将全面深化课程教学改革作为“双减”工作的 一项重大任务，坚持源头治理，促进课堂提质增效。如“赏 数学之美，享设计之趣”是都江堰市嘉祥外国语学校五年级 数学实践活动，孩子们利用轴对称、平移和旋转的知识，动 手剪一剪、贴一贴，用数学的眼光发现美、欣赏美，用数学 的思维探寻美、创造美。奎光小学开展《指向深度学习的小 学问题解决教学范式的实践研究》，依托“六课一研”的活 动架构， 提炼 “创设情境—提出问题， 自主探究—感悟问题， 合作交流—形成共识，总结反思—共同提高”的问题解决教 学范式，形成了小学语文、数学、英语学科问题解决教学范 式基本操作策略和课堂观察评价量表。 （三）优化作业设计，进一步减轻学生作业负担 一是加强学科组、年级组作业统筹布置，作业确保难度 不超国家各学科课程标准。开展义务教育段作业公示的学校 111 占比100%，作业时间控制达标的学校占比100%，不给家长布 置作业或不要求家长批改作业学校占比100%。二是聚焦常态 作业的设计与实施，实现全员专题培训。如团结小学开展全 体教师作业管理专项研究学习，让教师能及时更新作业设计 的理念和做法。三是深度挖掘教材，精准设计作业内容。市 中小学教育研究室制定《都江堰市义务教育阶段优化作业设 计工作的指导意见》 ， 学校结合学科、 学情制定 “一校一案” 。 如北街小学二年级美术学科指导学生设计最喜欢的季节，并 完成绘画，然后再由语文教师在托管当天指导学生完成文字 的表述。这种作业图文结合，大大激发了学生的学习兴趣。 四是规范作业管理，全面控制作业总量和时长。“控制作业 时间和作业总量”这个刚性要求，成为全市“双减”工作的 切入口和落地点。如灌州小学探索出“双线五级”作业管控 模式，“双线”的第一条线是周一到周五作业布置与管控， 第二条线是周未作业严格执行作业时长定量与分层。 “五级” 的一级管控是班级科任教师自查日志，协调每天作业时间； 二级管控是教研组内作业审批自查；三级管控是学科大组长 每周协调做到作业错峰布置；四级管控是校级层面对教研活 动的指导；五级管控是学校课程质量监控中心的调研指导。 由此，实现学校作业管理从自检自查到他测他评的良性循环。 （四）完善评价机制，助推减负增效 “双减”工作，减的是学生的额外负担，增的是学生的 综合素质。学业质量评价由形成性评价、专项能力评价和终 结性评价组成，开展全学科、全过程多元评价，促进教师对 学生全面发展的关注，充分发挥评价的导向和激励功能。如 112 永丰小学将五育并举和智慧评价相结合，搭建了“蒲公英五 育评价体系” ， 借助加载了24字关键词的蒲公英币， 将教师、 学生、家长三方卷入评价体系中来，对学生进行德智体美劳 五个维度的评价。 三、创新课后服务，满足学生个性化需求 推动课后服务全覆盖，提升质量，增强吸引力，与课堂 教学统筹推进，确保学生在校内学足学好。印发《关于做好 义务教育课后服务工作的通知》，对全市推进课后服务的时 间、范围、课程、师资及保障措施等做出全面部署。学校全 面实施“托管+拓展”的课后服务“5+2”模式。坚持家长自 愿参加、学校自管自办、成本补偿和非营利性原则，拓展服 务内容，拓宽服务渠道，全面提高课后服务水平，同时积极 探索课后服务的延伸工作，开展周末托管服务。如翠月湖学 校引进外部资源，在每周六上午开展了足球、象棋、舞蹈、 国学等课程，将课后服务“5+2”模式做成了“5+2+1+1”模 式（其中“1”为每周一天为无书面作业日），积极推动学生 的全面发展。 近日， 该校落实 “双减” 政策的文章 《 “锦囊” 伴成长 书包不回家》在《中国教育报》上刊登。 四、强化家校共育，引导家长履行教育责任 深入学习贯彻党的十八大以来就家庭教育做出的一系 列重要论述， 依托今年10月审议通过的 《家庭教育促进法》 ， 全市正在积极推进完善家校社协同育人体系。各校均建立了 家长学校，成立了三级家长委员会，充分利用家长会、网络 平台等指导家庭教育，适时培训家长，分享典型案例，让家 庭教育成为学校教育的有效助力。如锦堰中学采取线上线下 113 同步推进的方式召开家长会， 向广大家长解读 “双减” 政策， 交流学校在“双减”政策下的思考和措施，及时回应家长关 切的问题，缓解家长焦虑。 全面落实“双减”政策，营造绿色教育生态。都江堰市 将站在服务国家战略需求和建设高质量教育体系的高度，让 学生从过重作业负担和校外培训负担中解放出来，真正体验 到学习的愉快和成长的幸福，最终成长为德智体美劳全面发 展的社会主义建设者和接班人。 114 关于分层作业与家校协同育人的几点思考 都江堰外国语实验学校 孙洁 自九月实行“双减”至今，都江堰外国语实验学校为切 实落实双减，做到“五育”并举，从分层作业和家校共育两 个方面采取了相应措施。 晚自习两个小时的时间，但是只布置一个半小时的作业， 可以让学习能力较差的学生有足够的时间思考，而对于学习 有余力的学生这些作业只需要四十分钟就可以完成，有非常 多的空余时间。由于开学最开始两周，没有分层布置作业， 晚自习就特别吵闹。经过两周的调整后，各科教师就进行有 针对性的分层作业布置。对于能力较弱的同学，只要求完成 教师的基本作业，对于能力较强的同学，教师根据学生的具 体情况进行分层作业布置。语文推荐学有余力同学阅读《非 洲三万里》《历史的温度》《三体》《苏菲的世界》等；对 于学习能力很强，且阅读量已经很大的学生，要求每周背诵 古诗词等一至三篇不等，并了解诗词的背景知识。英语对于 能力较强的同学推荐英语报纸《二十一世纪英语报》等进行 摘录， 对于学习能力最强的同学推荐英语课外读物， 如牛津、 书虫系列的分级阅读丛书。数学对于能力较强的同学推荐使 用《名校题库》和《B卷必刷》等等。分层布置作业实施两个 月后的效果：一、学生的晚自习纪律好了，每个学生都有事 做，很少出现特别吵闹的情况；二、学习能力不足的学生可 以安心完成少量的基础作业；学习能力很强的同学又会有所 115 收获，学习习惯和学业成绩有明显提升；三、学生的学习兴 趣会更浓，学生在学习中会更加主动，且拓展了学生的知识 面。 但是在实际操作中，仍存在一些问题。一是学生的阅读 量非常少，小学阶段的总阅读量不超过两百万字，这不光会 影响语文的学习，还会影响其他科目的学习，这类学生经常 会读不懂题目的意思，或者是不能理解你的语言；小学阶段 相对学业较轻，但是很多家长只重视学业，周末不是做题就 是补课， 现在周末不补课了， 希望小学生能多一点时间阅读； 并且小学老师推荐的阅读书目一般是由语文老师推荐的文 学类书籍，所以能否由专业教师推荐更多的历史、科学、哲 学、心理学类的阅读书籍；二是教师对于分层作业的布置和 检查都非常难，首先学生是在完成基础作业的情况下才有时 间完成分层作业，所以每个学生的完成进度差距很多，其次 在检查上也只能以抽查为主；在实际操作中，要求学生写周 计划， 再按学号来抽查， 因为每个学生的计划都是不相同的， 检查难度很大，如果家长也能参与其中来督促，那样效果会 相对好很多，但是并不是每一个家长都有这个能力。 另外，家庭才是育人的重要场所，之前学生周末很多的 时间都花在了补习文化课上，能通过周末补课，取得良好的 效果的学生是很少。有的学生把学习希望全部寄托在补课上， 认为只要自己去补课，就会提升，而不是自己努力了，才会 提升；还有提前学的学生，大多数在学校学习就会不认真， 反正都学过；而且在补课中，上课的内容不可能是针对每一 个人的具体情况，所以效率并不高。而且小学、中学可以补 116 课，到了大学可以补课吗？周末本来就是亲子时间，但是之 前全部是围绕着分数转，不重视学生品德和劳动教育，亲子 之间疏于交流，最终导致亲子关系恶化，滋生很多学生的心 理问题，家长也非常苦恼。现在时间回归，家长们能否利用 好这些时间，需要一定的指导。学生在中学阶段最重要的任 务就是学会学习，其次才学会某项知识和技能，而周末的补 习很大可能会加重学生不会学习的情况，更加依赖于别人的 知识灌输，而不知道自己该怎么样做才能突破自己的不足。 比如一个学生在学习初一上期数学第二章时，对于绝对值的 化简掌握不好，就应该自己或者在教师、家长的协助下找到 相应资料上的题目，利用周末较长的时间段，重点练习、总 结，第二周再将自己练习的题目交给教师，再由教师适当地 指导。这样做学生才能学会怎么样克服困难，并且更有学习 的主动性。例如班上的小李同学，数学基础很弱，非常惧怕 数学，而且学习情况非常差，和家长沟通后，每周末给他适 当布置了一些基础计算题，经过几周的努力，他的计算能力 有所提升，对于学习数学也没有之前的那么强的抵触了。而 且周末也是学生发展美育和体育的重点时段，学生在周末的 空余时间可以学习舞蹈、绘画、球类等提高自己的审美和体 能。比如小张同学是一名基础好，能力强的同学，刚进入初 中时因为自主学习的时间较多，不知道怎样安排，所以在各 方面都有所下降。在和他的母亲交流后，根据他的爱好，让 他参加了乒乓球训练课，培养他坚韧的精神；并且给家长介 绍一些优质资源， 推荐了几款学习的APP， 让学习十分轻松的 小张可以拓展视野；并且小张还给自己制定了一个详细的学 117 习计划，包括课内的学习、课外知识的拓展、体育锻炼和家 务完成等等，并且每周将相应的拓展交给各科老师检查，周 末家长统一检查，培养他的自学能力和自主管理能力。经过 两个月的努力，小张在他相对较弱的语文学科中取得了一定 进步，利用在校课余时间打乒乓球，做到了劳逸结合。但是 这种家校合作，非常依赖于家长的责任性和教育艺术。一些 家长周末要上班，管不到孩子；有的家长除了打骂，不知道 还有其他的管教方式；有的家长只和孩子谈学习、分数，亲 子关系恶劣；有的家长无底线纵容孩子；甚至有的家长根本 不管孩子。让家长利用好周末的时间，既关心孩子的成长， 又激发孩子的学习兴趣，这对家长的要求非常高，但是实际 情况是很多家长并不能很好的胜任家长这份工作。所以利用 寒暑假督促和引导家长学习，为家长推荐一些学习资源。 双减之后对于教师来说，将会面临巨大的挑战，教会学 生学会学习，促进学生全面发展，这是很有难度的；对于有 教育意识的家长来说是一个机遇，自己的孩子可以更有大的 自主发展空间；而对没有教育意识的家长来说，会不会是沉 重的打击呢？教师们又改怎么样做才能帮助他们呢？这是 非常值得思考和做出尝试的一件事。 118 【广东珠海实验区】 提质减负，守正推新 广东省珠海市香山学校 2021年，国家层面颁布了“五项管理”“双减”等系列 减负举措，其力度之大、覆盖面之广，史无前例。学校作为 一线执行部门，正面临前所未有的挑战与压力。然而，坚决 贯彻执行的斗志，充盈每位教育人的内心，因为大家深深懂 得，此番政策，对人才长远发展意义深远。 一直以来，香山学校坚持质量立校，将之视为培养全面 发展人才的自然根基。学校努力实现基础课程规范化、特色 课程普及化、素养课程层级化的规范化办学思路，进而借助 “双减”政策，使办学规范趋向更加科学、合理，合乎教育 教学规律。 一、“三减、四加、两成立”，校本原则让教学规范张 弛有度 减轻过重课业负担，一直是学校教改工作的核心目标。 为更好地落实双减政策，学校提出“三减、四加、两成立” 等系列方案，要求全学科教师贯彻落实，为政策的落地赋能 增效。 119 珠海市香山学校“双减”“五项管理”背景下教学工作基本原则 “三减、四加、两成立”的核心任务，是落实国家立德 树人的根本任务，培养全面发展的人。围绕此项原则，学校 教学部门带领全体教师研究课堂，积极提高教学质量；研究 作业，全力提高教学效率。通过弹性增加晚辅时间以及加大 个别辅导力度的举措，在保障不过多增加教师负担的同时， 力求减轻家长负担。利用家长学校、公共交流平台等，向家 长宣讲课后活动的项目、类型，指引家长带领孩子开展阅读 活动、 体验活动、 实践劳动技能等， 全面培养学生核心素养。 成立“作业档案馆”与“书包减负营”，展示丰富多彩 的作业类型，拓宽作业实践广度，打通作业与实际学力的通 路，在实践中引导学生增广视野，提升个人底蕴。 学校允许对基础普及课程学有余力的学生将课本集中 放置专用教室，如音乐、美术、科学、道德与法治等书本， 均可统一回收，放置教室书桌抽屉或者专用篮筐。减轻书包 的重量，减轻孩子双肩的负重，更是向孩子传递学习不是负 担的理念。书包减轻了，喜欢的课外书、擅长的跳绳、小足 球都可以随时带来了，发展特长，培养兴趣，锻炼身心，健 康向上的学习生活方式建立了起来。 学校教学管理部门采取“每周公示”“每月一查”“家 “三减” 减作业量 课堂作业练习为主 规范执行一教一辅 严格控制作业总量 减作业时长 弹性延长晚辅时间 加大个别辅导力度 减轻家长负担 提升晚辅服务质量 指导课余活动项目 提升课堂教学质量 “四加” 加强阅读指导 加强体质健康 加强实践学习 加强劳动体验 “两成立” 成立作业“档案馆” 成立书包“减负营” 120 庭电访”的管理闭环。设置“作业展览馆”，在学校最显眼 的地方张贴学生的实践性和探究性作业，让作业成为作品， 让规范张弛有度。 二、 “冲突、52015、多元”，三位一体让教学质量趋向 深度 减少作业时间、 分层布置作业是 “双减” 工作落实到位与 否的重要指标。 运用结果导向思维方式， 提升课堂教学质量、 分层作业设计模块、多元评价手段，三位一体，保障着“双 减”工作顺利实施，让课堂教学质量走向深度。 向四十分钟要效益的课堂，注定要在各门学科之间触动 思维，打破界限，融合思想。香山学校教研部门带领教师们 翻转传统观念中对“冲突”一词的认知，利用其中积极、活 跃的因子， 接近元认知， 挑战课堂， 以期拓宽深度学习领域、 迈向终身发展的育人目标。 一学期来，全体教师在集体备课时间有保障的前提下， 加强科组集体备课的有效性，在学校提出的促发学生“思维 冲突”的课堂教学模式的改革思考中，不断打破封闭，互相 走入课堂，进行跨学科的课堂观摩。英语课、信息技术课利 用丰富的情境创设， 真实的运用需求， 促发积极的学习欲望， 诱发内驱力的自我“冲”“突”需求；语文课制造学生学习 渴望的情感，制造学生困惑的情感体验，制造学生之间意见 相对或相反的矛盾焦点，以情感冲突激发深入思考；科学实 验与已有的认知经验相矛盾，使得学生在科学、综合等实践 课中踌躇满志、大胆前行。 课堂教学极力变革的同时，作业设计分层与弹性则更显 121 得迫切和必要。在严格执行时间与总量控制的前提之下，各 学科利用集体备课时间开展作业设计讨论，将“实践性、综 合性、激励性”作为硬指标，以此为作业设计出发点开展作 业设计研究，定期公示。在这样的思考实践中，学校作业精 彩纷呈，种类繁多，适应不同学生发展需要。低年级的为作 业配图，是最走心的思维痕迹；中年级色彩丰富的读书卡， 记录着美妙的读书旅程；高年级动手动脑的实践作业，让科 学技术走进日常生活。教师们结合教授内容，充分利用现代 化信息技术平台中的趣味分类、超级分类、知识配对、分组 竞争等手段，设计不同层次的闯关游戏，让学生更积极地参 与到课堂活动中来。 学校向教师们提出 “52015” 的课堂时间 分配建议， 即利用5分钟的课堂导入， 20分钟的课程讲授， 再 加上15分钟的课堂练习，这样的课堂科学高效，更减轻了学 生的课后作业负担。 为了让学生更加适应未来，适应新时代人才的需要，香 山学校的教师们已经意识到，单一学科评价已经不能满足时 代所需。于是，教师们充分发挥集体智慧，反复推演方案， 反复修缮评价内容，以期跨学科、多角度、全方位地评价学 生，更是以评促发展，让评价活动成为培养能力的又一个天 然场域。在学期末进行的“核心素养背景下”学生综评活动 中，更是站在更高处，以更广阔的视野、更专业的思考，研 究教学目标的合理性、教学评价的科学性、教学手段的多样 性。当期末多元评价遇上核心素养，师生均在转化角色：教 师在这场考试中不再是监考者，他们时而站在师者高度，围 绕学科核心素养整合专题，界定专项能力试题；时而化身观 122 察者，在活动的过程中观察学生的行为特征、自主意识、学 科素养，从而不断反思教育教学行为的目标与策略的科学性、 合理性、有效性。而学生呢，不再只是被动地参与考试，而 成为活动的主体，在活动中的一个个有思想的成长个体。 三、多措并举、“三段式”教学服务，实现作业“堂堂 清” 学校通过“基本托管减负、博雅社团拔节、课后辅导夯 基”的举措，助力提质，落实减负。香山学校是一所大校， 有学生2494人，参加午托的有303人、晚托的有564人，参加 博雅社团的有200人， 二分之一的学生六点之后离校。 多维度 的考量后，全校教师积极参与，在学生自主选择的基础上， 力争丰富多彩、 富有成效。 基本托管保障学生课内完成作业， 有问题的、有困难的，通过教师帮扶助力，个性化地指导， 有效地完成作业。博雅社团激发兴趣、发挥特长，20个社团 满足了学生个性化的需求，让学有所长、尽情施展。完成作 业的学生，可以阅读、绘画、做手工，做自己喜欢的事。作 业减负和兴趣达成有效链接、有机融合。 四、“协同、合力、互补”，携手社区让家校共建发展 适度 香山学校所处山场社区，该社区一直重视开展家校社协 同育人工作。有效利用社区资源，可以为更多学生、家长提 供放学后提升个性发展、合理安排特长学习的便利。 学校与社区共建，定期委派党员教师为骨干先锋，利用 双休日，到社区开展特色教育专题讲座，如“书法”“绘本 故事” “手工游戏” 等， 为社区居民子女送去优质教育资源； 123 同时，山场社区也依据时令举办不同类型的学生专场活动， 如在暑假期间，开展夏令营活动，组织社区学生参加绘本阅 读、英语口语秀训练、金融理财小达人，还有心理成长体验 营等等，精心设计的活动类型，促进社区学生身心发展。合 力、合拍、合作的教育模式，对学校教育形成有力保障。 同时，香山学校依托家校共育的“加减乘除原则”“一 三六九模式”，与社区、与家长共建共育、同心协力，实现 立德树人为根本任务的减负增质。 从 “五项管理” 到 “双减” ， 再到 “考试规范指导意见” ， 香山人以促进儿童发展为理念，知行合一、守正推新，谱写 减负增质的新篇章。 124 学在乐中 乐在“考”中 ——“双减”下改进考试测评方式的实践探索 珠海市斗门区第二实验小学 黄美健 为了深入贯彻中共中央、国务院、教育部和珠海市教育 局关于“双减”政策相关精神，珠海市斗门区第二实验小学 以 “双减” 为契机， 充分发挥学校办学特色， 细化作业管理， 丰富课后服务课程， 强化课堂效率， 改进考试测评方式,让教 育遵循学习规律，合乎学生成长需求，让校园充满成长的快 乐，回归教育本质。 学校实施多元化评价改革，促进学生全面发展，探索基 于学科素养的全方位、 过程化、 综合性、 客观性的评价方式。 在关注学生学业结果的同时重视学习过程，关注学业水平提 高的同时关注学生日常活动中表现出来的情感和态度，帮助 学生认识自我，建立自信，增强学习兴趣，减轻学生的心理 压力和考试负担。让“乐学”与“乐考”成为教学与评价的 改进方向， 实现以评促教、 以评促学， 真正让学生在 “双减” 中享受学习的快乐。 一、以学生全面发展为根本，构建“六优”综合评价体 系 学校基于“五育并举”兼顾创新教育的理念，通过国家 课程校本化、学校课程本土化、实践活动生活化，并在 课程设置上全面体现，在学科活动上集中落实、在评价内容 和方式上多元并重，构建了“品优、学优、健优、艺优、劳 125 优、创优”的“六优”学生综合评价体系。根据“学生的品 德发展水平、学业发展水平、体质健康发展水平、艺术兴趣 特长养成、劳动技能和劳动习惯养成、创新能力综合表现” 六大板块 20多个小项指标， 依托 “争优” 的评价方式构建学 生“六优”综合评价结构框架。“六优”综合评价是一个开 放的系统，实现评价的关键在于全面性和可操作性，各年级 各班可根据不同年龄、学段以及评价目标，及时进行调整， 灵活实施。 二、以多方评价为依据，引入“三位一体”评价形式 在改革实践中，“乐考”评价系统逐步引入“单元、专 项、 表现” 三位一体的评价形式。 其中， “单元评价” 与 “专 项评价” 在教育教学过程中同步展开， 表现性评价主要在期 末结束时进行， 既对学生的知识掌握进行定期诊断， 又对学 生的能力发展进行持续监测， 而且让学生的思维、 创新实践、 情感态度、品格等得以呈现，三种形式共同构成“乐考”学 业综合评价系统。 在单元评价部分， 围绕学科教材的主题单元， 依据学科 教学计划，定期在单元学习结束之后进行书面诊断性检测， 教师能据此及时发现学生学习方面的问题， 改进教学活动安 排，保证单元目标的达成。 在专项评价部分， 将过程性检测贯穿整个教育教学过程， 把学生个体的过去与现在进行比较， 学科的各侧重点进行比 较， 进而了解学生的发展情况， 促使学生对学习的过程进行 积极地反思和总结。 在期末的表现性评价中， 各学科组教师深入研究本学期 126 教育教学目标， 设置情境， 让学生运用先前获得的知识与能 力， 探寻问题解决方法或用自己的行为表现来证明自己的学 习过程和结果，而不是选择答案。由此让学生的思维发展、 创新与实践能力、 情感与品格等目标最终落实到评价项目之 中。 期末综合评价活动中还注意将家长与学生纳入评价实施， 如家长作为考官协助， 能更加了解评价过程， 调整自己的教 育理念与方法； 高年级学生志愿者在带领低年级学生进行评 价的过程中，其组织、领导能力也能获得长足发展。 三、以多维评价为原则， 确定学业评价目标与评价标准 “乐考” 评价系统将以往单一纸笔测试中难以测量的指 标转换为可观察、 可量化的显性指标， 使评价目标由单一变 为多维，采用等级化的评价标准使评价更加趋于柔性。 （一）确定多维的评价目标 “乐考” 的项目设置以小学原有的学科课程为基准， 增 加基本生活常识及技能等项目。 评价的四维目标包括： 知识 与技能、过程与方法、情感与态度以及合作、专注、有序等 品格。以一年级“乐考”的部分项目内容与评价目标为例， 如语文学科的“小小朗读者”项目，主要考查课内外诵读内 容， 其评价目标在知识与技能方面引导学生积累丰富的语言； 情感态度价值观方面则是激发和培养学生热爱祖国语言文 字和优秀文化的思想感情。 （二）制定等级化的评价标准 “乐考” 评价标准以等级化代替分数化， 突出评价的发 展性功能。以一年级语文“乐考”评价标准设置为例，其中 “读书好少年” 项目的评价内容是课外阅读积累量和口语交 127 际能力。其评价标准为：文明有礼、大方自信、口齿清楚， 说出3本课外阅读的书名和喜欢理由，得“优秀”；文明有 礼，大致说出2本喜欢的书目名称及理由，得“良好”；文 明有礼，能基本说出1本喜欢的书名及理由，得“及格”。 等级化的评价标准能保护学生参与评价的积极性， 并让不易 量化的因素同样纳入评价系统中。 四、以多元评价为原则，改进学生评价内容和方式 一是通过以课堂、活动观察为评价平台，形成多样化的 学生综合评价方式，既有包括活动式、实践式、渗透式、互 动式等过程性评价，又有包括量化式、检测式、问卷式、总 结式的终结性评价。过程性评价和终结性评价灵活多样，突 出平行性和交叉性的评价功能，突现综合评价的易操作和易 接受。 二是 “六优” 综合评价体系与教育教学深度融合的实践， 涵盖学校为学生成长创建的所有活动。课程实施拓宽学习空 间，不局限于教室，不同的学习空间增加不同的“争优”项 目和评价。每学年学校都会开展“正而行，行而优”的德育 系列活动：入学礼、敬师礼、成人礼、国庆歌咏会、亲子趣 味运动会、 学生田径运动会、 足球文化节、 劳动能力大比拼、 英语节、数学节、艺术节、科技活动、汉服文化节、斗门本 地美食节、斗门传统文化节、国际嘉年华等。“评价—监测 —发展—评估—完善”的实践过程，为学生提供更多快乐有 效、自主选择的学习、评价平台，体现综合性评价的多元性 和实效性。 三是增设学业发展水平“乐考”评价方式。书面考试评 128 价过于关注学科知识的认知把握度，无法检测学生口头表达 能力、动手能力、创作能力、思维发散呈现和合作交流呈现 等能力。学校设置“乐考”检测方式，以单项、灵活的方式 呈现，更加全面地对学生的知识技能、学科思想方法、实践 能力、创新意识、情感态度价值观等关键性指标进行全方位 评价，立体把握本学期学生在学业发展水平的结果，体现综 合评价以人为本的引领性和发展性。 例如， 在三年级 “乐考” 检测中，教师根据各学科课标要求，科学地设计形式多样的 检测项目，语文学科的“诗词大会”“小小朗读者”“成语 大王”“小小书法家”等；数学学科的“口算小达人”“数 独大闯关”“数学小当家”“趣味二十四点”等。这些考查 项目有学生独立自主完成的，也有单元组合作完成的；有口 头表达，也有动手操作，从不同层面检测学生在本年级学业 发展水平方面的综合表现。 五、反思与展望未来 1.“乐考”不仅响应“双减”，更是教学评价方式的优 化 考试形式的变化，不能简单理解为减负，把严肃的考试 “开发”成既让学生动手动脑，又能培养团队协作意识的全 新形式，其实是教学评价方式的重大进步。通过评价反观自 身、查找不足，激发信心与兴趣，也是评价所承载的重要功 能。在良性的评价机制下，激发学校对于学生学习能力的培 养，激发学生全面发展，这才是未来考试改革的方向。 2.考试形式的变化，促使教学过程的改革与教学理念的 提升 129 “乐考”轻松了孩子，却对学校和教师的教学能力与教 学理念提出了更高要求，考试评价方式的变化，促使了教学 过程的改革与教学理念的提升。它要求教师必须更新教育观 念和教学策略，调整自己的教育教学行为；鼓励教师因材施 教，尊重学生的个性差异，关注学生作为“人”的发展，关 注学生综合素质的发展，关注学生的全面发展。 3. 将“乐学”“乐考”科学化和合理化的可持续推广 评价方式改革是一个复杂的系统工程，涉及方方面面， 决不是推行几种新的评价方法就可以大功告成的。如何运用 可操作的科学手段，通过系统收集有关教学的信息，结合教 学活动的过程和结果做出价值上的判断， 以实现 “乐学” “乐 考”科学化和合理化的可持续推广，是下一步需要努力的方 向。 130 【重庆江津实验区】 重庆江津：“双减”政策落地生根， 美好教育“减负提质” 重庆市江津区教育系统从坚守教育初心，牢记教育使命 的高度认识“双减”工作，以促进义务教育内涵发展和质量 提升为目标，念好严、导、质、研“四字诀”，以坚决的态 度、有力的举措稳步推进，保障“双减”出实效。 一、“严”字当头，“三个一批”促降温 江津区把规范校外培训机构发展作为“减负”的重要抓 手，构建“政府主导、教委牵头、部门协同、家校联合和社 会参与”的长效监管机制，坚持疏导结合、分类指导，实施 “关停一批、整改一批、转型一批”行动，深入开展校外学 科培训治理。 “关停一批”，即对无证无照的校外培训机构坚决予以 取缔，责令其停止办学、停止招生，同时纳入“黑名单”， 及时向社会公布。“整改一批”，即对“无证有照”超范围 经营的非学科类校外培训机构，提前介入、加强服务，督促 其改善办学条件、申请办学资质，让更多有条件、有意愿规 范办学的机构实现合法办学；“转型一批”，即对“有证有 照”的学科培训机构，指导其向素质教育转型发展。截至目 前全区校外培训机构已关停9家， 注销12家， 规范35家， 转型 15家。 131 二、“导”上发力，家校同心乐童心 “双减”不仅要减学科培训、减课外作业，减少家长和 孩子的焦虑更是重要的内在驱动。江津区积极引导家长真正 站在孩子成长的角度，考虑什么能让孩子终身受益的事，判 断教育改革的长远大势，逐步缓解家长焦虑，并引导教师做 好家校协同工作，构建以“儿童为中心”的组织化服务，赋 予“家校协同”强劲的组织力量，为家庭创设了相贴合的学 习成长环境。 江津区通过“点对点沟通”让家长了解政策，班主任利 用晚延时放学，约请家长交流“双减”精神，并与学生一起 做新学期“双减”政策下的学习与生活计划；通过“网状学 习” 让每个家庭行动起来， 将每个星期日设置为幸福家庭日， 策划组织 “我们家的地板时光” “爸爸妈妈玩过的游戏” “我 们一起读本书”等幸福家庭日的系列主题活动，增加亲子互 动时间，提高亲子陪伴质量。通过建立学生与家长的成长档 案，把学生的身心变化、学习成果等记录在“双减”的弹性 时间里，一段时间进行“回忆”，畅想美好时光。江津区依 托全国“三宽”教育数字平台和专家资源积极开展家校共育 工作，促进家校共育工作的针对性、系统性和有效性，以此 提升家校共育水平。 目前， 家长参加学习人数2.5万余人， 学 习次数7万余次，撰写学习笔记2万余篇。 三、“质”上提升，“三空间”课堂显活力 江津区找准 “双减” 背景下课改发力点， 注重因材施教、 学科育人和教师赋能，满足学生个性化学习需求。将学生在 132 校的学习、生活、交往、游戏、运动有机地结合起来，让社 会、校园和课堂“三空间”融合成整体的育人空间。 江津区坚持以生为本，全面滋养学生。各校积极组织学 生走进田间地头、 走向大街小巷、 走入博物馆、 纪念馆学习， 组织学生开展各种志愿服务。整合校园中空间，注重延时服 务提质增效，增添校园活力。将延时服务素质教育化，满足 学生多元化的成长需要， 学生可以自选课程， 有艺术修养课、 体育特色课、 阅读博览课等。 组织教研员团队通过区域培训、 校本教研等方式增强面向课堂的教研力度。通过有效提问教 研打磨问题的梯度、深度、广度、精准度，让学生在问题的 指引下体悟学习中思考的乐趣。通过项目学习的教研，让老 师和孩子一起建构学习内容，以自主探究为主渠道，让学生 在课堂里被激励、被唤起、被点燃，自己发现问题、解决问 题，感受探索之趣。截至目前，江津区累计开展区域培训21 次，项目学习教研16次，助力在“双减”大背景下的新时代 课程改革。 四、“研”中有创，作业设计出新意 “我们成为了延时服务的小主人！可以根据自己爱好选 择有趣的延时活动。”重庆市江津区四牌坊小学延时服务时 段除了每天两节作业答疑的小课之外，还有一节自主特长课 程走班选课。重庆市江津区珞璜实验小学建立适合每一个学 生的 “X+N作业超市” ， X是指学科， N指层次， N包含基础性、 拓展性、创造性作业，满足不同层次学生的需求。 “双减”之后，孩子的作业总量少了，但教学质量不能 133 减，基于此，如何设计作业，最大限度地提高课堂和作业的 “性价比”成了“双减”之下教师需要迫切研究的新课题。 江津区将深化教育教学改革和作业设计纳入教研体系，以集 体备课，课例教研，幼小、小初联合教研等多种形式的教研 提高教师的课堂教学水平；通过教研员、各学校的骨干教师 分角度、分层次、分任务开展教研，提高教师的作业设计能 力。以作息重塑为表、健康生活为里，以集中管理为表、多 元服务为核，以减量为表、提质为里，不断提高作业设计水 平。 下一步，江津区将进一步探索“双减”落细落实机制， 以分类专项实施，分层递进落实，构建横向全覆盖、纵向全 贯通的制度机制，将“双减”落实到底。同时着力于教师素 质的提升，切实提升育人水平，推进双减工作向深、向实开 展，为帅乡迈向美好教育赋能添彩。 134 作业巧设计 孩子多快乐 重庆市江津区珞璜实验小学校 凌荣 “书包轻了，作业少了，成绩好了，快乐多了！”珞璜 实验小学陈逸轩的家长最近在聊到孩子学习时，露出了开心 的笑容。本校在落实国家“双减”和“五项管理”政策时， 以江津区课程改革为契机， 以 “人文江津 美好教育” 为引领， 以“真适教育，草木芬芳”为办学愿景，让学习真实发生， 让教育适合发生，对学生作业进行了一系列的改革。巧妙的 作业设计旨在提高作业质量， 减轻作业负担， 让学习更高效， 让童年更美好。 一、设计层次性作业，让学生选择做 真实的是最美的，适合的是最好的。作为江津区课程改 革“种子学校”，珞璜实验小学把国家课程校本化作为课程 改革的先手和引擎，而设计适合学生的作业是国家课程校本 化的具体体现。每个学生的学情不同，接受作业的难易度也 不同，学校基于各学科课程标准和学生不同的学习基础，建 立“X+N作业超市”。X是指学科；N是指层次，包含基础型、 拓展型和创造型三个层次的作业。学生可以根据自己的学习 能力，选择适合自己的、有挑战难度的作业。 语文教师谢作维在讲授《遨游汉字王国》一课时，精心 设计了层次性作业。基础型作业：学习完汉字谜语建构方法 后，找一找用会意法、象形法、离合法等方法构建的汉字谜 语，并进行展示；拓展型作业：学生认识到制定《国家通用 135 语言文字法》的必要性，课后调查生活中汉字的规范使用， 并提出可行的修改意见；创造性作业：查找资料，深入研究 汉字、 鉴赏书法， 动手制作书签、 读书卡、 小报或者撰写 《汉 字研究调查报告》呈现你的发现。 层次性作业设计，可以满足不同层次学生的需求，给予 学生充分自主选择的权利和挑战难度的机会。适合学生的作 业，是最有效的作业，也是最受欢迎的作业。 二、设计趣味性作业，让学生喜欢做 好玩是儿童的天性，而传统的作业把做与玩分开，关注 的是“做”，而不是“玩”。简单枯燥、重复机械的作业成 为孩子的一种任务，而不是一种乐趣。让学生在玩中做、做 中玩，增强作业的趣味性，激发学生的好奇心和求知欲，成 为教师们在“双减”后共同的追求。大家都在思考如何让作 业变得鲜活，变得新颖，变得有趣。 数学教师罗安琴设计的作业总能“吊起”学生的胃口， 学生一看就跃跃欲试。她在执教《可能性的大小》后，设计 摸球、抽签、大转盘等游戏。让学生在游戏中感受到数学知 识的奥妙，激发学生探索的欲望，促进思维的活跃，保持学 习状态的持久。 体育教师邓发梅在教学 《跳绳》 后， 设计 “天 罗地网抓老鼠”的游戏作业，五名同学将绳子拉成五角星状 的网子，举高后让其他同学扮老鼠，在下面边唱边跳，违反 规则就被“惩罚”。孩子在开心玩耍时，既体验到游戏的乐 趣，又达到了锻炼的目的。 趣味性作业设计，变“苦学”为“乐学”，变“要我做” 为“我要做”，调动了学生学习的积极性和主动性。 136 三、设计实践性作业，让学生体验做 “这个班的孩子很能干，有灵气。”这是大家对六年级 一班的评价。该班教师紧密联系现实生活，引导学生实践探 究，提高解决问题的能力。在学习了《图形的面积》后，教 师布置的不是书面作业，而是让学生通过测量学校舞台和家 里客厅的地砖来求其面积。怎样才测得更准确？长和宽怎么 确定？孩子们分别用脚步丈量、软尺测量的方式进行估算和 计算。 根据《中小学生综合实践指导纲要》，学校充分利用地 处珞璜工业园区的资源优势，开发了《“管”好生活 创享未 来》校本课程。师生一起走进“伟星管业”，了解管材发展 历史，参观管材生产过程。课后学生依据所学知识通过项目 化方式， 进行创意物化， 制作自动浇花器、 自动供养鱼缸等。 在 《指向美好生活的小学财商校本课程》 中， 教师执教了 《认 识人民币》后，让学生在家长的带领下去超市查阅价目表、 规划自己的零用钱，体验“收银员”的角色。 结合国家课程和校本课程，设计实践性作业，让学生走 出课堂，走进生活，走向社会，提高学生的实践能力，培养 学生的创新精神，为生涯启蒙和未来生活打下基础。 四、设计养成性作业，让学生坚持做 著名教育家叶圣陶说过： “积千累万， 不如养个好习惯” 。 良好行为习惯的养成，对一个人的成长有着至关重要的作用。 然而，传统的学生作业往往只关注“智育、体育、美育、劳 育”，忽略了“德育作业”。因此设计养成性作业对于落实 立德树人至关重要。 137 学校根据《小学生守则》《小学生日常行为规范》，结 合学生的实际情况，编写了一套适合小学生边唱边跳的《一 日常规操》。各班班主任利用早读前十分钟、班会课组织学 生进行学习和行为习惯的训练，有针对性地布置相关行为习 惯作业，利用班会课对收拾课桌、整理书包、收纳雨伞等习 惯养成作业完成效果进行检查。对书本封面书写、书本整洁 度、坐姿、书写握笔姿势等学习常规进行评比……让学生更 加清楚怎样去规范自己的言行，大大地提高了班级管理水平， 形成了良好的班风、 学风。 在假期里， 学校德育处统一规划， 不同年级安排阶段性作业，如快乐规划师、快乐小读者、快 乐小厨师等。 培养学生的良好行为习惯，重在家校协同，贵在学生坚 持。 养成性作业设计， 丰富了作业的内容， 收到良好的效果。 学生在作业中实现“五育并举”和全面发展，成为恒心持久 的学习者。 五、设计互动性作业，让大家一起做 英语教师为了激发学生课堂和课后作业的兴趣，经常安 排对话作业。他们以课本为载体，以小组为单位，熟读、演 练、改编课本对话内容，充分发挥学生的创造性思维。教师 以分小组分角色进行情景对话表演，展示课后改编成果等方 式检验学生作业完成情况。学生在这样的作业中不仅学到了 知识，还提高了英语口头表达的能力。 学校还定期举办“科技节”“亲子故事会”等活动，让 学生完成项目化学习和合作表演等任务。互动性作业不局限 于课堂和校园， 学生走进社区， 参与 “争当小小志愿者” “小 138 手拉大手”“学雷锋”等活动。他们与同学、老师、父母、 社区工作人员一起参与完成的作业，是师生情感交流的润滑 剂，是同学间友谊的催化剂，更是亲子融洽关系的黏合剂。 互动作业的设计，可以提高学生与人交往、沟通、合作 的能力，增强团队意识和社会责任感，培养有情怀有担当的 现代小公民。 “双减”之后，作业成为全社会关注的一个热门话题。 “减”，是减轻过重负担，提高作业质量。近年来，学校秉 持“分层设计、反馈研讨、督查评价”的原则，建立作业管 理制度，优化传统作业，创新作业设计。年级组、教研组每 周开展教研活动，在作业目标、作业内容、作业数量、完成 方式等方面进行了大胆的探索和尝试，让作业更具适合性、 真实性、多样性和有效性。学校每月检查作业设计质量和完 成情况，每期末收集学生、家长、教师的意见，对作业设计 进行优化， 确保作业 “高质适量” ， 确保 “双减” 有效落实。 精心的作业设计，还孩子一个色彩斑斓的童年。教育的 最终目的，莫过于让生命自然生长。珞璜实验小学正在用自 己的行动，让学习真实发生，让教育适合发生，让我们一起 期待花开盛夏，草木芬芳。 139 【深圳龙华实验区】 “双减”背景下课堂提质增效的实践探索 深圳市龙华区教育局 深圳市龙华区坚持以学生发展为本,落实立德树人根本 任务，大力践行“积极教育”，主动作为，狠抓落实，从夯 实职业素养、加快专业成长、倡导科研强校、管控课堂质量 等方面促进提质增效，确保“双减”工作用实功见实效，从 而实现学校教育教学质量显著提升的目标。 一、提升教师专业素养，保障教育教学质量提升 培育师德师风。将师德师风培育纳入到教师全员培训， 通过教育政策法规解读、典型案例分析、签署承诺书、选树 典型等形式，强化教师文明从教、廉洁从教、规范从教的职 业意识，全面提高教师职业道德素养。 加强业务培训。积极开展新教师岗前、岗中培训及“新 教师汇报课例展播活动”，全面推进以课程教学实际问题为 核心、以优秀教学案例、项目式学习等为内容的区级教师培 训， 全力支持教师参加各类研修活动， 组织有关教师参加 “国 培”“省培”“市培”等各级各类培训，多途径引领教师专 业成长。学校大力开展板书、教学设计、作业设计等校本培 训，不断夯实教师教学基本功。 开展教学比武。为搭建教师展示自我、互相学习、共同 提高的平台，营造“你追我赶”的教学能力提升氛围，促进 教师专业成长，龙华区于2021 年9 月-11 月开展了“龙华区 140 141 能动性，鼓励并支持学校开展校际联合教研。本学期龙澜学 校、教科附属学校、观澜第二小学、民治小学、民新学校等 学校先后与周边学校联合开展了主题教研活动，有效地促进 了校际间教研质量的共同提升。 三、打造素养课堂，提升课堂教学质量 完善制度建设。 结合国家、 省、 市相关文件要求， 修订、 完善《龙华区中小学教学工作常规》和《龙华区中小学教学 管理常规》（简称为“两个常规”），引导学校结合实际制 定校本实施细则， 强化落实， 不断规范办学行为和教学行为， 促进学校教学工作的制度化、科学化和规范化。 推进专项研究。积极构建U-A-S（高校-区教研-学校） 行动研究模式，充分借助高校的专业理论研究优势，发挥区 教科院丰富的教学实践经验，以项目学校课堂教学改革，辐 射、引领全区中小学课堂教学改革，大力开展“素养课堂” 教学研究。该研究聚焦课堂教学设计与实施、作业设计等育 人关键环节，以“大作业观”为切入口，构建闭环的“素养 课堂”教学改革路径，推进基于核心素养的课程教学改革实 践，努力打造具有龙华特色的学科课堂教学模式。9 月份以 来， 先后在区内11 所基地项目校， 组织广州大学教育学院课 程与教学中心研究团队、区教科院学科教研员、项目研究种 子教师及项目校负责人开展课堂教学改革调研及学术研讨 活动，积极营造研究氛围，提升研究品质。 聚焦课堂教学。 创新开展教学视导工作， 深入课堂一线， 全面了解教学动态，及时诊断教学问题，指导教师精准分析 学情，优化教学方式和教学环节，开展研究型、项目化、合 142 作式学习，加强差异化教学和个别化指导，提高课堂教学效 率。稳步推进教研员到校蹲点、“名师进民校”、民办学校 教学质量提升行动、薄弱学校教学质量提升等工作，推动全 区教育质量的全面提升。 提升作业效能。制定《龙华区义务教育阶段学科作业设 计与实施整体推进工作方案》，建立健全作业管理机制，稳 步推进作业改革；出台学科作业设计指引，实现“课堂、作 业”联动优化。如行知实验小学语文科组的“作业类型选用 清单”和“作业超市”满足了不同学生的发展需求；民顺小 学的科学作业 “能不能把生态系统搬回家？” 紧密联系生活， 使学生感受自然界的奥秘，领悟生命之美；大浪实验学校的 英语手抄报、龙华区外国语学校的语文“辩论赛”、教科院 附属小学的“古诗新绎”等丰富了作业形式，激发了学生学 习兴趣；玉龙学校的劳动作业，不仅培养学生的生活能力和 实践能力，还有利于亲密和谐的家庭关系的形成。 四、构建积极评价，为提质增效保驾护航 不断深化教育质量评价改革，以“大数据” “互联网+” 为技术支撑， 积极构建以学业质量监测、 学生体质健康测试、 深度监测、专题监测、增值评价、教师学生成长档案为核心 的“六位一体”区域教育质量监测体系。通过“数据采集— 数据分析—数据反馈—反思改进—跟踪督导”的实施路径， 实现教育质量大数据的互联互通，再经过对数据挖掘与关联 分析，反映出教育质量的动态变化过程，实现监测结果智慧 化应用。 今年，在区教育局督导室的牵头下，组建了区义务教育 143 质量监测中心组，并开展了教育统计学、教育测量学及监测 报告案例撰写等系列培训。各中小学校也充分利用监测结果 诊断问题、改进行为，提升教育教学质量。如玉龙学校通过 挖掘深度监测数据， 优化学校德育评价， 提升德育工作品质； 行知小学成立校级质量监测中心，利用学业质量监测数据， 分析学科教学长短，实施教学管理提升行动，促进教学质量 提升。 教师也不断优化对学生的评价方式，有的教师批改作业 一改原来的“阅”“优”等单一评价方式，采用“等级、评 语”的双评方式，当学生有了明显变化时给予评语式评价。 还有一些教师做了特色印章，比如一朵小花，或是一个点赞 的大拇指等多样化方式来评价激励学生。面对面批改、定期 作业展评等形式也被越来越多的教师采用。在道德与法治学 科中有的学校尝试过程性评价方式，把学生的每一次好人好 事做成道德小银行，换取道德小达人称号。 144 1+1+1＞3 ——“双减”政策下的玉龙样态 深圳市龙华区玉龙学校 2021 年7 月，中共中央、国务院出台了《关于进一步减 轻义务教育阶段学生作业负担和校外培训负担的意见》，意 见进一步指出， “学校教育教学质量和服务水平进一步提升， 作业布置更加科学合理，学校课后服务基本满足学生需要， 学生学习更好回归校园。” 基于此，结合《深圳市义务教育阶段学校课后服务实施 意见》中“五育并举、因材施教、公益第一、质量惠普”的 课后服务实施要求，玉龙学校以“五项管理”为抓手，以减 负提质为总体要求，依据原有课程基础，设置了“1+1+1”课 后服务模式，即作业辅导+新生活特色课程+扬长课程，努力 做到“作业不回家，课程多样化，课后有收获”。玉龙学校 课后服务模式有助于满足学生成长的多样化需求，达成“双 减”效果，实现“四个有利于”，即：一是有利于解决家长 下午“4 点半”接送孩子的困难；二是有利于探索实现小学 生在学校完成作业；三是有利于利用公共资源发展学生的兴 趣特长；四是有利于提高家长的教育获得感。 一、让课后作业不回家 根据“双减”政策“教师要指导小学生在校内基本完成 书面作业，初中生在校内完成大部分书面作业”的要求，玉 龙学校制定了《深圳市龙华区玉龙学校课后服务工作实施方 案》（下称《方案》），《方案》强调要把辅导学生完成课 145 后作业放在首要位置，在学有余力的基础上再补充开设“新 生活系列”课程和“扬长课程”。 《方案》对课后服务做了课程规划。同时，对选课也提 出了要求，即课后服务必须是学生和家长自愿报名参加。课 后服务的第一课时必须安排为完成“课后作业”，再根据自 己的学习需求选择其他课程。 在学生自主完成作业的过程中，辅导教师主要完成以下 任务：一是指导学生自主完成作业。课后服务时段禁止教师 讲课，教师主要通过巡视学生学习情况，指导学生自主完成 作业；二是引导学生质疑和帮助学生释疑。学生除了完成学 业任务，还要学会质疑，通过寻求老师帮助将学习中的问题 及时解决，教师还可以指导学习能力较强的学生举一反三， 不断拓宽自主学习的边界（建议补充对学习基础弱的学生的 指导要点）；三是指导学生自主规划自己的学习。自主学习 的时间一般为40 分钟， 据观察， 很多学生 （尤其是小学生） 很快就完成了当天的作业，这时候，辅导教师就会有针对性 地指导学生自主阅读或学习其它内容，充分利用课后服务学 习时间。 为了更高效地开展课后服务，我校以学科组为单位开展 作业设计和作业辅导课活动，设计了不同类型的作业，主要 包含实验性作业、闯关类作业、影音展示类作业、分类矫正 作业和阅读性作业，并且充分利用作业备案结果，加强作业 分析，以评优教，更好地发挥作业检查的效用。为了更好地 服务学生课后作业辅导，我校合理安排不同学科教师，以 “小班化”的形式，精准辅导，真正做到“作业不回 146 家”“知识盲点不累积”，要求老师精心备好每一节课后辅 导课。由于课后服务安排合理，大大减轻了学生学业负担， 保证了学生充裕的睡眠时间。 二、让课后服务更丰富 因《方案》对学生作业量和完成时间进行严格控制， 并 要求老师进行作业辅导，所以在课后服务的第一课时，小学 生能够基本完成作业，中学生能够完成大部分作业。部分基 础较为薄弱的学生，若有需要，可以在第二课时继续完成作 业。为了全面提升学生的综合素养，我校秉承“生活为源， 发展为本”的办学理念，立足生活办教育，在课后服务第二 节课时间开设“新生活课程”供学生选择。 “新生活课程”是基于国家课程开发的学科拓展课程， 是我校长期探索的重要成果。学生可以在作业辅导课后，根 据自己的兴趣和特长自主选择课程。课程类型多样，主要围 绕“智慧生活、创意生活、健康生活和艺趣生活”四个方面 构建： （一）智慧生活夯实学生基础。智慧生活类课程面向全 体学生，指向学生的学习基础，开发语数英小初衔接拓展教 学课程。结合学生认知与年龄发展，语文学科以“首末文学 社”为阵地，每周开设课外阅读、吟诵、写作拓展课程；数 学学科拓展出绘本、24 点、魔方、理财、趣味数学、智慧数 学课程；英语学科拓展出自然拼读、绘本阅读，21 世纪英文 报、英文歌曲、英语话剧等。小学和初中均可选择。 （二）创意生活培养学生思维。新生活课程创新课堂模 式，从静态知识传播变为动手操作指导，将课后服务创新类 147 课程整合为科技、设计、艺术三大类，以社团为载体，激发 培养学生探索与发现的志趣与动力。譬如，依照课程体系开 设创客教育课程： 三、 四年级开设3D 打印、 机器人等课程； 五、 六年级开设编程、 无人机等课程， 鼓励学生探索小发明、 小创造和小制作，着力培养学生的创造性思维。创意生活课 程主要面向小学学段。 （三）艺趣生活润泽学生灵魂。我校坚持以艺术教育为 抓手开展生活美育普及教育。非课后服务时间，我校艺术课 程会按照年级普及民乐、剪纸、书法等艺术教育，要求学生 每人至少掌握一门乐器，学会一个画种，会写一手好字。课 后服务时间， 学生可根据兴趣选择相关艺术社团， 如合唱团、 书法社团、剪纸社团、美术社团、民乐团等加强艺术特长的 学习。艺术社团在课后服务时间均有专业老师指导，有专门 功能室练习，定期选拔成员参与国内大小艺术赛事，以赛促 学，成果卓著。艺趣生活课程面向全体学生开放选择。 健康生活滋养学生身心健康。健康生活系列课程，包含 整理、烘焙、烹饪、生态种植、茶艺、感统、足球等，从身 心两方面为学生的健康生活奠基。学校落实“阳光体育”项 目，学生在课后服务时间，可以选择足球、曲棍球、篮球等 体育社团强健体魄，增强身体素质；也可选择茶艺、烹饪、 烘焙、整理、生态种植等生活课程，进行劳动体验，形成良 好的劳动品质。这之中，劳动课程为我校特色课程。非课后 服务时间，我校面向1-9 年全面普及劳动教育课程：如：学 科融合生态种植课程、“一米阳光”主题种植项目等。课后 服务中的劳动课程，则更加注重感官体验和观察记录，进一 148 步提升学生的劳动素养。劳动教育课程面向全体学生，主要 以小学学段为主。 为了细化管理，学校采取了“一生一课表”的形式，以 每位学生的兴趣为起点，准备多样的学习内容供给学生选择， 尽量满足每个学生的不同需求。 为了让“新生活特色课程”真正落地，我校配套建设了 新生活课程实践基地，基地以开放、和谐、多功能为原则， 遵循整体规划，打造独具“玉龙风、中国味、国际范”的教 育场所。如：50 平米快乐厨房、70 平米烘焙乐园、90 平米 剪纸天地、90 平米的茶艺场、140 平米整理空间、1000 平米 的生活创客大列车、1200 平米生活农场、开放性地理园、生 活小磨坊、中草药园、开放性图书角、艺术大连廊、蝴蝶昆 虫博物馆等，一课一室，空间融通，实现环境育人。 三、让居家学习更自主 当学生在校完成了所有作业，并参加了新生活课程后， 在家是不是就不学习了呢？学生在家的时间该如何安排呢？ 虽然这不是课后服务的范畴，但这也是必须思考的延展问题。 这个问题不提前思考，又会派生出新的社会问题。 值得强调的是，学生居家学习必须服务于学生的作息时 间安排和兴趣爱好，要由学生自主决定和自主选择。我校根 据朱永新教授 《未来学校》 的一些思考， 开发了 “扬长课程” 资源库，供学生居家自主选择学习。 “扬长课程”由本校教师依据自己的特长和专业背景开 发、录制，并通过学校相关部门审核，经学校校长办公会审 定后，再予以实施。学有余力的学生可以通过“玉龙学校之 149 未来学校”服务号登录扬长课程平台，在任一时段选择感兴 趣的课程进行个性化拓展学习。 目前扬长课程已经先后上线两批， 共计48 门， 其中包含 12 门家长课程，课程形式多样，包含大美厨房、心理健康、 英语电影赏析、3D 建模、科学小实验、太极拳、健身操、音 乐鉴赏、劳动技能课程等。 自扬长课程开设以来，学生自主参与学习的热情高涨， 第二批课程在逐渐上线， 第一批已完结的课程总课时为2429 课时，共计72870 分钟。通过后台数据观测，扬长课程中各 门课程的平均进度基本都在80%左右，凡参加了课程学习的 学生，基本上都能够完成每节课时的全部学习，说明课程很 受学生欢迎，且学生学习“扬长课程”的效果好。 作为深圳市首批“课后服务”试点校，自玉龙学校开设 课后服务以来，收到了广大家长和学生的好评，课后服务家 长满意率达98.8%，学生满意率达98.6%。我校课后服务 “1+1+1”课程 （作业辅导+新生活特色课程+扬长课程） 成效 也得到了各级主管部门关注和肯定，先后在我校召开了市区 级两次现场研讨会， 中央电视台、 广东省电视台等12 家主流 媒体进行了报道。教育部基础教育司司长吕玉刚同志在调研 玉龙学校课后服务工作后，这样评价道：“老师关爱学生， 创造多种课程，让学生全面发展，全心全意为学生服务。” 150 “有效作业研究”牵引“减负提质”列车 鄂尔多斯市康巴什实验区 为全面落实《关于进一步减轻义务教育阶段学生作业负 担和校外培训负担的意见》精神，康巴什区围绕优化作业设 计、 助推减负提质进行积极探索。 通过区域牵头、 学校实践、 学生受益的“有效作业”研究机制，对义务教育阶段学校进 行了全面深入的“作业改革” 。 一、推进举措 （一） 区域牵头。 康巴什实验区自2020 年提出 “一校一 案”作业研究思路，对义务教育阶段学校的作业设计进行初 步规划与研究，并聘请相关专家对作业布置进行系统指导。 2021 年，明确推进“作业质量提升年” ，由区教育发展研究 中心牵头，确定“每年一主题、每月一目标”的核心思路， 以课题研究为抓手，全力推进作业研究。10 月23 日，康巴 什区家校社协同育人背景下有效作业实践研究课题立项启 动仪式暨培训会顺利召开，为开展“有效作业”研究提出了 明确的实施目标、研究路径与时间规划，指导并带动区域内 义务教育阶段学校开展作业优化设计研究。 （二）专家引领。在开展“有效作业”研究过程中，康 巴什区教育体育局、教育发展研究中心牵头做好中小学校长、 学科教研员的专项培训，组织有关人员认真学习上海、江苏 等地先行案例。特别邀请浙江省教育厅教研室语文教研员、 教育部基础教育教学指导专业委员会委员、中国教育学会中 151 【鄂尔多斯康巴什实验区】 学语文教学专业委员会常务理事章新其进行《指向核心素养 的作业设计》专题讲座，为学科教研员和学校相关人员对开 展作业研究答疑解惑。 （三）学校落实。在康巴什区教育体育局、教育发展研 究中心的指导和带动下，康巴什区各中小学及学科教师全面 开展“有效作业”研究。一是学校立足深度学习教学改进项 目和学生发展核心素养需求，充分结合本校工作实际，申请 立项“有效作业”研究课题，利用三年时间对本校全学科开 展作业研究。二是以学校课题为根本，各学科教师针对学科 核心素养和学生综合能力需求，探索研究本学科作业的合理 有效设计，通过实践帮助学生巩固课堂知识、检查学生知识 掌握程度、促进学生综合能力发展。 二、实践探索 （一）学校作业管理的研究 1.修订作业管理文件。康巴什区各中小学对学校作业管 理的有关内容进行整体设计和适当归并，提高作业管理的自 身水平， 在兼顾教师作业批改、 作业检查反馈等问题的同时， 重点关注前置性的作业规划与设计、内容目标是否符合教学 需求等问题。 2.关注作业总量和难度。一是控制作业总量，学校加强 对学科组、年级组的作业统筹，合理调控作业结构，借助问 卷、访谈、文本分析等方式定期了解学生作业量、完成度等 情况。二是把握作业难度。根据《课标》中“了解” “掌握” “运用” “体验” 等行为动词的层次把握好作业目标水平， 控 制作业设计难度，坚持面向全体与因材施教相结合，设计与 152 实施分层化、弹性化和项目化的作业。 3.探索校本作业体系建设。康巴什区教育发展研究中心 带领各中小学结合本校教学实际，选编、改编学科作业，基 于实践运用，逐步调整完善，建设校本作业体系。 （二）作业设计与实施的研究 1.研究作业内容。康巴什区各中小学立足教材，确定作 业的目标与内容，既注重学生应掌握的基本知识，又关注学 生需达到的能力目标和核心素养目标。因此，学校及各学科 根据教材特点和学生掌握知识实际，精选教材中的作业、根 据教材内容改编作业、根据教学内容拓展作业，让学生的作 业减少重复、 注重思考， 引导学生对所学知识的巩固和理解。 2.丰富作业形式。 ①抓好基础性作业：明确作业步骤，夯实学科功底。以 巩固课堂所学为主，整体考虑课前、课中、课后作业的关联 性、 系统性， 将作业从大水漫灌变为精准滴灌， 减量不减质， 让学生完成适量书面作业，夯实基础。 ②增加能力性作业：以创设情境梯度，提高作业效能。 在生活中寻找与所学知识相关的教学资源，设计利用所学知 识引导学生解决简单的实践问题，达到学以致用、提升学科 能力的目的。 ③开发探究性作业：实践中求创新，培养学生思维。以 能力为轴全学科开展从基础作业到拓展作业、从书面作业到 素质作业的研究。根据学科性质研发劳动作业、阅读作业、 锻炼作业、思维作业、鉴赏作业等。关注作业的情境性、真 153 实性、问题解决性和作业完成过程，鼓励学生采用自主、合 作、探究等多种方式完成作业，培养学生独立思考以及与他 人合作探究的习惯，激发学生的钻研精神。 ④关注项目化作业：拓展作业途径，激发综合素养。结 合国家大事、时事和节日等契机，与单元作业中实践性、综 合性强的内容进行整合，先确定项目主题，再分学科、分年 段确定项目化作业选材，研制项目作业实施指导意见和评价 准则。最后根据实施反馈优化作业内容和作业指导，形成单 元项目化作业范式。 （三）作业评价与反馈的研究 1.有效批改作业。指导学科教师认真、及时、规范地批 改作业，记录学生典型问题并做好梳理总结。针对作业明确 指出错误、提出建议，根据实际需求进行面批并给予学生针 对性指导。 2.深入研究分析。康巴什区各中小学带领学科教师运用 多种方式与手段，对学生作业结果进行归类整理，分析单元 作业目标的达成情况，关注不同学生的结果差异。学科组、 年级组借助个案分析、个别访谈和家校互动等方式，深入了 解学生的理解程度、思维过程、方法应用和态度习惯，探寻 问题产生原因，制定质量提升举措。 3.提高讲评效果。有效利用集中讲评和个别辅导的方式， 既做好全体学生的知识梳理、提炼思路和总结方法，也做好 个别学生的问题分析、习惯培养和提升帮助，同时适时开展 学生之间的成果展评，促进相互学习借鉴。 154 三、取得成果 在康巴什区家校社协同育人背景下有效作业实践研究 课题引领下，各义务教育阶段学校落实作业“减负提质”有 所突破，在作业数量管控、作业质量提升、特色作业留批、 分层作业指导等方面取得阶段性成果。 如康巴什区第一小学围绕“双减”政策，以“减作业增 体验、减时间强效率、减负担提质量”为目标，在作业管理 中实施“四全”管理。一是全领域关注，学校、年级组、备 课组、班级组成学校教育教学的全领域，作业分层、分类、 分段构成作业分层设计全领域，制定《康巴什第一小学有效 作业实施方案》，对作业形式、作业内容、作业难度、作业 效果进行统筹研究。二是全流程优化，在作业设计流程方面 进行“四个优化设计”：即过程性设计、弹性化设计、情境 类设计、融合性设计，有效提升作业质量和学生作业兴趣。 三是全过程监督，从教作业布置、批改、讲评、纠错、提升 等环节，对作业实施进行全程监控，确保有效减轻学生作业 负担。四是全人员成长，通过“作业分层策略改进”，促进 教师团队成长；基于有效作业研究的课堂教学改进，促进教 师成长；基于自我认知的作业选择改进，促进学生成长。 康巴什区第二中学从贯彻落实“双减”机制体制入手， 制定《关于切实减轻学生课业负担实施方案》《作业管理实 施方案》《作业布置与批改制度》《学生作业公示制度》等 一系列作业有关方案，健全作业管理机制，落实“双减”作 业新标准。在作业设计方面，统一制定并采用《康巴什区第 二中学书面作业设计模板（文科/理科）》，各个备课组集备 155 作业设计，发挥作业诊断、巩固提升、学情分析等功能，严 格杜绝重复性、惩罚性作业；在作业总量方面，建立作业统 筹制度， 实行班主任负责制， 确保每一班、 每一生都达到 “作 业总量不超过90 分钟” 的基本要求； 在作业内容方面， 积极 实行语数英书面必做作业、政史地生背诵记忆必做作业与各 学科拓展性选做作业相结合的弹性作业机制，音乐、体育、 美术、劳动、书法、心理等学科可布置适量的实践性作业， 实现学生巩固学科知识、提升综合能力和丰富课余生活的成 长目标。 156 “双减”之下开新局，精细管理谱新篇 鄂尔多斯市康巴什区第五小学 一、基本情况 康巴什区第五小学于2012年建校，学校占地面积33461 平米， 建筑面积15341平米， 学校办学条件达到自治区现代学 校建设标准。现有教师113人，班级37个,学生1671人。 学校依据自身特点，构建了特色鲜明的“问道”学习氛 围， 以 “学生的健康成长为本， 着眼于每个孩子的潜在智慧， 为孩子终身可持续发展奠基”为办学宗旨；以“学习从问号 开始”为办学理念，带领学生探究真善美。 二、课后服务工作系统套餐化 为落实中共中央办公厅、国务院办公厅印发《关于进一 步减轻义务教育阶段学生作业负担和校外培训负担的意见》 精神，按照各级教育行政部门的相关要求，康巴什区第五小 学积极行动，及时开展专题培训、实地调研、现场指导，严 格遵循 “一校一案” 原则， 制定课后服务实施方案， 确保 “双 减”工作有效落实开展。 （一）制度保障 全面落实 学校课后服务工作遵循 “以学生为中心， 以学校为主体， 以家长期盼为导向，以减负提质为目标”为原则，切实落实 减负提质。自实施以来，学校教育教学质量和服务水平明显 提升，课堂教学高效且有深度，作业布置数量和质量更加科 157 学合理，达到减轻学生和家长各方面负担、让学生学习更好 回归校园、突出学校在教育中的主体地位、促进学生全面发 展、满足学生个性多样化发展需求等目的。学校充分利用资 源优势，开设丰富多彩的社团活动，进一步培养学生兴趣特 长。同时, 对学习有困难的学生进行补习辅导与答疑，为学 有余力的学生拓展学习空间。 （二）多样安排 打造特色 学校充分利用教师资源和设备资源，开设服务内容有： 作业自修、兴趣培养、特长发展、课程实践等。每周一和每 周三下午的16:40—18:00为大社团时间，每周四个学时，共 计两小时四十分。大社团以培养全体学生的兴趣爱好为目标， 全校学生实行走班上课制，共开设舞蹈、管乐、绘画、足球 等36个社团，供学生自主选择。每周二、周四、周五下午 16:40—17:10为全校学生作业辅导时间，17:10—18:00为小 社团时间。小社团以培养学生特长、课程实践为目标。作业 辅导后，全校学生走班上课进行小社团，共开设女子足球、 男子足球、男子篮球、高段剪纸、油画、经典诵读、数学思 维、英语天地及习作乐园等57项内容。 课后服务工作与学校主题活动相结合，在主题课程统 一规划下，充分利用地区优势，坚持五育并举，与康巴什区 青少年活动中心携手，开发劳动实践活动，加强劳动技术教 育，共同为孩子们开发各类劳动实践课程。通过开展社区服 务、庄稼收获、卫生清理、志愿服务、植物栽培、动物饲养 等各类劳动实践，进一步提高学生的动手能力，磨炼学生的 158 意志品质。 （三）自愿参与 兴趣先导 课后服务的课程选择运用电子选课平台，通过引导家长 和孩子进行线上自主选课达到了兴趣先导、自愿参与的效果。 选课过程中，学校全程关注后台数据变化，根据孩子的选择 意向，及时调整社团数量和社团内容，按照“初选——复选 ——线下调整” 三个步骤， 竭力满足学生个性化多样化需求。 课后服务的具体流程如下： 学生参加课后服务具有充分的参与权和选择权，所有活 动采取家长学生自愿申报的原则，对于不愿参加的学生， 统一发放安全离校牌，指导学生按时安全有序离校。 三、作业管理规范之中求优化 康巴什区第五小学严格落实教育部办公厅《关于加强义 务教育学校作业管理的通知》精神，制定《学校有效作业管 理细则》，开展减负提质作业改革，发挥作业育人功能，促 进学校教学管理。为了更好的减轻学生课业负担，减轻学生 159 作业负担，学校以有效作业研究为课题，项目化推进，做到 了“七个规范”“五个优化”，达到作业“三清”。 “七个规范”是指规范作业功能、规范作业总量、规范 作业类型、规范作业设计、规范作业指导、规范作业批改、 规范作业布置、规范作业管理与评价；“五个优化”是指优 化作业设计、 优化作业类型、 优化作业研究、 优化作业批改、 优化作业评价反馈。此外，还建立了班内作业设计、作业评 比、作业反馈公示制度。 （一）减负提质，严控书面作业总量 一二年级不留书面作业， 三四年级学生书面作业时间不 超过30分钟， 五六年级学生书面作业时间控制在60分钟以内。 严禁布置机械性、重复性、惩罚性、随意性等低效作业。每 周三定为 “无作业日” ， 不安排书面作业。 双休日、 寒暑假、 法定假日同样控制书面作业总量， 适度安排个性化作业、 科 学设计探究性作业和实践性作业等非常规性作业。 （二）尊重差异，创新作业形式内容 1.分层作业 教师根据学生的不同学情和学习能力，设计“基础+变 式+拔高” 为内容的分层作业， 对学生布置不同层次的作业， 给学生自主选择的权利，学生根据作业“菜单”，选择“自 助餐”做到“因材施教 求同存异”。 2.特色作业 以教研组为单位创编符合学习规律、 适合学生发展的创 意钟表、购物清单、七巧板拼图、古诗配画、创意书签等特 色作业， 激发学生的兴趣， 使学生在实践中加深对知识的理 160 解。 3.实践作业 为学生定制了“党史”系列实践活动方案，设计了涵盖 阅读、写作、锻炼、劳动、科技、艺术等多门学科、多种类 型、 多个维度的实践作业。 学校对学生的每期实践作业进行 评价、展示和表彰。 （三）精细批改，作业管理良性循环 作业批改要做到“日批日改、及时订正、符号规范、批 改精准、等级评价、评语激励”，让作业成为师生共同成长 的“名片”。在学生自主选择作业内容的同时，班主任利用 课后看护服务时间， 对学生作业采取 “分组面批 个别指导” 的方法，保证每个学生每周都至少有一次面批作业的机会。 任课教师利用统一时间对学生作业中的共性问题进行集中 讲评，强化作业辅导的针对性、精准性。 （四）两个阵地，提高作业设计能力 1.课堂教学主阵地 学校提倡“向课堂要时间，向课堂要质量”，构建了以 教师为主导、学生为主体、学习为主线，探究“真实起点、 真实学习、真实成长”，追求“发现美、感受美、创造美”, 培养“敢问、好问、会问、会学、学会”的“问道”课堂实 践模式，导向课堂上的精讲多练，以提升教学效率。 2.校本教研阵地 针对作业布置及批改的具体问题，开展了作业设计的主 题论坛、校本研修、学生作业的展示评比等系列研讨活动。 在教研中，交流切磋，取长补短，明晰作业设计的思路和方 161 向。 四、注重总结 形成做法 注重反思,在总结中提炼出 “1234” 的做法， 即建立一个 “双减”工作体系，以学生为本，遵循教育规律，着眼健康 成长，积极推进“五育”并举，立足校内、校外两方面，创 新思路制定了形成体系的“双减”实施方案，确保“双减” 工作平稳有序扎实推进。搭建“两个载体”，用课后服务解 决学生作业无人辅导的问题，减轻家长经济负担；用社团活 动满足学生个性多样化需求。紧抓“三条主线”，紧抓“双 减” 工作管理主线、 教师队伍建设主线和课堂质量提升主线。 推行四项落实行动， 推行减轻作业负担， 提升教学质量行动； 推行课后延时服务夯实文化课基础，培养艺术特长行动；推 行双减不减责任担当，不减质量提升，家长满意行动；推行 双减下学生课外阅读增加行动。 总之，自“双减”政策实施以来，在各级教育行政部门 的正确引领下，康巴什区第五小学在教学质量、课程建设、 培养学生全面发展等方面取得了一定的成果，在接下来的工 作中， 将继续夯实和强化 “双减” 落实工作， 以更严、 更细、 更实的工作做法，推动“双减”工作落细落实，办家长和社 会满意的教育。 162 【广州黄埔实验区】 党建引领“双减” 助推学生全面个性发展 广州市黄埔区教育局 “双减”工作是贯彻落实习近平总书记关于教育的重要 论述的生动实践，是贯彻落实新时期党的教育方针的必然要 求，是教育发展理念的回归、教育生态的重塑和教育治理的 创新。广州市黄埔区深刻把握教育工作的政治属性和宗旨方 向，在全市率先出台首个党建引领“双减”工作8条“硬核措 施”，区教育局配套出台“23条”具体落实措施，形成“党 委政府领导、教育牵头、部门协同、上下联动”的“双减” 工作格局，实现教育发展理念的回归、教育生态的重塑和教 育治理的创新。黄埔区着力促进区域基础教育领域的深刻变 革，努力实现教育去功利化、回归公益化，去教育应试化、 回归素质化，去教育焦虑化、回归理性化，最终达成学生全 面而有个性地发展。 一、基本情况 （一）加强组织领导，深度调研摸准情况 黄埔区高度重视“双减”工作，区主要领导深入学校调 研推动 “双减” 工作。 区委常委会议专题研究 “双减” 工作， 成立由区主要领导任组长的“双减”工作领导小组，全面部 署双减工作。区教育局组织全系统深入学习贯彻习近平总书 记重要讲话精神，贯彻中办、国办《关于进一步减轻义务教 育阶段学生作业负担和校外培训负担的意见》精神和省、市 163 部署要求，准确把握“双减”目标和政策贯彻边界。区教育 局班子全员全天下沉学校， 问政问策， 摸准问题， 找准对策。 区委组织部印发《黄埔区、广州开发区党建引领“双减”工 作落实八条措施》，激励各级党组织、党员扛起责任、勇挑 重担、迅速行动，扎实有力地推进“双减”工作。 （二）优化服务保障，满足学生多样需求 落实课后服务两个全覆盖，统筹校内外资源，以学校为 主体、多方参与，促进学生回归校园。截至平台11月22日摸 查数据， 我区应开展课后服务小学60所、 小学生78290人， 初 中14所（不含寄宿超100人学校19所）、初中生8071人。实际 开展课后服务小学60所、小学生56224人（占比71.8%），初 中14所、 初中生6998人 （占比86.7%， 另寄宿制初中19171人） 。 所有实际参加课后服务学生63222人（占比73.2%），每周仅 一天参与课后服务的学生数1189人（占比1.8%），每周五天 均参与课后服务的学生54547人（占比86.3%），每周五天均 参与下午课后托管的学生41249人（占比65.2%）。参加午休 托管学生47683人（占比75.4%），其中午休能躺睡的学生 27607人（占比57.9%）。鼓励支持教职工参与课后托管，参 与教职工3964人，其他人员1889人。开设晚自习的初中24所 （非寄宿制3所），参与晚自习的初中生15061人。课后服务 全覆盖的实现，极大增强了课后服务吸引力，确保了学生校 内学足学好。 （三）深化督导力量，作业课程全面达标 将校内作业和课程开设情况纳入开学检查和日常督导 检查，局领导带队到片区学校开展飞行检查，组织片区督学 164 通过专项督导、飞行检查等方式，督导全区义务教育学校作 业课程全面达标。黄埔区现有义务教育阶段学校均做到开齐 开足国家课程，起始年级按照课程标准实施零起点教学，考 试次数符合有关要求和考试成绩等级呈现，85%及以上学生 均能在规定时间内完成作业，建立作业校内公示制度、作业 时间控制达标制度，不给家长布置作业或要求家长批改作业， 达标率为100%。区、校两级分别出台了中小学教育教学基本 要求、 基本规程、 基本规范， 净化教育教学秩序， 严格督查、 严肃处理超纲超进度教学等问题。 （四）治理培训机构，“双减”“维稳”两不误 全面停止学科类培训机构审批，积极指引学科类培训机 构转型。35所学科类机构已有19所按程序完成学科类办学内 容压减或转型非学科类，其余机构已进入相关审批程序。持 续加强对现有学科类机构的现场检查，密切掌控核查机构运 营动态、学生剩余学时、涉退费、投入资金、每月场租、运 行费用、 员工动态等基本情况。 区政法委统筹应对立尚教育、 树童英语、精锐教育等机构的维稳工作。各街镇对辖区培训 机构进行再次摸查。根据《教育部等六部门关于加强校外培 训机构预收费监管工作的通知》协调相关部门启动培训机构 资金监管工作，提高培训机构资金风险管控能力。坚持教育 公益属性，引导培训机构由营利性转为非营利性，不断地去 逐利化和去泡沫化，解除资本对教育绑架，斩断剧场效应， 降低家长教育消费负担。 二、具体措施 （一）强化打赢“双减”使命担当 165 加强学习贯彻对国家推进“双减”工作有关精神，针对 工作重点、难点和痛点，发挥教育统筹作用，推动各部门协 调联动，推动各街镇落实属地责任，形成强大治理合力，全 力推进“双减”工作取得更大进展。加快推动完成区专门工 作机构设置，把推动“双减”工作与党史学习教育相结合， 坚持“我为群众办实事”，研究制定干部联系学校制度，协 助解决学校发展过程中的问题。区内中小学校各级党组织每 年开展一次“双减”相关主题党日活动，各级党组织书记每 年开讲一次 “双减” 相关主题党课,设立党员教师落实 “双减” 工作示范岗，试行“双减”工作承诺制，开办“双减增效” 教育论坛，举办“学史力行·铁军答卷人开讲啦”“双减” 专场活动。充分利用国家、省、市平台，做好新教师培养， 打造素质过硬的黄埔教育铁军。 （二）强化优质教育资源供给 深入推进教育部基础教育课程改革实验区建设，争创教 育部数字与智能化教育装备创新与应用实验区，以“双区” 建设推动软硬件“双轮”驱动，促进教育“双高”发展。完 善优质教育保障机制，从需求侧和供给侧双向发力，增加义 务教育资源供给，加强区域教育统筹，以教育规划促规制。 优化资源配置，扩组基础教育集团化办学或扩联与师范大学 合作办学，推进义务教育高质量均衡发展。大面积大比例促 进干部教师交流轮岗，持续推动区域内校长交流轮换、骨干 教师均衡配置、 普通教师派位轮岗， 实现教育关键要素流动， 从根源上促进教育优质均衡。 166 （三）强化学校教育主体作用 深入开展提质增效研究，落实落细“五项管理”和五育 并举，提升教师作业设计和教学能力。充分利用区教研院实 验小学平台， 教研员深入一线， 扎根课堂， 从作业设计入手， 从课堂效果抓起，做出成果，打造黄埔课堂教学、作业设计 样板。 “四十分钟” 提质量， 改进教学方式方法， 应教尽教， 提升课堂教学质量，提高学生在校学习效率。“四十分钟” 外添能量， 改善学校课后服务质量， 让学生在校内完成作业， 提供丰富多彩的课后服务课程资源，更好地满足学生个性化 发展需求。吸取先进地区先进经验，提前做好新学校规划设 计，同时通过改扩建等方式提升现有学校硬件条件，为学校 午托“躺平睡”、学生就餐等方面提供条件。落实校内课后 服务补贴与收费等政策，优化第三方社会机构引入，推动建 立“线上＋线下”课后服务集约高效管理模式。各学校倡导 党员教师全员全程主动参与课后特色课程服务，扎实开展素 质教育， 着力培养学生的社会责任感、 创新精神和实践能力， 助推学生全面而有个性地发展。 （四）强化校外教育资源配套 提供优质的课余活动服务，加强网络游戏电子产品管控， 改革家校合作范式，积极发展社区托管，用好区青少年宫、 研学基地等公益性校外活动场所，培养学生自我管理能力和 综合素养。引导和协助校外培训机构转型升级，逐步将素质 教育、学习诊断、课后托管、生涯规划等问题解决纳入业务 范围，提供体育、音乐、舞蹈、美术、科技等个性化服务， 补齐学校教育的短板。落实好家庭教育指导服务规划，加强 167 网上家长学校和中小学家长课堂建设，依托“街镇吹哨、部 门报到”机制，发挥社区基层党组织在校外培训机构治理和 课后服务、暑期托管等方面的作用。以中小学研学基地为依 托，利用各街镇党群服务中心开展“第二课堂”，传承红色 基因， 拓展学生校外实践平台。 聘请赵宇亮院士等5位科学家 为首批校园“第一科学导师”，大力开展“大手拉小手，科 学进校园”系列科普讲座，推动全区人才资源与青少年科普 教育相结合， 并打造 “党员名师面对面” 空中课堂。 选聘 “健 康副校长”，推广课间体能拉伸操，保障学生每天校内、校 外各1小时体育活动时间。试点建立义务教育阶段学校与机 关事业单位党组织结对共建机制，依托新时代文明实践中心 （所、站）开展志愿服务宣讲。 （五）强化公共教育服务体系 整合“广州电视课堂”等线上精品课程，推进教学名师 同步在线授课，推动教育资源高效共享。加快教学研深度融 合，联手课程教材研究所、华南师范大学开发可满足学生学 习、 咨询、 答疑等相关需求的在线学习课程资源， 实现 “教、 学、评”良性互动。施行学校教育、校外教育和家庭教育协 同改革，实现各方主体多元合作共进共治，将课后服务、网 络教育、校外教育等纳入基础教育公共服务范畴，优化区域 公共教育服务体系。 （六）强化校外机构监管 根据省、市相关部门研究出台的培训机构监管办法（教 育），校外培训政府指导价（发改委），变相违规学科类培 训联合执法（市场监管、教育），非学科类培训机构审批标 168 准（体育、科技、文旅），转型非学科类证照办理流程、审 核标准及审批时效（教育、民政、市场、体育、文广旅、科 技局）等制度，加快“营转非”工作。建立高位协同、联防 联治的风险防范处置机制，从风险评估、监测预警、矛盾化 解、依法打击四个方面及时做好风险防控工作，注重利用信 息技术提升治理效率，建立培训机构管理平台，实现资金、 学生、教师、课程、材料等系统施治。充分发挥民办教育行 业协会和党组织在企业转型引导、行业自律自治、矛盾纠纷 化解、 困难企业帮扶等方面的作用， 做好校外培训机构登记、 转型、退出等各项服务，助力机构全面向非学科类公益性校 外培训机构转型发展。协同校外培训机构非公党组织保障党 群教师合法正当权益，协调教育行政部门认定有教师资格的 教师从教经历视同中小学教师从教经历，助力教培行业人才 转岗再就业。 各社区党员 “双报到” 教师努力斩断剧场效应， 示范与引导家长不安排孩子参加学科类校外培训，创建社区 党员“靠埔家长”主阵地。 169 平台+服务+课程+评价 完善校内课后托管 广州市黄埔区东荟花园小学 一、基本情况 东荟花园小学从 “幸福就像花儿一样” 的办学理念出发， 个性化制定《广州市黄埔区东荟花园小学校内课后托管服务 方案》，全面落实国家“五项管理”和“双减”政策的同时， 培养学生德、智、体、美、劳等方面的综合能力，强力助推 学校教育教学质量整体提升。 二、主要做法 1.课后服务实施有监管。为保证教育教学质量，充分用 好校内外教师资源，提升校内课后托管品质，学校根据广东 省教育厅、广州市教育局关于做好中小学生校内课后服务工 作的指导意见和《黄埔区教育局关于校内课后托管服务引入 第三方机构的指导意见（试行）》等指导精神，公开招标引 入第三方机构开展实施面向学生的校内课后服务，严格监管 服务内容、服务价格、教学计划、教学内容、教学质量及服 务人员资质。学校全面主导整个校内课后托课程的实施，并 结合学校“幸福之花”课程的建设，把校内课后托管素质教 育拓展服务作为校本课程实施的主要方式。第三方机构两名 专职驻校教务主任协助学校进行校内课后托管服务全流程 管理， 定期向学校书面汇报课后服务的课程设置、 社团报名、 分团编班、教学过程管理、教学效果评估。 170 2.课后服务平台有评价。学校创建了校内课后服务互联 网选课平台，实现在线实时动态开展课后服务教学质量评 价。学生和家长可以网络评价老师，学校后台管理全程监控 课后服务开展状况。课后服务平台开通教师线上签到签离、 学生出勤状态通知功能，确保精准评价每个师生的工作与学 习表现。除此之外，一个学期结束，学校在第三方机构进行 工作总自评的基础上，组织学生、家长和学校老师分别对基 础托管和特色托管的课程质量进行多元化评价，根据评价的 结果决定下一学期开设的课程内容。 3.课后服务课程有特色。学校积极探索校内课后托管特 色服务模式，基本托管服务的晚托主要是进行学生自主作 业、自主阅读、观看影片等活动和学困生辅导与答疑，素质 拓展服务主要是开展体育、艺术、科普、综合实践、非物质 文化遗产传习等素质教育特色课程。体育类开设篮球、羽毛 球、啦啦操等，提升学生身体素质。美术类开设创意美术、 素描、国画、漫画、硬笔书法、软笔书法等，塑造学生美感 想像力和创造力。语言艺术类开设小主持人、演讲与口才、 故事表演、朗诵与表演等，增强学生自信心和表达能力。科 技类开设科学实验、趣味编程、机器人搭建、3D 打印等，培 养学生爱科学和用科学意识。 4.课后服务资源有共享。学校广泛深入宣传课后服务实 施方案和特色课程，积极引导有需要的学生课后服务的全覆 盖。学校无偿提供给第三方机构课后服务功能场地及相应资 源。学校依法保障志愿参与课后服务教师权益，科学统筹安 排教师必要休息时间，实行弹性上下班制。 171 三、特色、亮点与成效 学校建立了 “平台+服务+课程+评价” 一体化模式， 实现 了课后服务 “校内校外+线下线上” 全场景覆盖， 构建了教育 生态六大课程体系，缓解了校内课后服务供给压力，全面提 升了学生发展核心素养。 172 【昆明市盘龙区教育体育局】 将“双减”落实到位 让教育回归本真 ——明通小学“双减”工作案例 昆明市盘龙区明通小学 今年以来，明通小学全面贯彻教育方针，认真落实立德 树人根本任务，以深化师德师风建设，强化“五项管理”、 课后托管服务为抓手，持续推进育人方式改革，切实减轻违 背教育教学规律、有损少年儿童身心健康的过重学业负担， 促进少年儿童健康成长，积极推进“双减”工作以促进义务 教育的内涵发展和质量提升。根据文件精神，按照“校外治 理、校内保障、疏堵结合、标本兼治”的总体思路，着力确 保部署到位， 扎实推进 “双减” 工作落实见效。 根据自愿性、 安全性、“轻负高质”性、作业辅导与特色班相结合、全员 参与性、公益服务性原则参与课后服务班，也根据兴趣爱好 选择特色兴趣班，促进学生个性发展。 一、打好基础 养成习惯 明通小学办学质量优良，社会反响良好,对当地的基础 教育发展起到了很好的引领和示范作用。为落实立德树人根 本任务，学校将学生的终生发展作为教育的终极目标，确立 了 “给予无条件的爱， 成就完整的人” 的学校精神， 并将 “读 书明礼、求知通达”定为校训。在此基础上，学校还提炼出 “打好基础、养成习惯、培养爱好、发展特长、锻炼体魄、 173 健全人格”的24 字育人目标。 在“双减”背景下，明通小学教师在原有的基础上，更 加注重优化课堂效能。以语文学科为例，在备课及实施教学 的过程中，教师们聚焦单元整体要素，抓住教学主线，紧扣 文本语言， 不仅教学生学习知识点， 更注重把方法教给学生， 提升学生的能力。学校以开展多元教研活动的方式，助力各 学科的教学提质增效。 在作业设计上，学校不断优化作业方式，让作业更贴近 生活。通过减“量”提“质”，优化学校作业生态。学校系 统化考量作业的设置，对各个学科的作业进行整合，从综合 性、育人效果上进行设计。学校将作业分为复习巩固类、拓 展延伸类、综合实践类三类，从整体内容和时长上进行统筹 设计，鼓励作业设计贴近学生生活，有趣又有质。每一份特 色作业就是一次创新、一份作品，学校深化改革作业的多元 化设计和评价方式，力争让每位学生都能在作业中获得满足、 愉悦和成功的体验。 明通小学的课后服务以集中管理辅导这一基本需求为 主，切实做到不上新课、不加重学生课业负担。同时，学校 积极开展德育、文体活动、科技制作、社会实践等项目，促 进学生个性发展。作业辅导与特色班相结合的原则，既满足 大多数学生家长需求，又丰富学生学习生活，进而培养学生 良好的学习习惯、阅读习惯、动手实践能力、创新思维能力 及学生的兴趣爱好。 二、培养爱好 发展特长 明通小学全面落实国家部署要求，以育人为核心，设置 174 以“打好基础、养成习惯”为旨的作业班，以“培养爱好、 发展特长、 锻炼体魄、 健全人格” 为旨的兴趣班， 多措并举， 减轻学生过重负担，促进小学生健康成长。学校四个校区在 课后服务开设的作业辅导班共计69 个、本校教师特色课程 班37 个、 第三方机构提供服务 （基于已经建立的网点、 培训 基地的项目，引入更高水准、专业化的俱乐部培训）共计34 个。丰富的课程内容成为学生全面发展的有力支撑。 （一）丰富课余生活，提高综合素质能力 丰富的校园生活，推动学校艺术教育蓬勃发展，明通小 学利用课后服务时间在四个校区同时开展合唱兴趣班，并同 时举行第三届 “我是小歌手” 比赛， 深受学生的支持与追捧。 “心有多大，舞台就有多大”，活泼好学、健康成长的明通 学子们，在学校“五育并举”的教育教导下，个个都是全面 发展的好少年，个性飞扬是他们的特色标签。明通小学在开 展多元教育的同时，把民族文化融入校园生活之中，小歌手 们在了解并喜爱云南民族之美的同时，爱国情怀更得到了升 华。因此在本次决赛中《唱歌给党听》《闪闪的红星》以及 《小背篓》《傈僳火旺旺》等红色歌曲和富有民族特色的曲 目比比皆是，唱出了小歌手们心中的家国情怀和家乡热情。 这样的课后服务极大地丰富了同学们的日常文化生活， 充实校园文化氛围，为同学们提供一块展示自己的舞台，发 挥学生特长，培养学生积极向上的心态和良好的竞争意识， 让广大同学在欣赏中提高鉴别美、创造美和发现美的意识。 明通“小歌手”比赛当日进行现场直播，在线观看比赛的观 众多达近五万人，比赛被春城晚报、昆明日报等多家媒体报 175 道，登上了11 月10 日的学习强国。此活动以每年一届的方 式固定下来，保证活动的连续性和教育效果的持续性，以此 丰富“三点半”课后服务质量。 （二）规范信息技术课程设置，培养学生自主学习能力 编程班是明通小学三点半特色课程之一，以“小学主题 编程课程开发与实践”为主体，开展编程教学活动，基于 steam 教育理念开设“编程+”系列主题课程，主要为“编程 +数学/语文/美术/音乐/动画/游戏/机器人” 等不同主题， 结 合软硬件开设基础课程和提高课程。通过一系列的课程学生， 培养学生的信息技术素养。在编程学习与教学中，注重学生 的数字化创新能力和计算思维的培养，提升学生自主学习的 能力。课程内容以编程学习为主，引导学生将编程作为创作 工具表达自己的想法并实现创作创造。课程前期以动手搭建 为主，结合生活中的问题通过积木搭建创造作品，并通过简 单的编程让作品动起来。课程中期以编程为主，围绕不同主 题创作作品，培养学生的编程能力和计算思维、数字化创新 能力，后期使用micro：bit 主板，加入一些扩展插件，让学 生围绕生活中的实际问题， 设计解决问题的方案， 创作作品， 在这个过程中，引导学生根据解决问题的需求，自主学习所 需知识，培养学生的创新创作能力和独立自主学习的能力。 （三）抓实行动，引导学生全面发展 课后服务中“基于城市生态建设背景下的小学生态教育 的实践研究”项目组，通过组织学生全过程、全方位地观察 白沙河在建校区，引导学生从建筑中的生态、海绵城市、园 林绿植、水土、生态循环等多方面去学习生态知识，将生态 176 建筑建设与学校生态文明教育相结合，从而全面普及青少年 生态环保意识，将生态文明教育融入教育教学全过程，更好 地推进生态校园建设。 这样的综合实践课程仅为“基于城市生态建设背景下的 小学生态教育的实践研究”的一个缩影，这样的研学实践活 动明通小学的学生每月都可以参加。 自2015 年起， 明通小学 就大力推行生态文明教育， 着力构建 “生态型学校” 。 对此， 生态教育专家吴程博士对明通小学积极开展的生态文明教 育赞赏有加， 他表示： “很感谢明通小学提供这么好的平台， 并积极推进关于生态文明教育的研究活动，能很好地唤醒小 朋友们的生态环境保护意识，从而让小朋友们从小养成爱护 环境的好习惯。” 在确保学生学业完成的基础上，明通小学积极开展的这 些生态文明研学活动及拓展训练，同时配合引进校外资源， 提供高质量的课程，成效明显，人民日报、中新社、昆明日 报等媒体多方报道，并吸引了社会积极的关注度。 三、锻炼体魄 健全人格 为了提高学生素质，使学校的阳光体育活动得到有效延 伸，学校的课后服务开设了足球、篮球、网球、乒乓球课等 体育特长课，对学生进行基础的球类动作和技巧的训练。教 师在课堂上采用讲解、示范、练习、指导等多种方法进行教 学，学生的身体灵活性、协调性得到了充分的锻炼，团队合 作意识也有所增强。 结合明通 “六艺” 特色课程紧扣 “五育” 并举的教育发展理念，在古代传统的“六艺”的基础上进行 拓展和延伸，并结合学校的生态文明教育课程理念深入挖掘 177 其内涵开设的射艺班，在保留其竞技特征的前提下，注入了 人文精神。明通小学射艺队把射箭作为一项文化与体育相结 合的运动项目，目的是培养读书明礼，健康向上的明通娃， 同时也为射箭运动培养后备人才。 在生态文明教育理念的影响下，以学生的自然成长为前 提，塑造学生的健全人格、聪明智慧、强健体魄、审美素养 和生活技能，并无条件地为学生提供自由成长和个性发展的 空间，让每一个学生如小树苗般在属于自己的生态花园里自 由生长。让他们在读书的过程中体验到求索的乐趣，在丰富 多彩的学习中感受到求知的快乐，在有趣的体育锻炼中体会 到拼搏的力量，在多元化的社会实践活动中感受到生活的多 彩。 总之，落实“双减”，最重要的是转变教育、教学观念 和学生成长观念，让学习回归学校，让学校教育承担起学生 全面发展的责任，让家庭教育与学校教育有机补充，让社会 机构与资源服务于学校教育。学校教育、家庭教育、社会教 育三者应形成有机整体， 各司其职， 又相互补充、 协同发展， 共同创建儿童教育健康、全面发展的良好生态。 178 以基于学情的分层作业设计促教学提质 ——盘龙区桃源小学落实“双减”工作案例 昆明市盘龙区桃源小学 一、案例提出的背景与概述 近年来，如何全面提升学校的育人质量，成为老师和家 长关注的重要话题，也是桃源小学老师研究的重点问题。在 反复的学习和研讨中，昆明市盘龙区桃源小学的教师们充分 认识到：作业的质量与学校的教育质量提升是正相关。随着 国家“双减”政策的出台，学校教师通过学习文件精神和研 讨，达成一种共识：“双减”政策要求全面压减作业总量和 时长，减轻学生过重作业负担，不是简单的作业减量，而是 通过作业设计实现教学效果的提质。这样就需要我们通过不 断地思考与实践来回答“教学中提供的作业如何满足全体学 生的不同学习需求？”这一核心问题。 学校结合教育部发布的《关于加强义务教育学校作业管 理的通知》，基于学情、校情实际因地制宜地开展了一系列 作业设计的研究，逐步强化了作业设计与学科分层目标达成 的教研活动，助力教师创设目标分层作业，为小学一线教师 重新思考作业的内涵与价值、开展作业改革提供了探索空间 与实践平台。 二、案例实践的内容与成效 1.分析学情，从“三维思考”的角度设计作业 高质量的作业建立在教师深入分析的基础之上。昆明市 179 盘龙区桃源小学地处昆明市中心，作为一所有着百年办学历 史的城区学校，现有19 个教学班，1006 名学生。其中外来 务工随迁子女占学生总数的68.7%。通过前期的问卷调查发 现，被调研的家长97%为学生的父母，家长的学历层次较为 参差，其中初、高中学历占比79.70%，有10.2%为大学以上 的学历。因受多种因素的影响，仅有15%的家长表示有能力 陪伴孩子完成家庭作业。 家长的文化程度、 学生的学习环境、 学生的认知基础、家校合作沟通的差异性突出。通过学生家 长的督促与指导来完成作业进行教学提质，几乎是不可能的。 因此，学校教师对于如何通过有效作业的设计与实施，来提 高教育教学质量，充满了研究的热情与效果的期待。 优化作业设计必然要求教师以严谨的态度进行三维分 析。“一维”指向学生群体差异分析。“二维”指向学生能 力要素分析。“三维”指向学生思维水平分析。为了全面了 解学生，学校从学习习惯、学科素养、学习能力、实践能力 四个维度对学生的学情进行了细致分析。结合各学科的课程 标准要求，各学科在教育质量监测中所反映的核心问题，由 点及面地探索分层分类作业设计的方法。在反复的研讨中教 师们认识到：作业设计是学校教师专业发展和学生立场的真 实反映。 2.尝试多学科协调发展，师生双向自主的有效应用 根据前期的研究和探索，学校结合各学科的特点，将作 业分为基础性作业、拓展性作业、创新型作业三个层次，提 倡作业的个性化设计与实施。结合教学实际，就“双减背景 180 下如何有效落实作业设计”畅所欲言，充分交流了想法，并 让每位教师提交一份和孩子们共同设计的作业。 桃源小学通过作业公示公开制度，提倡“作业超市”、 布置“作业自助餐”，根据学生的具体情况设计分层作业， 让不同学生可选择完成不同内容和数量的作业， 实现 “创新” 作业，满足不同层次学生学习需求。 3.梳理学科作业目标功能，统筹课程资源，尝试作业 系统“多效能”。 (1)数学学科作业体现分层特色 小学数学作业要体现通过真实的数学活动解决真实的 问题。通过作业设计为学生提供亲身体验、发现、解决生活 中数学问题的素材。 我校数学作业按照教学内容和目标设计作业单，分为： 必做题基础作业、选做题发展作业、思考题挑战作业3 种层 次。老师们把教材中的“做一做”“想一想”“说一说”等 课后同步习题作为课堂教学巩固性训练，教师面向全体学生， 做到“题题会，堂堂清”，及时反馈，提高课堂学习效果。 结合教材中的练习题，教师通过备课组的集体教研活动，对 教材提供的作业进行难度梳理，靶向学生的分层发展目标， 由点及面举一反三设计同步作业训练，可以由学生自主选择， 也可以由教师“分任务、务实效、重达成”地将作业进行分 配。 以五年级上册《平行四边形面积》一课为例，教学目标 是探索图形的面积公式，推导面积计算的方法，并能运用计 181 算方法进行面积计算。学生虽然在三四年级就已经学过用 “割补法”计算不规则图形的面积，但是“割补法”解决问 题是小学数学学习中的难点。为了引导不同层次的学生有不 同程度的能力提升， 我们设计了3 道题作为必做题基础作业： ①画一画，算一算。在方格纸上画出一个平行四边形， 想办法计算出平行四边形的面积。目的在于了解全体学生的 知识基础掌握情况。 ②填一填，想一想。用填空题的方式，引导学生回顾平 行四边形面积推导的思路， 掌握 “平行四边形的面积=底×高” 的方法，初步形成图形的转化思想。 ③做一做，比一比。从真实的问题情境入手，对比两组 生活中求图形面积的解决问题练习题，巩固学生面积计算方 法，形成从实物中抽象出图形的意识与能力，能通过问题比 较，灵活掌握解决问题的方法。 在此基础上设计选做题拓展作业，选择教材练习中解决 面积问题难度中等的几道题，进一步巩固面积计算方法，提 升分析问题和解决问题的能力。这部分题允许学生在完成必 做题改错的基础上，自主选择分几次完成。 而思考题挑战作业，则是让学生用不同的平行四边形或 者不同的方法验证平行四边形的面积计算方法，更注重对学 生综合素养训练，供学有余力的学生进行尝试。 三个层次作业的设计包括基础知识和基本技能的训练， 也提供了学生数学思考的素材，不仅保证了学习质量，还让 学生的学习积极性和自主性得到提高。 同时我们还设计一些阶段性作业，如：把知识点的整理 182 归类，对错题进行分析和反思，鼓励学生以思维导图、提纲 笔记等形式呈现出来进行交流。 (2) 英语学科体现诊断式作业设计特色 英语学科针对不同学生类型，进行诊断式作业设计研究。 学校聚焦英语作业问题，首先制作了作业满意度调查表， 针对作业量、作业难度、作业趣味、作业类型四个方面开展 调查， 反馈中， 发现92%的学生认为作业量符合自己的预期， 78.7%的学生认为作业难度适中，56.5%的学生认为作业趣味 性有待提高，71.9%的学生提出了自己喜欢的作业类型有什 么。通过调查，学校发现作业类型成为学生最大的期待，学 生提出的各式各样的作业类型，为教师作业设计提供了强大 的素材库。 其次，学校制定摸底作业单（书面+口语），对各个年级 学生的英语知识、 技能、 拓展能力进行摸底， 52%的学生能够 掌握四会单词与句型，但是情境交际和思维能力有待提升； 32%的学生能使用所习得的单词与句型进行自我表达与运用， 16%的学生能够进行英语知识的拓展表达及创作。通过摸底 发现，超过50%的学生在英语的情境交际和运用中，不能自 信的进行表达，结果表明，学校学生学习英语的瓶颈出现在 表达能力上的局限。最后，学校根据作业满意度调查、学生 课堂表现和作业单摸底进行综合数据分析，把学生分为知识 型学生、技能型学生和能力型学生三大类型。 教师根据诊断数据，对比学生喜欢的作业类型和学生的 实际学情，通过对作业设计、统筹、规划形成精准作业，按 思维层次性和拓展性设置不同难度的作业满足各类学生的 183 作业需求。以激发学生表达自信和思维形成为目标，能在基 础知识的基础上，提升学生英语情境交际能力和语用综合能 力。 针对知识型学生，教师着重培养该类型学生的语言表达 能力和表达自信，教师设计了每日一谚语的作业，学生每天 回家要教会家长一句英语谚语和家长 PK， 并记录在自己的谚 语本上，为英语写作也打下坚实的基础。学生开始从简单的 an apple a day keeps the doctor away 到 Two heads are better than one.学生掌握的谚语愈来愈多，学生能脱 口而出的英语句子与日俱增，家长能在家中听见孩子说英语 讲英语，家校互动性作业也得到家长的大力支持和赞扬。 针对技能型学生，这部分学生基础知识扎实，教师通过 拓展性作业提升该部分学生英语综合语用能力，教师推荐优 秀课外阅读读物，提倡这部分学生增加课外阅读量，通过大 量阅读进行绘本仿写与表达，学生的知识也得到拓展。 针对能力型学生，学生自己进行文本创作、小组合作， 开展舞台剧表演，英语组与学校的话剧社团联合，进一步的 进行排练和指导，更好的为学生提供了自我展示的舞台和机 会，激发了学生参加学校特色社团的积极性，让学生在学习 的过程中更能体会到成就感与满足感。 （3）科学学科作业体现实践探究 科学作业是学生课后的科学实践或活动。这不仅是一个 课外学习的过程，也是课堂教育的延续和补充。我校科学作 业摒弃了内容单一、形式单一、机械重复等作业，着重设计 与生活生产紧密结合的多元化的实践探究类作业，实践探究 184 又分为观察类、调查类、制作类、实验类、种植类和饲养类 等多种类型。不同种类的作业又根据学生认知情况分层设计： 以种植类为例，该类型教学内容在一年级时就有涉及并 贯穿整个学段，因此，学校从年级特点出发，设计开展全校 太空种子种植活动。 太空种子种植实践活动是桃源小学与北京市东城区分 司厅小学共同合作的项目， 2020 年 5 月 5 日桃源小学和北京 分司厅小学科技教育联盟共同推选的花油3 号油菜种子搭乘 长征五号 B 运载火箭成功发射，这是桃源小学和分司厅小学 及科技教育联盟科普活动的又一重大里程碑。太空油菜花已 经在 2020 年 10 月播种第一代，发现很多奇特的现象，通过 孩子们的研究现在已取得一定成果。结合学科教学，我们是 这样设计的： 一二年级以认知和观察为主，定期到太空种子种植基地 观察植物生长，并在观察过程发现问题，课后在教师指导下 学会查找问题的答案，同时还与美术学科融合，让孩子用画 一画的方式来认识太空种子，培养低段学生对科学的兴趣。 三四年级以种植实践为主，该年段的学生已具备基础的 种植知识，结合科学教材内容，将凤仙花种植活动迁移至太 空种子的种植。学校将太空种子发给每位同学，按照种子类 型分组合作探究， 形成科技种植小组撰写植物观察日记 ， 在 种植过程中遇到问题共同探讨，寻找解决方法，培养学生的 合作意识和科学思维。 五六年级以种植维护和撰写科技论文为主，通过前四年 185 的学习，高段学生具备较强的科学探究能力，在种植太空种 子的过程中能提出具有科学性和可研究性的问题，在活动中 收集学生提出的问题，引导学生选择一两个可研究的问题作 为研究目标， 设计对比实验， 通过实验数据撰写科技小论文。 此过程不仅培养学生的科学思维，更为学生的科技创造能力 提供了有效的平台和途径。 三、基于学情的分层作业设计实施后带来的变化 在桃源小学，实施作业设计管理到底改变什么？五年级 学生戴思诚说：“这学期的作业内容有了明显变化，由原来 单一的抄写课本内容或做练习题为主变成了‘菜单式’个性 化作业，老师会布置多项作业供我们选择，每名同学可以根 据自己的能力以及兴趣选择当天要做的作业内容，不再像以 前那样全班同学都做一模一样的作业，大家都能积极完成作 业。” 桃源小学数学教师杨林兵说：“‘双减’政策出台后， 教师不仅要用心设计作业的内容，还要关注到完成作业的时 间，要注意作业的有效性和指向性，不能让学生在题海战术 中厌倦学习。” 经过学校对作业设计的新一轮改革，系统化、统整性的 管理，进一步优化作业设计，学生在各个学科中的能力提升 较为明显。 数学中的解决问题一向是学生的薄弱环节，通过真实的 数学活动解决真实的问题，在作业中学生亲身体验、发现、 解决生活中数学问题，把抽象的数学问题生活化，学生的理 186 解能力大大提高。通过数据对比，学生解决问题类题型通过 率由77%提升至82%，图形操作类题型通过率由75%提升至 85%，学生计算类题型通过率由88%提升至95%。 英语的诊断式作业设计，为作业布置搭建数据依据与支 撑，为作业的精准性提供了技术支持。学生的英语表达和语 用能力的提高也逐步显现出来，学生单词速读优秀率提高 26%，看图表达及格率提高12%，综合表达及格率提高11%， 阅读和写作优秀率提升18%。诊断式作业的设计，真正满足 不同类型的学生获取不同维度的知识，更实现了所习得知识 的多方位输出。 科学太空种子种植特色作业，让学生主动探究能力有所 提升，特别是通过科学实践活动体现出实践能力的提升。高 段学生的提升情况更为突出，对比活动前和活动后问卷数据 反映出，活动前学生对“未知事物选择科学的探究方法”只 占6.2%，太空种子种植活动后问卷显示， “能够选用科学方 法探究未知事物”的占83.5%。 通过作业的设计研究，教师们充分感受到学习目标是学 习活动的出发点和归宿点。从促进学生发展的角度，制定与 各层次“最近发展区”相近的作业分层目标，实现特色作业 设计，满足不同学生的个性化需求，使每一层次的学生在完 成练习之前就有标可依，有章可循，为高质量的完成作业打 下良好基础。作业是检查学生对基础知识、基本技能掌握程 度的一种必不可少的有效手段。在作业设计中，要面向全体 学生，尊重学生个体差异，树立分层递进的教学观，在作业 设计时，要根据学生的不同层次需求设计不同的练习，使学 187 生在学习中达到事半功倍的效果。 四、下一步研究的方向和思考 2022 年是党的二十大召开之年， 基础教育工作要围绕中 心、 服务大局， 积极作出应有贡献。 学校的教育教学要提质， 要提高作业设计水平在 “压总量、 控时间” 的基础上注重 “调 结构、提质量”。学校将通过加强作业设计研究、完善作业 设计指南、开展优质作业展示交流、举办作业设计大赛等， 提升教师作业设计能力水平，提高作业针对性有效性。 立足“形成性评价+作业研究+课堂教学”联动优化，打 造 “规范+高效” 教学管理。 构建以青年教师风采展示活动等 为抓手的教学研究体系，深化研讨，构建高效课堂。立足作 业检测教学效果、 分析学情、 改进教学方法的功能， 开展 “形 成性评价+作业研究+课堂教学”三加联动式研究。研发单元 作业设计思维模型，构建“基础类—提升类—拓展类”分层 进阶作业体系。 教师应对学生作业进行持续的跟踪和关注，同时，教师 也应该在调整自己的教学过程中进一步反思，“我这样调整 之后是否有助于学生更好地理解”“作业设计的起点是否合 适？目的是否明确？”等。只有长久持续地跟踪和关注，才 能真正实现学生作业的发展性功能，从而全面提升学校教育 教学质量。 188 【井冈山市教育局】 打造高效课堂 减轻学生课业负担 宁冈中学 为了进一步落实中央办公厅、国务院办公厅发布的《关 于进一步减轻义务教育阶段学生作业负担和校外培训负担 的意见》，宁冈中学规范学校办学行为，切实减轻学生的课 业负担，科学有效地提高教育教学质量，促进学生全面健康 发展，使素质教育深入开展。 一、强化课堂教学结构改革，保证教学效益 改革课堂教学结构，是减轻学生课业负担的前提，是进 行素质教育的必要手段。 我们提出向45 分钟要质量， 制定了 上好每节课的原则和要求，提出教学中要做到“六为”“六 突”。“六为”即以教师为主导，以学生为主体，以课堂活 动为主线，以思维引导为核心，以教学方法为中介，以能力 培养为目标。“六突”：要突出重点、难点，不面面俱到； 要突出精讲巧练，不能以讲代练或以练代讲；要突出思维训 练，不满堂灌、满堂问；要突出学法指导和知识迁移，不加 重学生课业负担；要突出教学效率，优化教学方法；要突出 因材施教、分层教学，不搞千人一面，千人一法。教师的课 堂教学务必要处理好“三个关系”：即教与学的关系，知识 与能力的关系， 教书与育人的关系。 要确实抓好 “四个保证” : 即组织教学要保证学生纪律严明，听课专心认真；导入、检 查、复习力求简捷，保证尽快接触新知识；新课讲述努力做 189 到语言精练，减少旁征博引，保证学生注意力高度集中；巩 固练习紧紧围绕重点设计精巧练习题，保证对重点知识的深 刻理解与掌握，形成牢固的记忆。 二、精心设计弹性作业，保证学生足够休息时间 传统的作业布置是一刀切的， 其存在三种弊端： 优生 “吃 不饱”；差生“吃不了”；机械性的作业重复完成。弹性作 业与之不同，其对不同学生布置不同份量、不同质量、不同 要求的作业，能确实地减轻学生课业负担。学校要求教师在 备课时，要精心地设计数量少、质量高，大部分学生能在课 堂上完成的作业。 学校要求弹性作业要做到“二活”：内容活，具有启发 性、不机械、不重复；形式要灵活，或书面作业、或口头作 业、或操作性作业、或观察性作业、或游戏性作业、或视听 性作业。“三导向”：作业题的设计一部分是有指导学习方 法的练习，一部分是指导巩固性的练习，一部分是指导发散 性的练习。“四分层”：按学生学习程度好、中、差的层次 分层设计作业。一部分是三者共同完成的必做的基础性作业； 一部分是照顾优生“吃得好”的提高性作业；一部分是照顾 中等生“吃得饱”可以消化、加以巩固、散发思维的作业； 一部分是照顾差生“吃得了”的打基础、补漏补差的作业。 精心设计弹性作业，解决了学生差异的矛盾，充分发挥 了学生智力因素，激发了学生的非智力因素，也调动了学生 学习的积极性，促进了竞争意识，培养了学生各种技能，还 减少了学生用于重复、机械或耗尽脑汁也完不成的作业的时 间，从而保证了学生有足够的休息时间。 190 三、严格控制学生学习时间，保证学生全面发展 要使学生德、智、体、美、劳全面和谐发展，除了正常 的课堂教学以外，还必须开展多种形式的课外活动。为了让 学生有时间参加课外活动，让他们自由快乐地成长，学校要 求全体教师必须坚持做到“三不”：上课不拖堂、不随意调 课或占用其他学科课、不占用学生课外活动时间。下午第七 节课分年级开展书法、手工制作、乒乓球、田径等多种兴趣 小组，让学生根据自己的爱好与特长选择自己喜爱的兴趣小 组参加活动，让学生在各种活动中开阔视野、丰富知识、增 添学问、增长才干、发挥特长、发展个性。 四、严格控制订阅资料、控制考试次数，保证学生牢固 掌握基础知识和基本技能 学生订阅各种资料与频繁地考试，不仅加重学生课业负 担，也加重教师的工作负担，而且还打乱学校正常的教学秩 序，影响学生的课外活动和休息时间。学校规定教师根据课 堂需要， 结合学生实际， 科学设计课堂综合练习题和复习题， 使学生既能理解、掌握和运用课本知识，又不加重学生的负 担，达到全面提高教学质量的目的。此外，学校还把严格控 制考试次数作为教学改革和减轻学生负担的一项内容来抓， 要求教师加强平时作业的检查和评讲，努力提高课堂教学和 练习、作业训练的质量，加强平时的考查，建立“四过关” 制（即天天过关、单元过关、学期过关、年级过关），减少 不必要的考试。这样既减少了学生为考试加班加点进行复习、 练习、考试等的负担，也减轻了教师的工作负担，保证了学 生能牢固掌握基础知识和技能，全面提高各方面素质。 191 用好“加法”运算，破解“双减”难题 ——井冈山小学落实“双减”工作主要做法 井冈山小学 今年7月， 中央办公厅、 国务院办公厅印发了 《关于进一 步减轻义务教育阶段学生作业负担和校外培训负担的意见》， 要求减轻学生课业负担，严格控制书面作业总量，保证学生 睡眠时间；全面规范管理校外培训机构，坚持从严治理。这 充分体现了党中央对教育工作“培养什么人、怎样培养人、 为谁培养人”这一根本问题的深谋远虑和高瞻远瞩，充分体 现了党中央坚决防止侵害群众利益行为、构建教育良好生态 的坚强决心。这是利国利民的大事。 然而，要全面落实“双减”，意味着学校将承担更大的 责任和使命，也将面临来自家长和社会的更高期待。学校如 何转变观念和职能、创新思路，是主动作为还是被动应付， 将直接关系到“双减”政策实施的效果。在“双减”政策出 台后， 井冈山小学上下统一思想、 提高政治站位， 积极思考、 主动作为， 在做 “减法” 的同时不断自我加压， 用好各种 “加 法”运算，破解一道道“双减”难题，牢牢把握住学校教书 育人的话语权。 一、从五方面发力，在主责主业上做加法 1.广泛宣传认知，从“引导”上发力。层层传达国家、 省“双减”会议的精神，夯实学校教育这块主阵地。加大对 “双减”政策的宣传，引导广大家长树立正确的教育观和成 192 才观。 2.确保课堂高效，从“教学”上发力。打造高效课堂， 提高课堂效率；丰富学科活动，培养学生核心素养。 3.加强学科研讨，从“作业”上发力。形成各学科作业 布置的“1+1” 统筹机制，即常规作业加特色作业，让孩子 们启智练体提能。 4.加强教学管理，从“制度” 上发力。规范管理，建立 “晒作业”公示制度，强化延时服务管理。 5.完善课后服务， 从 “兴趣” 上发力。 课后服务采取 “4+X” 模式（“4”指的是每周一至周五下午放学后进行的4天作业 辅导班， “X”指的是每周一天开展的兴趣社团班）；作业辅 导严格执行“三不原则”（不上新课；一、二年级不布置书 面家庭作业，校内适当安排巩固练习，三至六年级书面作业 每天控制在1小时以内； 在校完成了相关作业， 不得另外布置 作业） ； 整合校内外资源开设兴趣社团， 课后服务内容丰富， 注重学生兴趣培养与能力发展，赢得了家长的广泛赞誉。 二、狠抓教师队伍成长，在教育教学质量上做加法 1.扎实做好学校党建引领。学校党支部充分发挥党员教 师先锋模范作用，提高党员的党性意识、服务意识和责任意 识。积极开展党员执行校长周活动，扎实推进“三个一”： 一名党员走近一个班级、一名学生、一位教师，让党员在师 德师风建设上站起来、亮出来。 2.严明师德师风纪律。开展中小学有偿补课和教师违规 收受礼品礼金问题专项整治，建立健全工作机制，制定实施 方案、建立领导小组、工作专班、建立工作台账、设立专门 193 举报电话和信箱。工作开展不走过场，不流于形式，扎实推 进专项治理工作，并及时上报相关材料，切实落实好两项工 作要求，确保各项工作取得实效。 3.构建教师成长大格局。井冈山小学以“培养名教师、 造就名校长、打造名学校”为重点，横向上拓宽校际交流渠 道，延伸教师成长平台，搭建资源共享、捆绑发展、休戚与 共的“教师成长共同体”。同时，纵向上专注个人专业素养 的自身提升，学校专门拿出经费建立教师成长档案，通过正 面引导，促使每个教师找到努力的方向，挖掘内在的潜能， 遇见最好的自己。 4.实现科教研落地。以课题研究为依托，健全了“学校 —科研处—课题组（教研组）—教师”四级研修工作管理网 络， 确立了 “主题式” 研修活动的模式； 打造 “名师工作室” ， 形成名师学术引领、骨干辐射带动的教研路径。目前，学校 2个国家级课题、4个省级课题结题后在学校推广，一大批教 师在各级各类竞赛中获奖。 5.打造高效课堂。设计科研规划，以课促教，组织“五 课”大赛（新教师过关课、青年教师成长课、中年教师展示 课、骨干教师示范课、名师的引领课），提升综合素能，实 现一年合格、两年优秀、五年骨干的目标。就在近期，各学 科组在两个多月筹备的基础上，全面发动40岁以下教师参加 以演讲、书法、写作、上课为主要内容的青年教师大比武， 反响热烈、效果显著，一大批青年教师脱颖而出，有效促进 双减下的课堂活力提升。 194 三、促进工作有机结合，在五项管理上做加法 为了强化学校教育主阵地作用，深化课堂教学改革，学 校将“五项管理”要求与课堂教学常规管理紧密结合。以提 高课堂教学质量为核心，以“家校社”协同育人为抓手，严 控书面作业总量。为了提升作业质量，本学期开始各年级各 学科组统一设计作业，从基础、提升、实践三个层次着手， 实行作业评价弹性制，做到量体裁衣、全面育人。通过发布 “睡眠令” ,建设书香校园, 坚持开展 “六年读一百本书” 等 各类活动，开设涵盖运动会、乒乓球、课后服务、大课间等 内容的体育节活动，确保学生有充足的体质训练、阅读和睡 眠时间， 切实提高学生综合素质， 满足学生多样化发展需求。 坚持疏堵结合原则，制定学生手机校园内统一保管制度，为 每个年级配备“手机储存箱”，在校园内安装公用电话，既 解决了小学生玩手机乱打电话的问题，又便于学生与家长之 间进行必要联系。 四、落实五育并举，在学生全面发展上做加法 1.坚持做精红色德育。学校充分发挥“红色”优势，紧 紧围绕井冈山精神大力开展“红色德育”活动，即：人人会 唱一首红色歌曲、会讲一个红色故事、会背一首红色诗词、 会做一道红色菜肴、会介绍一个红色景点、会表演一个红色 剧目、会参与一次红色实践活动。把“红色引领”融入到学 校教育的整个过程，融入到学生的日常学习生活。“红色德 育七个会”让学生在参与中体验，在体验中感悟，在感悟中 升华，知行合一，促进了道德、行为、知识的全面成长，为 孩子打上了“井冈烙印”。 195 2.全面推进素质教育。学校稳步推进课程改革，探索出 系列红色课程、安全课程、艺体课程等。利用课后服务开设 40多个兴趣社团，着力打造抖空竹等传统文化项目，2020年 在吉安市田径运动会中荣获小学组团体总分第一名，并获评 全国百所乒乓球特色项目学校。积极举办各类爱国爱家、文 明守礼主题系列活动，引导学生形成知礼守礼的高尚品格。 经过自上而下的努力，学校育人质量不断提升。 学校将继续坚持落细落实“双减”政策，构建良好教育 生态，让孩子从此告别沉沉的书包，摘下厚厚的眼镜，还孩 子们自由、快乐、奔跑的童年。 196 【北京市西城区三教寺幼儿园】 “双减”背景下幼儿园育人质量提升的实践探索 北京市西城区三教寺幼儿园 随着《关于进一步减轻义务教育阶段学生作业负担和校 外培训负担的意见》（以下简称“双减”）的颁布，促进了 学校管理者和每位教师对教育生态、学校教育教学改革的深 度思考。“双减”政策的实质是向在校的教育教学质量提出 了高标准、严要求，激发教育者不断审视自己的教育初心。 “双减”政策对幼儿园教育的影响主要体现：一是落实立德 树人任务，明确育人目标；二是避免小学化倾向，建构育人 途径；三是探索幼小协同，科学有效衔接。围绕这一思路， 北京市西城区三教寺幼儿园明确了幼儿园教育以“立德树人” 为根本目标，以“传统文化课程建设”和“幼小衔接贯通培 养”为两大抓手，明确育人目标、建构育人途径、科学有效 衔接，切实提升幼儿园教育质量，回应“双减”政策下对幼 儿园育人质量提升的时代吁求。 一、明确育人目标，落实“立德树人”根本任务 （一）立德树人，培养现代中国人 教育是培养人的事业，是为国家和社会培养德才兼备的 建设者和可靠接班人的事业。“双减”政策背景下，更要从 整个教育场景来思考幼儿园课程内涵、价值建构。立德树人 是教育的根本任务，而立德树人必须从中华优秀传统文化中 197 汲取精神营养。中华优秀传统文化教育“从娃娃抓起”，是 为儿童的终身发展“培根铸魂”的远大过程。解决好培养什 么人、怎样培养人和为谁培养人等重大问题，也是做好“双 减”工作幼儿园阶段的重要任务。 （二）建立育人目标，推进课程研究 在 “传统文化课程研究” 中围绕 “传承什么?” 进行思考, “社会主义核心价值观”承载着中华民族的精神追求，它从 国家、社会、公民三个层面分别阐述了价值目标、取向和准 则。这也为“中华优秀传统文化课程研究”育人目标指明了 方向。以“社会主义核心价值观”为指引，站在终身教育发 展的高度上建立了“生命自觉”育人目标体系，以修身、处 世、爱国三维度进行建构。修身维度注重人格修养教育，引 发幼儿关注自己；处世维度注重社会关爱教育，引发幼儿关 注他人；爱国维度注重家国情怀教育，引发幼儿关注社会和 国家（具体见下表）。 198 传统文化教育目标 “生命自觉” 三维度 修身 与自己的关系 处世 与他人的关系 爱国 与国家的关系 教育目标 人格修养教育 社会关爱教育 家国情怀教育 育人目标具体内涵 基于： 社会主义核心价值观 宣扬本土文化 幼儿园和合文化 身心健康、坚强乐观、 举止有礼、诚信自省、 自信自主、勤奋好学、 良好习惯…… 礼貌待人、善良乐群、 尊重他人、乐善好助、 孝老爱亲、包容理解、 勇于承担…… 遵守规则、爱惜物品、 节约资源、家庭团圆、 民族认同、热爱祖国、 爱好和平…… 在实施传统文化课程中，我们引导教师通过聚焦性综合 主题活动 “大处着眼” ， 深度挖掘传统文化中优秀精神价值， 渗透传统文化课题育人目标。以“霜降摘柿子”主题价值分 析为例，可以看出地聚焦性的综合主题活动中教师能够更好 地向深层挖掘其中蕴含的传统文化价值，将育人目标进行落 实。 二、构建育人途径，“以文化人”避免小学化倾向 幼儿园教育小学化现象一直是国家重点治理的内容，解 决“小学化”问题是提升幼儿园教育质量、落实“双减”政 策的重要实施路径。我园注重从综合主题活动、区域游戏活 动、一日生活活动中的形式入手，遵循幼儿直接感知、实际 操作、亲身体验的学习方式，以“支持幼儿有意义的学习过 程”为核心，通过产生兴趣、主动体验、深度探究、分享合 作、 联想创意五步路径， 将中华优秀传统文化融入活动主题， 让孩子在游戏与真实生活中培养学习品质、建构关键经验、 浸润传统文化，从而也避免了幼儿园教育小学化现象的发生。 （一）坚持合力育人，打造传统文化和谐生态 在课程建设中，坚持“合力育人”，通过携手家长、联通 社区，将孩子的发展置身于其生活的社会生态系统环境，构 建起教育的命运共同体，带给幼儿互助互爱、安定有序的精 神体验。如非遗传承人进幼儿园活动，一方面，把“文化” 变成实物，把看不见摸不着的“传统文化”转变成看得到摸 得着的“载体”带到幼儿园。孩子能通过学习，学到具体的 技艺，体会在学习记忆的过程中品格的养成，同时，幼儿的 199 学习品质能得以涵养， 在和传承人学习、 体验项目的过程中， 会产生好奇和兴趣，要坚持，要专注，要学习解决困难，要 敢于探究和尝试，要乐于想象与创造…… （二）打造四季主题，为课程融入传统文化的内容 三教寺幼儿园课程以优秀传统文化资源为载体，适宜幼 儿年龄特点与学习方式，尊重自然规律，依托社区资源，借 力家长资源，融合节日活动、节气特点，逐步形成了中华优 秀传统文化四季主题课程。课程内容以“春夏秋冬”四季为 线索，依据时节变化，从幼儿真实可感的生活中选取主题活 动的元素，围绕传统文化中与幼儿生活联系密切的两大主题 “节气、节日”，以民间游戏、民间习俗、民间文学、传统 艺术、传统美食为具体内容，涵盖饮食起居、物候特征、地 域特色和文化象征四个方面， 以 “春耕种植” “夏长纳凉” 、 “秋收团圆”“冬藏祈福”为每个季节文化象征。挖掘传统 习俗蕴含的积极教育价值， 促进幼儿在语言表达、 社会交往、 科学素养、创造合作、艺术审美等多方面的发展。在实施的 过程中，我园形成了园级庆典活动和班级主题活动两大路径。 两条线索互为补充和支持，又有各自具体的实施方法。园级 的庆典活动中有环境浸润、游园庙会、非遗传承和非凡舞台 四大实施方法，为幼儿提供具有仪式感、参与性、互动性、 体验性的活动氛围。如秋月节的粗粮美食品尝会、冬雪季的 太狮舞和冬奥项目体验。班级主题活动的推进又包含了游戏 活动、教学活动、生活活动和家园共育等具体途径，让儿童 在每日的一粥一饭、四时的一草一木中，感受“天人合一” 200 的哲学。 三、打造育人生态，幼小协同做好入学准备 “双减”政策中特别强调了要规范校外培训机构行为， 其目的在于扭转家长对于课外培训班的片面认知，激发学校 内部活力，打造家校携手互相信任的良好生态，解决家长迫 切关注的教育问题。在幼儿园阶段同样存在此类问题，有不 少家长忽视幼儿认知发展特点与教育规律，在不了解幼儿园 与小学真实差异和科学衔接方式的情况下，盲目依赖“幼小 衔接”课外机构，对幼儿园、家庭和社会的教育生态造成了 不影响。 为改变这一现状， 帮助家长理解教育规律， 助力幼儿科 学做好入学准备，我园在“和合文化”的引领下，充分发挥 自身优势，与集团小学充分开展合作，做好贯通培养，与小 201 学展开双向互动的幼小衔接工作， 将幼小衔接从 “幼儿园向小 学的单向靠拢”走向“幼小无限交流”的双向对接。 （一）探索实施机制，保障衔接工作力度 幼小合作的关键在于合作机制的探索，具体可以概括为 幼小之间的深度合作和家园之间的携手共育。 幼小之间的深 度合作包括幼儿和学生间的共同活动、 幼儿老师和小学老师 之间的互相了解、 联合教研。 幼小衔接家园共育机制创新包 括大班“幼小衔接”专题家长会、线上互动、合力小课堂微 课等方式，从理念更新、方法传授、实施监督等方面，确保 幼小衔接工作实施的效果。 （二）建构幼小双向互动模式，推进科学衔接 在“双减”背景下，我们统筹传统文化课程做好幼小科 学衔接。 以传统文化课程完成幼儿的一般能力准备， 以幼儿 园和小学双向合作完成专门准备。 一般能力准备是在传统文 化课程过程中，涵盖幼小衔接日常课程内容，包含生活、体 智能、社会性、语言、认知、学习方式六大方面目标和内容 的融入，为老师们日常开展教学明确方向、提供依据。同时 在日常的贯通培养过程中开展幼小一贯制过渡活动， 专门从 幼儿进入大班起， 就将大班阶段划分为四个小阶段， 每个阶 段设置幼小衔接的小目标， 旨在为幼儿减缓入学准备的坡度， 搭建小台阶， 让幼儿在和小学生朋友的互动中， 愉快地为步 入小学做好心理上的准备。 专门准备活动分为“学习适应” “环境适应” “生活适 应”“人际适应”等四个方面，活动形式采用年级组活动、 202 班级集体教学、 班级区域游戏三个形式完成。 在专门准备活 动中， 以年级组开展的专门活动为重点， 与小学教师共同制 定幼小衔接活动方案， 努力将幼儿园单方的衔接工作向幼小 双方共同协作开展， 打破了原有的衔方模式， 取得了一定的 成效。 幼小双方协同活动方案图 “双减”政策对学校教育教学改革具有非常的意义，充 分发挥学校教书育人的主体功能， 强化学校教育的主阵地作 用也是各教育阶段的责任与担当， 积极探索管理策略， 落实 “双减”政策,也是教育管理者义不容辞的责任。 203 【成都师范附属小学】 自主共享 个性飞扬 ——“1+X”课后服务模式助力“双减”政策落地 成都师范附属小学 为解决 “三点半” 难题， 成都师范附属小学于2020年4月 起，积极探索课后服务方式。“双减”政策落地，成都市作 为全国九大试点城市之一，迅速印发《关于深化中小学课后 服务提高课后服务水平的通知》，以期引导全市中小学深入 贯彻落实“双减”政策。我校立即响应，充分发挥学校育人 主阵地的作用，全面考虑学生学习和成长的需求，把课后服 务作为切实减轻学生学业负担的重要一环，推出了契合国家 要求、 彰显学校特色、 满足学生个性化发展需要的 “5+2” “1+X” 的课后服务模式。 一、充分研判 契合需求 为了保障课后服务顺利进行，学校进行了课后服务参与 意愿和需求的调查。 调查结果显示， 90%以上的学生愿意参加 课后服务，并期望在完成当天作业之后，学生能有兴趣拓展 类的课程可以参与。基于此，学校在“双减”工作部署时就 做了充分的准备，帮助全体教师理清认识，达成共识，全员 参与课后服务。同时结合家长、学生的需求和老师的优长， 以年级组、备课组为单位，双线并进开发课程内容，满足学 生发展个性化需求。 204 二、“1+X”课后服务模式革新 充分发挥课后服务对学生自我管理、时间规划、兴趣培 育的重要作用，立足于满足学生多样化学习和发展需求，丰 富课后服务内涵， 提升课后服务质量。 学校采取 “5+2” “1+X” 模式，即每周5天，平均每天2小时，1小时自主课程加上1小 时的X共享课程。 （一）自主作业 学习增效 第1个小时为自主课程， 在老师的指导下， 三至六年级学 生完成当天的作业，一、二年级学生则进行一些校内巩固练 习、课外阅读或体育活动。 1.优化单元作业设计，助力自主学习 为了提升学生的学习效率，学校高度重视作业在学生自 主学习中的积极功能，以单元作业为依托，实现从知识巩固 到思维进阶的转变。 首先引入设计工具。去年三月，申报成为成都市国家优 秀教学成果的推广应用示范校，借鉴作业设计工具，增强单 元作业的关联性、丰富作业类型。 其次注重思维进阶。作业的本质是学生自主学习的过程。 基于布卢姆的教育目标分类学，我校从三个方面进行了拓展 创新，以此激发学生学习的自主性，促进学生思维品质的发 展。首先以学生为主体，注重学生对于学习过程的计划、监 控与反思；其次注重长周期的实践性作业设计，关联学生的 学习与生活，指导学生学以致用；再次注重思维进阶。 再次整合课内外作业。教师设计作业时，将课内外作业 充分整合，并邀请学生依据自己对教学目标和内容的理解， 205 与教师共同制定作业类型、作业内容、完成方式等。 四是关照学生差异。从课堂学习实际出发，设计有差异 性的作业供学生自主选择，即70%的A类基础题和20%的B类能 力提升题，学生必须完成；10%的C类拓展挑战题，学生可以 自主选择完成与否。针对表现进步大、积极思考交流、作业 准确率高的学生，派发“作业免写卡”，可免写一次书面作 业，学期内有效，作业免写卡推行后，深受学生喜爱。 2.落实作业每日公示，提升管理效能 实践行动中，学校在原有作业管理基础上与时俱进，建 立作业公示制度， 保障常态推进。 首先， 各备课组开展研讨、 集体备课，统一单元作业设计。其次，配备“班级作业公示 栏”，做好作业每日公示。第三，班主任每日统计学生完成 各学科作业的预计时长，统筹安排，并提醒作业量过多的教 师，合理调控，不给学生造成过重作业负担。 （二）内容共享 激趣培优 第2个小时为“X”共享课程，根据学生需求和教师特长， 设置音乐舞蹈、 体育竞技、 手工折纸、 棋类学习、 思维训练、 心理课堂、朗诵辩论等多种多样的内容。共享课程分为兴趣 培养和特长发展两类。 1.兴趣培养课 兴趣培养课由教师走班，结合自身优长轮流为各班提供 多样化课程，其种类多，旨在为学生拓宽眼界，拓展兴趣。 以一年级组的兴趣培养课程“我们的身体”为例。对于 一年级学生来说，研究自己的身体和器官是最直观的，同时 能引发他们对自我深层次探究的兴趣。一年级组全体教师在 206 确定了项目的主题内涵后，设计出16个课时的兴趣培养课教 学课程。 表1 “我们的身体”兴趣培养课程 序 号 主题 主要目的 1 提出问题形 成思维导图 在游戏中学习， 了解身体的构造， 完成思维导图； 学生提出关于身体的问题。 2 网络搜索与 语音询问 初步懂得问题的简单分类；让学生了解一些解决 问题的基本途径， 初步了解网络搜索身体的步骤。 3 认识我们的 身体 学生能够独立的指出大部分身体器官的位置；简 单描述出主要器官的主要作用；初步养成用读书 的方式来解决问题的意识；初步养成爱护身体， 保持健康的良好意识。 4 绘本分享 了解身体 学生在家长的支持下能够独立地进行绘本阅读； 养成用读书的方式来解决问题的意识。 5 our bodies 学习各种身体部位的单词并比本掌握各个单词的 发音；培养学生热爱自己的身体，在日常生活中 讲卫生，爱干净，同时增强学生的责任心和自信 心。 6 身体比较与 测量 让学生能观察自己和他人的身体特征；让学生能 了解引起高、 矮、 胖、 瘦的原因； 培养学生关注身 体和健康的意识， 培养正确的生活习惯。 7 不一样的我 们 学生在家长的支持下能够独立地进行绘本阅读； 主动养成用读书的方式来解决问题的意识；通过 207 8 手形的联想 9 为什么有人 会戴眼镜 分享， 让学生学会提问分享获得问题答案的方法， 培养学生理解、表达、倾听等的良好学习意识。 感知手形的变化与组合，结合身边熟悉的事物进 行随形想象， 表现出有趣的画面； 通过作品欣赏、 课堂游戏、体验探究、师生演示等多样的教学方 法；培养学生的观察能力、形象思维能力和想象 力。 学生能够意识到眼睛的重要性； 通过讨论和学习， 能够在日常生活中正确用眼，学会保护眼睛。 10 体育与健康 了解运动损伤的原因；掌握预防运动损伤的主要 措施；认识体育运动中预防运动损伤的重要性， 增强保护身体的意识和责任感。 11 别伤着自己 让学生认识到我们身边有哪些安全隐患；使学生 明白危险是怎么发生的；让学生明白遇到了危险 应该怎么做。 12 牙齿为什么 会掉？ 了解和正确对待换牙现象；知道牙齿的重要性、 如何保护牙齿；积极思考并能大胆提问。 13 怎么长更高 了解各种食物所含的营养；了解我们的身体健康 和生长与营养和锻炼的关系；体验自己健康成长 的快乐。 14 我们的身体 从哪儿来？ 了解自己由爸爸的精子和妈妈的卵子结合并妈妈 肚子里孕育而来的；发现自己出生是因为战胜了 其他竞争者，是唯一的冠军，树立自信；培养学 生对父母给予自己生命和身体的感恩之心。 208 15 身体里面的 洞 认识身体上的“洞洞”，了解有关“洞洞”的作 用； 自主阅读绘本， 对身体的 “洞” 感到好奇， 大 胆地猜测想象并能清楚地表达；知道身体里有些 “洞洞”要藏起来，懂得如何保护。 16 戏剧表演 通过表演的形式来创造性地展示对于身体的认识 和理解。 2.特长发展课 特长发展课的课程重在培优，着力于加强各类艺体学科 代表队的日常训练， 培养艺术体育的优秀苗子,包括合唱、 木 笛、管乐、篮球、足球、田径、国际象棋、绘画、编织、剪 纸等、空竹、语言表演、劳动手工、团辅游戏等近30余个门 类。特长发展课打通了优长学生个性发展的渠道，为学生提 供专业化发展的机会和平台。两类课程相辅相成，以“服务” 为核心，凸显自愿性、选择性、个性化、延伸性，关注学生 的参与体验和过程性成长。学校实行“学校行政每课巡视、 教师每日反馈、师生每周总结”的管理方案，不断优化课后 服务管理、提升课后服务质量。 三、即时反馈 家校共育 建立多主体多方式的评价方式，对课后服务的质量进行 反馈总结。一方面，教师会在班级群里对每日课程学习情况 做出反馈，让家长知晓学生在校的课后服务学习情况，及时 帮助学生进行状态调整，力图形成家校教育的合力，提升课 后服务质量。另一方面，每期课后服务结束后，学生会对任 课教师进行星级评价，评选出他们心目中的“星级教师”。 同时，学校还会在相应的学科设置节庆活动的相关内容，例 209 如美术节中的美术作品展览会、音乐节的合唱比赛、体育节 的各项赛事，都为学生提供了展示自己的机会，这也是对学 生兴趣类学习内容的一次评价。 四、保障跟进 提升服务质量 为了确保课后服务高质量实施，学校在师资和经费以及 管理制度方面提供充分保障。一是师资保障。组建学校课后 服务教师资源库，合理分工，确保全校教师100%参与延时服 务。邀请有专业特长的家长资源、有正规资质的非学科类机 构进校参与部分共享课程。二是经费保障。课后服务经费专 款专用，确保教师参加课后服务的待遇。充分调动教师参与 课后服务的积极性，鼓励教师积极为高质量的课后服务献计 献策，将教师参与课后服务与评优选模、绩效考核等工作有 机结合起来。三是制度保障。建立作业公示制度、课后服务 反馈评价制度等，建立服务工作长效机制。 实施“双减”政策，既要做好减轻学生过重作业和校外 培训负担的“减法”，又要做好促进儿童全面发展、个性发 展的“加法”。课后服务，是校内教育教学的有效延伸，是 满足学生多样化学习与发展需要的重要抓手。提升课后服务 质量，为学生营造健康良好的教育生态，就是实实在在地为 民解忧，满足民众对美好教育的需求，促进学生综合素质提 升和全面健康发展。 210 【北京第一师范学校附属小学】 促进深度学习的实践性作业优化设计 ——以“制作数字故事”为例 北京第一师范学校附属小学 刁善玉思佳 近期，随着《关于进一步减轻义务教育阶段学生作业负 担和校外培训负担的意见》等相关文件的出台，多样化作业 成为研究的焦点。 北京第一师范学校附属小学在响应国家“双减”政策的 工作中，本着培养创新型、复合型人才的原则，在实际教学 中，以实践性作业——“制作数字故事”作为学校“双减” 背景下作业设计推广案例，展开实践研究。 一、制作数字故事的具体实施 数字故事是指以数字技术为主要载体和制作手段的一 种新型教育工具，可以容纳图像、文字、音频、视频等多样 化的教学素材，具有高度直观性、生动性、形象性及趣味性 特征。学生通过记录自己的研究过程，重新建构对知识、对 自我、对生活的认知，并在信息技术的支持下，以故事的形 式重新组合素材，形成完整视频。 下面以六年级数学“位置与方向”单元的数字故事作业 为例，介绍具体实施： 1.前期准备 在前期准备阶段，教师首先要让学生了解什么是数字故 211 事，并提出制作数字故事的要求。 （1）联想相关知识，提出真问题。 （2）用各种方式记录研究的过程和探究的方法，关 注组内遇到困难或争议时，克服和解决问题的过程。 （3）最终以小视频的形式，完成本次研究。内容要 体现知识的形成和应用过程，要体现故事性。 在研究之前引导学生发现生活中和“位置与方向”有关 的丰富素材，如环球影城游览图、北京动物园游览图、校园 平面图等；让学生思考“作为小导游，怎样清晰地向大家介 绍游览路线？”；引导学生将生活素材与课本所学内容建立 联系，产生应用知识解决问题的欲望，让此次的实践作业真 正起到解决真实问题的作用。 2.中期探究 在探究环节中， 教师首先要引导学生制作 “位置与方向” 研究脚本。 脚本包含组员角色分工 （制定计划、 多媒体制作、 录制、画图等工作）。探究过程中，教师应在重点环节，提 供探究支架，提出有助于学生思维走向深入的核心问题。比 如，要想准确描述一个地点的位置，需要从哪些方面考虑问 题呢？在解决问题中，你需要用到什么测量工具呢？在研究 的过程中，你们遇到什么问题了？怎么解决的？ 3.后期分享 小组完成数字故事以后，在全班进行展示。分享过程， 观看者可以向作品小组提出知识性问题，比如如何能够准确 测量出角度？量角器怎样摆放？这类问题直指学生空间观 念的培养，也是“位置与方向”这部分内容的难点。教师也 212 可以从学科之外的角度，引导学生从社会参与、合作沟通、 技术应用等层面进行评价，给予作品制作组后期反思指导建 议。 二、实践性作业特征解析 基于开展数字故事作业的经验，学校提炼了实践性作业 的特征，并结合具体案例，进行解析： 1.创设真实问题情景 注重学习体验 杜威的作业观指出，作业是儿童的一种活动方式，通过 这种活动儿童实现了学校生活与社会生活的联结。学生直接 接触现实世界中的相关素材，经历“数学化”的过程，感受 数学的应用价值。 2.经历知识形成过程 聚焦高阶思维 促进深度学习的实践性作业要引导学生经历知识的形 成过程，更加注重在解决问题中高阶思维的培养。比如，如 何把生活问题转化为数学问题？面对复杂问题的不确定性， 如何梳理出相对较为理想的解决路径？如何把各种零散的 素材以故事的形式，向观众阐述清楚？遇到失败和错误尝试 后，如何进行调整，获取新的经验？这些问题都属于理解、 应用、迁移、反思、创造类型的问题，围绕这些问题进行思 考，更加凸显学生的高阶思维。 3.建构跨学科思维模式 形成综合素养 制作数字故事的实践性作业将静态的课本知识变成了 一种学习活动。比如学生在描述两个地点的相对位置时，不 单是解决书本上两个点相对位置的问题，而是还原其知识本 身的价值，以问题解决的形式呈现作业题目。而学生在解决 213 现实问题的作业时，一定是需要跨学科知识的，或许还会涉 及到一些生活经验、个人感悟等综合素养。具有复杂性、不 确定性、综合性特点的数字故事作业，正是符合跨学科解决 问题的思维模式，学生需要有清晰地解决问题的整体规划思 路，用以保证作业的顺利完成；还要有对学科知识方向、角 度、距离的灵活应用，用以能够正确地解决问题；以及较强 的绘图功底、丰富的语言表达能力，助力数字故事更流畅、 生动、严谨、美观。一个具有长周期的实践性作业，能够在 不同的环节培养学生多方面的能力。学生在整个实践作业中， 综合素养得以提升。 4.营造恰当育人环境 开展多元评价 习近平总书记强调，要努力形成有利于创新人才成长的 育人环境。在进行实践性作业的过程中，有的小组同学会因 为商讨“谁来当组长？”这个问题，而发生争执；有的小组 同学会因为工作量分配不均而使作业处在停滞状态；有的小 组会探讨一个具有多种答案的问题，而出现意见不统一的现 象。通过处理各种人际交往过程中可能遇到的问题，学生积 累了解决问题的活动经验，并利用作业评价功能，进行自我 反思。 而创造多元的评价主体 （自我、 组员、 组间、 老师等） ， 能够丰富评价的角度与内容，尤其是关于思维品质的过程性 评价更能在实践性作业中发挥必要的价值，帮助学生形成正 确的价值观。 三、实践性作业是促进“双减”政策与“深度学习” 理念落地的有效途径 1.实践性作业促进“双减”政策的有效落实 214 长期以来，中小学作业被赋予了更多功利性色彩，大部 分纸笔作业都侧重知识点的强化与训练，以便更好地适应不 同阶段的应试测评。即便是开展“综合与实践”类的作业， 也常常因为教学评价不完善而流于形式，被教师所忽视。正 是对作业认识的偏差与价值判断的局限性，导致作业成为学 生、教师、家长之间的矛盾焦点。 面对这样的复杂社会问题，或许以开展实践性作业为突 破口，可以有效缓解学生过重的作业负担。学校的制作数字 故事实践性作业不同于纸笔作业，它将知识融入问题解决中， 更加侧重学生亲身经历、动手实践、交流合作，从而获得一 些经验， 形成相应的学科知识和分析问题、 解决问题的能力。 深入挖掘实践性作业的价值功能，有利于缓解义务教育阶段 学生的课业负担，进而为达到减负增效的理想状态提供实践 观照。 2.实践性作业成为课外延伸的深度学习 随着教育改革的不断深化，深度学习理念成为落实核心 素养的关键依托。深度学习，它强调学生围绕具有挑战性的 学习主题，获得有意义的学习过程。而这过程不仅仅只停留 在课堂的40分钟内，40分钟以外的作业同样是必不可少的学 习活动。因此，深度学习既是课堂教学的理念支撑，同样也 是实践性作业的价值追求。教师在设计主题式实践性作业过 程中，可以参考深度学习的内涵，真正使得实践性作业成为 课外延伸的深度学习。 四、实践性作业未来的改进方向 教师在观察和指导作业过程中，也发现了一些问题，针 215 对问题提出了应对的调控策略。 1.信息技术 学生在完成实践性作业的同时，会借助各种智能工具来 服务实践性作业的开展。但学生技术水平参差不齐，完成作 业效果差距颇大。在作业实践中，学校应提供硬件保障，教 师应统一对学生进行基本技术培训，加强学生信息素养的培 养。 2.评价内容 在实践性作业的总结评价环节，评价主体主要以作业视 频的整体效果好坏为判断依据，主观性较强。今后再开展实 践性作业，应制定实践性作业评价量规，更加科学理性地围 绕实践性作业特征与育人目标展开评价。 总之，“双减”的落实并非意味着教育质量的降低。相 反，更要借助“双减”政策，重新思索如何在轻松的氛围、 有限的时间、技术的支持下继续开展深度学习活动，创建符 合学校办学理念的快乐教育生态环境。 216 【北京光明小学】 北京光明小学课后服务工作的研究性实践 ——学生兴趣·爱好·特长拓展课程 北京光明小学 2021年7月， 中共中央办公厅、 国务院办公厅印发 《关于 进一步减轻义务教育阶段学生作业负担和校外培训负担的 意见》。8月，中共北京市委办公厅、北京市人民政府办公厅 印发《北京市关于进一步减轻义务教育阶段学生作业负担和 校外培训负担的措施》。一系列指导性文件的出台，从有效 减轻学生过重作业负担、提升学校课后服务水平、提升校内 教育教学质量等多个方面进行了工作要求和具体部署，指导 学校有效、高效地落实双减政策，全面提升学校教育教学质 量。 北京光明小学认真学习贯彻落实市区相关文件精神，坚 持以学生发展为本，坚持立德树人，把课后服务作为五育并 举的重要抓手，整体规划课程内容，满足学生多样化需求， 拓宽学生成长途径。 一、充分调研，了解需求 学校充分调研学生和家庭的真实需求，由学科教师负责 研发课程，满足学生多样化发展需要。以往每个学年学生参 与课后服务的比例都高达85%以上，刚刚过去的学年由于疫 情带来的一些影响，家长考虑到班级间学生的交叉接触而选 择放学后直接离校。即便如此，学生参与课后服务的比例也 217 在70%以上。但仍存在有的家长因为工作等的原因无法在 15:30或者16:30到学校接走学生，不少家庭对课后服务的需 求呈现出刚性需求。学校于9月3日召开新学期线上家长会， 向家长就中办国办以及北京市下发的“双减”文件精神进行 了宣传并就学校双减工作的具体举措以及家长普遍关心的 课后服务工作进行了介绍和说明。在家长会做好宣传、各校 区团队做好调研、与有个性化需求的学生家长做好对接后， 在9月6日—17日两周的运行中，全校参与课后服务的学生占 比接近93%，特别是高学段学生放学后留在学校参与课后服 务和综合素质拓展活动的人数较往年有较大提升。 二、五育并举，统筹规划 新学年的课后服务工作的设计筹备从刚一放暑假就已 经开始了。各校区团队认真阅读了上学期课后服务工作满意 度调研的数据，对上学期课后服务工作中开设的各类课程和 托管自习等工作的情况进行了分析，对新学期相关工作有了 初步计划，特别是初选了部分课后兴趣爱好类的课程。随着 市区双减政策的落地，以及学校双减工作整体推进方案的研 究与制定，课后服务工作的校本方案也随之进行多次研讨与 修改，在做好教育教学活动和教师资源的调研及统筹的基础 上，逐步形成以学校教师开发课程为主、社会资源提供课程 为辅的课后服务课程整体架构。 （一）统筹安排学生在校时间。课后服务工作覆盖到周 一到周五，每天15:30—17:30，每天的课后服务时段分两个 阶段进行整体规划和系统设计。第一阶段完成体育锻炼，保 障学生每日1小时体育锻炼时间；第二阶段开展课业辅导和 218 综合素质拓展类活动，结束时间原则上不早于17：30。两个 阶段相互衔接，满足学生多样化需求。 （二）统筹安排课后服务内容。每天的课后服务的一部 分是体育锻炼和课业辅导，当日没有体育课的班级由体育老 师统一组织学生进行体育锻炼，保障学生每日1小时体育锻 炼时间，其他班级由任课老师指导学生完成当日学科书面作 业。另一部分是关注综合素质拓展与特需个性辅导的内容。 综合素质拓展活动有校内教师自主开发的兴趣爱好特长类 课程，也有部分社会资源单位提供课程作为补充，更有针对 个性化特需学生开展的学业辅导、“小壮壮”训练等。这部 分基于学生学段特点、围绕德智体美劳五育内容开展。学校 根据学生学段特征， 分别组建了兴趣组、 爱好队、 特长团 （详 见下表）。 兴趣班 爱好队 特长团 德 光明少年成长营 快乐成长法律相伴 等 光明红领巾通讯团 等 光明童心鼓号团等 智 语文朗诵社、英文歌 曲赏析、少儿影视欣 赏、海洋探秘等 数学思维游戏、数 独、英语阅读与戏剧 表演、数学小课题研 究、纸折飞机等 电子技术、机器人编 程、人工智能、英语 戏剧社等 体 田径、武术、啦啦操 （基础队）、体能训 练等 田径、 武术、 啦啦操、 篮球 （梯队） 、 足球、 跆拳道等 田径、 武术、 啦啦操、 篮球（社团）等 219 美 弓弦乐团（基础班） 合唱（基础班） 水墨基础班、泥塑、 彩绘葫芦等 弓弦乐团（梯队班） 合唱（梯队班）硬笔 书法、软笔书法、班 级合唱、儿童版画等 弓弦乐团（二胡团） 合唱（正式团） 水墨社团 剪纸、书法社团等 劳 创意木工坊、竹艺、 编织、微缩景观等 旧衣新生、丝网工艺 等 目前五个团队的课后服务课程中， “德” 类课程7门， “智” 类课程50余门， “体” 课程7个项目， “美” 课程50余门， “劳” 课程10余种。此外还特别开设了学科类特需辅导，如数学基 础辅导、英语特需辅导等，为学科学习有个性需求的同学提 供针对性帮助和指导，解决部分学生的学科学习困难。 三、自主选择，满足需求 学校从9月1日起开始课后服务工作，9月8日开放线上选 课平台，将不同团队的围绕德智体美劳的兴趣、爱好、特长 课程的课程介绍、上课时间、授课教师、班级人数等基本信 息在平台上呈现出来，由学生及家长根据自己的时间及需求 进行选择。在经过初选和二次补选，以及团队进行选课数据 汇总后，9月16日全面启动包含各类课程活动的课后服务工 作。 截止到9月底的试运行期内，参与学生人数占全校学生 数接近93%， 满足了确有需求的学生和家长。 有些综合素质拓 展类课程受同学们的欢迎，如，硬笔书法、篆刻、田径队、 合唱队等都是报名人数爆满，各学科、团队也是积极响应， 针对报名学生的具体情况进行平行班的增加，或者开设梯队 220 班，以满足学生的课程需求。再有，针对特需学生开设的学 科学业辅导，也是解决了一些学科学习上有困难的同学的实 际需求，从而减轻了学生及家长的焦虑情绪，打造和谐的亲 子关系和家校协作。 本学期，全校2824名学生，参加课后服务2614名学生， 占学生总数92.47%，1480名学生五天均参加课后服务，其中 633名学生每天均参加2小时。 本学期教师在编人数218人， 在 岗209人， 均参与到课后服务各项工作中， 其中市区骨干教师 26人， 含特级教师1人。 本学期在课后服务中开设综合素质拓 展类课程 （班） 共计259个， 其中学校教师自主开发及组织实 施的课程为162门，占总课程数的62.5%。 四、评价反馈，提高质量 （一）成果展示。学校每学期结束前都采用作品成果展 示的方式，把同学们的学习收获展出来与更多的同学和教师 进行分享。同学们最喜欢在作品展示空间驻足，认真欣赏和 观看同学们的作品的同时，也在认真学习他人的学习成果。 （二）调研分析。学校每学期末都将针对课后服务组织 工作、课程质量面向全体学生和家长开展满意度调研，根据 调研结果进行充分总结，并将师、生、家长对课后服务工作 的意见与建议，作为新一学年的课程设计的重要依据。推动 学校进一步完善课后服务工作机制，优化课后服务工作质量， 用丰富、高效的课程吸引学生，使学生在课余时间里获取最 大化的课程体验和能力提升。 基于学生发展需求和目标的课后服务工作，为双减政策 的扎实落地提供了坚实的保障，为学生在校园生活中的主体 221 性发展创造了条件。课后服务的开展过程有助于师生从另一 个侧面观察和欣赏彼此，不仅仅帮助学生拥有兴趣、爱好、 特长，还为教师提供平台发挥自身的兴趣爱好，激发了教师 参与课程建设的热情，加深对教育教学本质的理解。 222 【重庆市南川区隆化第一小学】 增强作业管理透明度，助力“双减”落地见实效 重庆市南川区隆化第一小学校 庞涌洪 蒋明权 2021年，“双减”政策落地，强化作业管理既是及时回 应社会关切的热点问题所需， 也是落实 “双减” 的应是之举。 对此， 重庆市南川区隆化第一小学校立足校情， 在作业设计、 实施、评价等重点环节增强其“透明度”，优化作业管理， 促进学生全面发展、健康成长，打通减轻学生过重作业负担 的“最后一公里”。 一、设计透明，提升作业质量 作业质量很大程度上取决于设计的质量，而设计质量则 取决于思想的统一、研讨的深度、资源的保障等因素。在实 践中学校通过系列“团队协作制度”推动作业设计质量、作 业品质的提升，增强作业设计的透明度，让每个教师都能站 在“巨人”的肩膀上看到诗和远方。 首先， 统一设计思想。 为了让每个学科能基于课程标准、 学科核心素养、编者意图等开展作业设计，在政策学习的基 础上，先后开展了语文、数学、英语等学科作业系列培训， 市区级教研员、本校学科骨干等分别作为主讲人，开展专业 引领。通过系列培训，教师对五育融合的作业设计理念、作 业的育人价值、作业形式的选择、作业设计明细表的制作、 作业情景生活化等有了准确的把握。 对课时作业、 单元作业、 223 综合作业、假期作业等有了系统的认识。从管理导向到专业 引领，帮助教师渡过了“思想关”，也提升了教师对作业丰 富样态的多维认知与自觉追求。 其次，共建作业内容。在统一设计思想、把握作业设计 原则和方法的基础上，各教研组积极思考、热烈讨论，力求 所设计的作业无论从内容上，还是形式上，既能激发学生作 业兴趣，又能培养学生学科素养和综合能力的创新性，还能 充分体现“教—学—评的一致性”，实现作业与课堂教学的 有效衔接。因此，各教研组结合学校课堂深度变革中学习工 具“三单”（《引学单》《问学单》《拓学单》）的研发为 主要载体，自主设计符合校情和本班学情，在课前、课中、 课后使用的系统化作业，强化作业设计质量，把握好作业设 计“质量关”。例如《引学单》主要围绕 “我能阅读解答课 文内容或课本例题吗？我的收获是什么？我有什么疑问”等 4个主问题设计预习作业， 培养学生自主学习力。 《问学单》 主要在引学单真实呈现的学情基础上，围绕教学重难点而设 计，课中紧扣“问学单”中的问题主动学习、小组研讨、自 信展讲， 激发了兴趣， 夯实了基础， 突破了教学重难点。 《拓 学单》 则是结合课堂学情和学生学习规律， 设计分层、 弹性、 个性化作业，确保学生在课后服务时间内完成书面作业的同 时，跨学科作业、实践性作业和弹性作业应运而生。例如去 体育馆“走一走、跑一跑”深化对“千米”的认识；读、背、 书爱国主题的诗文、阅读中外名间故事；制作立方体、和家 人一起拼拼七巧板； 用一个月的时间观察身边的家人、 同学、 朋友，为通过具体事例描写人物作准备的长时作业等。 224 最后，共享设计成果。为了让之前、当下的作业设计服 务于作业管理，学校分别构建了线上线下两个平台，促进作 业设计成果的共享。 线上主要是各年级、 学科组在 “云空间” 建立了专门的作业文件夹，本校教师均可在这里找到历届优 秀作业设计和其它教师搜集的作业资源；线下主要是学校为 每个教研组专门订制了“作业存放台”，每个教师将自己使 用的“三单”放在里面，供其他教师结合本班学情自选后修 改所用，每天作业内容向组内人人开放。 二、实施透明，引导作业应用 “设计好”完成了作业管理的第一步，“实施好”则是 重要的第二步。在实践过程中，各教研组除了上交重在涵盖 “作业内容、作业形式、作业量、作业结构”等的《单元作 业设计规划表》并在校内线上公示外，主要还通过“当天公 示制度”“搭台展示亮点”等措施增强作业实施的透明度。 当天公示制度即教师将每天布置的作业内容、形式在黑 板上手写后拍照发到年级微信群里，让群内年级全体教师和 分管年级领导知晓，以此促进教师加强作业指导，年级分管 领导不定时通过访谈等方式督查作业公示的真实性。此举旨 在保证国家规定的 “作业红线” 不突破， 并对是否遵循了 “重 质轻量”“重选择轻统一”等学校制定的作业设计原则进行 有效的督查。分管年级领导在微信群里不时结合作业中的例 子引导教师“站在准确把握教育改革发展面临的新形势新任 务”的高度上认识作业管理，让教师知晓它绝不是教育中的 “小事”，是关系学生身心健康发展的大事，是新形势下教 育的“刚需”。 225 作业实施透明的目的不仅是发现问题，更为重要的是发 现“亮点”，让教师们在作业实施过程中的指导、批改、反 馈等环节的亮点充分“暴露”。在教研组长、年级组长等各 类工作会上，引导大家积极发现教师作业中的“亮点”，促 进优秀作业的交流，并且为这些“亮点”作业提供了诸多展 示平台； 组内研讨中专题交流、 学校 “阳光论坛” 汇报展示、 校刊“优秀作业”汇编等。 三、评价透明，助推作业增效 作业管理优化的受众是学生， 实施的关键在教师。 因此， 学校明确各学科教师是学生作业管理的第一责任人，教研组 长、年级组长、教科室学科主任、年级分管领导分别是作业 管理的共同责任人。学校在以往的作业管理中更注重第一责 任人的作用，却忽视了其他责任主体参与评价，除了在实施 中卷入各层主体责任人外，还通过“作业展评制度”“评价 量规融入作业过程”等增强作业评价的透明度。 作业展评制度即分别开展班级、年级、全校三个层次的 作业展评活动， 让每个学科作业在班级学科间、 年级班级间、 学校年级间透明。学校在开学初便制定了第3、9、18周分别 开展三次作业展评活动。目前通过“4Q预习”“引学单”的 展评活动，让教师从被动地接受检查走向了主动地展示，有 效地促进了相互间对作业内容、形式、批改的了解。作业展 评的过程更是促进了学科大教研的自然展开，实现了一至六 年级学情的深度了解。以班级作业展评为例，学生之间相互 学习自不必说，更为重要的是让同班级不同学科教师看到了 同一学生的差异。教师不同的批改方式对学生作业质量的不 226 同效果也启发了同班教师的思考，促进了更多教师让作业批 改从单一的对错判定走向了心灵对话，提升了学生学习动力。 同时，也让班级内教师间都能站在学生立场统筹语数英学科 书面作业时间。 评价量规融入作业过程即在布置作业的同时，向学生发 布本项作业的评价量规。例如学校一位六年级教师在教学完 第二单元《七律·长征》《狼牙山五壮士》等课文后，围绕 “重温革命岁月，把历史的声音留在心里”，设计了这一道 综合性作业：“选择最感兴趣的革命故事，自由选择同学组 成剧组， 编排一个情景剧， 第9周在年级魅力展示时表演。 ” 在布置的同时， 提供了如下 《革命故事情景剧表演评价量规》 （具体见表一）： 实践证明，它不仅引导学生自主学习，自主评价，对照 量规，实现自我提升；也促进家校结合，引导家长在业余时 表一：革命故事情景剧表演评价量规 评 价 要 素 等级描述 优秀 良好 合格 内 容 表演的故事 最能体现革命烈 士的特点。 表演的是 革命烈士的故 事。 表演的不是 革命烈士的故 事。 表 演 能让在场的 所有人听到声音， 且吐字清晰，表演 时动作、情感与声 音等融入一体。 能让在场 的所有人听到 声音且吐字清 晰，但动作和情 感配合不好。 声音能让大 多数同学听到， 缺少动作和情 感。 道 具 有情景需要 的全部道具和音 乐。 有部分情 景的道具和音 乐。 完全没有道 具和音乐。 227 间也能结合各个等级表现性描述，准确评价，不拔高要求； 更为重要的是教师则通过量规的设计，对作业内涵、层次有 了更精准的把握，在设计表现性描述的过程中，为学生指明 了达到目标层级的方向、路径，实现了过程性评价与结果性 评价的有机统一。 作业是推进减轻学生过重作业负担的“最后一公里”。 而打通这“最后一公里”，需要从设计、实施、评价等完善 学校作业管理体系。学校整体引领、明确方向，教师组集体 协作、共建共享，教师勇于挑战、积极探索，为“减负不减 责， 减量不减质” 助力， 让 “轻负担、 高质量” 与学生相伴， 从而提高学校育人水平。 228 【成都市盐道街小学】 将课后服务课程融入学校整体课程体系之中 ——成都市盐道街小学的新探索 成都市盐道街小学 四川省成都市是全国9个 “双减” 试点城市之一， 做好课 后服务是落实“双减”政策的重要措施。成都市盐道街小学 （以下简称“盐小”）深入贯彻中共中央办公厅、国务院办 公厅《关于进一步减轻义务教育阶段学生作业负担和校外培 训负担的意见》及《教育部办公厅关于进一步做好义务教育 课后服务工作的通知》 等相关文件精神， 赓续区域办学初心， 担当高质育人使命。 2021年9月，盐小正式启动课后服务2.0工作，以“立足 整体 着力融合 构建课程”为思路，将课后服务融入到学校 整体课程新体系中进行开发和实施，真正发挥其育人功能和 社会效应。 一、课后服务课程的定位与目标 开展课后服务，是促进学生健康成长、帮助家长解决按 时接送学生困难的重要举措，是进一步增强教育服务能力、 使人民群众具有更多获得感和幸福感的民生工程。盐小课后 服务坚持“立德树人、全面发展、绿色质量”的原则，以融 合育人理念为引领，回应立德树人、全面发展的培养之声， 以“完成作业+提升拓展”为课后服务基本模式，开发了“适 融”课后服务课程。在学校课程目标的整体指引下，将课后 229 服务课程目标设置如下： 1.立道厚德：坚持五育并举，促进学生全面发展，让学 生拥有健康体魄、家国情怀、创新精神和国际视野。 2.基础保质：让大部分学生在校期间高质量完成作业， 培养学生自主规划学习进程的意识与能力。 3.有盐有味：通过具有盐小特色的“提升拓展”路径， 让学生炼一己之能，在项目学习中学会解决实际问题。 二、课后服务课程的结构与内容 学校开发的“适融”课后服务课程包括基础保质服务课 程、拓展扬彩服务课程和实践炼能服务课程，作为学校基础 课程、拓展课程和实践课程的延伸与补充，互相融通，共同 构成支撑盐小学子全面发展与个性发展的“土壤”，同时实 现学校“盐道课程5.0版本”的优化。 图1 盐道课程5.0 版本 板块一：基础保质服务课程着眼于减轻学生课业负担， 由语、数、外教师合理分工，以作业指导、个别辅导、自主 学习、综合阅读和习惯养成等内容为主，巩固、提升学业质 230 量，养成良好的习惯。在浓郁的学习氛围中，引导学生自主 学习、自主规划、自主完成作业。 图2 基础保质服务课程 板块二：拓展扬彩服务课程着眼于学有余力的学生，促 进其个性化、差异化发展，设置了红色基因、劳动教育、体 育健康、 国际理解、 艺术审美、 科学技术等6个领域的内容， 目前共计64项，为满足学生发展需要提供了丰富多彩的选择。 图3 拓展扬彩服务课程 231 板块三：实践炼能服务课程旨在培养全面发展的人，设 置了学科内项目式学习，以及跨学科和超学科项目式学习。 图4 实践炼能服务课程 比如，三年级开展的“一架‘梦’的飞行器”项目，以 “飞行器”为问题导向，语文教师带着孩子们学习飞行器相 关知识并进行主题分组，美术教师对孩子们的设计图提供帮 助；科学老师和孩子们一起准备材料，动手搭建。根据项目 的多学科融合特性，本年级的语文、美术、科学教师共同指 导，以不同学科的相关知识模块为载体，并合理设计在实施 阶段各科知识结合点，同时融合德育、智育、美育和劳育， 促进学生的全面发展。 三、课后服务课程的实施与评价 （一）课程实施特色 开展课后服务要“留人留心”，要让孩子们喜欢课程和 校园，就需要学校不断更新课程，提升课后服务质量，凝炼 课后服务特色。在盐道课程的引领下，适融课后服务课程做 到了“一生一课表”，延续了多种学习方式，并更加强调管 232 理的精细。 1. 一生一课表 盐小为不同学生设计不同的“跑道”，在课后服务期间 致力于打造 “一生一课表” ， 通过学生网上选课、 混龄走班、 教师走班的方式，尽可能满足所有孩子的发展需求。如，拓 展扬彩服务课程需要学生在开学前一周在网上自主选课，课 程内容打破年龄界限，以混龄走班的方式进行；实践炼能服 务课程采取同年级不同学科教师走班的方式开展，教师围绕 同一主题，以跨学科视角来组织学生发现问题、解决问题、 制作作品。因此，即便是同一个班的学生，其课后服务课表 也是各具特色的。 233 2. 多种学习方式 课后服务1.0版本重在基础保质，解决了家长接送孩子 难题和作业辅导问题。 2.0版本则在保质的基础上， 在课程实 施过程中，侧重通过自我指导学习、小组学习、大团体指导 和沉浸式探索等学习方式，最大限度满足孩子的多元学习需 求，让学生在解决实际问题的过程中提升学习能力与团队合 作能力。 3. 精细管理 按照四川省教育厅办公室对进一步做好义务教育课后 服务工作的相关规定，精心制定课后服务实施方案、巡课、 排课等相关制度。管理实施“课课巡、日日结、周周评”。 管理员根据《五维考核表》，每天至少3轮巡视，从5个维度 入手， 做好1日小结， 保障绿色质量。 同时做好课程保障工作。 （二）课程评价 在课后服务课程评价方面，盐小在原有课程评价的基础 上，开发出更聚焦于课后服务课程评价方式。不仅强调评价 主体多元化和评价方式多样化，更关注“以评价促发展”， 使评价的价值回归到对学生五育全面发展、融合发展、整体 发展的关照，强调对教师课程设计的评价和学生五育发展情 况的评价。 1.教师评价 开发“课程星”评价系统，通过信息技术，以课程获得 的“星星数”对其进行等级划分，以优化和改进课程。同时 将“适融”课后服务课程的设计与实施纳入每学期的绩效考 核中并给予专项奖励。 234 2.学生评价 在盐道课程的“五育币”和“成长秀”评价系统中开辟 课后服务课程学生评价板块，应用信息技术手段全面获取评 价数据，建立学生课后服务成长档案，记录学生个人成长轨 迹，生成学生成长数字画像。比如，在评价学生基础保质服 务课程中，以“作业完成态度、作业完成数量”为主要评价 维度；在评价学生实践炼能服务课程中，以“积极参与、主 动思考、合作探究、作品质量”为主要评价维度。 四、课后服务课程的成效与反思 （一）课程实施成效 1. 服务成效 在课后育人过程中，做到了教师参与百分百、家长知晓 百分百、选修自愿百分百、学生愿留尽留百分百，通过分批 235 设定课后服务结束时间，实现了与家长接送时间无缝对接， 解决下午“三点半”问题，解决校内“吃不饱”的现象。 【场景：放学时校门口】 “孩子更开心、 更轻松了！ ” 盐小三年级一班饶若茜的妈 妈开心地告诉值周行政， “ ‘双减’政策自己举双手赞成，很 放心把孩子交给学校，以前放学回家，孩子在小区都找不到 小朋友，现在吃完饭大家都聚到小区里玩了！ ” 她笑着补充道： “孩子现在成长发育需要大量活动时间。 以前大家‘卷’得厉害，孩子不是在补习班，就是在去补习 班的路上，小朋友都没有了玩伴。如今孩子有了更多的个人 时间，亲子关系也在不知不觉中更好了。 ” 2. 育人成效 有盐有味的课后服务课程，培养了学生解决问题的能力， 促进了学生综合素养能力提升。开展课后服务以来，我校学 生个人获奖增加15%，亲子关系改善92%，课后服务参与人数 达90%，通过开展调查问卷了解到家长的满意率达到了98%。 一架 “梦” 的飞行器、 未来城市等项目被央视、 新华社报道。 （二）课程实施反思 课后服务课程实施百日有余，在实施的过程中还存在一 定的问题，比如受学校场地的限制，供社团用的空间不足； 课后服务课程建设还可以更加吸引学生的兴趣等。 育人先育德， 育才先育心， 教育才有厚度、 深度与温度， 才有激情、真情与温情。接下来，成都市盐道街小学还将深 入思考课后服务课程如何与学校既有课程体系相互融通，做 新的体系化思考和建构，以多彩的素拓活动为杠杆，高质量 236 达成“双减”目标，让真情的课后服务适度、适切、适融， 促进学生全面发展。 237 【成都天府新区实验小学】 三“点”赋能 融合育人 四川天府新区实验小学 赵玉如 余悦 侯毅 2021年9月， 中共中央办公厅、 国务院办公厅印发 《关于 进一步减轻义务教育阶段学生作业负担和校外培训负担的 意见》（以下简称《意见》）指出“要坚持以习近平新时代 中国特色社会主义思想为指导，全面贯彻党的教育方针，落 实立德树人根本任务，坚持学生为本、多方联动……强化学 校教育主阵地作用，深化校外培训机构治理，有效缓解家长 焦虑情绪，构建教育良好生态，促进学生全面发展、健康成 长。” 四川天府新区实验小学（以下简称“实小”）依据区域 整体教育发展规划，结合学校现有资源优势，探索高品质、 有内涵、内生型的高质量发展之路。落实“双减”过程中， 实小坚持学生为本、 全面育人， 探索服务 “入心” 、 提质 “精 心”、育人“守心”三条路径，在“双减”下为孩子的成长 赋能。 一、 以细化教学管理为逻辑起点， 牢牢抓住减负提质 的“牛鼻子” （一）减负的本质是提质。学校优化教学方式，强化教 学管理，提升学生在校学习效率，全面提升教学质量。实小 以 “让学校课程贴近儿童需求， 为儿童打通博学雅行的行程” 的全课程理念，以《基于“博雅教育”的学校课程设计与开 238 发研究》专题研究引领学校整体课程架构，以“博雅”课程 为基础，以教研和作业管理为抓手实现提质增效。 （二）备课详实。组织各科教师在开学前统一进行课标 学习和教材解读，明确目标。学期中落实“211”，即：2次 备课(开学前集体备课，课前二次备课)，形成1篇反思、1篇 精品教案。 （三） 教研扎实。开学前， 全校进行全学科教材解读展 示。各年级各学科在全校进行深度的教材解读。学期中，每 周定时举行教研，采取“线下+线上”的模式，教师上现场教 研课，听课教师利用醍莫豆信息技术实时线上评课议课。除 教研课外，以年级组为单位，在每单元起始课前统一进行本 单元的教材解读和单元作业设计优化。 （四）课堂落实。学校利用“常规课+大单元走班课” 相结合的模式，实现课堂的提质增效。语文以大单元概念整 合教学内容，在详实备课和扎实教研的基础上，凝练落实。 数学运用课堂观察量表，关注每位孩子落实掌握效果。语数 分别以年级教研组为单位，一位教师选择一个单元中的一堂 有代表性的课进行走班上课。通过共同设计，让同一个教师 在不同班级进行教学实践，更好地把握适合不同教学对象的 教学方法，发现平时教学中的一些低效无效的教学方式。通 过一次又一次的打磨，形成精品教案课例，延续使用，实现 课堂的提质增效。 以数学课为例： 课程选取：每个单元选最为典型的一课作为走课内容， 239 共同精心设计每一课的内容。 备课方法： 备教材：结合教参、统一教案。确定本课的重难点和教 学方法，细化教学过程，着重于突破重难点的方法以及设计 亮点。 备学生：因材施教（分层教学），各班老师对各班学生 的学情做一个具体的分析，针对各班学情制定相应的教学方 法以及教学风格，最终体现在教学设计中。 备教师：根据不同的课型确定不同的风格，根据教师自 身上课的优缺点，提出能规避的方法。 具体实施： 走课前：采用分工行动，主要有教学设计、作业设计、 课件制作、教具制作等。 走课时：听课老师根据自己的观测点进行详细记录，比 如本学期侧重观察学生6个方面： 听 （用心听老师讲以及同学 发言）、说（与同桌说以及在班上发言）、读（读数学概念 或方法）、写（课堂练习）、坐（坐姿是否端正）、小组合 作（小组交流时是否用心，开展是否顺利）。 走课后：当上完第一个班时，经过集体评课讨论出本堂 课的优缺点，根据情况对教学设计进行修改完善，在上第二 个班时投入使用，以此类推，逐步优化教学设计，形成最终 版本。 （五）作业做实。梳理作业门类， 如引入基础类、 提升 类、创新类作业概念。探索作业的差异化设计与总量控制。 240 以语文阅读作业为例： 教师设计出含金量高、灵活多样的作业，例如： 二、 以优化延时服务为突破重点， 精心用实素质教育 的“主阵地” “双减”后，学校教育主阵地作用被再一次强化。“减 负”的关键是“提质”，除课堂外，延时服务更是学生减负 提质的增长极。 学生在学校中不仅有学习， 更要有生活。 “双 减”之下，延时服务需要给学生留下一定的自我空间，让他 们有更多个体发展的选择与尝试。 在延时服务方面，实小始终坚持将“服务”二字入脑入 心，坚持把学生的身心健康和全面发展放在首位。 （一）机制创新，菜单式服务。实小遵循教育规律， 秉持个性化托管理念，即学生到学校不仅是学习技能，更是 来学校生活， 他们要在这里获得知识、 寻找友谊， 学会生活， 知识拓展作业 小太阳写话集、小森林诗集 艺体修身作业 绘心记录本 科学探究作业 自然笔记 生活技能作业 家庭劳动，寻味节气 课程融合作业 趣探西游、重走长征路 241 结识更多志趣相投的伙伴，这些也应成为教育的目标。 基于此，实小以延时服务改革为抓手，创新机制，依托 “托管+拓展”的模式，设置ABCD四个时段。A时段“绘心” 课程， 学生根据不同的学习层次、 特长兴趣进行美术、 音乐、 体育、阅读、知识日日清等不同的活动。以“固定+灵活”的 模式，灵活运用这30分钟的时段，照顾不同层次，不同特长 孩子的学习需求。B时段在教师指导下以班级为单位完成当 日学习任务。 C时段以校本课程为依托， 开展艺术教育、 劳动 教育、 国防教育、 体育锻炼等。 D时段携手天府新区共享中心， 以学生兴趣发展为导向，提供“菜单式”服务。学生可从围 棋、足球、篮球、中国舞、车模等个性化课程中进行选择， 实现每位孩子一生一课表。 既给予孩子自主学习的时间和空间，也充分利用和整合 校本资源， 有效实施多种育人活动， 给予孩子充分的选择权。 时段 课程 内容 A 绘心时段 第一层次：思维提升 第二层次：特长拓展 第三层次：知识日日清 B 学习时段 统一完成当日学习任务 C 自主时段 校本课程（劳动教育、国防 教育、体育锻炼等） D 拓展时段 “菜单式” 选修课程： 足球、 篮球、中国舞、车模、围棋…… 242 （二）用好灵活时段，关注童心，知识过手。A时段， 也叫“绘心时段”，是一个十分特别的时段。学生可以根据自 己的层次和爱好自由选择此时段的活动。 （三）绘出童心，融合育人。为了增加孩子们的艺术 修养，落实五育融合。实小以周为单位，每周的课间播放 经典音乐作品，让孩子聆听并感受。绘心时段给了想要用 画笔表达自己心理感受和音乐感受的孩子一个时间和平 台。他们可以在这个时段去到美术教室，在30分钟里充 足去感受和创作，用美术线条呈现内心的动态，画心绘情 。以此形成了学校的特色“绘心课程”，它打破学科壁垒， 融合了音乐、美术、心理等， 在全时全域的美育土壤中 让孩子的心灵绽放光芒。 （四）知识过手，素养过心。根据学生学习需求的不 同， 分层辅导让每一位学生的学习需求都被看见。分学科 对学习需求更大的孩子进行思维提升，对学习有困难的学 生进行辅导和答疑，做到知识日日清，基础过手。对艺体 有专长的孩子进行训练， 更好地满足不同学生的学习发展 需求。 三、以强化家校协同为战略支点，顺畅交好多元育人 的“接力棒” 要确保“双减”政策落到实处，还需要构建良好的教育 生态，有效缓解家长的焦虑情绪，促进学生全面发展、健 康成长。建设良好的教育生态不仅需要学校、教师、学生 的参与，更需要积极的家校互动和社会资源投入。 243 为架起学校与家庭协同教育的桥梁，形成教育合力， 提升教育成效，实小自建校起，就积极开展家访工作，做 到“三全”，即：全体教师参与，全学段覆盖，全体学生范 围内走访。 教师在家访前统一制定家访计划， 对家访的时 间、对象、路线、内容进行详细规划。教师们利用每月第 一周周一下午展开家访工作， 走进每个家庭， 走进孩子的 生活空间， 摸索共育时机， 在交流中直面“双减”中的问题， 用爱与真诚筑起坚固的家校桥梁。在“双减”政策出台后， 实小及时召开线上家长会， 为家长们解读政策， 交流学校 教师的做法， 提出相关建议和育人指导， 得到家长们的一 致认可。 实小以人为本，提质精心，服务入心，育人守心。以 细化教学管理为逻辑起点，以优化延时服务为突破重点， 以强化家校协同为战略支点赋能孩子成长。“双减”从“宏 观”到“微观”再到落地，从课程走向课堂，从课堂走向延 时服务， 一步一步走实， 坚定地走出一条高品质、 有内涵、 内生型的发展之路。 244 【山东省青岛二中】 发展核心素养，培育时代新人 青岛二中“十个一”项目助力“双减”落地 山东省青岛第二中学 2018年，青岛市教育局推出《青岛市促进中小学生全面 发展 “十个一” 项目行动计划》 ， 把立德树人作为根本任务， 实施一项体育技能、一项艺术才能、一本书、一篇日记、一 次劳动、一支歌、一首诗、一次演讲、一次研学、一次志愿 服务等 “十个一” 项目， 让每一位学生立足基础， 培养兴趣， 开发潜能，养成习惯，受益终身。 三年多来，岛城素质教育的活力全面迸发，不同学段、 不同校情的孩子们都因此获得全面的激发与锻造。山东省青 岛第二中学作为素质教育的倡导者与实践者，坚决贯彻落实 教育局的指导方针，树立科学的教育质量观、学生中心观、 全面成才观，制定并实施适合本校学生发展需求的青岛二中 “十个一+N” 行动计划， 呈现出 “系统化、 项目式、 个性化” 的显著特色， 构建起适宜学生全面发展的校园新生态， 为 “双 减”落地搭建有效的学生发展平台与评价机制。 一、坚持系统教育观，系统化推进“十个一+N”项目 1.系统化组织架构改革。坚持问题导向和系统化育人思 维， 充分调研学生发展的实际需求， 不断深化教育教学改革， 从学校管理、学科管理、学生管理、学生组织四大层面进行 颠覆性再造，构建起独具二中特色的组织架构体系。学校管 245 理层面增设创新发展处和学生发展中心，统筹设计学生发展 路径；将学科教研组升级为学科学生发展研究室，立足学科 核心素养开设多元课程、开展学科活动支持学生发展；学校 架设四大学生组织， 分别为学生联合会、 城市化发展委员会、 社团联合会、民主议事会，全面激发学生自我教育、自主管 理、 自我提升。 以这一系列组织架构为支撑， 将 “十个一+N” 行动计划层层分工、人人有责、落实到位。 2.系统化课程建设与改革。二十多年来，学校以“人” 的全面发展为价值指向，着力建设满足学生多元化发展需求 的、多层次的课程集群，不断推动国家课程实施层面的改革 和学校课程建设方面的创新，并配套以多元化的课程评价机 制，打造青岛二中“星光课程”体系，旨在激发每名学生独 特的发展潜质、让每个学生散发出独特的也是最耀眼的光芒。 2020年青岛二中“全人发展课程建构的实践研究”获青岛市 市级教学成果特等奖。 青岛二中星光课程体系 246 基于多年课改的经验，自“十个一”项目实施以来，二 中继续深化课程育人建设，将阅读、演讲、写作、劳动、研 学、志愿服务等主题教育融入课程、带进课堂。全学科开展 阅读工程， 各学科每学期向学生推荐书目， 两本必读、 N本选 读，并定期开展读书沙龙、读书笔记评选、将阅读题目纳入 模块考试中等，极大地提升了学生的读书热情与品质，政治 学科荐读的 《苏菲的世界》 、 历史学科荐读的 《曾国藩家书》 、 生物学科荐读的《生命是什么》等名作，在学校图书馆的阅 读大数据统计中连年排名前十，全校学生人均年阅读量达30 本。地理、物理、化学学科每年组织学生开展海洋主题研学 旅行活动，深挖青岛本土海洋教育资源，走到海边去、走进 研究所去，研究潮汐、新能源、环保、新型制药等小课题。 语文、外语学科指导学生天天记日记、周周练写作、月月做 展评、年年汇编学生文集，用写作浇灌学生思想之花。数学 学科每年组织学生在校园和社区开展建筑测绘与义务劳动， 指导学生用数学技巧测绘建筑、用辛勤劳动保护建筑。艺术 学科每年组织学生开展演出下乡、演出进福利院、书画进养 老院活动，将艺术的美和欢乐带给真正需要的人，培养学生 的公益心和社会责任感。这一系列的课程改革探索与实践， 将 “十个一” 有效融入课程， 保证素质教育常态化、 浸润式、 持续化开展，很好的回答了“怎样培养人”这一重要课题。 3.系统化评价体系构建。学生层面，学校实施《青岛二 中学生综合素质评价办法》，形成一套面向学生未来发展的 综合评价体系，围绕学生的人文道德素养、科学文化素养、 247 健康身心素质、人际交往能力、自我认知和生存能力五项基 本素质及学生卓越的领导能力、自主研究的能力、执着的创 新精神、开阔的国际视野五项特色素质对学生进行全面评价。 学生人手一本《青岛二中学生发展DIY自评手册》，涵盖“十 个一+N”、品德荣誉、综合实践、探究创新、最强技能等十 一个评价维度，指导学生每周自评、每学段互评，并将这一 评价结果作为学生评优、 评先的核心指标， 每学期在推选 “五 好学生”的基础上，推选不同领域的“校园之星”。教师层 面，学校实施《青岛二中教师“贡献度”评价办法》，将教 师课程育人、 担任全员导师、 开发校本课程、 指导学生社团、 为学生提供增值服务等项目列入教师考核和评优评先的重 要指标，激发教师全员育人、全程育人、全方位育人。通过 评价体系改革与实施，极大激活师生全面发展的意识、有力 保证“十个一+N”项目的有效实施。 248 二、项目式学习聚焦生活真样态，项目式运行提升育人 全效能 基于高中学生勇于批判、敢于创新的思维特质，学校深 挖教育教学的基础资源，调动广大师生交流合作，碰撞出若 干具有现实意义、 综合效益的课程创意， 开展具有任务驱动、 跨学科融合、学生为主体等特征的项目化学习，并在实践中 将“十个一”理念贯彻落实。 创新开展项目式校园文化活动，融合“一项艺术才能、 一本书、一篇日记、一支歌、一首诗、一次演讲”等内容。 “美术经典的哲学演绎”是由青岛二中美术、音乐、政治、 历史学科联袂打造的独创校园文化活动，已经历经七届，成 为学生口口相传的经典回忆。戏剧大赛的剧本来自学生对美 术经典作品的自主创编， 各科教师通力合作， 指导学生选题、 立意、搜集史料、撰写剧本、表演、提升艺术表现力和提炼 精神价值等，经过层层改进，以戏剧展演的形式表达美术经 典名作中蕴含的丰厚哲思，展现青年学子对世界的深入理解、 对人性的严肃反思、对真理的炽热渴望。“古风新韵，和诗 以歌”大赛为语文学科、美术学科、音乐学科、学生组织合 办的文化赛事，每一届比赛设定一个主题，引导学生精选一 部唐诗、宋词、元曲经典作品，深挖其文学内涵，编创适合 的情景，进行音乐旋律上的全新创作，选用适当的乐器现场 伴奏，编排优美的舞蹈、朗诵、短剧等现场展演，将文学之 韵与艺术之美一气呵成，深受师生、家长和社会各界好评。 249 三、打造个性化“十个一+N”项目集群，“一生一策” 精准推进 学校在“十个一”项目基础上，盘活校内外资源，增设 了“职业体验、生涯规划、课题研究、科技创新、活动组织 策划、戏剧编创演绎、设计与构建、生存挑战”等N个项目集 群，满足学生个性化发展需求。为了让学生做出适合自己的 选择，开设《“二中人”是怎样造就的》主题学生发展指导 课程，指导学生选择适配的课程、学生组织、社团等作为自 主发展的平台， 并从 “十个一+N” 项目集群中选择若干模块， 与志趣相投、能力互补的同学一起设计、组织、参与项目， 学校为孩子们提供支持与指导，让学生真正成为“十个一” 的创造者和享有者。 立德树人是教育之根本，终身发展是教育之使命。青岛 二中必将不忘教育初心、牢记教育使命，继续探索科学育人 的策略，不断坚持教育改革与创新，在实践中不断丰富和发 展新时代育人理念，培养全面发展的、无愧于时代的合格人 才。 250 【杭州师范大学东城中学】 基于教学改进视角下的作业减负行动 杭州师范大学东城中学 杨国军 一、作业设计的价值和意义 长期以来，教育行业关于作业还存在一系列的误解：误 以为作业越多越好，导致“用作业数量替代作业质量”，学 生睡眠时间越来越少，学习效率越来越低，身心健康状况越 来越差；误以为教师不需要自己设计作业，只要依靠校外教 辅或别人设计的试卷即可， 导致 “依赖心理” 和 “拿来主义” 盛行，很多教师作业的设计能力、命题能力每况愈下……关 于作业的两个误解直接导致作业问题积重难返，让教师和学 生每天都必须面对的作业成为教育研究中亟须开垦的“荒 原”。 其实，作业是学生自主学习的重要组成部分，是促进学 生掌握知识、形成能力、发展素养的重要手段，更是教师诊 断教学成效的重要依据。 基于以上阐述，杭州师范大学东城中学从提升作业设计 质量，加强作业实施过程两方面推进基于教学改进视角下的 作业减负行动。 二、提升作业设计质量 作业设计的质量是整个作业体系的命脉，围绕作业设计 的时间、难度、结构、形式、分层等五个方面改进因作业质 量低下而出现学生作业负担加重的情况。 251 （一）统筹作业时间，关注学生持续发展 作业量多是造成学生作业负担的主要原因，守住书面作 业班级总量不超过90分钟底线和实施作业限行管理机制是 控制作业量的重要方式。从班主任、年级组、教导处三方面 统筹学生作业完成时间，从关注完成时间走向关注学生持续 发展。以班主任牵头通过班级内部学科间的动态协调，总体 控制班级作业总量不超过90分钟，平均每门学科不超过25分 钟，并利用教室黑板记录当天作业内容，这不仅方便学生记 录作业内容，也有助于各学科协调作业量。学生用家校联系 本记录自己完成作业时间，年级组每天早上进行巡视，若某 科当天作业完成平均时长超30分钟，班主任和课代表介入跟 踪调查具体情况及时告知科任老师，从而达到学生作业完成 时间的及时监控和改进。同时，以教导处牵头学校层面实施 作业限行机制，作业限行科目每天一门，周一语文、社会， 周二英语，周三科学，周四数学，限行日除在当堂课内完成 的作业外该学科不留任何其他作业（包括口头作业）。 （二）降低作业难度，进阶学生能力养成 仅从减少作业量并不能完全减轻学生的作业负担，作业 难度是造成学生作业时长的另一个重要原因。为了切实合理 调整学生的作业难度，提升作业品质与效益，学校实施教师 提前作业 “下水” 机制， 将卡点和难点问题进行阶梯式分解、 针对性落实。比如，备课组在集体备课时聚焦研讨主题“如 何优化设计赋能素养提升”，利用课中作业和课堂卡点学情 分析， 精准科学实施课后作业设计， 切实减轻课后作业负担， 助力学生在完成作业过程中实现进阶式发展。 252 （三）丰富作业类型，注重学生素养培育 长期布置单一的书面作业会造成学生兴趣下降，不利于 学生能力的发展与素养的形成。学校提倡以发展学生核心素 养为作业设计的重点，鼓励教师积极探索项目式作业、跨学 科作业、拓展性作业、实践性作业的设计与实施。比如，科 学学科“制作肥皂项目式学习内容”，数学学科“美术与数 学的跨学科整合作业” ， 语文学科 “话剧表演的实践性作业” ， 英语学科“配音、演讲等拓展性作业”。通过丰富作业的类 型，激发学生对作业的兴趣，培养学生探索的能力，开阔学 生学习的视野，注重分析解决问题、学科核心素养的形成。 （四）优化作业结构，促进学生个性化学习 学校开展家庭作业“1+1”结构模式，即1本校本作业和 1本辅导资料， 校本作业作为学生学习的共性素材， 辅导资料 体现学生学习的个性选择。为了提高校本作业的质量，努力 实现学生课堂中的真学习和真练习，备课组提前两个月制定 下学期校本作业的修改方案，提前两个星期递交校本作业电 子稿到学校教导处审核。整个校本作业编修工作经历排稿、 审稿、 议稿、 修稿、 定稿等流程， 内容涉及新授课、 专题课、 单元复习、期末复习等，聚焦并全面涵盖教材知识内容，服 务于全体学生的发展与教、学、评一致。在个性作业方面， 学生自主完成并利用自主学习卡、自主学习共同体来促进学 生个性化学习。 （五）推进作业分层，兼顾学生思维提升 在作业分层方面，学校推行作业难度分层、作业评价分 层机制。在作业难度分层上，学校利用编制的校本作业，在 253 课后练习上设置A、 B、 C等级作业题， 将同一知识点设置成不 同的思维含量题，兼顾不同学生的思维发展。当然这样的分 层是动态的，也就是一段时间后根据学生发展情况的异同， 让学生进行 A、 B、 C 不同层级作业的轮换。 借助这样的方式 让学生在自己的“最近发展区”发展，从而使每个学生得到 适性而幸福的教育。在作业评价分层上，教师要注重评价内 容的分层处理，重视评价语言的激励性。对于学生个人进步 情况、 解题方法情况、 解题思路情况、 作业完成态度情况等， 通过科学分析和灵活处理，对进步情况进行评价，并且对不 足之处进行激励式评批。 三、健全作业实施体系 学校组建作业管理“三环反馈”机制。第一环由学校教 导处进行全校层面的大数据调查（包括学习压力、作业负担 等方面），将调查结果及时反馈给年级组和教研组。第二环 由年级组层面的大数据二次反馈（主要内容包括班级学科作 业协调、学生学习状况等方面），进一步的落实到年级备课 组和班集体，从而个性化地反馈到教师个体和学生个体。第 三环由教研组层面的大数据二次反馈（主要内容包括教研探 讨、 作业设计等方面） ， 进一步个性化的备课组和教师个体， 有效的改进教师的教， 促进学生的学。 最终从细化作业批改、 利用数据分析、精准推送错题、加强评价反馈四个层面加强 作业管理。 （一）细化作业批改 在作业批改方面，落实学生作业提前“下水”，真正解 决作业难、繁问题，减少重复性、机械性和无效作业。严格 254 执行有布置必有批改、有批改必有反馈、有反馈必要二次复 批促进学生主动订正，基于学生暴露的问题改进课堂教学。 同时，教导处每月通过学生调查、作业抽查等方式反馈校本 作业使用率，对使用率低的备课组加强集体备课、校本资源 建设的督促指导。 在作业复批率方面， 对复批率低的备课组， 教导处进一步巡查中午个别辅导的落实情况。在作业下水、 错误统计方面，教导处也及时进行跟进处理，教师的作业下 水是有效减轻学生作业负担的前提，错误统计是进行作业讲 评精准辅导的基础。 （二）利用数据分析 对学生作业错误的统计分析是进行精准讲评的基础，每 日的作业教师通过画“正”字、错题收集卡、智能学习平台 等方式，统计试题错答人数，分析错因，从而在课后或课前 及时纠正学生的问题。针对阶段性作业，学校利用智能学习 平台制作双向细目表，分析各个知识点学生的得分率，从而 调整教师的教学侧重，在复习阶段以达到针对性“补救”的 目的。 （三）精准推送错题 在每周五下午集体备课期间，备课组研讨本周作业中学 生暴露出的典型问题，编制15分钟的随堂知识纠错卷，并在 下周一中午或课堂时间进行二次巩固。对于阶段性的知识点 问题，利用智能学习平台，形成个性化发展手册，在周末推 送到学生端进行自主纠错，同时这样的错题推送在期末复习 阶段更有利于学生的精准复习。 255 （四）加强评价反馈 评价和反馈是改进教师、学生、班级作业管理的驱动机 制。在作业检查方面，推进践行作业负担监控机制、推进优 秀作业评选机制、规范教研组考核机制、实践常规作业评价 机制四项管理机制并驾齐驱。通过家长在线问卷调查、学生 问卷询查、教学处巡视督查等多维度反馈，了解学生就寝时 间以及课堂学习状态，及时公示公开各学科作业质量、作业 批改、作业负担问题。此外，利用智能学习平台等整合探索 实施作业的精准化分析，从个体、班级、教师、备课组、年 级对比，推送个性化手册优化校本作业。 “双减”呼吁学校回归教育本质，修炼内功，确保学校 教育的高质量发展。这意味着教师、学校和各级教育管理部 门在思考“减”的同时，也应该思考“提”，提升作业设计 质量， 健全作业实施体系。 “双减” 需要学校、 教师沉下心、 俯下身设计出既能巩固课堂知识，培育学科核心素养，又能 发挥学生的创造能力和实践能力，并且让学生有兴趣做的学 科作业，这也应该成为我们教育工作者长期研究的课题。 256 【江苏省锡山高级中学】 “双减”背景下以“优雅生活者”为导向的 趣味英语作业布置 江苏省锡山高级中学 李一丹 【事件】 本案例以基于高中英语新教材的一堂公开课及其作业 布置为例， 试验并讨论了两会大火的教育理念 “优雅生活者” 如何在英语新教材模块教学中进行应用，而作业又如何成为 课堂与生活联系的桥梁，高效而有趣地成为学习与自我发展 的一个环节，体现学科育人价值。 本案例是以选修一Unit3 Extended reading Qingming Scroll为材料，开设的一节阅读欣赏课。所读文本是一篇结 构清晰的说明文，介绍了中国古代名画《清明上河图》。在 课堂教学设计上，教师使用了有关《清明上河图》的更多图 片和细节，融入了指向跨文化交际的学生活动，从以下五个 方面培养学生对文本和画卷的理解：对美的初探、对美的细 节品味、基于冰山理论对美的深挖、对美的批判性思考和对 美的迁移创新。 通过五个环节的学习，学生不仅可以以文化传承人的身 份用英语介绍清明上河图，还能更进一步，以优雅生活者的 态度选取1-2个自己喜欢的细节侃侃而谈自己的个性化认识， 甚至能在老师的引导下利用冰山理论深挖其折射的社会文 化背景。因此，在作业环节，教师希望学生能以美的迁移创 257 造者的身份将课上所学加以趣味应用。 在这堂课的复习阶段， 学生回顾了他们对 《清明上河图》 的了解。清明上河图描述了17世纪的汴京，不仅宏观上大气 恢弘，在细节上更是颇具匠心，体现了当时社会环境下的人 文特色，是研究当时习俗文化历史的绝佳材料，也是每个细 节都值得推敲的杰作。 因此，在课后作业中，教师要求学生根据本堂课所学， 形成一个A4 纸尺寸大小的mini scroll，描绘他们印象最深 刻的无锡一景。学生对这个家庭作业都甚感兴趣。在下一节 课上， 教师留出时间， 给学生用英语展示与介绍自己的作品。 在展示环节， 学生不再畏畏缩缩， 都比较踊跃地想进行表达。 【分析】 “双减”在减少学生的学习负担之余，也应该重视提高 学生学习内驱力与效率。因此，作业与其说在于“少”，不 如说在于“精”。好的作业，应该与教学目标和内容紧密勾 连，是环环相扣的课堂教学的衍生，是对学生思维的一次拔 高性挑战。当一项作业真正激发起了学生的兴趣，与学生的 学习、生活息息相关，它就不会再是负担，而是想要追求的 乐趣。 本堂课介绍了《清明上河图》，从纵观、细节、深挖、 思辨等层面带学生了解并赏析了这幅风俗画之美。因此，在 作业环节，教师要求将美迁移创新，为他们所朝夕生活的无 锡一景画一幅小小的卷轴，将所学在真实情境下应用，对美 进行个性化输出。在自己作画的过程中，学生会将课上对画 轴的学习融会贯通，包括如何点面贯通、如何把握细节、如 258 何体现当代风俗文化以及如何暗藏深意等。通过作画的过程， 对美的欣赏不再是枯燥的文字，艺术作品也不再只是打卡一 看，而是能够真正体会揣摩的审美对象。 相信这次作业之后，学生再看到古画名画，会有更多的 兴趣和抓手去开展赏析与讨论。由于是自己感兴趣的事情， 等下节课用英语介绍自己的作品时，也会更有动力，更乐于 分享和表达。“画是无言的诗”，画中所蕴才是其内核与精 髓。 学生的生成还是非常令人惊喜的。最出色的是以下一张 作业： 这张无锡市中心迷你卷轴既有全景，又有细节，既有 人物，又有不同的建筑物。图片中既有高楼大厦，又不忘 地铁、公交车、出租车等各种交通工具，甚至在公交车上 和站台上还画上了广告，生活气息扑面而来。几笔勾勒， 将21世纪无锡的市井、建筑、商业、交通传神地表现了出 259 来。再跟着同学的介绍细看，在描写人的时候，每一个角 色都有他特有的服饰和动作细节，最传神的是边走边低头 玩手机的低头族，体现了21世纪所特有的人物形象。据同 学介绍，这也暗含着对当代社会沉迷于虚拟世界的担忧。 图画中还有近年来才流行的密室逃脱、滑板等等，将现代 风俗习惯文化也表达得很好。能画出这样一幅画的同学， 一定是一个生活中善于观察、善于发现美的优雅生活者。 同学在用英语介绍这幅画时，也引发了全班阵阵惊呼。 Hello, everyone. I’m presenting you my scroll of Wuxi’s city center. You must be very familiar with it. So let’s see if I have drawn the city center in your heart. Firstly, I’d like to draw your attention to the structure and buildings of it. I’ve drawn many high-rise buildings in the picture to show the prosperity of downtown scenery. According to what I learned about Qing Ming Scroll, I know details are very important for each building. So I designed some distinctive details representing 21stcentury, such as the subway station, the billboard of “密室逃脱”, widespread advertisements and our popular restaurants. Can you see your familiar ones? (interaction) As for people, I’ve designed details too. Can you find any? Student A: They’re playing skates. Student B: They’re looking at the mobile phone while walking. a typical “低头族” 260 Student C: Most of them are wearing glasses. Yes, you find all the details. Also, according to what I learned in the QingMing Scroll, different persons will have some relationships and interactions in the scene, making it more vivid. and I’m hoping that it will show the century features in Wuxi in my scroll. Thank you for listening. (applause) 【指导】 “双减”背景下，在英语学科中适度地将作业趣味化、 去操练化，与生活交融，也许可以有事半功倍的效果。 英语新教材囊括了生活的各个方面，就像本课以了解画 幅为主，作业就让他们画自己生活的城市，再用英语介绍， 将作业游戏化、生活化。还有很多单元的主题也可以有这样 生活化、趣味化的应用。比如，新教材中讲美食的单元，作 业就可以布置同学成为一日“美食vlog主”，回家录制一个 自己做美食的视频，边做边用英语介绍步骤、味道等。其实 在录制视频的时候，同学往往对自己要求很高，会不断自我 调整与提升，在这样的自我调整中，很多需要死记硬背的东 西就刻在了脑海中。 相信这样适当的趣味化作业，能让学生更好地内化单元 所学，也更能在真实情境中开展语用实践，让学习不再成为 象牙塔里的操练，而是他们对于生活的理解与升华。 261 【潍坊广文中学】 优化教学 迎双减政策落地 潍坊广文中学 孙宽正 曹文霞 夏宁 中共中央办公厅、国务院办公厅印发了《关于进一步减 轻义务教育阶段学生作业负担和校外培训负担的意见》，文 件中对作业时长等提出明确要求。 双减政策落地，首要问题就是课堂教学与课后作业的 “提质增效”。如果不能做到这一点，就会陷入一种两难的 困境， 要么作业时间长， 有违双减初衷， 要么教学质量下降， 不利学生发展。要做到提高各教学环节的效率，就要先分析 传统教学哪些环节是低效的。 传统教学，在概念新授课中，教师通过日常生活中的例 子引入新知。如果做过多背景扩充，会冲淡本节课的重点， 导致概念掌握不扎实；如果仅以一个简单的例子引入，学生 可能无法感受到所学概念的实际用处。即，学生在学习时经 常有“我为什么要学这个概念”的疑惑。 理想状态下，在每节新授课课后布置作业，几节新授课 后开设习题课，布置完成小专题练习。这样的安排可以帮助 学生巩固基本知识，操练基本技能。然而在实际教学中，学 生面对繁多的作业，往往是疲于应付，作业效果不佳。习题 课上，教师的着眼点本是“让学生知其然并知其所以然”， 而实际上学生只是机械模仿老师的解题过程，并没有吸收吃 透教师所授的解题方法。这导致所学内容在学生脑海中的留 262 存时间短、留存率低。于是，师生寄希望于周考、月考、单 元考这样频繁的测验，检测出遗忘的知识与技能，然后反复 重复，以求达到高分。但这样的重复，学生只是强化了程序 性知识，而策略性知识提升甚微。 程序性知识需要持续使用，才能保持记忆。单章节强化 好的程序性知识，如果后续章节使用较少，那么到初三，学 生的知识会出现大面积遗忘。而策略性知识则不然，对于问 题的分析策略是从根本上改善学生分析能力。如果学生应有 的策略性知识完备，即使知识点有遗忘，学生也可以根据自 己的分析快速回忆相关知识。 因此学生到初三时，需要再次复习多遍。这就是学生机 械重复性训练，不提高策略性知识的弊端，这些环节学生在 反复重复中浪费了大量精力，但思维素质却少有提升。 每单元末，进行单元复习检测等。学生被动接受各类总 结，其效果与前文所讲习题课并无区别。章末检测仅仅是为 学生进行结果性评价，对本单元的回顾存在一定意义，但毕 竟本章已经结束，对于掌握不佳的学生并不能很好改变现状。 总结上述分析，传统教学可以提升效率的方面有：1.概 念课更贴近生活，提高学生主动吸收知识的动力；2.习题课 更考察分析、评价、创新等策略维度，促进学生核心素养形 成良性循环；3.评价以量规形式贯穿教学，让学生在学习过 程中自我完善。 基于以上考虑，大概念项目化教学，成为了优化课堂的 可行方向。下文以青岛版初二上册《数据分析》为例，展示 基于大概念项目化教学，优化课堂环节。 263 步骤一：提炼大概念 《数据分析》 这章含 《加权平均数》 《中位数》 《众数》 《数据离散程度》《方差》《用计算器计算平均数和方差》 六节。每节可以形成每节的核心概念。结合各节，提炼出整 章教给学生的大概念是“以数据做决策”。 每一节教会学生在不同情景下如何做出决策。具体来讲， 第一节解决多评价维度并存，如何做决策的问题；第二节解 决存在极端数据，如何做出决策；第三节解决如何依照喜好 做出决策；第四节和第五节解决在数据有波动时，如何做出 决策；第六节解决实际应用中计算量过大问题。 以解决实际问题，提高学生“以数据做决策”的意识为 目标，分成了《“权”衡》 《“平均”探源》 《应对“极端” 数据》《投其所好》《“求稳”vs“冲一把”》《方程思想 与分类讨论应对未知数据》六个专题。 步骤二：对标核心素养 本章对标数学六大核心素养，可以提升学生数据分析、 数学运算、 数学建模三个方面。 数据分析， 是在实际情境中， 训练学生读取并应用数据的能力。数学运算，既要训练本章 第六节用计算器进行计算，也要训练纸笔运算。数学建模， 也要基于真实情景，让学生提出用数学方法解决实际问题的 方案。基于这三方面的考虑，选择以实际应用情景贯穿教学 始终。 步骤三：设定教学目标和学习目标 学习目标既要涵盖课程标准的基本要求，也要解决学生 心中的疑惑。首先，教学前一周末，让学生通读本章课本， 264 梳理自己疑惑。同时，教师精研课程标准，抓住“理解平均 数的意义”，“能计算中位数、众数、加权平均数，了解它 们是数据集中趋势的描述”，“体会刻画数据离散程度的意 义，会计算简单数据的方差”四个基本要求。将学生的困惑 与课程标准中提出的要求相结合，形成教学目标和学习目标。 其中，教学目标是教师根据课程标准和教材具体章节要 求提炼出的必须让学生掌握的内容。教学目标需要保障学生 所学内容无遗漏。 而学习目标， 是学生心中想要知道的内容， 它源于学生阅读课本的困惑，接轨于教学目标。学习目标需 要立足学生视角，以学生为中心，甚至不同的学生根据困惑 不同，可以有不同的学习目标。二者殊途同归。 学习目标以量规形式呈现，学生在学习过程中可以不断 校验自己达到的水平。 步骤四：设计项目化教学环节 基于学生分析教材：教材第一节为加权平均数。学生对 本节的困惑集中在两点：1.什么是“权”，为什么要加权； 2.什么是“平均数”，学生小学学习的“平均数”只有“相 加除以二”。本节换了一种算法，又叫某某平均数，学生从 感性认识上并不认同“加权平均数”这个名称。为此，本节 课拆为《“权”衡》和《“平均”探源》两节。 具体以《“权”衡》为例，设计项目化教学。 首先，分析加权思想在日常生活中的应用本质。日常决 策中，很多问题属于多维度并存进行决策。这类问题较为普 遍的解决方法是加权做决策。 其次，选择合适的背景展开项目化教学。通过与学生沟 265 通，发现他们周末要出去聚餐。便因势利导设置选餐厅这个 实际情境作为项目背景。 然后基于项目背景设置探究子任务： 子任务一：给出具体加权方案选择餐厅。用这个子任务 讲解加权平均的含义。 子任务二：换一种加权方案选择餐厅。用这个子任务让 学生体会不同的加权方式，有不同的结果产生。 子任务三： 根据特定目的， 自己设置加权方案选择餐厅。 初步引导学生在实际问题中，根据自己的目的使用加权平均 数。 子任务四：自己设定目的，设置加权方案选择餐厅。锻 炼学生根据实际需求，用加权平均得到自己最佳的选择。 这样四个问题，从应用到创新，逐步锻炼学生解决问题 的能力。在解决问题中，自然形成并逐步强化对“权”的认 识。基于任务情景把握概念，使得概念更加清晰。基于任务 目的进行计算，使得计算训练更有实效。 子任务五：基于学校综合素质评价方案，训练学生批判 思维和求同思维。“学校综合素质评价、分为心理与行为、 自我实现、体育与健康、艺术素养、创新与实践、标志性成 果六个维度。 每个维度给出了20%， 20%， 10%， 15%， 30%， 5%， 这样的权衡分量（权重）。请以小组为单位探讨这样设计综 合素质评价方案的思路是什么？这样设计有什么合理之处 及不合理之处？你能设计出更合理的评价方案吗？”这一系 列的问题， 旨在借助学生熟悉的情境， 从分析评价创新层面， 训练学生的策略性知识。 266 最后，在作业设置中，创设一个情境，训练学生用加权 的方式做决策： 假如你是班主任，请你用加权思想，设计三好学生评价 方案，为组内同学打分评价。 姓名 方面 + 权重 方面 + 权重 方面 + 权重 方面 + 权重 方面 + 权重 总得分 整个教学过程，围绕加权平均数在日常生活中的应用展 开。将生活中需要抉择的问题，变为需要用加权平均数辅助 决策的项目。 步骤五：与学生密切交流，指导学生使用量规 一切教学实验都存在不确定性。为保障教学实验切实可 行，可定期选取一部分学生沟通交流。内容可以分为三个层 面：1.对课堂和作业的变化是否适应；2.根据量规对学习成 果自我评价；3.通过对知识的具体提问，评价教师的教与学 生的学。 在沟通过程中，学生初期对于课堂与作业的变化可能感 到不适。随着时间的推移和不断调整，逐步实现学生适应的 合理变革。 通过以上探索，我们在有限时间内，基于大概念的项目 267 化教学，将夯实双基、提升思维、面向实用等进行了高效的 一体化整合，在合理减轻学生课业负担的同时，为学生实践 探究扩宽天地，推动落实数学学科核心素养。 268 【成都市七中育才学校】 五育融合视野下落实双减的育才行动 成都市七中育才学校 吴明平 臧玲 陈开文 詹滢 叶德元 “为谁培养人”“培养什么人”以及“怎样培养人”， 是我国教育改革必须回应的重要问题。 在坚持 “五育” 并举、 全面发展素质教育的时代背景下，学校聚焦学生核心素养， 以“深度学习”为课堂教学变革方向，进行了系列研究，从 教材解读到学情分析、从信息技术融合到前置学习、从聚焦 学科核心素养的培育到五育并举，学校始终坚持以学生的整 合性发展为核心目标来实施课堂教学改革。“五育并举”的 教育方针，强调学校必须构建德智体美劳全面培养的教育体 系，协同推进德育、体育、智育、美育、劳动教育，从而促 进学生更高质量的全面发展。 2021年， “五项管理” “双减” 等文件先后出台，这就要求学校思考：如何以小切口推动大 改革，在真正实现“减负”的同时，促进学生多方面、整体 性、多样态发展，为学生的社会适应及终身发展奠基，大力 提升学生的核心竞争力和综合能力。“五育融合视野下的育 才行动”是落实“双减”的具体措施，学校通过课程育人、 课堂教学、作业设计、协同育人等方面的深入思考，以落实 “双减”政策为“五育融合”的着力点，更加突出行动与实 践，建构“五育融合”背景下新的育人模式，促进学生更高 质量的发展。 269 一、优化课后服务，以“五育融合”的课程促进全面发 展 1.“五育融合”的学校课程体系 五育融合，课程先行。课程是学校教育最重要的载体， 学校在 “多元建构、 自主选择” 课程理念的基础上构建了 “聚 焦学生核心素养培育”的课程体系，不断完善和丰富“三大 类别”“六大模块”的百门课程，为学生搭建可供选择的、 立体开放的课程体系，满足学生从基础到高阶、从全面到个 性差异的需求。 2.“分层分类”的课后服务课程 “双减”要减负，更要增效。课后服务不仅是学校教育 在时间上的延伸，更是教育回归学校的重要表现。课后服务 270 是拓展学生学习空间、 全面推进素质教育、 促进 “五育融合” 的最佳时空。因此，学校对课后服务进行整体谋划和系统优 化， 将课后服务纳入 “五育融合” 的课程体系中， 形成了 《七 中育才学校课后服务课程方案》，并形成了全校学生“课后 服务一人一课表”的“课程风景线”。 根据学生学业基础和发展需求的层次差异，学校对课后 服务课程进行分层分类，包括：卓越课程、思拓课程、进阶 课程三个系列，其中“卓越课程”为拔尖创新人才的早期发 现与培育量身定做，着力于为基础扎实、学有专长和富有潜 力的学生搭建的发展平台，着力培养综合素质高、实践能力 强、 具有开拓和创新能力的初中毕业生。 “思拓课程” 和 “进 阶课程”则是满足不同学科基础的同学查漏补缺、提升学科 学习能力的需求。以我校在学业成绩反馈上“四级七档”的 标准衡量， “卓越课程”主要提供给A1 层次的同学， “思拓 课程”适合A2、A3、B1、B 2 层次的同学，为他们拓展视野 和提升学科学习能力； “进阶课程”适合C、 D 层次基础比较 薄弱的同学。 271 3.“个性特色”的选课走班课程 选课走班课程是根据国家《课程方案》，基于《课程标 准》和学校办学理念，根据本校学生个性发展的不同需求， 充分利用校内外的各种课程资源，组织本校教师并整合社会、 社区力量共同开发研制的校本课程。选课走班课程内容丰富， 囊括了艺术、体育、语言、人工智能等等；课程时长富有弹 性，有80分钟的长课，也有40分钟的短课；课程设置套餐组 合，文理搭配、大小学科搭配、动手动脑搭配、理论与实践 搭配，充分考虑学生兴趣，满足学生对课程的多元需求；菜 单式目录让课程选择自由， 学生自主选择符合自己 “口味儿” 的个性化课程。 此外，学校还设置了每周一节劳动教育课程、每周一节 体育锻炼课程、每周一个班级服务学习课程、每月至少一次 多彩梦想课程，从不同维度促进学生五育齐发展。课程与信 息技术的融合，更加拓展了学校的育人空间，学校借助自主 研发的在线平台“泛在育才”，不断丰富线上课程，如网校 录播课程、在线选修课程、主题微课课程，心理辅导课程、 艺术鉴赏课程、居家运动课程等。 总之，学校围绕培养全面发展的人这一目标，不断优化 课程结构，融通课后服务课程与现有课程，强化学校课程的 综合性与实践性，满足学生多样学习需求，充分发挥课程的 育人价值，促进学生全面发展和个性发展。 二、持续深度学习，以“五育融合”的课堂提升教学质 量 课堂是学校教育的主阵地。“双减”要求减轻学生过重 272 的学业负担，要求提升课堂教学效益。学校始终积极探索和 尝试课堂教学变革，鼓励和倡导以学生为主体的课堂教学模 式，将基于深度学习的课堂改革不断推向纵深。今天，学校 将研究纵深推进到“五育融合”视野下的深度学习，目的是 以“整体的教育培养整体的人”，实质是促进学生更高质量 的整合性学习。那么如何在课堂上实现“融合”？四川师范 大学李松林教授提出：“要以实践活动求融合、以核心问题 求融合、以更大概念求融合”，对此观点学校非常认同。以 实践参与为根本途径，核心问题为基本工具，大概念为深层 纽带，广泛适应力为目标导向，将五育融合真正落实到课堂 教学，就找到了推进学校质量提升的重要抓手： 明确一个目标： 以 《学科核心素养之育才表达》 的编写， 将学科教学、深度学习有机结合起来，着眼于促进学生核心 素养的积淀和学生的全面发展，解决了深度学习“为何做” 的问题。 用好两个工具：一是《五育融合视野下的深度学习设计 模板》（附件一），解决“怎么做”的问题；二是《五育融 合视野下的深度学习评价量表》（附件二），解决“做得怎 样”的问题。 实现深度学习需要进行深度整合，大概念就是深度整合 最重要的抓手。教育者要通过寻找“更上位、更深刻、更中 心”的概念，设计核心问题，推动学生深度学习；创设与生 活高度关联的真实生活场景， 激发学生深层动机、 深度体验。 “双减”政策之下， 呼唤课堂质量做加法， 教育品质再提升。 提升学校课堂质量，就需要用好上述两个工具，让教师明晰 273 “怎么做”以及“做得怎样”。学校要引领教师在每堂课的 核心目标确定、 核心内容选择、 核心问题设定、 核心任务 （活 动）设计、评价体系建构等方面持续深入探索；学科集体备 课强调“七备”：备课标的深刻领会、备教材的深度解读、 备学情的深入剖析、 备核心目标、 备核心知识、 备核心问题、 备核心任务（活动）。每一位教师在集体备课的基础上进行 二次备课，进一步明确每一课时的学习目标、学习过程和作 业设计。 持续抓牢大概念、 核心问题、 实践参与这三根纽带， 彰显“深度学习”课堂应有的样态，促进学生掌握核心知识， 形成建构能力，就能实现德智体美劳全面发展，最终形成必 备品格和关键能力。 三、创新作业设计，以“五育融合”的作业实现减负 增效 “作业设计”是推动“双减”落地的重要载体，是影响 教育质量的关键，对作业的研究与探索是促进课程改革内涵 发展的核心问题之一。 在“作业减负”问题上，不仅要聚焦“减量”，更重要 的是要关注“提质”，学校在依托课标定位目标、挖掘教材 梳理逻辑、瞄准素养确定方向的基础上，提出作业与课堂应 紧密呼应，关注五育整合设计的要求。以“把握功能、严控 总量、提高质量、强化管理”为作业设计总体思路，在作业 优化与创新上做加法，在作业量上做减法。（附件三） 1.“分层分类”作业创新设计 按照国家对作业管理的要求，学校根据学段、学科特点 及学生实际需要和完成能力进行作业设计，合理布置基础作 274 业、个性作业和特色作业。基础作业为学生必做作业，主要 是对所学知识的巩固消化，根据学生不同的学科基础，在内 容上分层、数量上分层、难度上分层。个性作业为学生在完 成必做作业之后，根据自身发展需求，在教师的指导下个性 化自主选择完成的作业，教师参与学生作业选择的过程，为 学生提供专业的个性作业建议，甚至一对一设计适合学生的 订制作业。特色作业为学生在周末或者假期进行的综合实践 作业，侧重培养学生的创新实践能力。各学科设计的特色作 业，最能体现五育融合的理念，特色作业多以项目学习的方 式，体现跨学科融合、五育融合。如我当三天家课程、博物 馆课程、 劳动服务课程等。 以前各学科分别布置的多项作业， 通过特色作业得以整合，更有挑战性、新颖性、学生也更愿 意完成的一项综合性任务，符合减负提质的要求。 学校的作业设计体现了科学性与针对性、层次性与趣味 性、适度性与多样性，通过精心设计与布置，及时评价与反 馈， 在减轻学生作业负担的同时， 充分挖掘作业的育人功能， 激发学生的学习兴趣，培养学生的核心素养，从而更好地提 升学习的效益。 同时， 学校倡导 “每周三为学生自主作业日” ， 要求所有学科不布置书面作业和软性作业，完全由学生自主 进行合理安排，充分调动学生在作业完成上的自主性，充分 发挥学生的创造力。 2.“科学规范”作业管理制度 为了落实“双减”让学生减少作业负担的要求，校长牵 头制定作业管理规范，建立教导处、年级组（备课组）、班 级三级管理制度：教导处建立作业量问卷监管制度；年级组 275 建立班级作业公示制度；备课组建立作业集体备课制度；班 主任建立统筹“协调制”；作业监管员“提醒制”；家长监 督反馈制。多方齐心协力，调控作业结构，监督作业总量， 形成 “班班有监控、 周周有反馈、 月月有调查、 期期有评价” 的监控体系。 作业管理事关学生健康成长，事关学校教育质量，事关 教育发展方向。在“双减”政策背景下，学校、教师、家长 更应该在思想认识上和实际行动上保持一致，站在贯彻党和 国家的教育方针、培养社会主义建设者和接班人的高度对待 工作，把“双减”工作做到实处，真正减轻学生作业负担。 四、践行家校共育，以“五育融合”共同体营造育人生 态 “双减” 实施后， 学生回家作业少了， 无所事事怎么办？ 周末不补课了，孩子成绩会不会下滑？对于家长的困惑和焦 虑，学校、教师也给予了高度关注与深切思考，给出了“居 家自习”和“居家周末生活”两份建议及指导，指导中有模 板示例、案例分享和明晰的要求，让家长和学生感受到很实 在，愿意积极尝试。（附件四） 学生在教师的指导下，在家长的陪伴下，自主规划和制 订“居家自习”和“周末生活”作息表，将时间板块化，将 抽象的学习目标转化为一个个具体的任务。通过这两份计划 的制定，引导学生学会合理安排学习内容和学习节奏，高效 利用时间，进而学会安排自己的课余生活，让体育锻炼、拓 展阅读、 亲子时光也走进他们的视野， 丰盈他们的居家时光。 在实施过程中，学校坚持过程管理，通过“云自习”“五项 276 管理自主规划本”“班级大事每月记”等多种方式保持家校 沟通，引导家长更科学地管理孩子，协同落实“五项管理” 和“双减”要求。可以说，推行居家自习和居家周末生活是 家校协同落实双减的重要途径和方式。 此外，为了更好地践行“家校共育”的教育理念，促进 家校联系，学校以多元路径、多维方式展开对家长的指导。 学校召集三级家委会成员签订《落实“五项管理”宣言》， 协同促进 “双减” 在家庭的落地。 通过学校官方微信公众号， 特别推出 “家长学校” 栏目， 每周定期推送最新的教育资讯、 政策法规，邀请教育大咖进行政策解读和优秀家长分享家庭 教育经验。目前已推出“暑期家书推荐” “首席班主任说” “育才名师说双减” 等多个系列， 不断更新家长的教育理念， 提升家庭教育的品质， 在家校协作中推动 “双减” 真正落地。 无论是 “五项管理”还是“双减”，目的都是使教育回 归学校主阵地、学习回归课堂主阵地，从根本上实现教育回 归生命原点的本真追求。学校希望通过“五育融合”拓宽教 育的通道，搭建五育融通的课程“立交桥”，通过课程的融 合、深度学习的持续发生、作业设计的持续优化、家校协作 的有效开展，为学生的成长做加法，为学生的负担做减法， 让“五育融合”成为破解双减难题的有益探索。 277 附件一： “五育融合视野下的深度学习”教学设计模板（2021版） 学科 课型 习题式□ 课题式□ 项目式□ 课题 教师 班级 深度 解读 教材 精选教材中最能体现学 科本质的知识，充分挖 掘教材内容中的五育育 人点。 准确 把握 学情 根据学生在学科学习及 五育发展上的需要、缺 失、困难和障碍深度分 析学情，精选最能促进 学生全面发展的知识。 五育融合 维度 智育□ 德育□ 体育□ 美育□ 劳动教育□ 大概念 核心 目标 Know（知道什么） Understand（理解什 么） Do（能做什么） Be（成为什么） 核心问题 (核心任务) 深度 学习 过程 环节设计 活动设计 设计意图 教学反思 （课后完 成） 278 附件二： “五育融合视野下的深度学习”课堂评价量表（2021版） 执教者 日期 年 月 日 班级 学科 课 题 课型 听课人 融合维度 □智育 □德育 □体育 □美育 □劳动教育 评价指标 评价内容及分值 评价结果 每项得分 总 分 教学 目标 1. 是否在目标设置上体现五育融合，层级分明、清晰具体、科学适切? （10分） 教学 内容 2.是否对教材进行深度解读，精选最能体现学科本质的知识，挖掘教材 内容中的五育育人点？（10分） 3.是否根据学生在学科学习及五育发展上的需要、缺失、困难和障碍深 度分析学情，精选最能促进学生全面发展的知识？（10分） 4.是否深度分析学习与生活的联系，精选最能促进学生适应社会发展的 知识？（10分） 教学 过程 5.是否准确提炼出大概念，并设计出具有挑战性的核心问题（核心任 务），进而引导学生围绕核心问题（核心任务）展开深度学习？（10 分） 6.是否注重设计并有效开展基于真实、复杂情境的实践参与活动，促进 学生对事物或知识本质的理解，对规律、方法的概括、提炼，培养学生 分析、综合、评价等高阶思维能力？（10分） 7.是否注重与现代教育技术的适切融合并展开深度互动，激发学生深层 动机，促进学生深切体验？（10分） 发展 评价 8.是否促进学生掌握学科核心知识，理解知识的关系与结构，明确知识 的作用与价值，达成深度理解？（10分） 9.是否促进学生面对真实、复杂情境，形成知识的建构能力和问题的解 决能力，实现实践创生？（10分） 10.是否促进学生德智体美劳多个方面的融合发展，并汇聚生成学生后 续学习、终身发展必备的广泛适应力？（10分） 点评 （亮点、不足、建议等） 【说明】总分为100分，请逐项评分，并本着促进老师专业发展的目的，做出精要、客观而有意义的点 评。 279 附件三： “五育融合视野下的深度学习”作业设计参考模板（2022版） 作业内容 作业类型 基础作业 （必选必做） 个性作业 （自主拓展） 特色作业 （实践创新） 卓越型(A1) 思拓型(A2\A3\B1\B2) 进阶型(C\D) ■五大学科：重点探索基础作业设计，积极开展个性作业指导，适时尝试特色作业设 计。 ■其他学科：重点探索特色作业设计。 ■基础作业：每天自选；与分层分类课程结合；与学业水平等级评价结果结合（四级 七档）。 ■个性作业：每周学生自主拓展；教师指导不同层次学生根据学情个性化拓展。 ■特色作业：适用于节假日/项目学习/跨学科/博物馆课程/劳动教育课程等长周期作 业。 280 附件四： 281 282 【北京市二十一世纪国际学校】 做好课后服务，实现“五育并举” 北京市二十一世纪国际学校 北京市二十一世纪国际学校是一所集小学、初中、高中 为一体的十二年一贯制寄宿制国际学校。学校一直以来重视 学生的课后服务工作，为学生提供课外兴趣班、社团、教师 课业答疑辅导、自主阅读等丰富多样的课后服务项目，供学 生自愿选择， 每周5天面向全员开放， 满足学生学习、 锻炼、 发展兴趣爱好等多种需求，促进学生德智体美劳全面发展。 一、开展劳动教育 劳动教育具有“树德、增智、强体、育美”的综合育人 价值，因此学校充分利用课后服务时间，开展了多种形式的 劳动教育。 （一）开设多门劳动教育选修课 在课后服务时间里，学校开设了多门劳动课程供学生选 修，如：厨艺、服装设计、茶艺、木工与精工等。 厨艺课程， 既能帮助学生掌握烹饪技能， 提高动手能力， 在毕业前学会近36道中国菜；又能帮助学生了解中华饮食文 化，尤其是饮食习惯背后的价值取向，让学生成为中华美食 传承者。课程遵循南北融合、中西结合、从易到难的原则， 安排课程内容， 编写读本 《厨艺汇》 两册， 供5-12年级选修； 有“名厨品尝会”“为你点赞”“荣誉晚宴”等展示活动。 服装设计课程，旨在让学生学习了解中华服饰文化，会 283 为自己和他人制作或选择服饰，掌握缝纫等技能。“汉服之 旅”是该课程的展示活动之一。从汉服文化追根溯源到面料 设计、版型制作、模特选拔、妆容造型，再到导演组策划、 服装海报宣传、灯光摄影、服装秀呈现和后期剪辑等，全部 由学生自己完成。这对学生来说是一次记忆深刻的劳动体验。 茶艺是中国特色的劳动课。 1-3年级， 侧重引导学生学茶 礼、品茶香、爱国饮；4-6年级，侧重引导学生学茶俗、展茶 艺、悟茶趣；7-9年级，侧重引导学生懂茶技、练工艺、明传 承；10-12年级，侧重引导学生知茶史、赏茶韵、铸茶魂…… 学生动脑、动手、用心、注情，在沏泡好每一杯茶的过程中 体味中国茶的魅力。 学校的劳动课程不仅仅培养学生的劳动技能，更与德育、 美育相融合，传承中华传统文化，培养学生的文化自信，把 学生的根留在中国， 这也是在有效落实学校的育人目标： “培 养具有中国灵魂、国际视野和跨文化交流能力的社会主义接 班人”。 （二）开展多样劳动教育活动 1.美化环境活动 每周四下午4点半到5点半，学校会统一组织学生开展劳 动大扫除活动。班级里的动植物等实施“认养制”，由学生 照料。此外，学校提供轮换制的劳动岗位，如：图书角管理 员、黑板美容师等。学校提倡用孩子们的劳动作品、劳动照 片美化教室，班主任老师会用积分卡奖励爱劳动的学生。 284 2.生活技能提升活动 充分发挥寄宿制学校优势开展劳动教育。生活老师都是 兼职的劳动教师，教学生一些生活劳动技能，如：叠衣服、 餐后收拾桌子、 洗衣服等； 同时， 经常开展 “内务最棒宿舍” “叠被小达人” 等评比活动， 提高学生的劳动积极性。 其中， “大手拉小手，新生入学教育先从劳动开始”是学校的传统 特色活动，高年级的同学会用一个月的时间，陪伴一年级新 生，手把手教他们整理内务，直到他们可以独立照顾自己。 3. 种植类、销售类等职业体验活动 学校专门开辟了一片“航天育种”小菜园。孩子们春天 学习播种，夏天学习施肥、浇水，秋天学习收割，最后再到 厨艺教室里将这些变成美食。同时，邀请专家开设《小种子 大科学》等科普讲座，让学生体验种植劳动，学习农业科学 知识。 此外，校内还开设有小书屋、咖啡屋等。在课后服务时 间里，安排学生开展职业体验活动，让学生体验采购、售卖 商品等流程。在每学期一次的“跳蚤市场”上，学生们还能 互相售卖闲置物品，并将所获收益捐给慈善事业。 二、增强学生体质 学校在充分利用体育课、早操、课间操增强学生体质的 基础上，在课后服务时间里也为学生提供了多种体育类的选 修课或社团活动，包括网球、篮球、足球、少年高尔夫、击 剑、排球、棒球、围棋、象棋、五子棋、跆拳道等十余种体 育项目； 定期开展 “世纪杯” 篮球联赛、 趣味体育赛等活动。 285 三、提升艺术修养 为有效落实美育，学校在课后服务时段开设了多门艺术 类选修课程。如：乐器类的二胡、小提琴、吉他等；舞蹈类 的民族舞、古典舞、啦啦操等；美术类的水粉画、油画等； 手工类的陶艺、布艺等；书法类的毛笔、硬笔；还有京剧、 T台秀、魔术表演等。 四、培养科技素养 为进一步响应“科技强国”的号召，学校开设了多门科 技类选修课程， 如： 少年科学实验、 环球地理、 化学DIY、 STEAM 课程， 还有信息技术类的FLASH设计、 3D设计与打印、 编程机 器人；航模类的航模基础课、VR无人机等。 五、开展阅读活动 学校十分重视阅读，启动了“阅读工程”，引导学生读 好书、好读书；每年精选大批优秀图书，放在图书馆和各班 教室中，制定《学校读物管理办法》，指导学生正确阅读。 小学每天下午4点半都会开展 “悦读书社” 活动， 由教师 组织社团学生到图书馆阅读；在晚上8点会安排半小时的睡 前阅读时间，由导育教师组织学生在宿舍读书。中学在每天 的18点20也有半小时的阅读时间。 学校定期开展 “书香班级” “金银铜牌书童”评选活动。 六、提升作业质量 在课后服务时间里，学生要完成当天作业。如何充分提 升作业质量，实现作业的减负增效？学校总结出了五条路径。 第一，邀请专家培训。如邀请专家为全校教师开展了题 为《指向核心素养的单元设计与实施》线上培训。 286 第二，每位教师至少读一本有关作业设计的书。在课程 教材研究所推荐书目的基础上， 教师们自己补充、 选择书目， 学校统一购买。读完一本书后，开展图书漂流活动。 第三，开展关于作业设计的课题研究。每年都会有60多 个校级课题立项，研究时长为一年。在今年的校级课题研究 中， 鼓励教师将作业设计作为研究重点。 《一年级 “非纸笔” 作业设计与实施研究》 《节假日语文长线作业优化设计策略》 《初中地理作业优化设计研究》等十几个课题成功立项。 第四，将作业设计列为学科组、教研组备课、教研重点 研讨的内容。学校要求学科组长、教研组长制定作业设计研 讨计划，通过学科研讨提升作业质量。 第五，组织教师分享作业设计经验。学校一贯重视作业 设计，在双减政策出台前，教师已积累了一些作业设计方面 的经验。在本学期，组织优秀教师开展作业设计分享，让好 的成果得到推广应用。 七、优化课业辅导 对有需求的学生，学校统筹安排课业辅导。该项工作由 年级组和学科组配合完成。年级组确定各年级辅导教室，学 生在规定时间内进入相应教室进行学习。学科组统筹教师资 源，选择骨干教师根据学生情况开展一对一、一对多辅导。 每月，辅导老师会对近期辅导的内容进行回顾和梳理， 通过游戏和PK等方式，帮助学生进行新一轮的系统复习。学 校鼓励教师为学生开展多种形式的个性化学习辅导。 八、做好保障措施 为提供高质量的课后服务，学校总结出四点保障措施。 287 第一，提前准备。暑假前，组织教师商讨课后服务具体 安排，确保在新学期开始之前合理安排好各类课后服务的时 间、场地、器材、人员等。 第二，规范管理。每项课后服务都要求有具体的教学计 划、时间安排及教学评价展示方案，学部每日安排专人检查 课后服务的实施情况。 第三，做好评价。在每学期末，请全体学生填写调查问 卷，对课后服务中的课程和社团进行评价。学校从中发现问 题，改进方案，提升质量。 第四，完善教师管理与考核办法，将课后服务纳入教师 绩效考核，体现学生和家长对课后服务的评价权重。 课后服务已经成为了学校教育教学的重要组成部分，充 分利用好这段时间，能促进学生德、智、体、美、劳全面发 展。未来，北京市二十一世纪国际学校将继续提升课后服务 质量，让学生和家长更满意，获得更多幸福感和成就感。 288 【北京市十一学校】 “双减”之下作业设计提质增效的实践探索 北京市十一学校 李素娟 “双减”背景下，学生作业问题凸显。作业设计问题， 可以说关乎学科核心素养，关乎人的全面发展。如何设计既 能减时减负又能提质增效的作业，是每一位教师都应该具备 的一项专业技能，也是每一所学校教学变革的切入点。 北京市十一学校在“从教到学”的教学变革中，不仅从 课堂上创新教学组织形式，更从作业设计中探索落实学科核 心素养的途径，主要做法如下。 一、设计情境性作业，激发学生探究兴趣 “双减”是要减量不减质。在紧扣学习目标的前提下， 要减少数量过大、质量不高的作业，关注学生完成作业的经 历，强化学生完成作业的体验和质效。情境性作业就是以学 生的真实生活为载体，伴随学生实践、探究等过程，增强学 生生活体验的作业。这类作业能更好地衔接课程与生活，促 使课堂教学不断动态地向生活延伸。它不仅让作业变得更生 动有趣，拓宽了学生的学习空间，也有助于激发学生主动探 究的兴趣和热情。而且这种问题解决的学习方式，还能使学 生把在自主探究中习得的知识和经验有效迁移应用到生活 中。这种从学到用，从知识的获取到能力的提升，再到素养 的形成的过程，正是真正有效促进“人”全面发展的过程。 289 在《童话节》单元，教师可以设计这样的情境性作业： 你是年级狂欢节策划人，在北京市十一学校一年一度的狂欢 节上，请你将自己的一位同学或教师装扮成一个童话人物， 邀请他参加学校狂欢节巡游。学校每年都举办狂欢节，由学 生策划团队选择年级师生要扮演的角色，为师生设计出场服 饰和动作。 这种与学生真实生活衔接的体验性、 实践性作业， 极大地激发了学生完成作业的热情。为了选出自己心仪的童 话人物，学生会主动深入阅读文本，探究童话人物形象，比 较不同人物的特点，撰写并修改人物形象解读稿。为了能够 说服老师或同学接受邀请，他们会反复修改润色劝说信、出 场动作设计和出场解说词，以增加邀请成功率。有的学生在 邀请遭拒时，还会积极修改完善自己的劝说信，调动自己以 往的知识和经验，并向他人请教，以调整自己的邀请行为， 直至邀请成功。这种在真实生活体验中学到的人际沟通技巧， 尤其是说服别人的策略，可以有效地迁移到与同伴或长辈的 交往中。学生不仅在学科中学习，更是在生活中汲取。 二、设计选择性作业，满足学生个性化需求 建构主义认为，学习知识是学生主动建构的过程，作业 作为学生学习的一个环节，其设计也应以学生为中心，满足 学生的个性化需求，发掘学生的潜能。允许学生根据自己的 能力，以自己认为舒服的节奏进行学习，并且取得相应的进 步，这就需要在聚焦目标的作业设计过程中，关注学情，考 虑到作业的弹性、分层和个性化。这种选择性作业可以起到 个性化学习方案的作用，使学生在完成作业的过程中扬长补 短，在自己的最近发展区得到充分发展，从而体验学习的快 290 乐和进步的自豪，并形成持续的学习动力。 在设计《四世同堂》整本书阅读作业时，依据单元学习 目标设计了A、B、C三个层级作业供学生选择（见下图）。 这部百万字的小说，阅读难度较大。如果作业一刀切， 则容易出现针对同一能力点重复训练的作业和“学生跳起脚 来也够不着”的无效作业。鉴于此，分层设计作业能给不同 能力层级的学生以自主选择和建构的空间。本项作业设计首 先是能力层级的选择， A、 B属于必选层级， C属于可选层级。 其次是同一层级不同任务的选择。学生可以根据自己的阅读 兴趣，从相应的能力层级中选择至少一项任务完成。一共完 291 成不少于5项即可。 语文能力较弱的学生可以在A、 B两个层级 中选择，语文能力较强的学生可以从三个层级中分别选择， 而语文能力特别强的学生业可以放弃A级，直接从B级和C级 中依据自己的探究兴趣选择更具综合性、有难度的作业。学 生无论做了何种选择，都能达成梳理分析、鉴赏品评的目标 要求。作业分层较好的解决了学生“消化难”“吃不饱”等 现象。学生语文能力和阅读兴趣点有差异这是客观事实，选 择性作业能让每一类学生基于自身情况在自己的最近发展 区获得个性化发展。 三、设计作业支架，让作业成为学习过程 要想切实减轻学生的作业负担，在让作业“精”起 来、“亮”起来的同时，也要使学生作业完成的过程成为 高效开展自主学习的过程。“作业支架”就是帮助学生实 现自主学习的一个有效路径。这里“支架”指的是教师通 过对学生做作业过程中最可能遇到的“困境”进行预估判 断，对其中难度较大、综合性强的部分给予思考路径、操 作方法、关键步骤等指导。有了学习“支架”的引领，学 生就可以在“教师不在场”，但“教师指导时时在场”的 情况下打开思维，找到突破“困境”的方向，在“支架” 的指引下，实现自主建构，形成个性化的学习经历和体 验。 《沙乡年鉴》对初二的学生来说阅读难度较大。为了 达成“通过多种阅读方式，了解《沙乡年鉴》全书内容” 这一阅读子目标，设计了拟副标题、做读书笔记和思维导 图等作业。由于此前在《寂静的春天》整本书阅读时，教 292 师已经在作业中设计了做读书笔记和思维导图的学习支架 指导，所以预估此项作业中学生最可能遇到的困境是不清 楚如何拟好小标题，于是在作业中设计了相应的支架。 有了支架的引导，学生在面对作业困境时可谓利器在手， 披荆斩棘。 学生在完成作业的过程中， 自主构建了拟小标题、 做摘录、 列提纲、 做批注、 写心得、 画思维导图等知识体系。 自主学习在完成作业的过程中真实发生，教会学生学习不仅 可以为学生减负，还可以培养学生的目标意识和学科思维力， 课堂和作业由此形成了学习的闭环，真正做到了为“人”的 终身发展服务。 四、设计作业量规，使作业自带评估功能 “我的作业哪些方面做得不错？” “哪些方面做哪些努力 可以获得提高？”这是学生完成作业后经常会有的困惑。要 想减少学生“我还要怎样做才能做得更好”的困惑，并且让 学生在作业之前就明确作业应达到的高度，作业之后能够自 主评估和升格，需要作业自带评估功能。而作业量规的设置 恰好可以满足这一需求。 量规设计要包含评价维度 （指标） 、 等级和相应等级要求的具体描述。它不仅可以作为教师考查 学生学习过程、学习进展以及最终学习结果的重要依据，同 293 时还清楚地告知了学生学业优异的具体标准，能使学生明确 作业的期望和达到这种期望的途径。量规的设置有助于学生 在做作业前做好心理准备，并激发学生在完成作业的过程中 不断向更高的标准迈进。 在《童话节》单元中设计了这样一项作业：围绕文本提 出三个你希望在本周“童话沙龙”中进一步讨论的好问题。 学生对这项作业最感到困惑的地方是：什么样的问题是“好 问题”？于是设计了《好问题量规》： 这个量规从“目标”“价值”“文本”三个维度明确了 “好问题”的标准，并给出了“优秀”“良好”“合格”三 个层级。为了让学生能友好对待自己或同学提出的问题，我 们分别用了“问题大帝”“问题老君”“问题仙童”等与刚 刚读完的《西游记》相关的要素代替了“优秀” “良好” “合 294 格”三个层级。有了量规，学生在提出问题前可以先明确作 业要求，在提出问题过程中可以通过深入阅读文本，从三个 维度反复斟酌；问题提出后，学生还可以对照量规评估自己 提出的问题，尝试对问题进行升格。此外，这个量规的意义 不仅指向本单元的学习，也不仅局限于语文学科学习，更重 要的还在于给了学生思考的方向：无论在学科学习还是未来 职场，如果想提出有价值的问题，必须思考目标、价值、情 境（文本）。好的作业设计解决的不仅是学科问题，更重要 的是培养全面发展的人。 作业的提质增效对教师的专业能力提出了更高的要求， 尤其考验教师对课程标准和教材的把握能力、对学情的掌握 能力和与时俱进的学习能力。教师只有在作业设计过程中紧 扣课程标准，始终以培养和提升学科核心素养为目标，从学 生实际出发，紧密围绕学生“学”的过程进行高质量的作业 设计，才能帮助学生在减负增质的利好中获得更充分的发展。 295 【北京陈经纶中学】 减负与提质并举 有效与优质共存 北京市陈经纶中学 徐琳 发展出题目，改革做文章。“双减”政策出台以来，北 京市陈经纶中学深刻认识，深化教育改革阶段性的新特点、 新任务，从高效课堂运行机制、有效作业管理机制、课后服 务协调机制三个机制入手，解决“双减”中的突出问题。 一、高效课堂运行机制 课堂教学效率提高了，学生的作业负担、考试压力才能 减轻，睡眠、体质等也会得到改善，学生在校生活体验就会 轻松愉快。因此，学校构建了“高校课堂”运行机制。 296 1.高效的“课堂时间管理” 最大限度地减少课堂时间损耗，提高课堂学习实用时间 和学习时间，是减负的重要途径。 一节课40分钟，学生处于最紧张的脑力劳动时间段是上 课后的5-20分钟，这也是教师最佳的教学时间段，高效利用 学生专注学习的时间，完成知识内容主要部分的教学，是高 效课堂的关键。 因此， 我们通过 “课堂诊断中心” 诊断机制， 促进教师打造高效课堂。 （1）“干部组长共听一节课”制度 （2）打破学科界限“专家教师约课”制度 （3）同一备课团队“合作伙伴互听课”制度 （4）聚焦同一教师“全面跟进式听课”制度 通过上述听课制度，促进了备课组整体的备课质量，提 升了教师课堂教学的研究力，在一定程度上逐渐转变了教师 的教学观念。通过系列评课及诊断课堂，让教师建立了课堂 的“时间节点”意识，从而让教师能够围绕着“高效”，设 计更加行之有效的教学策略，真正让“高效课堂”落地。 2．转变方式 创优提升 （1）优化教与学，提升课堂有效性 只有每个教师的应教尽教，才能保证每个学生在校学足 学好。我们以课堂常规为底线，优化教学方式。学校出台教 师课堂10条规范， 确保教师守住底线。 落实 “课堂教学1+3” ： 坚持一个基本原则，即以学生为主体，采取三种策略： 297 （2）优化学习过程，让深度学习真发生 长期以来，教师对学生的学习内在机制研究不够，这是 学生课业负担重的主要原因。学校致力于加强学习过程研究， 提高课堂学习效率，促进深度学习真实发生。在变革教学方 式的基础上，实现学习过程质的变化。 一是制定清晰的学习目标，让学生在课堂上有明确的学 习方向，激发学生学习动力。 二是创设真实的学习情境，寓教学内容于具体真实的情 境之中，唤醒学生学习兴趣。 三是设计优质的课堂问题，让学生思维在问题链的引领 下层层深入，让深度理解真实发生。 四是搭建重难点学习支架，清晰呈现学习进阶，为学习 “更上一层楼”搭建“脚手架”。 五是基于素养总结反思，在回顾与归纳中实现由量变到 质变、由知识到能力的转化。 二、有效作业管理机制 “双减”后，我们一直在思考：如何让作业成为激活课 堂教学的引擎？如何通过作业满足学生的获得感？如何通 过优化作业设计做到“有加有减”，在作业总量上做减法， 减掉机械、重复性作业；在品质上做加法，增加实践类、融 合性作业？为此，我们构建了“有效作业管理机制”，包含 298 三个部分：作业设计机制、作业量化机制、作业监控机制。 1.作业量化机制 （1）“四时机制” “双减”中首要提到的就是减轻学生的“作业负担”。 针对这项要求， 我们首先从 “量” 上作文章， 出台了 “四时” 机制，严把作业时长关。 299 (2)“时量标准” “四时机制”将作业时间进行了整体统筹和划分，聚焦 到作业布置的过程中，要将作业时间精细化落到每一位教师 身上， 作用于学生身上， 可以通过三种方式： 一是试做体验， 教师亲自试水做题，如：一道应用题、实验题大概的时间总 量； 二是抽样检验， 即通过抽取学生样本,在规定时间内， 以 班级为单位，半小时为计量，统计学生完成作业所需时长， 获得时间参考值；三是精准对标，对标期中、期末及中考试 题中的时间分配，如：一篇英语阅读在考试过程中的完成时 间应在5—6分钟。结合以上三种方式，最终形成“作业时量 标准表” ， 让作业量精准对标、 精准聚焦， 让减量落到实处。 2.作业设计机制 （1）确立设计原则 有了作业量化机制的时间标准，那第二部分就是作业的 核心——内容设计。教学处作为管理层，提出了作业设计的 原则： 两个三不： 不重复 不机械 不惩罚 不超时 不超量 不超纲 两个优先： 设计优先（科学性、分层性、实践性） 实效优先（全批、全改、全讲） （2）落实有效分层 作业设计可以依据学科特点分为基础落实、巩固应用、 拓展提升三个梯度。基础题适合基础相对薄弱的学生，通过 练习加以夯实；巩固应用难度适中，适合所有学生；拓展提 升适合具有学有余力的学生，帮助其进一步拓展视野和思路。 300 （3）创新作业维度 日作业：源于教材、课堂的作业，学生务必要会的作业 内容，要求少而精； 周作业：源于学生个性化的自我总结、梳理、反思的作 业内容，强调综合性； 月作业：源于生活的作业内容，鼓励跨学科整合，培养 学生思考、合作、探究、实践能力。 作业设计主要依托备课组，学校将作业设计固定为备课 组每周的例行项目，在备课组长的带领下，经过全体组员的 共同研讨，保障作业设计品质，同时也解决一些新进教师的 相关困惑。在组内整体作业设计完毕后，推行“一微调”原 则，即：教师可根据本班学情，进行相应微调。 3.作业监控机制 闭环管理是任何一项工作得以有效落实的重要保障，在 作业机制当中，作业监控机制是落实作业减负的防火墙，建 立三层监控机制，相互制约，相互监督。 一级监控：教学处确立作业监控总方针，每日通过计量 表监控作业布置是否落实作业量的标准，每周通过“作业小 管家”与“作业时量表”对比，反馈学生实际作业与登统作 业是否一致，落实有效监督。 二级监控：班主任作为二级监控，负责每日在群内监督 本班的作业量是否落实“四时”原则，尤其在专时及无课学 科上做好监控，无问题在群内签字确认。 三级监控：学习委员作为三级监控，每天负责填写好自 己的“作业小管家”记录册，周五上交教学处，由教学处对 301 标作业时量表统一检查反馈。 三级监控相互监督，共同将作业减负落到实处。 三、课后服务协调机制 本学期，在课后服务工作中，主要遵循让学生“动”起 来的工作思路，激发学生兴趣，提高学生素质，提升学习成 绩。主要任务概括起来涉及四个方面，即：全面覆盖、保证 时间、提高质量、强化保障。 1.在课后作业辅导过程中体现“三动” （1）一要让学生“心动”。达到情感认同，激发学生内 驱力。教师要走近学生，亲其师信其道，补的不只是知识、 不只是作业，而是在维系情感、和谐关系。 （2）二要让学生“脑动”。激发创造思维，发展学生学 习力。教师要把知识教“活”，不是就错论错，就题改题， 要激发学生兴趣，主动思考。 （3）三要让学生“行动”。引导知行合一，提高学生行 动力。教师要善用评价与鼓励，激发学生的行动力，提高学 习效率。 2.在课后活动中也要体现“三动” （1）一要让学生“劳动”。重视劳动教育，加强学生劳 动力。构建了劳动活动框架，其中包含：劳动课程、劳动岗 位、劳动周、劳模讲堂、智慧劳动、劳动评价六个方面培养 学生劳动习惯与劳动品质。 （2） 二要让学生 “运动” 。 强化体育运动， 创设60加20、 40加20两种课堂模式，保证每天至少一小时，结合“个人吉 尼斯挑战赛”“课间一小时竞赛”“课上班班赛”“运动队 302 高水平赛”等，激发学生运动积极性，提升学生身体素质。 （3）三要让学生“活动”。发展社团活动，关注特长培 养。继续以“普及+特长”的模式推进，既有学校教师的“走 班制”发展学生的兴趣爱 好， 也有外聘教练的专业化 特长训练， 确保课后活动的 丰富性、全面性和专业性。 “双减” 并不只是 “减 量” ，还要“提质”。因此， 从四个层面做到“底线”不 减。学校层面：不减规范、 不减质量、不减成长；教师 层面： 不减钻研、 不减实效、 不减奉献；学生层面：不减 态度、 不减努力、 不减勤奋； 家长层面：不减责任、不减沟 通、不减陪伴。 在“双减”政策下，北京市陈经纶中学力争做到让课堂 专心、操场开心、活动赏心、作业静心，为办人民满意的教 育不断探索与贡献。 303 【天津市五十中学】 历史学科作业优化设计 天津市五十中学 一、作业管理制度设计 （一）设计理念 教育要以人为本，以学生的发展为本。天津市五十中学 历史学科作业管理制度的设计本着遵循教育规律和学生发 展规律，全面贯彻“以学生为本”的教育理念，积极探索过 程性评价和发展性评价相结合，提高学生学习的有效性，提 升历史教师设计和实施作业的能力，落实立德树人的根本任 务。 （二）制度保障 根据中共中央办公厅、国务院办公厅《关于进一步减轻 义务教育阶段学生作业负担和校外培训负担的意见的通知》 （中办发〔2021〕40 号），教育部办公厅《关于加强义务教 育学校作业管理的通知》（教基厅函〔2021〕13 号）和天津 市教委《关于进一步加强和改进义务教育学校作业管理的若 干措施》 （津教中小学函〔2021〕2 号）等文件精神，并结合 本校《天津市五十中学“五项管理”工作实施方案》《天津 市五十中学作业管理制度》，制定了《天津市五十中学历史 学科作业管理方案》，旨在建立历史学科作业长效管理机制 和监督检查机制， 实现 “轻负担、 高质量” 的作业建设目标。 304 （三）历史学科作业管理机制 学校历史学科组制定了“一提高、两控制、三注重、四 严禁”制度。 “一提高”：提高作业设计质量。历史教师认真研究作 业类型、内容、时长、作业批改和教学质量，系统设计符合 学生年龄特点和认知规律，体现素质教育导向的基础性作业， 鼓励因材施教，布置分层、弹性和个性化作业，坚决克服机 械、无效作业，杜绝重复性、惩罚性作业。 “两控制” ： 控制作业难度， 确保难度不超过国家课标； 控制书面作业总量。要在考虑大多数学生作业能力的基础上， 对作业进行分年级量化。七、八年级历史每天书面作业总量 平均完成时间不超过10 分钟，九年级不得超过15 分钟。 “三注重”：注重作业的层次性和趣味性，照顾不同层 次学生的学习需求，激发他们的学习兴趣；注重作业的自主 性和探究性，教师可将课后作业与时事热点巧妙地结合起来， 让学生在课后作业中从更深更广的角度去对比探究问题，做 到以史为鉴，开创未来；注重作业的实践性和创新性，培养 学生勤于思考、 乐于创新的良好习惯， 从中体验学习的乐趣。 “四严禁” ： 严禁布置要求学生或者家长通过手机APP及 网络下载并打印的作业， 严禁布置机械性、 重复性、 惩罚性、 随意性等低效作业，严禁给家长布置或变相布置作业，严禁 要求家长批改作业。 二、作业类型 （一）校内、课内作业 校内、课内以基础型和巩固型作业为主，夯实学习基础 305 与质量。课堂教学是学校的主阵地，教师要做到每节课准备 充分、 知识储备丰富、 能力训练到位， 才能打造有趣、 有味、 有度的高效课堂。作业是课堂教学的一部分，作业设计也应 精选精设，减少总量，提升质量。 在初中历史教学中培养学生以唯物史观为基础的批判 性思维至关重要，这就要求学生具备高效且逻辑清晰的能力。 因此教师可以在课堂小结环节中布置一项课内作业，即由学 生在规定时间内完成对本课内容的思维导图制作并找同学 做课堂小结。这样一来，锻炼了学生逻辑关系能力的同时还 能夯实基础。 教师在讲完本节课内容后，可组织学生在课上完成针对 本节课重难点内容的课堂作业。这能使教师及时地、准确地 发现学生存在的问题，也便于学生高质高效地巩固所学知识。 课堂作业时长控制在五分钟以内。 （二）课后作业 课后作业以拓展性、 探究性、 实践性、 合作性作业为主， 注重拓展学生学习空间， 鼓励学生灵活运用知识， 走进生活， 走向社会，在调查、寻访、研学、观察、实验等活动中主动 发现、主动学习、主动实践、主动探究，培养学生创新精神 和实践能力。 1.选择型 根据学生的差异，设计适合不同基础学生需要的多层次 作业，供学生自由选择。 ①说，即把自己当天学到的内容向家长或与同学相互 “说一遍”。如，学《秦始皇统一中国》后，可以设计这样 306 的作业给学生：请同学们课后向家长或同学口头复述一遍秦 始皇统一和巩固国土的措施，并口头谈谈你对统一的认识。 ②读，即让学生把上节课所学的重点段落读一遍。如， 学《敌后战场的抗战》后，可布置这样的作业：请再读一遍 毛泽东《论持久战》和抗日根据地的建立与发展。这样既复 习了所学的重点内容，又有助于提高学生的阅读水平。 ③问，即由学生和老师互相提出历史问题并解答。根据 学习内容，历史教师把“提问”作为作业布置给学生，具体 操作如下：学生课下给老师提几个问题，由老师回答；或老 师课下找学生提问，由学生回答；或由学生之间进行问答。 这种互相提问的模式，不仅可以激发学生主动复习历史知识 积极性，还能促进师生、生生之间的情感沟通。 ④写，即进行课后练习。学生有一本与教材相配套的练 习册，因此教师要求每节课后，学生完成全部选择题，并挑 选一道大题完成。第二天课代表将作业收上来，教师及时批 改、分析，对所发现的共性问题在课堂上进行集体讲评，个 性问题进行个别辅导，这对学生巩固历史知识起到良好的作 用。 2.实践型 主要指调查、访问、参观、制作等活动性作业，这类作 业有利于培养学生观察能力与动手能力。例如：在讲五四运 动时，可以建议学生去参观天津本地的觉悟社，通过对周恩 来、马骏、邓颖超等人所在的先进青年组织——觉悟社的实 地了解，深刻把握由北京爱国学生发起的五四运动。 307 3.小组合作型 主要指由小组合作完成的作业，这类作业有利于培养学 生的合作探究能力。例如：在讲八国联军侵华战争时，可以 布置由小组代表分享聂士成将军奋力抵抗八国联军的事件。 三、评改方式与诊断反馈 1.严格遵循“布置必批改”的原则，书面作业要做到全 批全改，不得要求学生自批自改。 2.采取全员批阅和重点面批相结合、等级制评价和激励 性评语相结合的方式，不断提高作业的针对性和有效性。 ①等级制评价。 采取 “优+-” 的方法， 主要针对书面作业。 学生作业的完成率、 正确率高即可以获得优+， 连续获得10个 优+，学生还会收到一张“最佳作业”奖状作为奖励。这种奖 励既是对学生学习成果和学习态度的一种肯定，也是一种激 励。 ②小组积 “花” 制。 在作业完成方面采取小组检查积分的 制度。历史课代表会到同学中间去检查大家的作业完成情况。 若本组所有人都完成， 加5朵小红花， 若有一人没完成， 则没 完成的学生扣5朵小红花， 这是对作业完成率的把控。 在作业 完成质量上，积分制也在起着重要作用。比较重要的作业， 老师们会收上来批改， 在批改过程中看学生完成质量。 此外， 还可以采用互批互讲的方式，调动学生积极性，让学生成为 评价的主人，也让他们在互批的过程中取长补短。 ③激励性评语。 在批改作业时， 学校鼓励教师除了运用恰 当的评价符号、小印章等表示鼓励外，还提倡教师运用激励 性点评式语言进行一种“私聊”。它既是一种评语，更是一 308 种心灵对话， 它能成为学生们的一种期盼。 如 “思维敏捷” “望继续保持”“未来可期”等短语激励，促其奋发上进。 同时还可以采用“再细心些”“再想想”等惋惜语，以调整 学生的心理差距，激励学生更进一步。这样因人而异的短语 评价，使各类学生在下次完成作业时能做得更规范、准确。 3.可以根据作业的类型和特点，采用多样化的评价方式， 除了教师批阅以外，可采取作品展评、展示交流、讨论分享 等多样化的方式进行作业评价。 4.作业讲评要及时有效，共性问题全班讲评，个性问题 个别指导。 5.要督促学生养成及时纠错的好习惯。引导学生根据作 业批改结果分析学习存在的问题，更好地开展后续学习。同 时，教师还要认真分析学生作业错误的原因，及时调整教学 进程、教学重难点和教学策略等。 四、亮点与成效 （一）亮点 1.对标课程标准，落实历史学科核心素养。在新课标背 景下，历史作业不仅仅旨在了解学生对历史知识的掌握情况， 也要能够检测学生学科核心素养的发展情况。因此，学校历 史教师在作业设计的过程中，要有意识地引导学生运用唯物 史观分析和评价历史事件、历史人物，总结历史规律，帮助 学生构建正确的历史观；通过布置撰写历史小论文或历史电 影观后感，开展历史辩论赛、历史小故事演讲、历史剧表演 等活动，通过形式多样的作业落实学科核心素养培育。 2.与天津地方史相结合，培养学生的家乡情。教师结合 309 初中历史的地方教材 《天津历史》 ， 引导学生通过检索网络、 查阅图书、访谈长辈、参观考察等方式搜集资料，了解天津 各区著名的故居旧址、名胜古迹、文化遗产和具有历史文化 风貌的街区等，探寻其背后的历史背景，以培养学生对家乡 的热爱之情，培养爱国主义和集体主义精神，在探究过程中 培养合作精神、实践能力和探究能力。 3.坚持分层作业和课后辅导相结合。课后服务时间，教 师带领学生认真完成分层作业，为学有困难的学生进行辅导， 帮助其巩固基础知识，为学有余力的学生拓展学习空间，补 充课外知识，培养兴趣爱好。 （二）成效 一是改善了师生关系。教师们在作业的评价中更多地关 注学生完成作业的态度，不斤斤计较一道题的正确与否。教 师的鼓励性评价让学生看到了希望，师生关系融洽、宽松。 二是激发了学生学习历史的积极性，增强了学生学习历 史的自信心，让他们及时找到改进的方向，体验到了“我能 行”的成就感，树立了民族心、家国情，增强了学习兴趣。 三是端正了学生的学习态度，培养了良好的学习习惯。 通过对作业评价的改革，学生完成作业的态度更认真了，因 为学生明白，只要认真对待每天的作业都会得到老师的肯定， 从而形成学习内驱力。 四是增进了家校的联系。家长对学生作业的形式、内容 非常认可，自觉地参与作业的评价，家长成了教师可以依靠 的力量，形成了教育的合力。 310 |
3491 | https://www.combinatorics.org/files/Surveys/ds6/ds6v27-2024.pdf | A Dynamic Survey of Graph Labeling Joseph A. Gallian Department of Mathematics and Statistics University of Minnesota Duluth Duluth, Minnesota 55812, U.S.A.
jgallian@d.umn.edu Submitted: September 18, 1996; Accepted: November 14, 1997 Twenty-seventh edition, November 15, 2024 Mathematics Subject Classifications: 05C78 Abstract A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the mid-1960s.
In the intervening years over 350 graph labelings techniques have been studied in over 3600 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey, I have collected everything I could find on graph labeling. For the convenience of the reader, the survey includes a detailed table of contents and index.
This edition has 34 additional pages and 110 new references that are identified with the reference number and the word “new” in the right margin.
the electronic journal of combinatorics (2023), #DS6 1 Contents 1 Introduction 3 2 Graceful and Harmonious Labelings 7 2.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 2.2 Cycle-Related Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 2.3 Product Related Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20 2.4 Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 2.5 Disconnected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24 2.6 Joins of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27 2.7 Miscellaneous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40 Table 1: Summary of Graceful Results . . . . . . . . . . . . . . . . . . . .
40 Table 2: Summary of Harmonious Results . . . . . . . . . . . . . . . . . .
44 3 Variations of Graceful Labelings 48 3.1 α-labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48 Table 3: Summary of Results on α-labelings . . . . . . . . . . . . . . . . .
63 3.2 γ-Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64 3.3 Graceful-like Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 Table 4: Summary of Results on Graceful-like labelings . . . . . . . . . . .
76 3.4 k-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76 3.5 Skolem-Graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . .
80 3.6 Odd-Graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82 3.7 Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87 3.8 The Friendly Index–Balance Index . . . . . . . . . . . . . . . . . . . . . . 105 3.9 k-equitable Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.10 Hamming-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4 Variations of Harmonious Labelings 115 4.1 Sequential and Strongly c-harmonious Labelings . . . . . . . . . . . . . . . 115 4.2 (k, d)-arithmetic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.3 (k, d)-indexable Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4 Elegant Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.5 Felicitous Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.6 Odd Harmonious and Even Harmonious Labelings . . . . . . . . . . . . . . 129 5 Magic-type Labelings 138 5.1 Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Table 5: Summary of Magic Labelings . . . . . . . . . . . . . . . . . . . . 149 5.2 Edge-magic Total and Super Edge-magic Total Labelings . . . . . . . . . . 150 Table 6: Summary of Edge-magic Total Labelings . . . . . . . . . . . . . . 174 Table 7: Summary of Super Edge-magic Labelings . . . . . . . . . . . . . . 176 the electronic journal of combinatorics (2023), #DS6 2 5.3 Vertex-magic Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . . 179 Table 8: Summary of Vertex-magic Total Labelings . . . . . . . . . . . . . 187 Table 9: Summary of Super Vertex-magic Total Labelings . . . . . . . . . 189 Table 10: Summary of Totally Magic Labelings . . . . . . . . . . . . . . . 189 5.4 H-Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.5 Magic Labelings of Type (a, b, c) . . . . . . . . . . . . . . . . . . . . . . . . 194 Table 11: Summary of Magic Labelings of Type (a, b, c) . . . . . . . . . . . 197 5.6 Sigma Labelings/1-vertex magic labelings/Distance Magic . . . . . . . . . 198 5.7 Other Types of Magic Labelings . . . . . . . . . . . . . . . . . . . . . . . . 203 6 Antimagic-type Labelings 216 6.1 Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Table 12: Summary of Antimagic Labelings . . . . . . . . . . . . . . . . . 229 6.2 (a, d)-Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Table 13: Summary of (a, d)-Antimagic Labelings . . . . . . . . . . . . . . 239 6.3 (a, d)-Antimagic Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . 240 Table 14: Summary of (a, d)-Vertex-Antimagic Total and Super (a, d)-Vertex-Antimagic Total Labelings . . . . . . . . . . . . . . . . . . . 252 Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings . . . . . . . 253 Table 16: Summary of (a, d)-Edge-Antimagic Vertex Labelings . . . . . . . 254 Table 17: Summary of (a, d)-Super-Edge-Antimagic Total Labelings . . . . 255 6.4 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1) . . . . 256 Table 18: Summary of Face Antimagic Labelings . . . . . . . . . . . . . . 260 Table 19: Summary of d-antimagic Labelings of Type (1,1,1) . . . . . . . . 260 6.5 Product Antimagic Labelings . . . . . . . . . . . . . . . . . . . . . . . . . 261 7 Miscellaneous Labelings 263 7.1 Sum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Table 20: Summary of Sum Graph Labelings . . . . . . . . . . . . . . . . . 273 7.2 Prime and Vertex Prime Labelings . . . . . . . . . . . . . . . . . . . . . . 274 Table 21: Summary of Prime Labelings . . . . . . . . . . . . . . . . . . . . 288 Table 22: Summary of Vertex Prime Labelings . . . . . . . . . . . . . . . . 289 7.3 Edge-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Table 23: Summary of Edge-graceful Labelings . . . . . . . . . . . . . . . 300 7.4 Radio Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 7.5 Product and Divisor Cordial Labelings . . . . . . . . . . . . . . . . . . . . 307 7.6 Other Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 7.7 Edge Product Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . 324 7.8 Difference Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . 326 7.9 Prime Cordial Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 7.10 Mean Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 7.11 Pair Sum and Pair Mean Graphs . . . . . . . . . . . . . . . . . . . . . . . 363 7.12 Irregular Total Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 the electronic journal of combinatorics (2023), #DS6 3 7.13 Geometric Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 7.14 Strongly Multiplicative Graphs . . . . . . . . . . . . . . . . . . . . . . . . 382 7.15 Line-graceful Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 7.16 k-sequential Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 7.17 IC-colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 7.18 Minimal k-rankings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 7.19 Set Graceful and Set Sequential Graphs . . . . . . . . . . . . . . . . . . . 387 7.20 Vertex Equitable Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 7.21 Representations of Graphs modulo n . . . . . . . . . . . . . . . . . . . . . 393 7.22 Sequentially Additive Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.23 Difference Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 7.24 Square Sum Labelings and Square Difference Labelings . . . . . . . . . . . 395 7.25 Permutation and Combination Graphs . . . . . . . . . . . . . . . . . . . . 400 7.26 Strongly -graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 7.27 Triangular Sum Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 7.28 Divisor Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 7.29 Other Kinds of Labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 References 419 Index 687 the electronic journal of combinatorics (2023), #DS6 4 1 Introduction Most graph labeling methods trace their origin to one introduced by Rosa in 1967, or one given by Graham and Sloane in 1980. Rosa called a function f a β-valuation of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, . . . , q} such that, when each edge xy is assigned the label |f(x) −f(y)|, the resulting edge labels are distinct. Golomb subsequently called such labelings graceful and this is now the popular term. Alternatively, Buratti, Rinaldi, and Traetta define a graph G with q edges to be graceful if there is an injection f from the vertices of G to the set {0, 1, . . . , q} such that every possible difference of the vertex labels of all the edges is the set {1, 2, . . . , q}. Rosa introduced β-valuations as well as a number of other labelings as tools for decomposing the complete graph into isomorphic subgraphs. In particular, β-valuations originated as a means of attacking the conjecture of Ringel that K2n+1 can be decomposed into 2n+1 subgraphs that are all isomorphic to a given tree with n edges. Independently, Keevash and Staden in April 2020 and Montgomery, Pokrovskiy, and Sudakov in January in 2021 proved Ringel’s 1963 conjecture that any tree with n edges packs 2n + 1 times into the complete graph K2n+1 for large n.
Keevash and Staden used an embedding algorithm in which the various subroutines are analyzed by a wide range of methods, some of which are adaptations of existing methods whereas, Montgomery et al. used probabilistic methods. Although an unpublished result of Erdős says that most graphs are not graceful (see ), most graphs that have some sort of regularity of structure are graceful. Sheppard has shown that there are exactly q! gracefully labeled graphs with q edges. Rosa has identified essentially three reasons why a graph fails to be graceful: (1) G has “too many vertices” and “not enough edges,” (2) G “has too many edges,” and (3) G “has the wrong parity.” The disjoint union of trees is a case where there are too many vertices for the number of edges. An infinite class of graphs that are not graceful for the second reason is given in . As an example of the third condition Rosa has shown that if every vertex has even degree and the number of edges is congruent to 1 or 2 (mod 4) then the graph is not graceful.
In particular, the cycles C4n+1 and C4n+2 are not graceful. Knuth has observed the more general condition that in any graceful labeling of a graph with the number of edges congruent to 1 or 2 (mod 4), the number of vertices with an odd degree and an odd label is always odd. Knuth proved by way a computer search that all cubic graphs on 4, 6, 8, 10, 12, or 14 vertices, except 2K4 and 3K4, which was proved by Kotzig, are graceful. He conjectures that that every connected cubic graph is graceful.
It has been known since 1975 that rooted symmetric trees are graceful. However, the proofs that have been presented for this fact are either indirect inductive proofs or algorithmic descriptive proofs showing the many separate steps involved in labelling the vertices. In 2021 Rofa provided a graceful labeling for any given rooted symmetric tree in the form of a direct algebraic function that algebraically maps each vertex to a unique label. The function is a generalization of the way a path is canonically gracefully labeled as algorithmically described by Rosa in his 1967 classic paper . Rofa uses his function to show that a class of rooted symmetric trees that contains the class of the electronic journal of combinatorics (2023), #DS6 5 binomial trees has weakly α-labeling (see Section 3.1) and it can provide a concise prac-tical, computational way of producing graceful labelings of large rooted symmetric trees in relatively minimal time.
Acharya proved that every graph can be embedded as an induced subgraph of a graceful graph and a connected graph can be embedded as an induced subgraph of a graceful connected graph. Acharya, Rao, and Arumugam proved: every triangle-free graph can be embedded as an induced subgraph of a triangle-free graceful graph; every planar graph can be embedded as an induced subgraph of a planar graceful graph; and every tree can be embedded as an induced subgraph of a graceful tree. Sethuraman, Ragukumar, and Slater show that every tree can be embedded in a graceful tree (see also ) and pose a related open problem toward settling the Graceful Tree Conjecture.
Rao and Sahoo proved that every connected graph can be embedded as an induced subgraph of an Eulerian graceful graph thereby answering a question originally posed by Rao and mentioned by Acharya and Arumugum in . As a consequence they deduce that the problems on deciding whether the chromatic of a graph number is less than or equal to k, for k ≥3, and deciding whether the clique number of a graph is greater than or equal to k, for k ≥3 are NP-complete even for Eulerian graceful graphs.
Sethuraman and Ragukumar provided an algorithm that generates a graceful tree from a given arbitrary tree by adding a sequence of new pendent edges to the given arbitrary tree thereby proving that every tree is a subtree of a graceful tree. They ask the question: If G is a graceful tree and v is any vertex of G of degree 1, is it true that G −v is graceful? If the answer is affirmative, then those additional edges of the input arbitrary tree T introduced for constructing the graceful tree T by their algorithm could be deleted in some order so that the given arbitrary tree T becomes graceful. This would imply that the Graceful Tree Conjecture is true. These results demonstrate that there is no forbidden subgraph characterization of these particular kinds of graceful graphs.
Harmonious graphs naturally arose in the study by Graham and Sloane of modular versions of additive bases problems stemming from error-correcting codes. They defined a graph G with q edges to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x) + f(y) (mod q), the resulting edge labels are distinct. When G is a tree, exactly one label may be used on two vertices. They proved that almost all graphs are not harmonious. Analogous to the “parity” necessity condition for graceful graphs, Graham and Sloane proved that if a harmonious graph has an even number of edges q and the degree of every vertex is divisible by 2k then q is divisible by 2k+1. Thus, for example, a book with seven pages (i.e., the Cartesian product of the complete bipartite graph K1,7 and a path of length 1) is not harmonious. Liu and Zhang have generalized this condition as follows: if a harmonious graph with q edges has degree sequence d1, d2, . . . , dp then gcd(d1, d2, . . . dp, q) divides q(q −1)/2. They have also proved that every graph is a subgraph of a harmonious graph. More generally, Sethuraman and Elumalai have shown that any given set of graphs G1, G2, . . . , Gt can be embedded in a graceful or harmonious graph. Determining whether a graph has a harmonious labeling was shown to be NP-complete by Auparajita, Dulawat, and Rathore in 2001 (see ).
the electronic journal of combinatorics (2023), #DS6 6 In the early 1980s Bloom and Hsu , , , , extended graceful labelings to directed graphs by defining a graceful labeling on a directed graph D(V, E) as a one-to-one map θ from V to {0, 1, 2, . . . , |E|} such that θ(y) −θ(x) mod (|E| + 1) is distinct for every edge xy in E. Graceful labelings of directed graphs also arose in the characterization of finite neofields by Hsu and Keedwell , . Graceful labelings of directed graphs was the subject of Marr’s 2007 Ph.D. dissertation . In and Marr presents results of graceful labelings of directed paths, stars, wheels, and umbrellas. Hegde and Kumudakshi we use complete mappings to construct graceful labelings of two directed cycles. Siqinbate and Feng proved that the disjoint union of three copies of a directed cycle of fixed even length is graceful.
Over the past five decades in excess of 3,000 papers have spawned a bewildering array of graph labeling methods. Despite the unabated procession of papers, there are few general results on graph labelings. Indeed, the papers focus on particular classes of graphs and methods, and feature ad hoc arguments. In part because many of the papers have appeared in journals not widely available, frequently the same classes of graphs have been done by several authors and in some cases the same terminology is used for different concepts. In this article, we survey what is known about numerous graph labeling methods. The author requests that he be sent preprints and reprints as well as corrections for inclusion in the updated versions of the survey.
Earlier surveys, restricted to one or two labeling methods, include , , , , and . In Shivarajkumar, Sriraj, and Hegde provided a 2021 survey arti-cle graceful labeling of digraphs. The book edited by Acharya, Arumugam, and Rosa includes a variety of labeling methods that we do not discuss in this survey. In 2002 Eshghi wrote a 65 page paper providing an introduction to graceful graphs. The re-lationship between graceful digraphs and a variety of algebraic structures including cyclic difference sets, sequenceable groups, generalized complete mappings, near-complete map-pings, and neofields is discussed in and . The connection between graceful label-ings and perfect systems of difference sets is given in . The computational complexity of the gracefulness of a graph is not known, but the complexity of finding a harmonious la-beling of a graph is in the NP-class . Labeled graphs serve as useful models for a broad range of applications such as: coding theory, x-ray crystallography, radar, astronomy, cir-cuit design, communication network addressing, data base management, secret sharing schemes, cryptology, models for constraint programming over finite domains, ultrasound screens, , , , , , , , , , , and net-work passwords–see , , , , , , , and for details. Applications of graph labelings to encryption and decryption schemes are given in , , , and . According to Wang, B. Yao, and M. Yao , graph labelings are used for incorporating redundancy in disks, designing drilling ma-chines, creating layouts for circuit boards, and configuring resistor networks. In Hsieh, Chen, Jiang, Liaw, and Shin use graph labelings in algorithms for image processing schemes for monitoring air quality. Zhang, Ye, Zhang, and Yao investigated the use of graph colorings and graph labelings for designing topological passwords that resist attacks. Sivakumar, Vidyanandini, Sreedevi, Nayak, and Bhoi demonstrated how the electronic journal of combinatorics (2023), #DS6 7 the notion of total edge irregularity strength of complete tripartite graphs can be used in anti-theft networks. Nithya and Anitha investigated how graph labelings can be applied to the study of computer networks. In Medini, D’souza, Nayak, and new Bhat apply graph labelings and the complements of a graphs to create a new type of data-sharing mechanism.
Terms and notation not defined below follow that used in and .
the electronic journal of combinatorics (2023), #DS6 8 2 Graceful and Harmonious Labelings 2.1 Trees The Ringel-Kotzig conjecture (GTC) that all trees are graceful has been the focus of many papers. Kotzig has called the effort to prove it a “disease.” Among the trees known to be graceful are: caterpillars (a caterpillar is a tree with the property that the removal of its endpoints leaves a path); trees with at most 4 end-vertices , and ; trees with diameter at most 5 and ; symmetrical trees (i.e., a rooted tree in which every level contains vertices of the same degree) , , ; rooted trees where the roots have odd degree and the lengths of the paths from the root to the leaves differ by at most one and all the internal vertices have the same parity ; rooted trees with diameter D where every vertex has even degree except for one root and the leaves in level ⌊D/2⌋; rooted trees with diameter D where every vertex has even degree except for one root and the leaves, which are in level ⌊D/2⌋; rooted trees with diameter D where every vertex has even degree except for one root, the vertices in level ⌊D/2⌋−1, and the leaves which are in level ⌊D/2⌋; the graph obtained by identifying the endpoints any number of paths of a fixed length except for the case that the length has the form 4r + 1, r > 1 and the number of paths is of the form 4m with m > r ; regular bamboo trees (a rooted tree consisting of branches of equal length the endpoints of which are identified with end points of stars of equal size); and olive trees , (a rooted tree consisting of k branches, where the ith branch is a path of length i); Bahls, Lake, and Wertheim proved that spiders for which the lengths of every path from the center to a leaf differ by at most one are graceful. (A spider is a tree that has at most one vertex (called the center) of degree greater than 2.) Jampachon, Nakprasit, and Poomsa-ard provide graceful labelings for some classes of spiders. Panpa and Poomsa-ard showed that all spider graphs with at most four legs of lengths greater than one admit graceful labeling. In , , , , and Panda and Mishra and Panda, Mishra, and Dash give graceful labelings for some new classes of trees with diameter six. Pradhan and Kumar proved that all combs Pn ⊙K1 with perfect matching are graceful. In Varadhan and Guruswamy give a method for combining caterpillars in a specific way such that the resulting tree is graceful. Venkatesh1 and Balasubramanian also create graceful trees by recursively merging caterpillars. In 2006 Wilf and Yoshimura defined an ordering on the set of all rooted trees of a fixed number of vertices that leads to fast ranking and unranking algorithms. As an application to the graceful tree conjecture, they showed how their method can eliminate repeated isomorphism testing. They investigated graphs with at most 10 vertices. In 2022 Brankovic and Reynolds published a survey of various computer search algorithms for finding graceful labeling of trees.
In 2018 Montgomery, Pokrovskiy, and Sudakov proved that every tree is almost-harmonious. That is, every n-vertex tree has an injective Γ-harmonious labeling for any Abelian group Γ of order n + o(n). In 2022 Gnang posted a paper with a proof on arXiv of the Graceful Tree Conjecture. See for a newer proof by Gnang. In 2022 the electronic journal of combinatorics (2023), #DS6 9 Gnang and Williams posted a proof on arXiv of the long standing Graham-Sloane conjecture that every tree admits a harmonious labeling.
Motivated by Horton’s work , in 2010 Fang used a deterministic back-tracking algorithm to prove that all trees with at most 35 vertices are graceful. In 2011 Fang used a hybrid algorithm that involved probabilistic backtracking, tabu search-ing, and constraint programming satisfaction to verify that every tree with at most 31 vertices is harmonious. In Mahmoudzadeh and Eshghi treat graceful labelings of graphs as an optimization problem and apply an algorithm based on ant colony opti-mization metaheuristic to different classes of graphs and compare the results with those produced by other methods. In Suparta and Agus Ariawan provide two methods for expanding graceful trees from certain graceful trees.
Aldred, Širáň and Širáň have proved that the number of graceful labelings of Pn grows at least as fast as (5/3)n. They mention that this fact has an application to topological graph theory. One such application was provided by Goddyn, Richter, and and Širáň who used graceful labelings of paths on 2s + 1 vertices (s ≥2) to obtain 22s cyclic oriented triangular embeddings of the complete graph on 12s + 7 vertices. The Aldred, Širáň and Širáň bound was improved by Adamaszek to (2.37)n with the aid of a computer. Cattell has shown that when finding a graceful labeling of a path one has almost complete freedom to choose a particular label i for any given vertex v. In particular, he shows that the only cases of Pn when this cannot be done are when n ≡3 (mod 4) or n ≡1 (mod 12), v is in the smaller of the two partite sets of vertices, and i = (n −1)/2. In Wang enumerated the nonequivalent graceful trees and obtained a closed formula for the number.
Using an algorithm to run through all n! graceful graphs on n + 1 vertices Anick proves that the average number of graceful labelings grows super exponentially. He provides a simple criterion to predict which trees have an exceptionally large number of graceful labelings and gives evidence that trees with an exceptionally small number of graceful labelings fall into two already known families of caterpillar graphs. Over the full set of graceful labelings for a given n, Anick shows that the distribution of vertex degrees associated with each label is very close to Poisson, with the exception of labels 0 and n.
A graph is said to be k-ubiquitously graceful (also called ‘‘k-rotatable’’) if for every vertex there is a graceful labeling which assigns that vertex the label k. He also gives two new families of trees that are not k-ubiquitously graceful and includes questions suggested by his results. Pegg proved that a graceful graph with edges 0 to m can always be constructed with the nearest integer to p 3m + 9/4 + E vertices, where the excess E is a 0 or 1 value. For m < 51, E = 0. In and Eshghi and Azimi discuss a programming model for finding graceful labelings of large graphs. The computational results show that the models can easily solve the graceful labeling problems for large graphs. They used this method to verify that all trees with 30, 35, or 40 vertices are graceful. Stanton and Zarnke and Koh, Rogers, and Tan , , gave methods for combining graceful trees to yield larger graceful trees. In Wang, Yang, Hsu, and Cheng generalized the constructions of Stanton and Zarnke and Koh, Rogers, and Tan for building graceful trees from two the electronic journal of combinatorics (2023), #DS6 10 smaller given graceful trees. Rogers in and Koh, Tan, and Rogers in provide recursive constructions to create graceful trees. Burzio and Ferrarese have shown that the graph obtained from any graceful tree by subdividing every edge is also graceful.
and trees obtained from a graceful tree by replacing each edge with a path of fixed length is graceful.
The binomial tree B0 consists of a single vertex. The binomial tree Bk consists of two binomial trees Bk−1 that are linked together: the root of one is the leftmost child of the root of the other. Ragukumara and Sethuraman proved that all binomial trees are graceful. Sethuraman and Murugan introduced a new method of combining graceful trees called the recursive attachment method and showed that the recursively attached tree Ti = Ti−1 ⊕T Ai−1 is graceful for i ≥1, where the base tree T0 is a caterpillar and the attachment tree T Ai−1 is any caterpillar. Here Ti−1 ⊕T Ai−1 represents a tree obtained by attaching a copy of T Ai−1 at each vertex of degree at least two in Ti−1, for i ≥1. Sethuraman and Murugan proved that any acyclic graph can be embedded in an unicyclic graceful graph.
It 1999 Broersma and Hoede proved that an equivalent conjecture for the grace-ful tree conjecture is that all trees containing a perfect matching are strongly graceful (graceful with an extra condition also called an α-labeling–see Section 3.1). Wang, Yang, Hsu, and Cheng showed that there exist infinitely many equivalent versions of the graceful tree conjecture (GTC). They verify these equivalent conjectures of the graceful tree conjecture are true for trees of diameter at most 7.
In 1979 Bermond conjectured that lobsters are graceful (a lobster is a tree with the property that the removal of the endpoints leaves a caterpillar). Morgan has shown that all lobsters with perfect matchings are graceful. Krop proved that a lobster that has a perfect matching that covers all but one vertex (i.e., that has an almost perfect matching) is graceful. Ghosh used adjacency matrices to prove that three classes of lobsters are graceful. Broersma and Hoede proved that if T is a tree with a perfect matching M of T such that the tree obtained from T by contracting the edges in M is caterpillar, then T is graceful. Superdock used this result to prove that all lobsters with a perfect matching are graceful. Mishra, Panda, and Dash gave a class of graceful lobsters with an even number of branches incident on the central path.
They also provided graceful labelings for a family of lobsters in which one end vertex of the central path is attached to an even number of branches and the remaining vertices are attached to the combinations of branches.
Mishra and Panda and , Mishra and Bhattacharjee , and Mishra, Rout, and Nayak gave graceful labeling for a general classes of lobsters by applying component moving transformations on graceful caterpillars. More result results on graceful labeling of lobsters are in .
Sathiamoorthy, Natarajan, Ayyaswamy, and Janakiraman proved that the splitting graph of a caterpillar is graceful.
In Pakpahan, Mursidah, Novitasari, and Sugeng showed that a super caterpillar new constructed from several caterpillar graphs of the same size with each caterpillar having uniform pairs is graceful.
Simarmata, Sandy, and Sugeng used an adjacency new matrix to generalized the result of Pakpahan et al. by showing that a super caterpillar the electronic journal of combinatorics (2023), #DS6 11 constructed from several star graphs of different sizes is graceful.
A Skolem sequence of order n is a sequence s1, s2, . . . , s2n of 2n terms such that, for each k ∈{1, 2, . . . , n}, there exist exactly two subscripts i(k) and j(k) with si(k) = sj(k) = k and |i(k) −j(k)| = k. A Skolem sequence of order n exists if and only if n ≡0 or 1 (mod 4). Morgan has used Skolem sequences to construct classes of graceful trees.
Morgan and Rees used Skolem and Hooked-Skolem sequences to generate classes of graceful lobsters.
Mishra and Panigrahi and found classes of graceful lobsters of diameter at least five. They show other classes of lobsters are graceful in and . In Sethuraman and Jesintha explores how one can generate graceful lobsters from a graceful caterpillar while in and (see also ) they show how to generate graceful trees from a graceful star. More special cases of Bermond’s conjecture have been done by Ng , by Wang, Jin, Lu, and Zhang , Abhyanker , and by Mishra and Panigrahi . Renuka, Balaganesan, Selvaraju proved spider trees with n legs of even length t and odd n ≥3 and lobsters for which each vertex of the spine is adjacent to a path of length two are harmonious.
A tree in which all internal vertices have degrees r+1 except one, is called an full r-ary tree. A uniform full r-ary tree is a full r-ary tree in which all of its leaves are at the same level. A tree that is obtained from copies of a full r-ary tree by identifying each vertex of a fixed path with each vertex of the tree of degree r is called a uniform-distant tree.
Suparta and Ariawan gave methods for constructing graceful classes of caterpillars, lobsters, and uniform trees that generalize results in and .
Barrientos defines a y-tree as a graph obtained from a path by appending an edge to a vertex of a path adjacent to an end point. He proves that graphs obtained from a y-tree T by replacing every edge ei of T by a copy of K2,ni in such a way that the ends of ei are merged with the two independent vertices of K2,ni after removing the edge ei from T are graceful.
Sethuraman and Jesintha , , and (see also ) proved that rooted trees obtained by identifying one of the end vertices adjacent to either of the penultimate vertices of any number of caterpillars having equal diameter at least 3 with the property that all the degrees of internal vertices of all such caterpillars have the same parity are graceful. They also proved that rooted trees obtained by identifying either of the penul-timate vertices of any number of caterpillars having equal diameter at least 3 with the property that all the degrees of internal vertices of all such caterpillars have the same parity are graceful. In , , and (see also and ) Sethuraman and Jesintha prove that all rooted trees in which every level contains pendent vertices and the degrees of the internal vertices in the same level are equal are graceful. Kanetkar and Sane show that trees formed by identifying one end vertex of each of six or fewer paths whose lengths determine an arithmetic progression are graceful.
Chen, Lü, and Yeh define a firecracker as a graph obtained from the concatenation of stars by linking one leaf from each. They also define a banana tree as a graph obtained by connecting a vertex v to one leaf of each of any number of stars (v is not in any of the stars). They proved that firecrackers are graceful and conjecture that banana trees are the electronic journal of combinatorics (2023), #DS6 12 graceful. Before Sethuraman and Jesintha and (see also ) proved that all banana trees and extended banana trees (graphs obtained by joining a vertex to one leaf of each of any number of stars by a path of length of at least two) are graceful, various kinds of bananas trees had been shown to be graceful by Bhat-Nayak and Deshmukh , by Murugan and Arumugam , and by Vilfred .
Consider a set of caterpillars, having equal diameter, in which one of the penultimate vertices has arbitrary degree and all the other internal vertices including the other penul-timate vertex is of fixed even degree. Jesintha and Sethuraman call the rooted tree obtained by merging an end-vertex adjacent to the penultimate vertex of fixed even degree of each caterpillar a arbitrarily fixed generalized banana tree. They prove that such trees are graceful. From this it follows that all banana trees are graceful and all generalized banana trees are graceful.
Jeba Jesintha and Subashini proved the following graphs are graceful: the cycle of vertex switching of even cycles ; the path union of vertex switching of odd cycles ; the path union of vertex switching of even cycles in increasing order ; the path union of vertex switching of odd cycles the path union of vertex switching of even cycles ; the path union of PmθSn ; and the cycle of PmθSn .
In they prove the cycle of caterpillar trees is graceful and as a corollary the cycle of comb graphs, cycle of paths, and the cycle of coconut trees are graceful. In Jeba Jasintha and Subashini proved the two quadrilateral snake graphs connected by a path, two alternate quadrilateral snake graphs connected by a path, two double quadrilateral snake graphs connected by a path, and two alternate double quadrilateral snake graphs connected by a path, In Jeba Jesintha, Subashini, and Rashmi Beula proved that the series of isomorphic copies of a star graph connected between two ladder graphs is graceful. In Jeba Jesintha, Subashini, and Sabu proved that two complete bipartite graphs connected by an arbitrary path of length n is graceful. In Jeba Jesintha and Subashini, and Sabu proved that the twig diamond graph with pendant edges is graceful.
In Jeba Jesintha, Subashini, and Siddiqa proved that the path union of vertex switching of odd and even cycle graphs alternately is graceful.
Zhenbin has shown that graphs obtained by starting with any number of identi-cal stars, appending an edge to exactly one edge from each star, then joining the vertices at which the appended edges were attached to a new vertex are graceful. He also shows that graphs obtained by starting with any two stars, appending an edge to exactly one edge from each star, then joining the vertices at which the appended edges were attached to a new vertex are graceful. In Jesintha and Sethuraman use a method of Hrnciar and Havier to generate graceful trees from a graceful star with n edges.
Aldred and McKay used a computer to show that all trees with at most 26 vertices are harmonious. That caterpillars are harmonious was by Graham and Sloane .
Ramya and Meenakshi gave graceful labelings, harmonious labelings, and Zumkeller labelings for ladders, banana trees, and firecrackers.
Vietri utilized a counting technique that generalizes Rosa’s graceful parity con-dition and provides constraints on possible graceful labelings of certain classes of trees.
He expresses doubts about the validity of the graceful tree conjecture. In Vietri the electronic journal of combinatorics (2023), #DS6 13 introduced a family of homogeneous polynomials (mod 2), one for every degree, having as many variables as the number of vertices, for any fixed graph; a so-called “graceful poly-nomial” that vanishes (mod 2) that may be useful for proving that the related graph is non-graceful (the degree 1 case dates back to Rosa’s work). He also classified graphs whose graceful polynomials vanish for degrees 2 to 4, thereby obtaining some new non-graceful graphs.
Using a variant of the Matrix Tree Theorem, Whitty specifies an n × n matrix of indeterminants whose determinant is a multivariate polynomial that enumerates the gracefully labeled (n + 1)-vertex trees. Whitty also gives a bijection between gracefully labelled graphs and rook placements on a chessboard on the Möbius strip. In Buratti, Rinaldi, and Traetta use graceful labelings of paths to obtain a result on Hamiltonian cycle systems.
In Brankovic and Wanless describe applications of graceful and graceful-like labelings of trees to several well known combinatorial problems including complete graph decompositions, the Oberwolfach problem, which asks for a decomposition of Kv into copies of a given 2-regular graph F, and coloring.
They also discuss the connection between α-labeling of paths and near transversals in Latin squares and show how spectral graph theory might be used to further the progress on the graceful tree conjecture.
In Burgess, Danziger, and Traetta show that Oberwolfach problem has a solution whenever F has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of F. Furthermore, they prove analogous results for the minimum covering version and the maximum packing version of the problem. They also show a similar result when the edges of Kv have multiplicity 2, but in this case they do not require that F be single-flip. Their approach allows them to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms. Their constructions use graceful labelings of 2-regular graphs with a vertex removed. They show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an α-labeling of such graphs to exist.
Arkut, Arkut, and Basak and Basak proposed an efficient method for managing Internet Protocol (IP) networks by using graceful labelings of the nodes of the spanning caterpillars of the autonomous sub-networks to assign labels to the links in the sub-networks. Graceful labelings of trees also have been used in multi protocol label switching (MPLS) routing platforms in IP networks , , and .
Despite the efforts of many, the graceful tree conjecture remains open even for trees with maximum degree 3. More specialized results about trees are contained in , , , , , , and . In Edwards and Howard provide a lengthy survey paper on graceful trees. Robeva provides an extensive survey of graceful la-belings of trees in her 2011 undergraduate honors thesis at Stanford University. Alfalayleh, Brankovic, Giggins, and Islam survey results related to the graceful tree conjecture as of 2004 and conclude with five open problems. Alfalayleh et al.: say “The faith in the [graceful tree] conjecture is so strong that if a tree without a graceful labeling were the electronic journal of combinatorics (2023), #DS6 14 indeed found, then it probably would not be considered a tree.” In his Princeton Univer-sity senior thesis Superdock provided an extensive survey of results and techniques about graceful trees. He also obtained some specialized results about the gracefulness of spiders and trees with diameter 6. Arumugam and Bagga discuss computational efforts aimed at verifying the graceful tree conjecture and we survey recent results on generating all graceful labelings of certain families of unicyclic graphs. Sethuraman and Murugan construct a graceful unicyclic graph G from every graceful tree T with V (G) = V (T) such that the graceful labeling of G is derived from the graceful labeling of T.
2.2 Cycle-Related Graphs Cycle-related graphs have been a major focus of attention. Rosa showed that the n-cycle Cn is graceful if and only if n ≡0 or 3 (mod 4) and Graham and Sloane proved that Cn is harmonious if and only if n is odd. Wheels Wn = Cn+K1 are both graceful and harmonious – , , and . As a consequence we have that a subgraph of a graceful (harmonious) graph need not be graceful (harmonious). The n-cone (also called the n-point suspension; the 1-cone is the wheel; the 2-cone is also called a double cone of Cm) Cm + Kn has been shown to be graceful when m ≡0 or 3 (mod 12) by Bhat-Nayak and Selvam . When n is even and m is 2, 6 or 10 (mod 12) Cm+Kn violates the parity condition for a graceful graph. Bhat-Nayak and Selvam also prove that the following cones are graceful: C4+Kn, C5+K2, C7+Kn, C9+K2, C11+Kn and C19+Kn. The helm Hn is the graph obtained from a wheel by attaching a pendent edge at each vertex of the n-cycle. Helms have been shown to be graceful and harmonious , , (see also , , , , ), and .
new Koh, Rogers, Teo, and Yap, define a web graph as one obtained by joining the pendent points of a helm to form a cycle and then adding a single pendent edge to each vertex of this outer cycle. They asked whether such graphs are graceful. This was proved by Kang, Liang, Gao, and Yang . Yang has extended the notion of a web by iterating the process of adding pendent points and joining them to form a cycle and then adding pendent points to the new cycle. In his notation, W(2, n) is the web graph whereas W(t, n) is the generalized web with t n-cycles. Yang has shown that W(3, n) and W(4, n) are graceful (see ), Abhyanker and Bhat-Nayak have done W(5, n) and Abhyanker has done W(t, 5) for 5 ≤t ≤13. Gnanajothi has shown that webs with odd cycles are harmonious. Seoud and Youssef define a closed helm as the graph obtained from a helm by joining each pendent vertex to form a cycle and a flower as the graph obtained from a helm by joining each pendent vertex to the central vertex of the helm. They prove that closed helms and flowers are harmonious when the cycles are odd. A gear graph is obtained from the wheel Wn by adding a vertex between every pair of adjacent vertices of the n-cycle. In 1984 Ma and Feng proved all gears are graceful while in a Master’s thesis in 2006 Chen proved all gears are harmonious. Liu has shown that if two or more vertices are inserted between every pair of vertices of the n-cycle of the wheel Wn, the resulting graph is graceful. Sethuraman and Sankar the electronic journal of combinatorics (2023), #DS6 15 showed that the subdivisions of wheels are graceful for even values of n ≥4. Liu has also proved that the graph obtained from a gear graph by attaching one or more pendent edges to each vertex between the vertices of the n-cycle is graceful. Pradhan and Kumar proved that graphs obtained by adding a pendent edge to each pendent vertex of hairy cycle Cn ⊙K1 are graceful if n ≡0 (mod 4m). They further provide a rule for determining the missing numbers in the graceful labeling of Cn ⊙K1 and of the graph obtained by adding pendent edges to each pendent vertex of Cn ⊙K1.
Kumar, Mishra, Kumar, and Kumar proved the following: Cn ⊙K1, n ≡0 (mod 4) possesses an alpha labeling with the missing number 3n/2; the one-point union of C4n and a path possesses an alpha labeling with an identifiable missing number; and the graphs obtained by joining two isomorphic copies of the one-point union of C4n and a path posses an alpha labeling with identifiable missing numbers.
Abhyanker has investigated various unicyclic (that is, graphs with exactly one cycle) graphs. He proved that the unicyclic graphs obtained by identifying one vertex of C4 with the root of the olive tree with 2n branches and identifying an adjacent vertex on C4 with the end point of the path P2n−2 are graceful. He showed that if one attaches any number of pendent edges to these unicyclic graphs at the vertex of C4 that is adjacent to the root of the olive tree but not adjacent to the end vertex of the attached path, the resulting graphs are graceful. Likewise, Abhyanker proved that the graph obtained by deleting the branch of length 1 from an olive tree with 2n branches and identifying the root of the edge deleted tree with a vertex of a cycle of the form C2n+3 is graceful. He also has a number of results similar to these. In Bagga, Fotso, Max, and Arumugam investigate the gracefulness of unicyclic graphs with pendent caterpillars at two adjacent vertices of the cycle, and pendent edges at some other vertices of the cycle. In Bagga and Heinz give some properties of graceful graphs obtained by adding pendent edges at each vertex of a cycle.
Delorme, Maheo, Thuillier, Koh, and Teo and Ma and Feng showed that any cycle with a chord is graceful. This was first conjectured by Bodendiek, Schumacher, and Wegner , who proved various special cases. In 1985 Koh and Yap gener-alized this by defining a cycle with a Pk-chord to be a cycle with the path Pk joining two nonconsecutive vertices of the cycle. They proved that these graphs are graceful when k = 3 and conjectured that all cycles with a Pk-chord are graceful. This was proved for k ≥4 by Punnim and Pabhapote in 1987 . Chen obtained the same result except for three cases which were then handled by Gao . In 2005, Sethuraman and Elumalai defined a cycle with parallel Pk-chords as a graph obtained from a cycle Cn (n ≥6) with consecutive vertices v0, v1, . . . , vn−1 by adding disjoint paths Pk, (k ≥3), between each pair of nonadjacent vertices v1, vn−1, v2, vn−2, . . . , vi, vn−i, . . . , vα, vβ where α = ⌊n/2⌋−1 and β = ⌊n/2⌋+ 2 if n is odd or β = ⌊n/2⌋+ 1 if n is even. They proved that every cycle Cn (n ≥6) with parallel Pk-chords is graceful for k = 3, 4, 6, 8, and 10 and they conjecture that the cycle Cn with parallel Pk-chords is graceful for all even k. Xu proved that all cycles with a chord are harmonious except for C6 in the case where the distance in C6 between the endpoints of the chord is 2. The gracefulness of cycles with consecutive chords has also been investigated. For 3 ≤p ≤n −r, let the electronic journal of combinatorics (2023), #DS6 16 Cn(p, r) denote the n-cycle with consecutive vertices v1, v2, . . . , vn to which the r chords v1vp, v1vp+1, . . . , v1vp+r−1 have been added. Koh and Punnin and Koh, Rogers, Teo, and Yap have handled the cases r = 2, 3 and n −3 where n is the length of the cycle. Goh and Lim then proved that all remaining cases are graceful. Moreover, Ma has shown that Cn(p, n−p) is graceful when p ≡0, 3 (mod 4) and Ma, Liu, and Liu have proved other special cases of these graphs are graceful. Ma also proved that if one adds to the graph Cn(3, n −3) any number ki of paths of length 2 from the vertex v1 to the vertex vi for i = 2, . . . , n, the resulting graph is graceful. Chen has shown that apart from four exceptional cases, a graph consisting of three independent paths joining two vertices of a cycle is graceful. This generalizes the result that a cycle plus a chord is graceful. Liu has shown that the n-cycle with consecutive vertices v1, v2, . . . , vn to which the chords v1vk and v1vk+2 (2 ≤k ≤n−3) are adjoined is graceful.
For the cycle Cn : v1v2v3 · · · vnv1 and a cycle with a Ck−chord Venkatesh and Sivagu-runathan let Cn,k denote the graph obtained from Cn by adding a cycle Ck of length k between the non-adjacent vertices v2 and vn. They define a cycle with a par-allel Ck chord as the graph obtained from a cycle Cn by adding a cycle Ck of length k between every pair of non-adjacent vertices (v2, vn), (v3, vn−1), . . . , (va, vb) where a = ⌊n 2⌋, b = ⌊n 2⌋+ 2, if n is even and a = ⌊n 2⌋, b = ⌊n 2⌋+ 3, if n is odd. They proved that Cn,4 and C+ n,4 are graceful for n ≡0 (mod 4) and that C+ n,6 is graceful for all odd values of n ≥5.
In Deb and Limaye use the notation C(n, k) to denote the cycle Cn with k cords sharing a common endpoint called the apex. For certain choices of n and k there is a unique C(n, k) graph and for other choices there is more than one graph possible. They call these shell-type graphs and they call the unique graph C(n, n −3) a shell. Notice that the shell C(n, n −3) is the same as the fan Fn−1 = Pn−1 + K1. Kuppusamy and Guruswamy show that the subdivision graph of K2,n is graceful for n ≥1 and the subdivision graph of the shell graph C(n, n −3) is graceful for n ≥4. Deb and Limaye define a multiple shell to be a collection of edge disjoint shells that have their apex in common. A multiple shell is said to be balanced with width w if every shell has order w or every shell has order w or w +1. Deb and Limaye have conjectured that all multiple shells are harmonious, and have shown that the conjecture is true for the balanced double shells and balanced triple shells. Yang, Xu, Xi, and Qiao proved the conjecture is true for balanced quadruple shells. Liang proved the conjecture is true when each shell has the same order and the number of copies is odd.
Jeba Jesintha and Hilda define a shell-butterfly graph as a one-point union of two shells of any order with two pendent edges at the apex. They prove that certain shell-butterfly graphs are harmonious. Jeba Jesintha and Ezhilarasi Hilda proved butterfly graphs with one shell of order m and the other shell of order 2m+1 are graceful and double shells in which each shell has the same order are graceful. Jeba Jesintha and Hilda define a bow graph as a double shell in which each shell has arbitrary order.
A bow graph in which each shell has the same order is called a uniform bow graph. They prove that all uniform bow graphs are graceful. Jeba Jesintha and Ezhilarasi Hilda proved that shell-butterfly graphs are graceful. In Jeba Jesintha and Hilda prove k copies of C(4, 1) ∪K2, and shellflowers (a double shell with shells of order m and 2m) the electronic journal of combinatorics (2023), #DS6 17 are graceful.
In Haviar and Kurtulík defined a k-enriched fan graph kFn, for integers k, n ≥2, as the graph of size (k + 1)n −1 obtained by connecting n copies of the star Sk of order k to the fan Fn such that one vertex of each copy of the star Sk is identified with one vertex of the main path Pn of Fn. They proved that k-enriched fan graphs are graceful and and provided characterizations of the k-enriched fan graphs among all simple graphs via Sheppard’s labeling sequences introduced in the 1970s, as well as via labeling relations and graph chessboards.
Sethuraman and Dhavamani use H(n, t) to denote the graph obtained from the cycle Cn by adding t consecutive chords incident with a common vertex. If the common vertex is u and v is adjacent to u, then for k ≥1, n ≥4, and 1 ≤t ≤n −3, Sethuraman and Dhavamani denote by G(n, t, k) the graph obtained by taking the union of k copies of H(n, k) with the edge uv identified. They conjecture that every graph G(n, t, k) is graceful. They prove the conjecture for the case that t = n −3.
For i = 1, 2, . . . , n let vi,1, vi,2, . . . , vi,2m be the successive vertices of n copies of C2m.
Sekar defines a chain of cycles C2m,n as the graph obtained by identifying vi,m and vi+1,m for i = 1, 2, . . . , n −1. He proves that C6,2k and C8,n are graceful for all k and all n. Barrientos proved that all C8,n, C12,n, and C6,2k are graceful.
Truszczyński studied unicyclic graphs and proved several classes of such graphs are graceful. Among these are what he calls dragons. A dragon is formed by joining the end point of a path to a cycle (Koh, et al. call these tadpoles; Kim and Park call them kites). This work led Truszczyński to conjecture that all unicyclic graphs except Cn, where n ≡1 or 2 (mod 4), are graceful. Guo has shown that dragons are graceful when the length of the cycle is congruent to 1 or 2 (mod 4). Lu uses C+(m,t) n to denote the graph obtained by identifying one vertex of Cn with one endpoint of m paths each of length t. He proves that C+(1,t) n (a tadpole) is not harmonious when a + t is odd and C+(2m,t) n is harmonious when n = 3 and when n = 2k+1 and t = k−1, k+1 or 2k−1.
In his Master’s thesis, Doma investigates the gracefulness of various unicyclic graphs where the cycle has up to 9 vertices. Guruswamy and Varadhan proved that any acyclic graph can be embedded in a unicyclic graceful graph. Because of the immense diversity of unicyclic graphs, a proof of Truszczyński’s conjecture seems out of reach in the near future. In Biatch, Baggab, and Arumugam gave a survey of results related to Truszczynski’s conjecture on the gracefulness of unicyclic graphs and provided a new class of graceful unicyclic graphs.
Cycles that share a common edge or a vertex have received some attention. Murugan and Arumugan have shown that books with n pentagonal pages (i.e., n copies of C5 with an edge in common) are graceful when n is even and not graceful when n is odd.
Lu uses Θ(Cm)n to denote the graph made from n copies of Cm that share an edge (an n page book with m-polygonal pages). He proves Θ(C2m+1)2n+1 is harmonious for all m and n; Θ(C4m+2)4n+1 and Θ(C4m)4n+3 are not harmonious for all m and n. Xu proved that Θ(Cm)2 is harmonious except when m = 3. (Θ(Cm)2 is isomorphic to C2(m−1) with a chord “in the middle.”) Nurvazly and Sugeng proved that Θ(C3)n graphs (n copies of C3 that share an edge) have graceful labelings.
the electronic journal of combinatorics (2023), #DS6 18 A kayak paddle KP(k, m, l) is the graph obtained by joining Ck and Cm by a path of length l. Litersky proves that kayak paddles have graceful labelings in the following cases: k ≡0 mod 4, m ≡0 or 3 (mod 4); k ≡m ≡2 (mod 4) for k ≥3; and k ≡1 (mod 4), m ≡3 (mod 4). She conjectures that KP(4k + 4, 4m + 2, l) with 2k < m is graceful when l ≤2m if l is even and when l ≤2m + 1 if l is odd; and KP(10, 10, l) is graceful when l ≥12. The cases are open: KP(4k, 4m + 1, l); KP(4k, 4m + 2, l); KP(4k + 1, 4m + 1, l); KP(4k + 1, 4m + 2, l); KP(4k + 2, 4m + 3, l); KP(4k + 3, 4m + 3, l).
Let C(t) n denote the one-point union of t cycles of length n. Bermond, Brouwer, and Germa and Bermond, Kotzig, and Turgeon ) proved that C(t) 3 (that is, the friendship graph or Dutch t-windmill) is graceful if and only if t ≡0 or 1 (mod 4) while Graham and Sloane proved C(t) 3 is harmonious if and only if t ̸≡2 (mod 4). Koh, Rogers, Lee, and Toh conjecture that C(t) n is graceful if and only if nt ≡0 or 3 (mod 4). Yang and Lin have proved the conjecture for the case n = 5 and Yang, Xu, Xi, Li, and Haque did the case n = 7. Xu, Yang, Li and Xi did the case n = 11. Xu, Yang, Han and Li did the case n = 13. Qian verifies this conjecture for the case that t = 2 and n is even and Yang, Xu, Xi, and Li did the case n = 9. Figueroa-Centeno, Ichishima, and Muntaner-Batle have shown that if m ≡0 (mod 4) then the one-point union of 2, 3, or 4 copies of Cm admits a special kind of graceful labeling called an α-labeling (see Section 3.1) and if m ≡2 (mod 4), then the one-point union of 2 or 4 copies of Cm admits an α-labeling. Bodendiek, Schumacher, and Wegner proved that the one-point union of any two cycles is graceful when the number of edges is congruent to 0 or 3 modulo 4. (The other cases violate the necessary parity condition.) Shee has proved that C(t) 4 is graceful for all t.
Seoud and Youssef have shown that the one-point union of a triangle and Cn is harmonious if and only if n ≡1 (mod 4) and that if the one-point union of two cycles is harmonious then the number of edges is divisible by 4. The question of whether this latter condition is sufficient is open. Figueroa-Centeno, Ichishima, and Muntaner-Batle have shown that if G is harmonious then the one-point union of an odd number of copies of G using the vertex labeled 0 as the shared point is harmonious. Sethuraman and Selvaraju have shown that for a variety of choices of points, the one-point union of any number of non-isomorphic complete bipartite graphs is graceful. They raise the question of whether this is true for all choices of the common point.
Another class of cycle-related graphs is that of triangular cacti. The block-cutpoint graph of a graph G is a bipartite graph in which one partite set consists of the cut vertices of G, and the other has a vertex bi for each block Bi of G. A block of a graph is a maximal connected subgraph that has no cut-vertex. A triangular cactus is a connected graph all of whose blocks are triangles. A triangular snake is a triangular cactus whose block-cutpoint-graph is a path (a triangular snake is obtained from a path v1, v2, . . . , vn by joining vi and vi+1 to a new vertex wi for i = 1, 2, . . . , n−1). Rosa conjectured that all triangular cacti with t ≡0 or 1 (mod 4) blocks are graceful. (The cases where t ≡2 or 3 (mod 4) fail to be graceful because of the parity condition.) Moulton proved the conjecture for all triangular snakes. A proof of the general case (i.e., all triangular cacti) seems hopelessly difficult. Liu and Zhang gave an incorrect proof that triangular snakes the electronic journal of combinatorics (2023), #DS6 19 with an odd number of triangles are harmonious whereas triangular snakes with n ≡2 (mod 4) triangles are not harmonious. Xu subsequently proved that triangular snakes are harmonious if and only if the number of triangles is not congruent to 2 (mod 4).
A double triangular snake consists of two triangular snakes that have a common path.
That is, a double triangular snake is obtained from a path v1, v2, . . . , vn by joining vi and vi+1 to a new vertex wi for i = 1, 2, . . . , n−1 and to a new vertex ui for i = 1, 2, . . . , n−1.
Xi, Yang, and Wang proved that all double triangular snakes are harmonious.
A hexagonal snake is obtained from a path p1, p2, p3, . . . , pn by joining pi, pi+1 to new vertices xi and yi respectively and adding edges xiyi for i = 1, 2, . . . , n −1 and replacing every edge with a 6-cycle; an alternate hexagonal snake is obtained from a path p1, p2, p3, . . . , pn by joining pi, pi+1 to new vertices xi and yi (alternatively) and adding edges xiyi, where 1 ≤i ≤n −1 for even n and 1 ≤i ≤n −2 for odd n and replacing each alternate edge with a 6-cycle; a double hexagonal snake is obtained from two hexagonal snakes that share the n-path; a double alternate hexagonal snake is obtained from two alternative hexagonal snakes that share the n-path.
Pattabiraman, Loganathan, and Rao provided graceful labelings for double hexagonal snakes, alternate hexagonal snakes, odd alternate hexagonal snakes, and double alternate hexagonal snakes.
For any graph G defining G-snake analogous to triangular snakes, Sekar has shown that Cn-snakes are graceful when n ≡0 (mod 4) (n ≥8) and when n ≡2 (mod 4) and the number of Cn is even. Gnanajothi [1143, pp. 31-34] had earlier shown that quadrilateral snakes are graceful. Grace has proved that K4-snakes are harmonious.
Rosa has also considered analogously defined quadrilateral and pentagonal cacti and examined small cases. Yu, Lee, and Chin showed that Q2-snakes and Q3-snakes are graceful and, when the number of blocks is greater than 1, Q2-snakes, Q3-snakes and Q4-snakes are harmonious.
Barrientos calls a graph a kCn-snake if it is a connected graph with k blocks whose block-cutpoint graph is a path and each of the k blocks is isomorphic to Cn. (When n > 3 and k > 3 there is more than one kCn-snake.) If a kCn-snake where the path of minimum length that contains all the cut-vertices of the graph has the property that the distance between any two consecutive cut-vertices is ⌊n/2⌋it is called linear. Barrientos proves that kC4-snakes are graceful and that the linear kC6-snakes are graceful when k is even. He further proves that kC8-snakes and kC12-snakes are graceful in the cases where the distances between consecutive vertices of the path of minimum length that contains all the cut-vertices of the graph are all even and that certain cases of kC4n-snakes and kC5n-snakes are graceful (depending on the distances between consecutive vertices of the path of minimum length that contains all the cut-vertices of the graph).
Badr defines a linear cyclic snake (m, k)Cn as the graph consisting of k copies of Cn with two non-adjacent vertices in common where every copy has m copies of Cn and the block-cutpoint graph is not a path. He proves that the linear cyclic snakes (m, k)C4-snake and (m, k)C8-snake are graceful and conjectures that all the linear cyclic snakes (m, k)Cn-snakes are graceful for n ≡0 (mod 4 ) or n ≡3 (mod 4).
Several people have studied cycles with pendent edges attached. Frucht proved the electronic journal of combinatorics (2023), #DS6 20 that any cycle with a pendent edge attached at each vertex (i.e., a crown) is graceful (see also ). If G has order n, the corona of G with H, G ⊙H is the graph obtained by taking one copy of G and n copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H. Barrientos also proved: if G is a graceful graph of order m and size m −1, then G ⊙nK1 and G + nK1 are graceful; if G is a graceful graph of order p and size q with q > p, then (G ∪(q + 1 −p)K1) ⊙nK1 is graceful; and all unicyclic graphs, other than a cycle, for which the deletion of any edge from the cycle results in a caterpillar are graceful.
For a given cycle Cn with n ≡0 or 3 (mod 4) and a family of trees T = {T1, T2, . . . , Tn}, let ui and vi, 1 ≤i ≤n, be fixed vertices of Cn and Ti, respectively. Figueroa-Centeno, Ichishima, Muntaner-Batle, and Oshima provide two construction methods that gen-erate a graceful labeling of the unicyclic graphs obtained from Cn and T by amalgamating them at each ui and vi. Their results encompass all previously known results for unicyclic graphs whose cycle length is 0 or 3 (mod 4) and considerably extend the known classes of graceful unicyclic graphs. Khairunnisa and Sugeng let A(m,n) denote the graph obtained from Cm by connecting each two adjacent vertices with Pn+1. They prove that the graphs A(3,1) ⊙Kr are graceful.
In Barrientos proved that helms (graphs obtained from a wheel by attaching one pendent edge to each vertex) are graceful. Grace showed that an odd cycle with one or more pendent edges at each vertex is harmonious and conjectured that C2n⊙K1, an even cycle with one pendent edge attached at each vertex, is harmonious. This conjecture has been proved by Liu and Zhang , Liu and , Hegde , Huang , and Bu . Sekar has shown that the graph Cm ⊙Pn obtained by attaching the path Pn to each vertex of Cm is graceful. For any n ≥3 and any t with 1 ≤t ≤n, let C+t n denote the class of graphs formed by adding a single pendent edge to t vertices of a cycle of length n. Ropp proved that for every n and t the class C+t n contains a graceful graph.
Gallian and Ropp conjectured that for all n and t, all members of C+t n are graceful.
This was proved by Qian and by Kang, Liang, Gao, and Yang . Of course, such graphs are just a special case of the aforementioned conjecture of Truszczyński that all unicyclic graphs except Cn for n ≡1 or 2 (mod 4) are graceful. Sekar proved that the graph obtained by identifying an endpoint of a star with a vertex of a cycle is graceful. Lu shows that the graph obtained by identifying each vertex of an odd cycle with a vertex disjoint copy of C2m+1 is harmonious if and only if m is odd.
Sudha proved that the graphs obtained by starting with two or more copies of C4 and identifying a vertex of the ith copy with a vertex of the i + 1th copy and the graphs obtained by starting with two or more cycles (not necessarily of the same size) and identifying an edge from the ith copy with an edge of the i + 1th copy are graceful.
Sudha and Kanniga proved that the graphs obtained by identifying any vertex of Cm with any vertex of degree 1 of Sn where n = ⌈(m −1)/2⌉are graceful.
For a given cycle Cn with n ≡0 or 3 (mod 4) and a family of trees T = {T1, T2, . . . , Tn}, let ui and vi, 1 ≤i ≤n, be fixed vertices of Cn and Ti, respectively. Figueroa-Centeno, Ichishima, Muntaner-Batle, and Oshima provide two construction methods that gen-erate a graceful labeling of the unicyclic graphs obtained from Cn and T by amalgamating the electronic journal of combinatorics (2023), #DS6 21 them at each ui and vi. Their results encompass all previously known results for unicyclic graphs whose cycle length is 0 or 3 (mod 4) and considerably extend the known classes of graceful unicyclic graphs.
Solairaju and Chithra defined three classes of graphs obtained by connecting copies of C4 in various ways.
Denote the four consecutive vertices of ith copy of C4 by vi,1, vi,2, vi,3, vi4. They show that the graphs obtained by identifying vi,4 with vi+1,2 for i = 1, 2, . . . , n −1 is graceful; the graphs obtained by joining vi,4 with vi+1,2 for i = 1, 2, . . . , n −1 by an edge is graceful; and the graphs obtained by joining vi,4 with vi+1,2 for i = 1, 2, . . . , n −1 with a path of length 2 is graceful.
Venkatesh showed that for positive integers m and n divisible by 4 the graphs obtained by appending a copy of Cn to each vertex of Cm by identifying one vertex of Cn with each vertex of Cm is graceful.
2.3 Product Related Graphs Graphs that are Cartesian products and related graphs have been the subject of many papers. That planar grids, Pm × Pn (m, n ≥2), (many authors use G □H to denote the Cartesian product of G and H) are graceful was proved by Acharya and Gill in 1978. In 1980, Maheo clarified the complicated-appearing construction of Acharya and Gill for Pm × P2 that readily extends to all grids. Liu, T. Zou, Y. Lu proved Pm × Pn × P2 is graceful. In 1980 Graham and Sloane proved ladders, Pm × P2, are harmonious when m > 2 and in 1992 Jungreis and Reid showed that the grids Pm × Pn are harmonious when (m, n) ̸= (2, 2).
A few people have looked at graphs obtained from planar grids in various ways. Kathiresan has shown that graphs obtained from ladders by subdividing each step exactly once are graceful and that graphs obtained by appending an edge to each vertex of a ladder are graceful . Barrientos and Minion showed that a graceful graph is obtained when every step of a ladder is subdivided an even number of times. In addition, they proved that when each edge of a ladder is subdivided exactly once, the resulting graph is graceful.
Acharya has shown that certain subgraphs of grid graphs are graceful. Lee defines a Mongolian tent as a graph obtained from Pm × Pn, n odd, by adding one extra vertex above the grid and joining every other vertex of the top row of Pm × Pn to the new vertex. A Mongolian village is a graph formed by successively amalgamating copies of Mongolian tents with the same number of rows so that adjacent tents share a column.
Lee proves that Mongolian tents and villages are graceful. A Young tableau is a subgraph of Pm × Pn obtained by retaining the first two rows of Pm × Pn and deleting vertices from the right hand end of other rows in such a way that the lengths of the successive rows form a nonincreasing sequence. Lee and Ng have proved that all Young tableaus are graceful. Lee has also defined a variation of Mongolian tents by adding an extra vertex above the top row of a Young tableau and joining every other vertex of that row to the extra vertex. He proves these graphs are graceful. In and Solairaju and Arockiasamy prove that various families of subgraphs of grids Pm × Pn are graceful.
Sudha proved that certain subgraphs of the grid Pn ×P2 are graceful. Knuth the electronic journal of combinatorics (2023), #DS6 22 proved that Kn × P3 is graceful if and only if n ≤6.
Prisms are graphs of the form Cm × Pn. These can be viewed as grids on cylinders.
In 1977 Bodendiek, Schumacher, and Wegner proved that Cm × P2 is graceful when m ≡0 (mod 4). According to the survey by Bermond , Gangopadhyay and Rao Hebbare did the case that m is even about the same time. In a 1979 paper, Frucht stated without proof that he had done all Cm × P2. A complete proof of all cases and some related results were given by Frucht and Gallian in 1988.
In 1992 Jungreis and Reid proved that all Cm × Pn are graceful when m and n are even or when m ≡0 (mod 4). They also investigated the existence of a stronger form of graceful labeling called an α-labeling (see Section 3.1) for graphs of the form Pm × Pn, Cm × Pn, and Cm × Cn (see also ).
Yang and Wang have shown that the prisms C4n+2 ×P4m+3 , Cn ×P2 , and C6 ×Pm (m ≥2) (see ) are graceful. Singh proved that C3 ×Pn is graceful for all n. In their 1980 paper Graham and Sloane proved that Cm × Pn is harmonious when n is odd and they used a computer to show C4 ×P2, the cube, is not harmonious. In 1992 Gallian, Prout, and Winters proved that Cm × P2 is harmonious when m ̸= 4.
In 1992, Jungreis and Reid showed that C4 ×Pn is harmonious when n ≥3. Huang and Skiena have shown that Cm × Pn is graceful for all n when m is even and for all n with 3 ≤n ≤12 when m is odd. Abhyanker proved that the graphs obtained from C2m+1 × P5 by adding a pendent edge to each vertex of an outer cycle is graceful.
Torus grids are graphs of the form Cm × Cn (m > 2, n > 2). Very little success has been achieved with these graphs. The graceful parity condition is violated for Cm × Cn when m and n are odd and the harmonious parity condition [1190, Theorem 11] is violated for Cm × Cn when m ≡1, 2, 3 (mod 4) and n is odd. In 1992 Jungreis and Reid showed that Cm × Cn is graceful when m ≡0 (mod 4) and n is even. A complete solution to both the graceful and harmonious torus grid problems will most likely involve a large number of cases.
There has been some work done on prism-related graphs. Gallian, Prout, and Winters proved that all prisms Cm×P2 with a single vertex deleted or single edge deleted are graceful and harmonious. The Möbius ladder Mn is the graph obtained from the ladder Pn × P2 by joining the opposite end points of the two copies of Pn. In 1989 Gallian showed that all Möbius ladders are graceful and all but M3 are harmonious. Ropp has examined two classes of prisms with pendent edges attached. He proved that all Cm × P2 with a single pendent edge at each vertex are graceful and all Cm × P2 with a single pendent edge at each vertex of one of the cycles are graceful. Ramachandran and Sekar proved that the graph obtained from the ladder Ln (Pn × P2) by identifying one vertex of Ln with any vertex of the star Sm other than the center of Sm is graceful.
Another class of Cartesian products that has been studied is that of books and “stacked” books.
The book Bm is the graph Sm × P2 where Sm is the star with m edges. In 1980 Maheo proved that the books of the form B2m are graceful and conjectured that the books B4m+1 were also graceful. (The books B4m+3 do not satisfy the graceful parity condition.) This conjecture was verified by Delorme in 1980.
Maheo also proved that Ln × P2 and B2m × P2 are graceful. Both Grace the electronic journal of combinatorics (2023), #DS6 23 and Reid (see ) have given harmonious labelings for B2m. The books B4m+3 do not satisfy the harmonious parity condition [1190, Theorem 11]. Gallian and Jungreis conjectured that the books B4m+1 are harmonious. Gnanajothi has verified this conjecture by showing B4m+1 has an even stronger form of labeling – see Section 4.1.
Liang also proved the conjecture. In 1988 Gallian and Jungreis defined a stacked book as a graph of the form Sm × Pn. They proved that the stacked books of the form S2m×Pn are graceful and posed the case S2m+1×Pn as an open question. The n-cube K2 × K2 × · · · × K2 (n copies) was shown to be graceful by Kotzig —see also .
Although Graham and Sloane used a computer in 1980 to show that the 3-cube is not harmonious (see also ), Ichishima and Oshima proved that the n-cube Qn has a stronger form of harmonious labeling called an α-labeling (see Section 3.1) for n ≥4.
In 1986 Reid found a harmonious labeling for K4×Pn. In 2003 Petrie and Smith investigated graceful labelings of graphs as an exercise in constraint programming satisfaction. They determined that Kn × P2 is graceful for n = 3, 4 and 5; K4 × P3 is graceful; K4 × C3 is graceful; (Cn ∪Cn) + K1 (double wheel) is graceful for n = 4 and 5; and (C3 ∪C3) + K1 is not graceful. That K3 × K3 is not graceful follows from the parity condition given in the introduction. Using significantly better methods in 2010, Smith and Puget obtained the results about graceful labelings for Km × K1, Km × Pn, and Km × Cn given in Table 1. Their labeling for K5 ×P2 and K6 ×P3 are the unique graceful labelings for those graphs. Redl proved that K4 × Pn is graceful for n = 1, 2, 3, 4, and 5 using a constraint programming approach and asked if all graphs of the form K4 × Pn are graceful Vaidya, Kaneria, Srivastav, and Dani proved that Pn ∪Pt ∪(Pr × Ps) where t < min{r, s} and Pn ∪Pt ∪Kr,s where t ≤min{r, s} and r, s ≥3 are graceful. Kaneria, Vaidya, Ghodasara, and Srivastav proved Kmn ∪(Pr × Ps) where m, n, r, s > 1; (Pr × Ps) ∪Pt where r, s > 1 and t ̸= 2; and Kmn ∪(Pr × Ps) ∪Pt where m, n, r, s > 1 and t ̸= 2 are graceful. Xie, Zhao, and Yao proved that graphs of the form Cn ⊙T where T is a graceful tree are graceful.
The composition G1[G2] is the graph having vertex set V (G1) × V (G2) and edge set {(x1, y1), (x2, y2)| x1x2 ∈E(G1) or x1 = x2 and y1y2 ∈E(G2)}. The symmetric product G1 ⊕G2 of graphs G1 and G2 is the graph with vertex set V (G1) × V (G2) and edge set {(x1, y1), (x2, y2)| x1x2 ∈E(G1) or y1y2 ∈E(G2) but not both}. Seoud and Youssef have proved that Pn ⊕K2 is graceful when n > 1 and Pn[P2] is harmonious for all n. They also observe that the graphs Cm ⊕Cn and Cm[Cn] violate the parity conditions for graceful and harmonious graphs when m and n are odd.
2.4 Complete Graphs The questions of the gracefulness and harmoniousness of the complete graphs Kn have been answered. In each case the answer is positive if and only if n ≤4 ( , , , ). Both Rosa and Golomb proved that the complete bipartite graphs Km,n are graceful while Graham and Sloane showed they are harmonious the electronic journal of combinatorics (2023), #DS6 24 if and only if m or n = 1.
Aravamudhan and Murugan have shown that the complete tripartite graph K1,m,n is both graceful and harmonious while Gnanajothi [1143, pp. 25–31] has shown that K1,1,m,n is both graceful and harmonious and K2,m,n is graceful.
Some of the same results have been obtained by Seoud and Youssef who also observed that when m, n, and p are congruent to 2 (mod 4), Km,n,p violates the parity conditions for harmonious graphs. Beutner and Harborth give graceful labelings for K1,m,n, K2,m,n, K1,1,m,n and conjecture that these and Km,n are the only complete multipartite graphs that are graceful. They have verified this conjecture for graphs with up to 23 vertices via computer.
Beutner and Harborth also show that Kn−e (Kn with an edge deleted) is graceful only if n ≤5; any Kn −2e (Kn with two edges deleted) is graceful only if n ≤6; and any Kn −3e is graceful only if n ≤6. They also determine all graceful graphs of the form Kn −G where G is K1,a with a ≤n −2 and where G is a matching Ma with 2a ≤n.
The windmill graph K(m) n (n > 3) consists of m copies of Kn with a vertex in common.
A necessary condition for K(m) n to be graceful is that n ≤5 – see . Bermond has conjectured that K(m) 4 is graceful for all m ≥4. The gracefulness of K(m) 4 is equivalent to the existence of a (12m + 1, 4, 1)-perfect difference family, which are known to exist for m ≤1000 (see , , , and ). Bermond, Kotzig, and Turgeon proved that K(m) n is not graceful when n = 4 and m = 2 or 3, and when m = 2 and n = 5. Stones proved that K(3) 5 and K(4) 5 are graceful. In 1982 Hsu proved that K(m) 4 is harmonious for all m. Graham and Sloane conjectured that K(2) n is harmonious if and only if n = 4. They verified this conjecture for the cases that n is odd or n = 6. Liu has shown that K(2) n is not harmonious if n = 2apa1 1 · · · pas s where a, a1, . . . , as are positive integers and p1, . . . , ps are distinct odd primes and there is a j for which pj ≡3 (mod 4) and aj is odd. He also shows that K(3) n is not harmonious when n ≡0 (mod 4) and 3n = 4e(8k + 7) or n ≡5 (mod 8). Koh, Rogers, Lee, and Toh and Rajasingh and Pushpam have shown that K (t) m,n , the one-point union of t copies of Km,n, is graceful. Sethuraman and Selvaraju have proved that the one-point union of graphs of the form K2,mi for i = 1, 2, . . . , n, where the union is taken at a vertex from the partite set with exactly 2 vertices is graceful if at most two of the mi are equal. They conjecture that the restriction that at most two of the mi are equal is not necessary. Sudha proved that two or more complete bipartite graphs having one bipartite vertex set in common are graceful. Mitra and Bhoumik proved that K2n,2n ⊙K2 is graceful.
Koh, Rogers, Lee, and Toh introduced the notation B(n, r, m) for the graph consisting of m copies of Kn with a Kr in common (n ≥r). (We note that Guo has used the notation B(n, r, m) to denote the graph obtained by joining opposite endpoints of three disjoint paths of lengths n, r and m.) Bermond raised the question: “For which m, n, and r is B(n, r, m) graceful?” Of course, the case r = 1 is the same as K(m) n .
For r > 1, B(n, r, m) is graceful in the following cases: n = 3, r = 2, m ≥1 ; n = 4, r = 2, m ≥1 ; n = 4, r = 3, m ≥1 (see ), .
Seoud and Youssef have proved B(3, 2, m) and B(4, 3, m) are harmonious.
Liu has the electronic journal of combinatorics (2023), #DS6 25 shown that if there is a prime p such that p ≡3 (mod 4) and p divides both n and n −2 and the highest power of p that divides n and n−2 is odd, then B(n, 2, 2) is not graceful.
Smith and Puget has shown that up to symmetry, B(5, 2, 2) has a unique graceful labeling; B(n, 3, 2) is not graceful for n = 6, 7, 8, 9, and 10; B(6, 3, 3) and B(7, 3, 3) are not graceful; and B(5, 3, 3) is graceful. Combining results of Bermond and Farhi and Smith and Puget show that B(n, 2, 2) is not graceful for n > 5. Lu obtained the following results: B(m, 2, 3) and B(m, 3, 3) are not harmonious when m ≡1 (mod 8); B(m, 4, 2) and B(m, 5, 2) are not harmonious when m satisfies certain special conditions; B(m, 1, n) is not harmonious when m ≡5 (mod 8) and n ≡1, 2, 3 (mod 4); B(2m + 1, 2m, 2n + 1) ∼ = K2m + K2n+1 is not harmonious when m ≡2 (mod 4).
More generally, Bermond and Farhi have investigated the class of graphs con-sisting of m copies of Kn having exactly k copies of Kr in common. They proved such graphs are not graceful for n sufficiently large compared to r. Barrientos proved that the graph obtained by performing the one-point union of any collection of the complete bipartite graphs Km1,n1, Km2,n2, . . . , Kmt,nt, where each Kmi,ni appears at most twice and gcd(n1, n2, . . . , nt) = 1, is graceful.
Sethuraman and Elumalai have shown that K1,m,n with a pendent edge attached to each vertex is graceful and Jirimutu has shown that the graph obtained by attaching a pendent edge to every vertex of Km,n is graceful (see also ). In Sethuraman and Kishore determine the graceful graphs that are the union of n copies of K4 with i edges deleted for 1 ≤i ≤5 and with one edge in common. The only cases that are not graceful are those graphs where the members of the union are C4 for n ≡3 (mod 4) and where the members of the union are P2. They conjecture that these two cases are the only instances of edge induced subgraphs of the union of n copies of K4 with one edge in common that are not graceful.
Renuka, Balaganesan, Selvaraju proved the graphs obtained by joining a vertex of K1,m to a vertex of K1,n by a path are harmonious. Sethuraman and Selvaraju have shown that union of any number of copies of K4 with an edge deleted and one edge in common is harmonious.
Clemens, Coulibaly, Garvens, Gonnering, Lucas, and Winters investigated the gracefulness of the one-point and two-point unions of graphs. They show the following graphs are graceful: the one-point union of an end vertex of Pn and K4; the graph obtained by taking the one-point union of K4 with one end vertex of Pn and the one-point union of the other end vertex of Pn with the central vertex of K1,r; the graph obtained by taking the one-point union of K4 with one end vertex of Pn and the one-point union of the other end of Pn with a vertex from the partite set of order 2 of K2,r; the graph obtained from the graph just described by appending any number of edges to the other vertex of the partite set of order 2; the two-point union of the two vertices of the partite set of order 2 in K2,r and two vertices from K4; and the graph obtained from the graph just described by appending any number of edges to one of the vertices from the partite set of order 2.
A Golomb ruler is a marked straightedge such that the distances between different pairs of marks on the straightedge are distinct. If the set of distances between marks is every positive integer up to and including the length of the ruler, then ruler is a called the electronic journal of combinatorics (2023), #DS6 26 a perfect Golomb ruler. Golomb proved that perfect Golomb rulers exist only for rulers with at most 4 marks. Beavers examines the relationship between Golomb rulers and graceful graphs through a correspondence between rulers and complete graphs.
He proves that Kn is graceful if and only if there is a perfect Golomb ruler with n marks and Golomb rulers are equivalent to complete subgraphs of graceful graphs.
2.5 Disconnected Graphs There have been many papers dealing with graphs that are not connected. For any graph G the graph mG denotes the disjoint union of m copies of G. In 1975 Kotzig investigated the gracefulness of the graphs rCs. When rs ≡1 or 2 (mod 4), these graphs violate the gracefulness parity condition. Kotzig proved that when r = 3 and 4k > 4, then rC4k has a stronger form of graceful labeling called α-labeling (see §3.1) whereas when r ≥2 and s = 3 or 5, rCs is not graceful. In 1984 Kotzig once again investigated the gracefulness of rCs as well as graphs that are the disjoint union of odd cycles. For graphs of the latter kind he gives several necessary conditions. His paper concludes with an elaborate table that summarizes what was then known about the gracefulness of rCs.
M. He has shown that graphs of the form 2C2m and graphs obtained by connecting two copies of C2m with an edge are graceful. Cahit has shown that rCs is harmonious when r and s are odd and Seoud, Abdel Maqsoud, and Sheehan noted that when r or s is even, rCs is not harmonious. Seoud, Abdel Maqsoud, and Sheehan proved that Cn ∪Cn+1 is harmonious if and only if n ≥4. They conjecture that C3 ∪C2n is harmonious when n ≥3. This conjecture was proved when Yang, Lu, and Zeng showed that all graphs of the form C2j+1 ∪C2n are harmonious except for (n, j) = (2, 1).
As a consequence of their results about super edge-magic labelings (see §5.2) Figueroa-Centeno, Ichishima, Muntaner-Batle, and Oshima have that Cn ∪C3 is harmonious if and only if n ≥6 and n is even. Renuka, Balaganesan, Selvaraju proved that for odd n Cn∪P3 (see also ) and Cn⊙Km∪P3 are harmonious. Ng, Alwie, Marjadi, and Sugeng proved: Cm∪P1 is harmonious if and only if m ̸= 2 mod 4, Cm∪P2 (m ≥3), and they conjectured that Cm ∪P3 is harmonious for all m ≥3. Youssef has shown that if G is harmonious then mG is harmonious for all odd m.
In 1978 Kotzig and Turgeon proved that mKn is graceful if and only if m = 1 and n ≤4. Liu and Zhang have shown that mKn is not harmonious for n odd and m ≡2 (mod 4) and is harmonious for n = 3 and m odd. They conjecture that mK3 is not harmonious when m ≡0 (mod 4). Bu and Cao give some sufficient conditions for the gracefulness of graphs of the form Km,n ∪G and they prove that Km,n ∪Pt and the disjoint union of complete bipartite graphs are graceful under some conditions.
Recall a Skolem sequence of order n is a sequence s1, s2, . . . , s2n of 2n terms such that, for each k ∈{1, 2, . . . , n}, there exist exactly two subscripts i(k) and j(k) with si(k) = sj(k) = k and |i(k) −j(k)| = k. (A Skolem sequence of order n exists if and only if n ≡0 or 1 (mod 4)). Abrham has proved that any graceful 2-regular graph of order n ≡0 (mod 4) in which all the component cycles are even or of order n ≡3 (mod 4), with exactly one component an odd cycle, can be used to construct a Skolem sequence the electronic journal of combinatorics (2023), #DS6 27 of order n + 1. Also, he showed that certain special Skolem sequences of order n can be used to generate graceful labelings on certain 2-regular graphs.
The graph Hn obtained from the cycle with consecutive vertices u1, u2, . . . , un (n ≥6) by adding the chords u2un, u3un−1, . . . , uαuβ, where α = (n −1)/2 for all n and β = (n −1)/2 + 3 if n is odd or β = n/2 + 2 if n is even is called the cycle with parallel chords. In Elumalai and Sethuraman prove the following: for odd n ≥5, Hn ∪Kp,q is graceful; for even n ≥6 and m = (n −2)/2 or m = n/2 Hn ∪K1,m is graceful; for n ≥6, Hn ∪Pm is graceful, where m = n or n −2 depending on n ≡1 or 3 (mod 4) or m ≡n−1 or n−3 depending on n ≡0 or 2 (mod 4). Elumali and Sethuraman proved that every n-cycle (n ≥6) with parallel chords is graceful and every n-cycle with parallel Pk-chords of increasing lengths is graceful for n = 2 (mod 4) with 1 ≤k ≤(⌊n/2⌋−1).
In 1985 Frucht and Salinas conjectured that Cs ∪Pn is graceful if and only if s + n ≥6 and proved the conjecture for the case that s = 4. The conjecture was proved by Traetta in 2012 who used his result to get a complete solution to the well known two-table Oberwolfach problem; that is, given odd number of people and two round tables when is it possible to arrange series of seatings so that each person sits next to each other person exactly once during the series. The t-table Oberwolfach problem OP(n1, n2, . . . , nt) asks to arrange a series of meals for an odd number n = P ni of people around t tables of sizes n1, n2, . . . , nt so that each person sits next to each other exactly once. A solution to OP(n1, n2, . . . , nt) is a 2–factorization of Kn whose factors consists of t cycles of lengths n1, n2, . . . , nt. The λ–fold Oberwolfach problem OPλ(n1, n2, . . . , nt) refers to the case where Kn is replaced by λKn. Traetta used his proof of the Frucht and Salinas conjecture to provide a complete solutions to both OP(2r + 1, 2s) and OP(2r + 1, s, s), except possibly for OP(3, s, s). He also gave a complete solution of the general λ–fold Oberwolfach problem OPλ(r, s).
Seoud and Youssef have shown that K5 ∪Km,n, Km,n ∪Kp,q (m, n, p, q ≥ 2), Km,n ∪Kp,q ∪Kr,s (m, n, p, q, r, s ≥2, (p, q) ̸= (2, 2)), and pKm,n (m, n ≥2, (m, n) ̸= (2, 2)) are graceful.
They also prove that C4 ∪K1,n (n ̸= 2) is not graceful whereas Choudum and Kishore , have proved that Cs ∪K1,n is graceful for s ≥7 and n ≥1. Lee, Quach, and Wang established the gracefulness of Ps ∪K1,n. Seoud and Wilson have shown that C3∪K4, C3∪C3∪K4, and certain graphs of the form C3∪Pn and C3 ∪C3 ∪Pn are not graceful. Abrham and Kotzig proved that Cp ∪Cq is graceful if and only if p + q ≡0 or 3 (mod 4). Zhou proved that Km ∪Kn (n > 1, m > 1) is graceful if and only if {m, n} = {4, 2} or {5, 2}. Knuth used a computer to show that K5∪K2 has a unique graceful labeling up to a complement. (C. Barrientos has called to my attention that K1 ∪Kn is graceful if and only if n = 3 or 4.) Shee has shown that graphs of the form P2 ∪C2k+1 (k > 1), P3 ∪C2k+1, Pn ∪C3, and Sn ∪C2k+1 all satisfy a condition that is a bit weaker than harmonious. Bhat-Nayak and Deshmukh have shown that C4t ∪K1,4t−1 and C4t+3 ∪K1,4t+2 are graceful. Section 3.1 includes numerous families of disconnected graphs that have a stronger form of graceful labelings.
For m = 2p + 3 or 2p + 4, Wang, Liu, and Li proved the following graphs are graceful: Wm ∪Kn,p and Wm,2m+1 ∪Kn,p; for n ≥m, Wm,2m+1 ∪K1,n; for m = 2n + 5, Wm,2m+1 ∪(C3 + Kn). If Gp is a graceful graph with p edges, they proved W2p+3 ∪Gp is the electronic journal of combinatorics (2023), #DS6 28 graceful.
In Roblee and Abueida proved that a disjoint union of an odd cycle with a new graph that consists of a central vertex that is adjacent to an endpoint of a certain number of fixed length paths are harmonious. They extend this result to include specifying that the central vertex in the tree be adjacent to different vertices in each of the t many s-paths.
In considering graceful labelings of the disjoint unions of two or three stars Se with e edges Yang and Wang permitted the vertex labels to range from 0 to e + 1 and 0 to e + 2, respectively. With these definitions of graceful, they proved that Sm ∪Sn is graceful if and only if m or n is even and that Sm ∪Sn ∪Sk is graceful if and only if at least one of m, n, or k is even (m > 1, n > 1, k > 1).
Seoud and Youssef investigated the gracefulness of specific families of the form G ∪Km,n. They obtained the following results: C3 ∪Km,n is graceful if and only if m ≥2 and n ≥2; C4 ∪Km,n is graceful if and only if (m, n) ̸= (1, 1); C7 ∪Km,n and C8 ∪Km,n are graceful for all m and n; mK3 ∪nK1,r is not graceful for all m, n and r; Ki ∪Km,n is graceful for i ≤4 and m ≥2, n ≥2 except for i = 2 and (m, n) = (2, 2); K5 ∪K1,n is graceful for all n; K6 ∪K1,n is graceful if and only if n is not 1 or 3. Youssef completed the characterization of the graceful graphs of the form Cn ∪Kp,q where n ≡0 or 3 (mod 4) by showing that for n > 8 and n ≡0 or 3 (mod 4), Cn ∪Kp,q is graceful for all p and q (see also ). Note that when n ≡1 or 2 (mod 4) certain cases of Cn ∪Kp,q violate the parity condition for gracefulness.
For i = 1, 2, . . . , m let vi,1, vi,2, vi,3, vi,4 be a 4-cycle. Yang and Pan define Fk,4 to be the graph obtained by identifying vi,3 and vi+1,1 for i = 1, 2, . . . , k −1. They prove that Fm1,4 ∪Fm2,4 ∪· · · ∪Fmn,4 is graceful for all n. Pan and Lu have shown that (P2 + Kn) ∪K1,m and (P2 + Kn) ∪Tn are graceful.
Barrientos has shown the following graphs are graceful: C6∪K1,2n+1; St i=1 Kmi,ni for 2 ≤mi < ni; and Cm ∪St i=1 Kmi,ni for 2 ≤mi < ni, m ≡0 or 3 (mod 4), m ≥11.
In Kaneria, Makadia, and Viradia proved that the union of three grid graphs, S3 l=1 (Pml × Pnl), is graceful, the union of finitely many copies of Pm × Pn is graceful, and provided two new graceful labeling for Pm × Pn.
Wang and Li use St(n) to denote the star Kn,1, Fn to denote the fan Pn ⊙K1, and Fm,n to denote the graph obtained by identifying the vertex of Fm with degree m and the vertex of Fn with degree n. They showed: for all positive integers n and p and m ≥2p + 2, Fm ∪Kn,p and Fm,2m ∪Kn,p are graceful; Fm ∪St(n) is graceful; and Fm,2m ∪St(n) and Fm,2m ∪Gr are graceful. In Wang, Wang, and Li gave a sufficient condition for the gracefulness of graphs of the form (P3 + Km) ∪G and (C3 + Km) ∪G.
Wei, Wang, and Sun provided graceful labelings for the unions of some families of wheels related graphs and complete bipartite graphs. They also gave graceful labelings for some graphs of the form G∪(C3+Km)∪Sn where G is wheel related. In Yu, Wang, and Song proved the following graphs are graceful: Kn,m ∪(K2 + Kn), Kn,m ∪(P3 + Kn), Kn,m ∪(P1 + P2n+2), and Kn,m ∪K1,2n. They proved the gracefulness of such graphs for a variety of cases when G involves stars and paths. More technical results like these are given in , , and .
the electronic journal of combinatorics (2023), #DS6 29 2.6 Joins of Graphs A number of classes of graphs that are the join of graphs have been shown to be graceful or harmonious. Koh, Rogers, and Lim proved G + H is graceful if G is a graceful tree and H is one of Kn, Pn ∪K1, or a star. Koh, Phoon, and Soh point out that some versions of this survey prior to 2017 incorrectly stated that Acharya proved that if G is a connected graceful graph, then G + Kn is graceful. Redl showed that the double cone Cn + K2 is graceful for n = 3, 4, 5, 7, 8, 9, 11. That Cn + K2 is not graceful for n ≡2 (mod 4) follows that Rosa’s parity condition. Redl asks what other double cones are graceful. Bras, Gomes, and Selman showed that double wheels (Cn ∪Cn) + K1 are graceful. Koh, Phoon, and Soh prove that K3 + Kn is graceful. Reid proved that Pn +Kt is harmonious. Sethuraman and Selvaraju and have shown that Pn + K2 is harmonious. They ask whether Sn + Pn or Pm + Pn is harmonious. As stated in an earlier section, wheels are of the form Cn + K1 and are graceful and harmonious.
In 2006 Chen proved that multiple wheels nCm + K1 are harmonious for all n ̸≡0 mod 4. She believes that the n ̸≡0 (mod 4) case is also harmonious. Chen also proved that if H has at least one edge, H + K1 is harmonious, and if n is odd, then nH + K is harmonious.
For n ≥t + 2 and t ≥1, Koh, Phoon, and Soh use P(n, t) to denote the graph of order n consisting of a path of length t and n −(t + 1) isolated vertices. For n ≥2t + 1 and t ≥1, they use I(n, t) to denote the disjoint union of tK2 and Kn−2t. They proved: Kp + P(n, t) is graceful for all p ≥1, n ≥t + 2 and t ≥1; Kp + I(n, t) is graceful for all p ≥1, n ≥2t + 1 and t ≥1; and for s, t ∈{1, 2}, P(m, s) + P(n, t) is graceful for all m ≥s + 2 and n ≥t + 2. In Koh, Phoon, and Soh ask “What can be said about the gracefulness of Cm + P(n, t) where n ≥t + 2” and is “Is P(m, s) + P(n, t) always graceful for all m ≥s + 2, n ≥t + 2, where s ≥3 or t ≥3?” In they state as problems about graceful graphs: Cm + Pn (m ≥3, n ≥3); Cm + Cn (m ≥3, n ≥3) and K1,p + P(n, t) and prove that C3 + P(n, t) is graceful for all n ≥t + 2, where 1 ≤t ≤3 and C5 + P(n, 1) is graceful for all n ≥3.
Shee has proved Km,n + K1 is harmonious and observed that various cases of Km,n + Kt violate the harmonious parity condition in . Liu and Zhang have proved that K2 + K2 + · · · + K2 is harmonious. Youssef has shown that if G is harmonious then Gm is harmonious for all odd m. He asks the question of whether G is harmonious implies Gm is harmonious when m ≡0 (mod 4). Yuan and Zhu proved that Km,n + K2 is graceful and harmonious. Gnanajothi [1143, pp. 80–127] obtained the following: Cn+K2 is harmonious when n is odd and not harmonious when n ≡2, 4, 6 (mod 8); Sn+Kt is harmonious; and Pn+Kt is harmonious. Balakrishnan and Kumar have proved that the join of Kn and two disjoint copies of K2 is harmonious if and only if n is even. Ramírez-Alfonsín has proved that if G is graceful and |V (G)| = |E(G)| = e and either 1 or e is not a vertex label then G + Kt is graceful for all t.
Sudha and Kanniga proved that the graph Pm + Kn is graceful.
Seoud and Youssef have proved: the join of any two stars is graceful and har-monious; the join of any path and any star is graceful; and Cn + Kt is harmonious for the electronic journal of combinatorics (2023), #DS6 30 every t when n is odd. They also prove that if any edge is added to Km,n the resulting graph is harmonious if m or n is at least 2. Deng has shown certain cases of Cn +Kt are harmonious. Seoud and Youssef proved: the graph obtained by appending any number of edges from the two vertices of degree n ≥2 in K2,n is not harmonious; dragons Dm,n (i.e., an endpoint of Pm is appended to Cn) are not harmonious when m + n is odd; and the disjoint union of any dragon and any number of cycles is not harmonious when the resulting graph has odd order. Youssef has shown that if G is a graceful graph with p vertices and q edges with p = q + 1, then G + Sn is graceful.
Sethuraman and Elumalai have proved that for every graph G with p vertices and q edges the graph G+K1+Km is graceful when m ≥2p−p−1−q. As a corollary they deduce that every graph is a vertex induced subgraph of a graceful graph. Balakrishnan and Sampathkumar ask for which m ≥3 is the graph mK2 + Kn graceful for all n.
Bhat-Nayak and Gokhale have proved that 2K2 + Kn is not graceful. Youssef has shown that mK2 + Kn is graceful if m ≡0 or 1 (mod 4) and that mK2 + Kn is not graceful if n is odd and m ≡2 or 3 (mod 4). Ma proved that if G is a graceful tree then, G + K1,n is graceful. Amutha and Kathiresan proved that the graph obtained by attaching a pendent edge to each vertex of 2K2 + Kn is graceful.
Wu proves that if G is a graceful graph with n edges and n+1 vertices then the join of G and Km and the join of G and any star are graceful. Wei and Zhang proved that for n ≥3 the disjoint union of P1 + Pn and a star, the disjoint union of P1 + Pn and P1+P2n, and the disjoint union of P2+Kn and a graceful graph with n edges are graceful.
More technical results on disjoint unions and joins are given in , , , , and .
2.7 Miscellaneous Results It is easy to see that P 2 n is harmonious while a proof that P 2 n is graceful has been given by Kang, Liang, Gao, and Yang . (P k n, the kth power of Pn, is the graph obtained from Pn by adding edges that join all vertices u and v with d(u, v) = k.) This latter result proved a conjecture of Grace .
Seoud, Abdel Maqsoud, and Sheehan proved that P 3 n is harmonious and conjecture that P k n is not harmonious when k > 3. The same conjecture was made by Fu and Wu . However, Youssef has proved that P 4 8 is harmonious and P k n is harmonious when k is odd.
Yuan and Zhu proved that P 2k n is harmonious when 1 ≤k ≤(n −1)/2. Selvaraju has shown that P 3 n and the graphs obtained by joining the centers of any two stars with the end vertices of the path of length n in P 3 n are harmonious.
Cahit proves that the graphs obtained by joining p disjoint paths of a fixed length k to single vertex are harmonious when p is odd and when k = 2 and p is even.
Gnanajothi [1143, p. 50] has shown that the graph that consists of n copies of C6 that have exactly P4 in common is graceful if and only if n is even. For a fixed n, let vi1, vi2, vi3 and vi4 (1 ≤i ≤n) be consecutive vertices of n 4-cycles. Gnanajothi [1143, p. 35] also proves that the graph obtained by joining each vi1 to vi+1,3 is graceful for all n and the generalized Petersen graph P(n, k) is harmonious in all cases (see also ). Recall the electronic journal of combinatorics (2023), #DS6 31 P(n, k), where n ≥5 and 1 ≤k ≤n, has vertex set {a0, a1, . . . , an−1, b0, b1, . . . , bn−1} and edge set {aiai+1 | i = 0, 1, . . . , n −1} ∪{aibi | i = 0, 1, . . . , n −1} ∪{bibi+k | i = 0, 1, . . . , n −1} where all subscripts are taken modulo n . The standard Petersen graph is P(5, 2).) Redl has used a constraint programming approach to show that P(n, k) is graceful for n = 5, 6, 7, 8, 9, and 10. In and Vietri proved that P(8t, 3) and P(8t + 4, 3) are graceful for all t. He conjectures that the graphs P(8t, 3) have a stronger form a graceful labeling called an α-labeling (see §3.1). The gracefulness of the generalized Petersen graphs is an open problem. Shao, Deng, Li, and Vese provide an backtracking algorithm that finds graceful labelings for all generalized Petersen graphs P(n, k) with n ≤75 within several seconds. The algorithm strongly outperforms the standard backtracking algorithm.
Rao and Sahoo prove that every connected graph can be embedded as an induced subgraph in an Eulerian graceful graph.
They also show that for an integer k ≥3, the problems of deciding whether the chromatic number is less than or equal to k and whether the clique number is greater than or equal to k are NP-complete even for Eulerian graceful graphs. Sethuraman, Ragukumar, and Slater proved that any tree with m edges can be embedded in a graceful tree with less than 4m edges and in a graceful planar graph. A conjecture in the graph theory book by Chartrand and Lesniak [706, p.
266] that graceful graphs with arbitrarily large chromatic numbers do not exist was shown to be false by Acharya, Rao, and Arumugam (see also Mahmoody ).
In Barrientos calculates the number of non-isomorphic harmoniously labeled graphs with n edges and at most n vertices. He provides harmonious labelings for certain unicyclic graphs obtained via the corona product and triangular grids obtained via edge amalgamation of copies of C3 in such a way that each copy of a cycle shares at most two edges with other copies. Moreover, he uses the edge-switching technique on C4t to generate unicyclic graphs with strongly felicitous labelings (see §4.4).
Bača and Youssef investigated the existence of harmonious labelings for the corona graphs of a cycle and a graph G. They proved that if G+K1 is strongly harmonious (that is, a harmonious labeling f for which the edge labels induced by f(x)+f(y) for each edge xy are 1, . . . , q. with the 0 label on the vertex of K1, then Cn ⊙G is harmonious for all odd n ≥3. By combining this with existing results they have as corollaries that the following graphs are harmonious: Cn ⊙Cm for odd n ≥3 and m ̸≡2 (mod 3); Cn ⊙Ks,t for odd n ≥3; and Cn ⊙K1,s,t for odd n ≥3.
Sethuraman and Selvaraju define a graph H to be a supersubdivision of a graph G, if every edge uv of G is replaced by K2,m (m may vary for each edge) by identifying u and v with the two vertices in K2,m that form the partite set with exactly two mem-bers. Sethuraman and Selvaraju prove that every supersubdivision of a path is graceful and every cycle has some supersubdivision that is graceful. They conjecture that every supersubdivision of a star is graceful and that paths and stars are the only graphs for which every supersubdivision is graceful. Barrientos disproved this latter conjecture by proving that every supersubdivision of a y-trees is graceful (recall a y-tree is obtained from a path by appending an edge to a vertex of a path adjacent to an end point). Bar-rientos asks if paths and y-trees are the only graphs for which every supersubdivision is the electronic journal of combinatorics (2023), #DS6 32 graceful. This seems unlikely to be the case. The conjecture that every supersubdivision of a star is graceful was proved by Kathiresan and Amutha . In Sethuraman and Selvaraju prove that every connected graph has some supersubdivision that is grace-ful. They pose the question as to whether this result is valid for disconnected graphs.
Barrientos and Barrientos answered this question by proving that any disconnected graph has a supersubdivision that admits an α-labeling (see §3.1). They also proved that every supersubdivision of a connected graph admits an α-labeling. Sekar and Ramachan-dren proved that an arbitrary supersubdivision of disconnected graph is graceful and supersubdivisions of ladders are graceful . Sethuraman and Selvaraju also asked if there is any graph other than K2,m that can be used to replace an edge of a connected graph to obtain a supersubdivision that is graceful.
Sethuraman and Selvaraju call supersubdivision graphs of G where every edge uv of G is replaced by K2,m and m is fixed an arbitrary supersubdivision of G. Barrientos and Barrientos answered the question of Sethuraman and Selvaraju by proving that any graph obtained from K2,m by attaching k pendent edges and n pendent edges to the vertices of its 2-element stable set can be used instead of K2,m to produce an arbitrary supersubdivision that admits an α-labeling (a stable set S consists of a set of vertices such that there is not an edge vivj for all pairs vi, vj in S).
Kathiresan and Sumathi affirmatively answer the question posed by Sethuraman and Selvaraju in of whether there are graphs different from paths whose arbitrary supersubdivisions are graceful.
For a graph G Ambili and Singh call the graph G∗a strong supersubdivision of G if G∗is obtained from G by replacing every edge ei of G by a complete bipartite graph Kri,si. A strong supersubdivision G∗of G is said to be an arbitrary strong supersubdivision if G∗is obtained from G by replacing every edge ei of G by a complete bipartite graph Kr,si (r is fixed and si may vary). They proved that arbitrary strong supersubdivisions of paths, cycles, and stars are graceful.
They conjecture that every arbitrary strong supersubdivision of a tree is graceful and ask if it is true that for any non-trivial connected graph G, an arbitrary strong supersubdivision of G is graceful?
In Sethuraman and Selvaraju present an algorithm that permits one to start with any non-trivial connected graph and successively form supersubdivisions that have a strong form of graceful labeling called an α-labeling (see §3.1 for the definition).
Kathiresan uses the notation Pa,b to denote the graph obtained by identifying the end points of b internally disjoint paths each of length a. He conjectures that Pa,b is graceful except when a is odd and b ≡2 (mod 4) and proves the conjecture for the case that a is even and b is odd. Liang and Zuo proved that the graph Pa,b is graceful when both a and b are even. Daili, Wang and Xie provided an algorithm for finding a graceful labeling of P2r,2 and showed that a P2r,2(2k+1) is graceful for all positives r and k. Sekar has shown that Pa,b is graceful when a ̸= 4r + 1, r > 1, b = 4m, and m > r. Yang (see ) proved that Pa,b is graceful when a = 3, 5, 7, and 9 and b is odd and when a = 2, 4, 6, and 8 and b is even (see ). Yang, Rong, and Xu proved that Pa,b is graceful when a = 10, 12, and 14 and b is even. Yan proved P2r,2m is graceful when r is odd. Yang showed that P2r+1,2m+1 and P2r,2m (r ≤7, and r = 9) are the electronic journal of combinatorics (2023), #DS6 33 graceful (see ). Rong and Xiong showed that P2r,b is graceful for all positive integers r and b. Kathiresan also shows that the graph obtained by identifying a vertex of Kn with any noncenter vertex of the star with 2n−1 −n(n −1)/2 edges is graceful.
For a family of graphs G1(u1, u2), G2(u2, u3), . . . , Gm(um, um+1) where ui and ui+1 are vertices in Gi Cheng, Yao, Chen, and Zhang define a graph-block chain Hm as the graph obtained by identifying ui+1 of Gi with ui+1 of Gi+1 for i = 1, 2, . . . , m. They denote this graph by Hm = G1(u1, u2)⊕G2(u2, u3)⊕· · ·⊕Gm(um, um+1). The case where each Gi has the form Pai,bi they call a path-block chain. The vertex u1 is called the initial vertex of Hm. They define a generalized spider S∗ m as a graph obtained by starting with an initial vertex u0 and m path-block graphs and join u0 with each initial vertex of each of the path-block graphs. Similarly, they define a generalized caterpillar T ∗ m as a graph obtained by starting with m path-block chains H1, H2, . . . , Hm and a caterpillar T with m isolated vertices v1, v2, . . . , vm and join each vi with the initial vertex of each Hi. They prove several classes of path-block chains, generalized spiders, and generalized caterpillars are graceful.
The graph Tn with 3n vertices and 6n −3 edges is defined as follows. Start with a triangle T1 with vertices v1,1, v1,2 and v1,3. Then Ti+1 consists of Ti together with three new vertices vi+1,1, vi+1,2, vi+1,3 and edges vi+1,1vi,2, vi+1,1vi,3, vi+1,2vi,1, vi+1,2vi,3, vi+1,3vi,1, vi+1,3vi,2. Gnanajothi proved that Tn is graceful if and only if n is odd. Sekar proved Tn is graceful when n is odd and Tn with a pendent edge attached to the starting triangle is graceful when n is even.
In and Begam, Palanivelrajan, Gunasekaran, and Hameed give graceful labelings for graphs constructed by combining theta graphs (that is, a collection of edge disjoint paths that have common endpoints) with paths and stars.
Khatun and Abu Nayeem prove that the zero divisor graph of the commutative ring of integers modulo n is graceful if n = pq, 4p or 9p, where p and q are prime numbers.
The torch graph On is defined by V (On) = {vi | 1 ≤i ≤n + 4}, E(On) = {vivn+1 | 2 ≤i ≤n −2} ∪{vivn+3 | 2 ≤i ≤n −2} ∪{v1vi | 2 ≤i ≤n + 4} ∪ {vn−1vn, vnvn+2, vnvn+4, vn+1vn+3}. Manulang and Sugeng showed that the torch graph is graceful.
For a graph G, the splitting graph of G, S′(G), is obtained from G by adding for each vertex v of G a new vertex v ′ so that v ′ is adjacent to every vertex that is adjacent to v.
Sekar has shown that S′(Pn) is graceful for all n and S′(Cn) is graceful for n ≡0, 1 (mod 4). Vaidya and Shah proved that the square graph of a bistar, the splitting graph of a bistar, and the splitting graph of a star are graceful graphs.
In Sudha and Kanniga proved that fans and the splitting graph of a star are graceful. Sudha and Kanniga proved that the following graphs are graceful: arbi-trary supersubdivisions of wheels; combs (Pn⊙K1); double fans (Pn⊙K2); (Pm∪Pn)⊙K1; and graphs obtained by starting with two star graphs Sm and Sn and identifying some of the pendent vertices of each. Sudha and Kanniga proved that the graphs obtained from Pn ⊙K1 by identifying the center of a Sn with the endpoint of a pendent edge at-tached to the endpoint of Pn are graceful; and the graphs obtained from a fan Pn ⊙K1 by deleting a pendent edge attached to an endpoint of Pn are graceful. Sunda provided the electronic journal of combinatorics (2023), #DS6 34 some results on graphs obtained by connecting copies of Km,n in certain ways. Sudha and Kanniga proved that the graphs obtained by joining the vertices of a path to any number isolated points are graceful. They also proved that the arbitrary supersubdivision of all the edges of helms, combs (Pn ⊙K1) and ladders (Pn × P2) with pendent edges at the vertices of degree 2 by a complete bipartite graphs K2,m are graceful.
The duplication of an edge e = uv of a graph G is the graph G′ obtained from G by adding an edge e′ = u′v′ such that N(u) = N(u′) and N(v) = N(v′). The duplication of a vertex of a graph G is the graph G′ obtained from G by adding a new vertex v′ to G such that N(v′) = N(v). Kaneria, Vaidya, Ghodasara, and Srivastav proved the duplication of a vertex of a cycle, the duplication of an edge of an even cycle, and the graph obtained by joining two copies of a fixed cycle by an edge are graceful.
For a graph G and a vertex v of G, a vertex switching Gv is the graph obtained from G by removing all edges incident to v and adding edges joining v to every vertex not adjacent to v in G. Boxwala and Vashishta show that the graph obtained by switching an arbitrary vertex of Cn (n > 3), the duplication of an arbitrary vertex on the rim of a wheel with an even number of vertices, and the mirror graph of a path are graceful. Jeba Jesintha and Subashini proved that the path union of vertex switching of even cycles in increasing order is graceful.
The join sum of complete bipartite graphs < Km1,n1, . . . , Kmt,nt > is the graph ob-tained by starting with Km1,n1, . . . , Kmt,nt and joining a vertex of each pair Kmi,ni and Kmi+1,ni+1 to a new vertex vi where 1 ≤i ≤k −1. The path union of a graph G is the graph obtained by adding an edge from n copies G1, G2, . . . , Gn of G from Gi to Gi+1 for i = 1, . . . , n −1. We denote this graph by P(n · G). Kaneria, Makadia, and Megh-para proved the following graphs are graceful: the graph obtained by joining C4m and C4n by a path of arbitrary length; the path union of finite many copies of C4n; and C4n with twin chords. Kaneria, Makadia, Jariya, and Meghpara proved that the join sum of complete bipartite graphs, the star of complete bipartite graphs, and the path union of a complete bipartite graphs are graceful.
Given connected graphs G1, G2, . . . , Gn, Kaneria, Makadia, and Jariya define a cycle of graphs C(G1, G2, . . . , Gn) as the graph obtained by adding an edge joining Gi to Gi+1 for i = 1, . . . , n −1 and an edge joining Gn to G1. (The resulting graph can vary depending on which vertices of the Gis are chosen.) When the n graphs are isomorphic to G the notation C(n·G) is used. Kaneria et al. proved that C(2t·C4n) and C(2t·Kn,n) are graceful. In and Kaneria, Makadia, and Meghpara prove that the following graphs are graceful: C(2t · Km,n); C(C4n1, C4n2, . . . , C4nt) when t is even and P t 2 i=1 ni = Pt i= t 2 ni; C(2t · Pm × Pn); the star of Pm × Pn; and the path union of t copies of Pm × Pn.
Kaneria, Viradia, Jariya, and Makadia proved the cycle graph C(t · Pn) is graceful.
The star of graphs G1, G2, . . . , Gn, denoted by S(G1, G2, . . . , Gn), is the graph ob-tained by identifying each vertex of K1,n, except the center, with one vertex from each of G1, G2, . . . , Gn. The case that G1 = G2 = · · · = Gn = G is denoted by S(n · G). In and Kaneria, Meghpara, and Makadia proved the following graphs are graceful: S(t · Km,n); S(t · Pm × Pn); the barycentric subdivision of Pm × Pn (that is, the graph obtained from Pm × Pn by inserting a new vertex in each edge); the graph obtained by the electronic journal of combinatorics (2023), #DS6 35 replacing each edge of K1,t by Pn; the graph obtained by identifying each end point of K1,n with a vertex of Km,n; and the graph obtained by identifying each end point of K1,n with a vertex of Pm × Pn. Kanani and Kaneria proved that the following graphs are graceful: the barycentric subdivision of Cn-snakes (that is, the graph obtained from the subdivision of Cn by inserting a new vertex in each edge); the barycentric subdivision of alternate Cn-snakes; and quadrilateral snakes.
Kaneria and Makadia and proved the following graphs are graceful: (Pm× Pn) ∪(Pr × Ps); C2f+3 ∪(Pm × Pn) ∪(Pr × Ps), where f = 2(mn + rs) −(m + n + r + s); the tensor product of Pn and P3; the tensor product of Pm and Pn for odd m and n; the star of C4n; the t−supersubdivision of Pm × Pn; and the graph obtained by joining C4n and a grid graph with a path. In Kaneria, Meghpara, and Makadia proved that the star of K1,n is a graceful tree.
The graph P t n is obtained by identifying one end point from each of t copies of Pn. The graph P t n(G1, G2, . . . , Gtn) obtained by replacing each edge of P t n, except those adjacent to the vertex of degree t, by the graphs G1, G2, . . . , Gtn is called the one point path union of G1, G2, . . . , Gtn. The case where G1 = G2 = · · · = Gtn = H is denoted by P t n(tn · H) . In and Kaneria, Meghpara, and Makadia proved P t n and P t n(tn · Km,r) are graceful. In Kaneria and Meghpara proved P t n (tn·Pr×Ps), P t n(tn·K1,m), S(t·C4n), and P t n(tn · C4m) are graceful.
A graph H is said to be a m-super subdivision of a simple graph G, if every edge of G is replaced by the complete bipartite graph Km,m with m > 2 in such a way that the end vertices of the edge are merged with any two vertices of the same partite set A or B of Km,m after removal of the edge of G. Srinivasan, Chidambaram, Devadoss, Pakkirisamy, and Krishnamoorthi proved that m-super subdivision of path and cycle are graceful.
Kanneria and Makadia define a step grid graph as the graph obtained by starting with paths Pn, Pn, Pn−1, . . . , P2 (n ≥3) arranged vertically parallel with the vertices in the paths forming horizontal rows and edges joining the vertices of the rows. In and they prove the following graphs are graceful: step grid graphs; one point union for a path of step grid graphs; cycles of step grid graphs; stars of step grid graphs; m−super subdivisions of the step grid graphs; open stars of step grid graphs; one point unions of paths of step grid graphs; and graphs obtained by joining C4m and step grid graphs with a path of arbitrary length.
For n even Kaneria and Makadia define a double step grid graph of size n (denoted by DStn) as the graph obtained by starting with paths Pn, Pn, Pn−2, Pn−4, . . . , P4, P2 arranged vertically parallel with the vertices in the paths forming horizontal rows and edges joining the vertices of the rows. They prove the follow-ing graphs are graceful: double step grid graphs; path unions of copies of DStn; cycles of r ≡0, 3 (mod 4) copies of double step grid graphs; and stars of double step grid graphs.
In Kaneria, Makadia and Viradia prove the following graphs are graceful: open stars of double step grid graphs; one point union of paths of double step grid graphs Pn t(tn · DStm); graphs obtained by joining C4m and a double step grid graph with a path of arbitrary length; and graphs obtained by starting with a cycle Cm + (m ≡2 mod 4) the electronic journal of combinatorics (2023), #DS6 36 with chords that form a triangle with an edge of the cycle and joining Cm + and a double step grid graph with a path of arbitrary length.
For even n > 2 Kaneria and Makadia define a plus graph of size n (denoted by Pln) as the graph obtained by starting with paths P2, P4, . . . , Pn−2, Pn, Pn, Pn−2, . . . , P4, P2 arranged vertically parallel with the vertices in the paths forming horizontal rows and edges joining the vertices of the rows. They prove plus graphs, path unions of copies of Pln, cycles of r ≡0, 3 (mod 4) copies of Pln, and stars of plus graphs are graceful.
In Kaneria and Makadia prove the following graphs are graceful: open stars of plus graphs; graphs obtained by joining C4m and a plus graph with a path of arbitrary length; graphs obtained from cycles Cm + (m ≡2 (mod 4)) with twin chords that form a triangle with an edge of the cycle by joining Cm + and a plus graph with a path of arbitrary length.
Kaneria and Makadia define a swastik graph as the graph obtained from four copies of C4n (n > 1) with vertices Vi,j ( i = 1, 2, 3, 4, j = 1, 2, . . . , 4n) and identifying V1,4t and V2,1, V2,4t and V3,1, V3,4t and V4,1, and V4,4t and V1,1. They proved that path unions of swastik graphs of the same size, cycles of r ≡0, 3 (mod 4) copies of swastik graphs of the same size, and the star of swastik graphs are graceful. In Kaneria and Makadia prove the following graphs are graceful: open stars of swastik graphs; one point unions for paths of swastik graphs; graphs obtain by joining C4m and a swastik graph with a path of arbitrary length; graphs obtained from cycles Cm (m ≡2 (mod 4)) with twin chords that form a triangle with an edge by joining Cm ⊙K1 and a swastik graph with a path of arbitrary length.
In and Kaneria and Jariya define a smooth graceful graph as a bipartite graph G with q edges with the property that for all positive integers l there exists a map g : V − →{0, 1, . . . , ⌊q−1 2 ⌋, ⌊q+1 2 ⌋+l, ⌊q+3 2 ⌋+l, . . . , q+l} such that the induced edge labeling map g⋆: E − →{1 + l, 2 + l, . . . , q + l} defined by g⋆(e) = |g(u) −g(v)| is a bijection.
Note that by taking l = 0 a smooth graceful labeling is a graceful labeling. Kaneria and Jariya proved the following graphs are smooth graceful: Pn; C4n; K2,n; Pm × Pn; and the graph obtained by joining a cycle C4m+2 with twin chords to C4n. They also proved that the graph obtained by joining C4m to Wn with a path is graceful. They proved that K1,n is semi smooth graceful, the star of K1,n is graceful, the path union of a smooth graceful tree is graceful, and the star of a smooth graceful tree is a graceful tree.
Kaneria, Makadia and Viradia proved the following: the star of a semi smooth graceful graph is graceful; Km,n, P(t · H) are semi smooth graceful where H is a semi smooth graceful graph; step grid graphs; and the cycle graphs C(t · H) are smooth graceful, when t ≡(mod 4), H is a semi smooth; Ct(m · Cn), P t(k · T), < Cn1, Pn2, Cn3, . . . , Pn2t, Cn2t+1 >, < Km1,n1, Pr1, Km2,n2, Pr2, . . . , Prt−1, Kmt,nt >, < Pn1 × Pm1, Pr1, Pn2 × Pm2, . . . , Prt−1, Pnt × Pmt > are graceful when T is semi smooth graceful tree.
Kaneria and Meghpara prove that Bm,n, the splitting graphs S′(Bm,n) and S′(Pn) are semi smooth graceful and if graphs obtained by joining semi smooth graceful graph and B2 m,n by an arbitrary path is graceful.
A komodo dragon is formed by attaching a path to a vertex of degree 3 in a cycle with a chord and attaching star graphs to the end points of the path. A komodo dragon with the electronic journal of combinatorics (2023), #DS6 37 many tails is formed by attaching many paths of length two to an endpoint of the path in a komodo dragon. In and Shahul Hameed, Palanivelrajan, Gunasekaran and Raziya Begam provide graceful labelings of various komodo dragon graphs and their extensions. In and Shahul Hameed et al. investigated the gracefulness of classes of graphs constructed by combining some subdivisions of certain theta graphs with stars.
For a bipartite graph G with partite sets X and Y let G′ be a copy of G and X′ and Y ′ be copies of X and Y . Lee and Liu define the mirror graph, M(G), of G as the disjoint union of G and G′ with additional edges joining each vertex of Y to its corresponding vertex in Y ′. The case that G = Km,n is more simply denoted by M(m, n).
They proved that for many cases M(m, n) has a stronger form of graceful labeling (see §3.1 for details).
The total graph T(Pn) has vertex set V (Pn) ∪E(Pn) with two vertices adjacent when-ever they are neighbors in Pn. Balakrishnan, Selvam, and Yegnanarayanan have proved that T(Pn) is harmonious.
For any graph G with vertices v1, . . . , vn and a vector m = (m1, . . . , mn) of positive integers the corresponding replicated graph, Rm(G), of G is defined as follows. For each vi form a stable set Si consisting of mi new vertices i = 1, 2, . . . , n (a stable set S consists of a set of vertices such that there is not an edge vivj for all pairs vi, vj in S); two stable sets Si, Sj, i ̸= j, form a complete bipartite graph if each vivj is an edge in G and otherwise there are no edges between Si and Sj. Ramírez-Alfonsín has proved that Rm(Pn) is graceful for all m and all n > 1 (see §3.4 for a stronger result) and that R(m,1,...,1)(C4n), R(2,1,...,1)(Cn) (n ≥8) and,R(2,2,1,...,1)(C4n) (n ≥12) are graceful.
For any permutation f on 1, . . . , n, the f-permutation graph on a graph G, P(G, f), consists of two disjoint copies of G, G1 and G2, each of which has vertices labeled v1, v2, . . . , vn with n edges obtained by joining each vi in G1 to vf(i) in G2. In 1983 Lee (see ) conjectured that for all n > 1 and all permutations on 1, 2, . . . , n, the permu-tation graph P(Pn, f) is graceful. Lee, Wang, and Kiang proved that P(P2k, f) is graceful when f = (12)(34) · · · (k, k + 1) · · · (2k −1, 2k). They conjectured that if G is a graceful nonbipartite graph with n vertices, then for any permutation f on 1, 2, . . . , n, the permutation graph P(G, f) is graceful. Fan and Liang have shown that if f is a per-mutation in Sn where n ≥2(m −1) + 2l then the permutation graph P(Pn, f) is graceful if the disjoint cycle form of f is Ql−1 k=0(m + 2k, m + 2k + 1), and if n ≥2(m −1) + 4l the permutation graph P(Pn, f) is graceful the disjoint cycle form of f is Ql−1 k=0(m + 4k, m + 4k + 2)(m + 4k + 1, m + 4k + 3). For any integer n ≥5 and some permutations f in S(n), Liang and Y. Miao, discuss gracefulness of the permutation graphs P(Pn, f) if f = (m, m+1, m+2, m+3, m+4), (m, m+2)(m+1, m+3), (m, m+1, m+2, m+4, m+ 3), (m, m+1, m+4, m+3, m+2), (m, m+2, m+3, m+4, m+1), (m, m+3, m+4, m+2, m+1) and (m, m + 4, m + 3, m + 2, m + 1). In Liang, Zhang, Xu, Ye, Fan, and Ge prove the permutation graphs P(Pn, f) where f is one of the permutations (12345), (2345), (234), (123456) and (23)(45) are graceful. Some families of graceful permutation graphs are given in , , and .
In Rofa defined generalized strongly graceful permutations and discovered two the electronic journal of combinatorics (2023), #DS6 38 new permutations in addition to the known permutation that is obtained by replacing each vertex label f(v) by q −f(v). He use these permutations to prove, by induction, that a lobster with a perfect matching that consists of the set of end edges of the lobster, is strongly graceful. He further showed that there exist strongly graceful labelings that assign the label 0 to four specific vertices of any tree belonging to this family of lobsters.
He provided a tractable way for proving an equivalent form of Bermond conjecture which states that all lobsters are graceful. Two out of a total of three cases of the proposed equivalent form of Bermond’s conjecture are completed leaving the third case open for refutation or completion. As an applications of his results, he provided in number systems that are representable by all vertices of a rooted symmetric tree in such a way that the number representation of each vertex depends on its distance from the root vertex.
In Bell provided methods to combine graceful bipartite graphs to create new graceful graphs.
These methods unify and generalize some well-known results in the graceful labeling literature. She also found a new class of graceful trees.
The power graph of a finite group G has the elements of G as its vertex set and two distinct vertices of G are adjacent if one is a power of other. Sehgal, Takshak, Maan, and Malik proved that the power graph of Zk−1 2 × Z4 has a graceful labeling.
A graph (p, q)-graph G(V, E) is said to be (k, d)-hooked Skolem graceful if there exists a bijection f from V (G) to {1, 2, . . . , p −1, p + 1} such that the induced edge labeling gf from E to {k, k+d, . . . , k+(n−1)d} defined by gf(uv) = |f(u)f(v)| for all uv in E is also bijective. Such a labeling f is called a (k, d)-hooked Skolem graceful labeling of G. Note that when k = d = 1, this notion coincides with that of hooked Skolem graceful labeling of the graph G. In Pereira, Singh, and Arumugam present some preliminary results on (k, d)-hooked Skolem graceful graphs and prove that nK2 is (2, 1)-hooked Skolem graceful if and only if n ≡1 or 2 (mod 4).
Gnanajothi [1143, p. 51] calls a graph G bigraceful if both G and its line graph are graceful. She shows the following are bigraceful: Pm; Pm × Pn; Cn if and only if n ≡0, 3 (mod 4); Sn; Kn if and only if n ≤3; and Bn if and only if n ≡3 (mod 4). She also shows that Km,n is not bigraceful when n ≡3 (mod 4). (Gangopadhyay and Hebbare used the term “bigraceful” to mean a bipartite graceful graph.) Murugan and Arumugan have shown that graphs obtained from C4 by attaching two disjoint paths of equal length to two adjacent vertices are bigraceful.
Several well-known isolated graphs have been examined. Graceful labelings have been found for the Petersen graph , the cube , the icosahedron and the dodecahe-dron. In [pp. 163-164] Gardner credits Ashenfelter and Chandra for showing the Platonic solids have graceful labelings.1 Gardner stated that the icosahedron has only five fundamentally different graceful labelings, whereas in 2021 Knuth determined the correct number to be 24. Graham and Sloane showed that all of these solids except the cube are harmonious. Winters verified that the Grőtzsch graph (see [615, p.
118]), the Heawood graph (see [615, p. 236]), and the Herschel graph (see [615, p. 53]) are graceful. Graham and Sloane determined all harmonious graphs with at most five vertices. Seoud and Youssef did the same for graphs with six vertices.
1D. Knuth called this to my attention.
the electronic journal of combinatorics (2023), #DS6 39 In 2009 Zak defined the following generalization of harmonious labelings. For a graph G(V, E) and a positive integer t ≥|E| a function h from V (G) to Zt (the additive group of integers modulo t) is called a t-harmonious labeling of G if h is injective for t ≥|V | or surjective for t < |V |, and h(u) + h(v) ̸= h(x) + h(y) for all distinct edges uv and xy. The smallest such t for which G has a t-harmonious labeling is called the harmonious order of G. Obviously, a graph G(V, E)with |E| ≥|V | is harmonious if and only if the harmonious order of G is |E|. Zak determines the harmonious order of complete graphs, complete bipartite graphs, even cycles, some cases of P k n, and 2nK3. He presents some results about the harmonious order of the Cartesian products of graphs, the disjoint union of copies of a given graph, and gives an upper bound for the harmonious order of trees. He conjectures that the harmonious order of a tree of order n is n + o(n). Hegde and Murthy proved Zak’s conjecture using the value sets of polynomials, which partially proves the cordial tree conjecture by Hovey that all trees of order less than a prime p are p-cordial. (See Section 3.7.) A graceful labeling of Pn is said to be an (a, b; n)-graceful labeling if one endpoint is labeled a and the other labeled b. A conjecture made in Gvozdjak’s PhD Thesis on the Oberwolfach Problem in 2004 is: “An (a, b; n)-graceful labeling of Pn exists if and only if the integers a, b, n satisfy (1) b −a has the same parity as n(n + 1)/2; (2) 0 < |b −a| ≤(n + 1)/2 and (3) n/2 ≤a + b ≤3n/2.” In Zhang, Zhang, and Wang showed that the conjecture is true for every n whenever it is true for n ≤4a + 1 and a is a fixed value. Moreover, they proved that the conjecture is true for a = 0, 1, 2, 3, 4, 5, 6.
For a graph with e edges Vietri generalizes the notion of a graceful labeling by allowing the vertex labels to be real numbers in the interval [0, e]. For a simple graph G(V, E) he defines an injective map γ from V to [0, e] to be a real-graceful labeling of G provided that P 2γ(u)−γ(v) + 2γ(v)−γ(u) = 2e+1 −2−e −1, where the sum is taken over all edges uv. In the case that the labels are integers, he shows that a real-graceful labeling is equivalent to a graceful labeling. In contrast to the case for graceful labelings, he shows that the cycles C4t+1 and C4t+2 have real-graceful labelings. He also shows that the non-graceful graphs K5, K6, and K7 have real-graceful labelings. With one exception, his real-graceful labels are integers.
The gamma-number (or gracefulness) of a graph G, denoted by γ (G), is the smallest positive integer n for which there exists an injective function f : V (G) →{0, 1, . . . , n} such that each uv ∈E (G) is labeled |f (u) −f (v)| and the resulting edge labels are distinct. The strong gamma-number of a graph G, denoted by γs (G), is defined to be the smallest positive integer n such that γ (G) = n with the additional property that there exists an integer λ so that min {f (u) , f (v)} ≤λ max {f (u) , f (v)} for each uv ∈E (G).
The strong gamma-number is defined to be +∞, otherwise. Ichishima and Oshima proved that if G is a bipartite graph, then γ (mG) ≤mγ (G) + m −1 for any positive integer m. They also show that γs (G) < +∞and γs (G) ≤2γ (G) + 1 for any bipartite graph G. Moreover, they provide a sharp upper bound for γ (G ∪H) in terms of γ (G) and γs (H) when G and H are graphs such that H is bipartite, and give formulas for the gamma-number of certain forests. In addition to these, they present strong gamma-number analogues to the gamma-number results and determine the exact values of the the electronic journal of combinatorics (2023), #DS6 40 gamma-number and strong gamma-number for all cycles.
A graph G with m vertices and n edges, is said to be prime graceful if there is an injection φ from the vertices of G to {1, 2, . . . , k} where k = min{2m, 2n} such that gcd(φ(vi), φ(vj)) = 1 and the induced injective function φ∗from the edges of G to {1, 2, . . . , k1} defined by φ∗(vivj) = |φ(vi)φ(vj)|, the resulting edge labels are distinct.
In Selvarajan and Subramoniam proved paths, cycles, stars, friendship graphs, bistars, C4 ∪Pn, Km,2, and Km,2 ∪Pn have prime graceful labelings.
In 2020 Zeen El Deen introduced a new type of labeling of a graph as follows.
For any positive integer δ an edge δ graceful labeling of a graph G(V, E) with p vertices and q edges is a bijective f from E to {δ, 2δ, . . . , qδ} such f[V ) = {f(u) = P f(uv) mod(kδ) over all edges v incident to u and k = max(p, q) are pairwise distinct. He proved the existence of an edge δ−graceful labeling, for any positive integer δ, for wheels, alternate triangular cycles, double wheels, Cn×P2, Wn×P2, gears, helms, butterflies, and friendship graphs. In Zeen El Deen and Elmahdy showed that for any positive integer δ there is an edge δ graceful labeling for the following graphs: the splitting graphs of cycles, fans, and crowns; the shadow graphs of the paths, cycles, and fans; the middle graphs and the total graphs of paths, cycles, and crowns; twigs; and snails.
In Byers and O’Mellan introduced a new concept that combines graceful and harmonious labelings as follows. A connected graph G with size m is said to be a graceful-harmonious labeling if there exits injection f : V (G) →{0, 1, 2, . . . , m} such that when each edge uv is assigned either |f(u)f(v)| or (f(u) + f(v)) (mod q). each edge receives a distinct label from the set {0, 1, 2, . . . , m}. They prove that cycles, friendship graphs, and double cones C4k + K2 admit graceful-harmonious labelings.
For a graph G(V, E) without isolated vertices, Pereira, Singh, and Arumugam defined the gracefulness, grac(G), of G as the smallest positive integer k for which there exists an injective function f : V →{0, 1, 2, . . . , k} such that the edge induced function gf : E →{1, 2, . . . , k} defined by gf(uv) = |f(u) −f(v)| is also injective. Let c(f) = {1, 2, . . . , i} denote the edge labels and let m(G) = max{c(f)}, where the maximum is taken over all injective functions f : V →N ∪{0} such that gf is also injective. This measure m(G) determines how close G is to being graceful. They determine m(G) for certain cycles and friendship graphs.
A number of authors have investigated the gracefulness of the directed graphs obtained from copies of directed cycles ⃗ Cm that have a vertex in common or have an edge in common. A digraph D(V, E) is said to be graceful if there exists an injection f : V (G) → {0, 1, . . . , |E|} such that the induced function f ′ : E(G) →{1, 2, . . . , |E|} that is defined by f ′(u, v) = (f(v) −f(u)) (mod |E| + 1) for every directed edge uv is a bijection. The notations n · ⃗ Cm and n −⃗ Cm are used to denote the digraphs obtained from n copies of ⃗ Cm with exactly one point in common and the digraphs obtained from n copies of ⃗ Cm with exactly one edge in common. Du and Sun proved that a necessary condition for n−⃗ Cm to be graceful is that mn is even and that n· ⃗ Cm is graceful when m is even. They conjectured that n · ⃗ Cm is graceful for any odd m and even n. This conjecture was proved by Jirimutu, Xu, Feng, and Bao in . Xu, Jirimutu, Wang, and Min proved that n −⃗ Cm is graceful for m = 4, 6, 8, 10 and even n. Feng and Jirimutu (see ) the electronic journal of combinatorics (2023), #DS6 41 conjectured that n −⃗ Cm is graceful for even n and asked about the situation for odd n.
The cases where m = 5, 7, 9, 11, and 13 and even n were proved Zhao and Jirimutu .
The cases for m = 15, 17, and 19 and even n were proved by Zhao et al.
in , and . Zhao, Siqintuya, and Jirimutu proved that a necessary condition for n −⃗ Cm to be graceful is that nm is even.
In a 1985 paper Bloom and Hsu say a directed graph D with e edges has a graceful labeling θ if for each vertex v there is a vertex labeling θ that assigns each vertex a distinct integer from 0 to e such that for each directed edge (u, v) the integers θ(v) −θ(u) mod (e + 1) are distinct and nonzero. They conjectured that digraphs whose underlying graphs are wheels and that have all directed edges joining the hub and the rim in the same direction and all directed edges in the same direction are graceful. This conjecture was proved in 2009 by Hegde and Shivarajkumar . Yao, Yao, and Cheng investigated the gracefulness for many orientations of undirected trees with short diameters and proved some directed trees do not have graceful labelings.
Hegde and Kumudkshi established the gracefulness of the directed graph that is an orientation of the planar grid graph Pm × Pn in which each cell is a unicycle of length four.
A graceful difference labeling of a directed graph G with vertex set V is a bijection f : V → {1, . . . , |V |} such that, when each arc uv is assigned the difference label f(v)f(u), the resulting arc labels are distinct. Hertz and Picouleau conjectured that all disjoint unions of circuits have a graceful difference labeling, except in two particular cases. They provided partial results that support this conjecture.
A survey of results on graceful digraphs by Feng, Xu, and Jirimutu is given in . Marr and summarizes previously known results on graceful directed graphs and presents some new results on directed paths, stars, wheels, and umbrellas.
In Shivarajkumar, Sriraj, and Hegde provided a 2021 survey results on graceful labeling of digraphs.
2.8 Summary The results and conjectures discussed above are summarized in the tables following. The letter G after a class of graphs indicates that the graphs in that class are known to be graceful; a question mark indicates that the gracefulness of the graphs in the class is an open problem; we put a question mark after a “G” if the graphs have been conjectured to be graceful. The analogous notation with the letter H is used to indicate the status of the graphs with regard to being harmonious. The tables impart at a glimpse what has been done and what needs to be done to close out a particular class of graphs. Of course, there is an unlimited number of graphs one could consider. One wishes for some general results that would handle several broad classes at once but the experience of many people suggests that this is unlikely to occur soon.
The Graceful Tree Conjecture alone has withstood the efforts of scores of people over the past four decades. Analogous sweeping conjectures are probably true but appear hopelessly difficult to prove. I thank Don Knuth for his correspondence about the results of Smith and Puget in Table 1 regarding the gracefulness Km × K, Km × Pn, and Km × Cn.
the electronic journal of combinatorics (2023), #DS6 42 Table 1: Summary of Graceful Results Graph Graceful trees G if ≤35 vertices G if symmetrical G if at most 4 end-vertices G with diameter at most 5 G? Ringel-Kotzig G caterpillars G firecrackers G bananas , G? lobsters cycles Cn G iff n ≡0, 3 (mod 4) wheels Wn G , helms (see §2.2) G webs (see §2.2) G gears (see §2.2) G cycles with Pk-chord (see §2.2) G , , , Cn with k consec. chords (see §2.2) G if k = 2, 3, n −3 , unicyclic graphs G? iff G ̸= Cn, n ≡1, 2 (mod 4) P k n G if k = 2 C(t) n (see §2.2) n = 3 G iff t ≡0, 1 (mod 4) , G? if nt ≡0, 3 (mod 4) G if n = 6, t even G if n = 4, t > 1 G if n = 5, t > 1 G if n = 7 and t ≡0, 3 (mod 4) G if n = 9 and t ≡0, 3 (mod 4) G if t = 2 n ̸≡1 (mod 4) , G if n = 11 Continued on next page the electronic journal of combinatorics (2023), #DS6 43 Table 1 – Continued from previous page Graph Graceful triangular snakes (see §2.2) G iff no. blocks ≡0, 1 (mod 4) K4-snakes (see §2.2) ?
quadrilateral snakes (see §2.2) G , crowns Cn ⊙K1 G Cn ⊙Pk G grids Pm × Pn G prisms Cm × Pn G if n = 2 , G if m even G if m odd and 3 ≤n ≤12 G if m = 3 G if m = 6 see G if m ≡2 (mod 4), n ≡3 (mod 4) Km × Pn G if (m, n) = (4, 2), (4, 3), (4, 4), (4, 5), (5, 2), (5, 3), (6, 3), (4, 6), (4, 7), (4, 8) not G if (3, 3), (m, 2) m = 6, 7, 8, 9, 10, 11,12 not G? for (m, 2) with m > 12 Km × Cn G if (m, n) = (4, 3), (3, 4), (4, 4), (4, 5), (3, 6), (4, 6) not G for (m, n) = (6, 3) Km ⊙K1 G if m = 3, 4, 5, 6, 7, 8, 9 not G if m = 10, 11, 12, 13, 14, 15 not G? if m > 15 Km,n ⊙K1 G Km ∪Kn (m, n > 1) G iff {m, n} = {4, 2} or {5, 2} St i=1 Kmi,ni G 2 ≤mi < ni torus grids Cm × Cn G if m ≡0 (mod 4), n even Continued on next page the electronic journal of combinatorics (2023), #DS6 44 Table 1 – Continued from previous page Graph Graceful not G if m, n odd (parity condition) vertex-deleted Cm × Pn G if n = 2 edge-deleted Cm × Pn G if n = 2 Möbius ladders Mn (see §2.3) G stacked books Sm × Pn n = 2, G iff m ̸≡3 (mod 4) , (see §2.3) , G if m even n-cube K2 × K2 × · · · × K2 G Kn × P3 G iff n ≤6 Kn G iff n ≤4 , Km,n G , K1,m,n G K1,1,m,n G windmills K(m) n (n > 3) (see §2.4) G if n = 4, m ≤1000 , , , G? if n = 4, m ≥4 G if n = 5, m = 4, 5 not G if n = 4, m = 2, 3 not G if (m, n) = (2, 5) not G if n > 5 B(n, r, m) r > 1 (see §2.4) G if (n, r) = (3, 2), (4, 3) , (4,2) G (n, r, m) = (5, 2, 2) not G for (n, 2, 2) for n > 5 , mKn (see §2.5) G iff m = 1, n ≤4 Cm ∪Pn G iff m + n ≥6 Cm ∪Cn G iff m + n ≡0, 3 (mod 4) Cn ∪Kp,q for n > 8 G iff n ≡0, 3 (mod 4) Continued on next page the electronic journal of combinatorics (2023), #DS6 45 Table 1 – Continued from previous page Graph Graceful G C6 × K1,2n+1 G C3 × Km,n iff m, n ≥2 G C4 × Km,n iff (m, n) ̸= (1, 1) G C7 × Km,n G C8 × Km,n Ki ∪Km,n G St i=1 Kmi,ni G 2 ≤mi < ni Cm ∪St i=1 Kmi,ni G 2 ≤mi < ni, m ≡0 or 3 (mod 4), m ≥11 G + Kt G for connected graceful G double cones Cn + K2 G for n = 3, 4, 5, 7, 8, 9, 11, 12 not G for n ≡2 (mod 4) t-point suspension Cn + Kt G if n ≡0 or 3 (mod 12) not G if t is even and n ≡2, 6, 10 (mod 12) G if n = 4, 7, 11 or 19 G if n = 5 or 9 and t = 2 P 2 n (see §2.7) G Petersen P(n, k) (see §2.7) G for n = 5, 6, 7, 8, 9, 10 , (n, k) = (8t, 3) Table 2: Summary of Harmonious Results Graph Harmonious trees H if ≤31 vertices H? H caterpillars ? lobsters Continued on next page the electronic journal of combinatorics (2023), #DS6 46 Table 2 – Continued from previous page Graph Harmonious cycles Cn H iff n is odd wheels Wn H helms (see §2.2) H , webs (see §2.2) H if cycle is odd gears (see §2.2) H cycles with Pk-chord (see §2.2) ?
Cn with k consec. chords ?
(see §2.2) unicyclic graphs ?
P k n H if k = 2 , k odd , H if k is even and k/2 ≤(n −1)/2 C(t) n (see §2.2) n = 3 H iff t ̸≡2 (mod 4) H if n = 4, t > 1 triangular snakes (see §2.2) H if number of blocks is odd not H if number of blocks ≡2 (mod 4) K4-snakes (see §2.2) H quadrilateral snakes (see §2.2) ?
crowns Cn ⊙K1 H , grids Pm × Pn H iff (m, n) ̸= (2, 2) prisms Cm × Pn H if n = 2, m ̸= 4 H if n odd H if m = 4 and n ≥3 Continued on next page the electronic journal of combinatorics (2023), #DS6 47 Table 2 – Continued from previous page Graph Harmonious torus grids Cm × Cn, H if m = 4, n ≥3 not H if m ̸≡0 (mod 4), n odd vertex-deleted Cm × Pn H if n = 2 edge-deleted Cm × Pn H if n = 2 Möbius ladders Mn (see §2.3) H iff n ̸= 3 stacked books Sm × Pn (see §2.3) n = 2, H if m even , not H m ≡3 (mod 4), n = 2, (parity condition) H if m ≡1 (mod 4), n = 2 n-cube K2 × K2 × · · · × K2 H if and only if n ≥4 K4 × Pn H Kn H iff n ≤4 Km,n H iff m or n = 1 K1,m,n H K1,1,m,n H windmills K(m) n (n > 3) (see §2.4) H if n = 4 m = 2, H? iff n = 4 not H if m = 2, n odd or 6 not H for some cases m = 3 B(n, r, m) r > 1 (see §2.4) (n, r) = (3, 2), (4, 3) mKn (see §2.5) H n = 3, m odd not H for n odd and m ≡2 (mod 4) nG H when G is harmonious and n odd Continued on next page the electronic journal of combinatorics (2023), #DS6 48 Table 2 – Continued from previous page Graph Harmonious Gn H when G is harmonious and n odd Cm ∪Pn H n = 1 iff m ̸= 2 mod 4 H n = 2 H (m, 3) odd m ≥3 , H? (m, 3) m ≥3 fans Fn = Pn + K1 H nCm + K1 n ̸≡0 mod 4 H double fans Pn + K2 H t-point suspension Pn + Kt of Pn H Sm + K1 H , t-point suspension Cn + Kt of Cn H if n odd and t = 2 , not H if n ≡2, 4, 6 (mod 8) and t = 2 Petersen P(n, k) (see §2.7) H , the electronic journal of combinatorics (2023), #DS6 49 3 Variations of Graceful Labelings 3.1 α-labelings In 1966 Rosa defined an α-labeling (or α-valuation) as a graceful labeling with the additional property that there exists an integer k so that for each edge xy either f(x) ≤k < f(y) or f(y) ≤k < f(x). (Other names for such labelings are balanced, interlaced, and strongly graceful.) It follows that such a k must be the smaller of the two vertex labels that yield the edge labeled 1. Also, a graph with an α-labeling is necessarily bipartite and therefore can not contain a cycle of odd length. Wu has shown that a necessary condition for a bipartite graph with n edges and degree sequence d1, d2, . . . , dp to have an α-labeling is that the gcd(d1, d2, . . . , dp, n) divides n(n −1)/2. Barrientos and Minion proved that any tree of size n and excess ϵ is a spanning tree of a graph of size n + ϵ that admits an α-labeling.
For a path with consecutive vertices v1, v2, . . . , vn a triangular tree is the tree obtained identifying each vi to an end vertex of the path Pi.
Barrientos proved that all triangular trees admit an α-labeling. He also presented several ways to combine this type of trees to construct new trees and unicyclic graphs that can α-labeled.
A common theme in graph labeling papers is to build up graphs that have desired labelings from pieces with particular properties.
In these situations, starting with a graph that possesses an α-labeling is a typical approach. (See , , , and .) Moreover, Jungreis and Reid showed how sequential labelings of graphs (see Section 4.1) can often be obtained by modifying α-labelings of the graphs.
Graphs with α-labelings have proved to be useful in the development of the theory of graph decompositions. Rosa , for instance, has shown that if G is a graph with q edges and has an α-labeling, then for every natural number p, the complete graph K2qp+1 can be decomposed into copies of G in such a way that the automorphism group of the decomposition itself contains the cyclic group of order p. In the same vein El-Zanati and Vanden Eynden proved that if G has q edges and admits an α-labeling then Kqm,qn can be partitioned into subgraphs isomorphic to G for all positive integers m and n. Although a proof of Ringel’s conjecture that every tree has a graceful labeling has withstood many attempts, examples of trees that do not have α-labelings are easy to construct (one example is the subdivision graph of K1,3 — see ). Kotzig has shown however that almost all trees have α-labelings. Sethuraman and Ragukumar have proved that every tree is a subtree of a graph with an α-labeling.
As to which graphs have α-labelings, Rosa observed that the n-cycle has an α-labeling if and only if n ≡0 (mod 4) whereas Pn always has an α-labeling. Other familiar graphs that have α-labelings include caterpillars , the n-cube , Möbius ladders Mn when n is odd (see §2.3) for the definition) , B4n+1 (i.e., books with 4n + 1 pages) , C2m ∪C2m and C4m ∪C4m ∪C4m for all m > 1 , C4m ∪C4m ∪C4n for all (m, n) ̸= 1, 1) , Pn × Qn , K1,2k × Qn , C4m ∪C4m ∪C4m ∪C4m , C4m ∪C4n+2 ∪C4r+2, C4m ∪C4n ∪C4r when m + n ≤r , C4m ∪C4n ∪C4r ∪C4s when m ≥n+r+s , C4m∪C4n∪C4r+2∪C4s+2 when m ≥n+r+s+1 , ((m+1)2+1)C4 for the electronic journal of combinatorics (2023), #DS6 50 all m , k2C4 for all k , and (k2 +k)C4 for all k . Abrham and Kotzig have shown kC4 has an α-labeling for 4 ≤k ≤10 and that if kC4 has an α-labeling then so does (4k + 1)C4, (5k + 1)C4, and (9k + 1)C4. Eshghi proved that 3C4k and 5C4k have an α-labeling for all k. In Eshghi and Carter show several families of graphs of the form C4n1 ∪C4n2 ∪· · · ∪C4nk have α-labelings.
In Eshghi provides an integer programming model and a Tabu search algorithm to generate α-labelings of the quadratic graphs mC4k) where 6 ≥m ≥10 and 2 ≥k ≥10.
(See also .) The computational complexity of the gracefulness of a graph is not known, but the complexity of finding a harmonious labeling of a graph is in the NP-class . Research on programming models for finding graceful labelings of graphs can be found in , , , , , , , , , and .
In Amini and Eshghi gave a new mathematical integer programming model for the graph labeling graphs of the form mCn (some authors use the notation Q(m, n)).
The advantages of this model are linearity and the existence of an objective function.
They also gave two constraint programming models and a meta-heuristics algorithm that generate feasible graceful labeling and α-labeling for special classes of quadratic graphs.
Their results include: mC4k with 1 ≤11 and less than 1000 vertices has an α-labeling with the exception of 3C4; 12C4k has α-labeling for 1 ≤k ≤19; and 13C4k has α-labeling for 1 ≤k ≤13. In and Eshghi and Salarrezaei proved that 7C4k has an α-labeling for all k. Lakshmi and Vangipuram proved that 4C4k is graceful.
Paterson and Stinson used α-valuations of the lexicographic product of cycles new and the complement of a complete graph to construct various circular external difference families.
In , Barrientos and Minion investigated series-parallel operations with graphs that admit α-labelings. They provided necessary conditions on the graphs G1 and G2 to obtain a new α-labeled graph G through each of these operations. As consequence of the series operation, they proved that the one-point union of three or four copies of Kn,n has an α-labeling, and that any tree with maximum degree four that can be decomposed into copies of the path of length eleven has an α-labeling when the distance between any pair of vertices of degree four is even. They also showed that any graph of order n + 1 and size n with an α-labeling is an induced subgraph of a graph of order n + 3 and size 2n + 1. Additionally, they presented an α-labeling for any graph of the form K2,n × Pm.
In Barrientos used vertex and edge duplications, replications of the entire graph, and k-vertex amalgamations to generate α-labeled graphs. He proved that for some families of graphs, it is possible to duplicate several vertices or edges. Using k-vertex amalgamations he obtained an α-labeling of a graph that can be decomposed into multiple copies of a given α-labeled graph as well as a robust family of irregular grids that can α-labeled.
Figueroa-Centeno, Ichishima, and Muntaner-Batle have shown that if m ≡0 (mod 4) then the one-point union of 2, 3, or 4 copies of Cm admits an α-labeling, and if m ≡2 (mod 4) then the one-point union of 2 or 4 copies of Cm admits an α-labeling.
They conjecture that the one-point union of n copies of Cm admits an α-labeling if and only if mn ≡0 (mod 4).
In Simarmata, Sandy, and Sugeng gave a new approach to showing graphs admit the electronic journal of combinatorics (2023), #DS6 51 an α labeling using an adjacency matrix. They use it to construct graceful labelings for superstars (rooted trees constructed from several star graphs by connecting the leaves from each star graph to a root vertex) and a super-rooted trees (rooted trees constructed from several rooted trees by connecting the root vertex of each rooted tree to a root vertex).
Pei-Shan Lee proved that C6×P2t+1 and gear graphs have α-labelings. He raises the question of whether C4m+2 × P2t+1 has an α-labeling for all m. Brankovic, Murch, Pond, and Rosa conjectured that all trees with maximum degree three and a perfect matching have an α-labeling. In Uma and Rajasekaran gave an α-valuation for new the tensor product of paths and cycles.
In his 2001 Ph. D. thesis Selvaraju investigated the one-point union of complete bipartite graphs. He proves that the one-point unions of the following forms have an α-labeling: Km,n1 and Km,n2; Km1,n1, Km2,n2, and Km3,n3 where m1 ≤m2 ≤m3 and n1 < n2 < n3; Km1,n, Km2,n, and Km3,n where m1 < m2 < m3 ≤2n.
Zhile uses Cm(n) to denote the connected graph all of whose blocks are Cm and whose block-cutpoint-graph is a path. He proves that for all positive integers m and n, C4m(n) has an α-labeling but Cm(n) does not have an α-labeling when m is odd.
Abrham and Kotzig have proved that Cm ∪Cn has an α-labeling if and only if both m and n are even and m + n ≡0 (mod 4). Kotzig has also shown that C4 ∪C4 ∪C4 does not have an α-labeling. He asked if n = 3 is the only integer such that the disjoint union of n copies of C4 does not have an α-labeling. This was confirmed by Abrham and Kotzig in . Eshghi proved that every 2-regular bipartite graph with 3 components has an α-labeling if and only if the number of edges is a multiple of four except for C4 ∪C4 ∪C4. In Eshghi gives more results on the existence of α-labelings for various families of disjoint union of cycles.
Jungreis and Reid investigated the existence of α-labelings for graphs of the form Pm × Pn, Cm × Pn, and Cm × Cn (see also ). Of course, the cases involving Cm with m odd are not bipartite, so there is no α-labeling. The only unresolved cases among these three families are C4m+2 × P2n+1 and C4m+2 × C4n+2. All other cases result in α-labelings.
Let v1,j, v2,j, . . . , vm,j be the consecutive vertices of the jth copy of Pm in Pm × Pn.
An elementary transformation of Pm × Pn is the graph obtained by replacing the edge vi,jvi+1,j by the new edge vix,jvi+1+x,j. A graph is said to be a grid-like graph if it is obtained through a sequence of elementary transformations. In Barrientos and Minion proved the existence of an α-labeling for any grid-like graph. As consequence of this result, they showed that the graphs C4t × Pn ∪Pn and C4t × Pn ∪Pt−1 × Pn admit α-labelings.
Balakrishman uses the notation Qn(G) to denote the graph P2×P2×· · ·×P2×G where P2 occurs n −1 times. Snevily has shown that the graphs Qn(C4m) and the cycles C4m with the path Pn adjoined at each vertex have α-labelings. He also has shown that compositions of the form G[Kn] (see §2.3 for the definition) have an α-labeling whenever G does (see §2.3 for the definition of composition). Balakrishman and Kumar have shown that all graphs of the form Qn(G) where G is K3,3, K4,4, or Pm have an α-labeling. Balakrishman poses the following two problems. For which the electronic journal of combinatorics (2023), #DS6 52 graphs G does Qn(G) have an α-labeling? For which graphs G does Qn(G) have a graceful labeling?
Rosa has shown that Km,n has an α-labeling (see also ). In Ichishima and Oshima proved that if m, s and t are integers with m ≥1, s ≥2, and t ≥2, then the graph mKs,t has an α-labeling if and only if (m, s, t) ̸= (3, 2, 2). Barrientos has shown that for n even the graph obtained from the wheel Wn by attaching a pendent edge at each vertex has an α-labeling. In Barrientos shows how to construct graceful graphs that are formed from the one-point union of a tree that has an α-labeling, P2, and the cycle Cn. In some cases, P2 is not needed. Qian has proved that quadrilateral snakes have α-labelings. Yu, Lee, and Chin showed that Q3-and Q3-snakes have α-labelings. Fu and Wu showed that if T is a tree that has an α-labeling with partite sets V1 and V2 then the graph obtained from T by joining new vertices w1, w2, . . . , wk to every vertex of V1 has an α-labeling. Similarly, they prove that the graph obtained from T by joining new vertices w1, w2, . . . , wk to the vertices of V1 and new vertices u1, u2, . . . , ut to every vertex of V2 has an α-labeling. They also prove that if one of the new vertices of either of these two graphs is replaced by a star and every vertex of the star is joined to the vertices of V1 or the vertices of both V1 and V2, the resulting graphs have α-labelings.
Fu and Wu further show that if T is a tree with an α-labeling and the sizes of the two partite sets of T differ by at most 1, then T × Pm has an α-labeling. Zhao, Ma, and Yao proved that a class of super lobster trees have α-labelings. Ghosh uses various methods of joining graceful graphs and graphs with α-labelings to obtain some classes of graceful lobsters. Lalitha and Tamilselvi proved that the hexagonal snake graph has an α-labeling.
Selvaraju and G. Sethurman prove that the graphs obtained from a path Pn by joining all the pairs of vertices u, v of Pn with d(u, v) = 3 and the graphs obtained by identifying one of vertices of degree 2 of such graphs with the center of a star and the other vertex the graph of degree 2 with the center of another star (the two stars needs need not have the same size) have α-labelings. They conjecture that the analogous graphs where 3 is replaced with any t with 2 ≤t ≤n −2 have α-labelings.
In Hafez, El-Shanawany, and El Atik proved that if G has an α-labeling, then new G[Kr], r ≥2 and Splm(G) have α-labelings. Let the vertices of Pn (n > 1) be v1v2 · · · vn.
Hafez et al. define K(n) = {Ks(n), s = 1, 2, . . .}, where Ks(n) is the graph obtained from Pn by joining the end vertices of the edge vivi+1 to every vertex in the complete graph Kri, i = 1, 2, . . . , n −1, of order ri, such that (a) if |V (Kri) ∩V (Kri+1)| = t, then t ≤min{ri, ri+1} and (b) Kri and Krj are disjoint when |i −j| > 2. They further prove that if G has an α-labeling, then the graph G ▽K(n) has an α-labeling (G ▽H has V (G) × V (H) as its vertex set and u = (x1, y1) is adjacent to v = (x2, y2) whenever y1 = y2 and x1x2 ∈E(G) or x1x2 ∈E(G) and y1y2 ∈E(H).
Makadia, Karavadiya, and Kanerian proved that the graph obtained by merging t consecutive vertices of two cycle C4r and C4s has an α-labeling when t ≤2min{r, s}.
They also proved that if G1 has an α-labeling and G2 is graceful then there exists a graceful labeling of the graph obtained by joining G1 and G2 by any path. Moreover, if both G1 and G2 have α-labelings then there exists an α-labeling of the graph obtained by the electronic journal of combinatorics (2023), #DS6 53 joining G1 and G2 by any path. Let Cn1, Cn2, . . . , Cnk be a collection of cycles. In , Barrientos and Minion say that a graph G is the coalescence of these cycles if for every 2 ≤i ≤k, the first ti vertices of Cni are identified with the last ti vertices of Cni−1, where ti ≤ni/2. They proved that the coalescence of these cycles admits an α-labeling when each ni ≡0( mod 4).
Lee and Liu investigated the mirror graph M(m, n) of Km,n (see §2.3 for the definition) for α-labelings. They proved: M(m, n) has an α-labeling when n is odd or m is even; M(1, n) has an α-labeling when n ≡0 (mod 4); M(m, n) does not have an α-labeling when m is odd and n ≡2 (mod 4), or when m ≡3 (mod 4) and n ≡4 (mod 8).
Kumar, Mishra, Kumar, and Kumar proved that the following graphs have alpha labelings: C4n ⊙K1, the graph obtained by joining any path to a vertex of C4n, and graphs obtained by joining two isomorphic copies of C4n ⊙K1.
Barrientos and Minion proved that the Cartesian product of two α-trees is an α-tree when both trees admit α-labelings and their stable sets are balanced. (A stable set S consists of a set of vertices such that there is not an edge vivj for all pairs vi, vj in S).
In addition, they present a tree that has the property that when any number of pendent vertices are attached to the vertices of any subset of its smaller stable set the resulting graph is an α-tree. They also prove of an α-labeling of three types of graphs obtained by connecting, sequentially, any number of paths of equal size.
Barrientos defines a chain graph as one with blocks B1, B2, . . . , Bm such that for every i, Bi and Bi+1 have a common vertex in such a way that the block-cutpoint graph is a path. He shows that if B1, B2, . . . , Bm are blocks that have α-labelings then there exists a chain graph G with blocks B1, B2, . . . , Bm that has an α-labeling. He also shows that if B1, B2, . . . , Bm are complete bipartite graphs, then any chain graph G obtained by concatenation of these blocks has an α-labeling.
The symmetric product G1 ⊕G2 of G1 and G2 is the graph with vertex set V (G1) × V (G2) and edge set {(u1, v1)(u2, v2)} where u1u2 is an edge in G1 or v1v2 is an edge in G2 but not both u1u2 is an edge in G1 and v1v2 is an edge in G2. A snake of length n > 1 is a packing of n congruent geometrical objects, called cells, such that the first and the last cell each has only one neighbor and all n −2 cells in between have exactly two neighbors. In Barrientos and Minion define a snake polyomino as a snake with square cells. They prove that given two graphs of sizes m and n with α-labelings, the graph that results from the edge amalgamation (identification of two edges) of the edges of weight 1 and n, also has an α-labeling. They use that result to prove the existence of α-labelings of snake polyominoes and hexagonal chains. The result about snake polyominoes partially answers the question of Acharya. In , they prove that the third power of a caterpillar admits an α-labeling and that the symmetric product G ⊕2K1 has an α-labeling when G does. In addition they prove that G∪Pm is graceful provided that G admits an α-labeling that does not assign the integer λ + 2 as a label, where λ is its boundary value. They ask if all triangular chains are graceful.
In Barrientos and Minion proved that under certain conditions, the union Cr ∪G of the cycle Cr and a caterpillar G admits a graceful labeling when r is odd, and an the electronic journal of combinatorics (2023), #DS6 54 α-labeling when r is even. They also proved the existence of an α-labeling for any tree obtained by connecting with a path of length two the central vertices of Gi and Gi+1, where Gi is a caterpillar of diameter 2d with bipartite sets Ai and Bi such that |Ai| = |Bi| + 1 and Ai contains the vertices of maximum eccentricity in Gi.
Let T1, T2, . . . , Ts be trees. A chain tree obtained by identifying, for every 1 ≤i ≤s−1, a vertex of Ti with a vertex of Ti+1. In , Barrientos and Minion prove that if every Ti admits an α-labeling, then there exists a chain tree that also admits an α-labeling. Let T be a tree of size n and v be a fixed vertex of T. The tree T +r v is obtained by connecting, with a path of length r, two copies of T, by identifying the end-point of this path with the vertices v of each copy of T. They give necessary conditions for the existence of an α-labeling for a tree T +2 v , where v is any of the vertices labeled λ, λ−1, . . . , λ−deg(v)−1 by an α-labeling with boundary value λ that assigns the labels λ+1, λ+2, . . . , λ+ deg(v) to leaves of T. In addition they proved that T +4 v has an α-labeling if there exists an α-labeling f of T, with boundary value λ, such that f(v) = λ −1. In , Barrientos and Minion prove the following.The tree ⊕(T1, T2, T3, T4) obtained by connecting to a new vertex w, the vertices labeled n in T1 and T3 and the vertices labeled n/2 in T2 and T4, where Ti is an α-labeled tree of even size n that has partite sets of cardinality n/2 and n/2 + 1. If G is a graph of order m and size n, with m < n, that admits an α-labeling, and H is any graceful graph of size t −1, then tG ∪H is a graceful graph. For every m ≥n, m ≥3, n ≥2, and t ≥2, tKm,n ∪Lt−1 admits an α-labeling where Lt−1 is any linear forest of size t −1. If G is a graph of order m and size n, with m < n, that admits an α-labeling, then tG ∪Lt−1 also admits an α-labeling when Lt−1 is a linear forest of size t −1. As a consequence of this result they prove that tG ∪Pt admits an α-labeling provided that G does.
Barrientos showed that all lobsters constructed with k copies of any caterpillar of diameter four by connecting the central vertices of all pairs of consecutive copies with an edge have an α-labeling. Additionally, he proved that any chain-tree formed by caterpillars and this type of lobsters admits an α-labeling. Barrientos and Minion say that a tree is regular when the cardinalities of its stable sets are equal or differ by one. They prove if S and T are regular trees that admit α-labelings then S × T also admits an α-labeling.
They use this result to prove that S × T admits a sequential labeling (see Section 4.1) as well as a harmonious labeling. They define a fence as the tree obtained by connecting an internal vertex of Pni with an internal vertex of Pni+1 by a path of length li for every 1 ≤i ≤t. They prove the existence of an α-labeling for any fence constructed with t copies of Pn, where li = 2. They define a 2-link fence as the graph obtained by connecting with an edge, two vertices of the ith copy of Pn, with the corresponding two vertices of the (i + 1)th copy of Pn. They prove that all such graphs admit α-labelings. In Barrientos says that a fence is irregular if two consecutive copies of Pn are connected by one or two pairs of corresponding vertices. He proved that all irregular fences have an α-labeling provided that all their Eulerian subgraphs have size divisible by four. In Barrientos and Minion study subfamilies of 2-link fences, a subfamily of column-convex polyominoes, and a subfamily of irregular cyclic-snakes. They prove that under certain conditions, an α-labelings of these graphs can be transformed into harmonious labelings.
the electronic journal of combinatorics (2023), #DS6 55 Barrientos and Minion provided new families of harmoniously labeled graphs built on α-labeled tress. Among them are P k n , the join of G and tK1 where G has a restrictive type of harmonious labeling and its order is different of its size by at most one, Km,n ∪K1,m−1, and G ∪T where G is a unicyclic graph and T is a tree built with α-trees.
They also showed that almost all trees admit harmonious labelings.
In Barrientos and Minion extend the concept of vertex amalgamation as follows.
The k-vertex amalgamation of G1 and G2 is the graph obtained by identifying k indepen-dent vertices of G1 with k independent vertices of G2. A t-fold of a graph G is obtained using t-copies of G, where the ith copy of G is k-vertex amalgamated with the (i + 1)th copy of G. They prove that if G admits an α-labeling, then any t-fold of G admits an α-labeling. They consider a more general version of this construction for the case where G is a tree. They also introduce a new family of trees that admit α-labelings; in particular, they prove that any tree of diameter 2n formed by identifying the end-vertices of four caterpillars admits an α-labeling.
Fronček, Kingston, and Vezina generalized snake polyomino graphs by intro-ducing straight simple polymonial caterpillars and proving that they also admit an alpha labeling.
This implies that every straight simple polymonial caterpillar with n edges decomposes the complete graph K2kn+1 for any positive integer k. In Fronček in-troduced a similar family of graphs called full hexagonal caterpillars and prove that they admit an alpha labeling. This implies that every full hexagonal caterpillar with n edges decomposes the complete graph K2kn+1 for any positive integer k.
Golomb introduced polyominoes in 1953 in a talk to the Harvard Mathematics Club. Polyominoes are planar shapes made by connecting a certain number of equal-sized squares, each joined together with at least one other square along an edge.
A graph G = (V (G), E(G)) is even graceful if there exists an injection f from the set of vertices V (G) to {0, 1, 2, 3, 4, . . . , 2|E(G)|} such that when each edge uv is assigned the label |f(u) −f(v)|, the resulting edge labels are 2, 4, 6, . . . , 2|E(G)|. Elsonbaty and Mohamed use even graceful labelings to give a new proof for necessary and sufficient conditions for the gracefulness of cycles. They extend this technique to odd graceful (in the obvious way) and super Fibonacci graceful labelings of cycle graphs (see §3.3). The polar grid graph Pm,n consists of n copies of Cm numbered from the inner most cycle to the outer cycle as C(1)m, . . . , C(n)m and m copies of paths Pn+1 intersected at the center vertex v0 numbered as P (1)n+1, . . . , P (m)n+1 In Elsonbaty and Daoud provided edge even graceful labelings for various classes of Pm × Cn. El Dean obtained an edge even graceful labeling for Y -trees, double stars Bn,m, ⟨K1,2n : K1,2m⟩, P2n1⊙K2m, K2+Pn, the cycle v1, v2, . . . , v2n with a chord from v1 to vn, P2 ⊙Cn, flags, and flowers. Zeen El Deen and Omar gave sufficient conditions for Km,n to have an edge even graceful labeling. They also provided edge even graceful labelings of the join of K1 with stars, wheels, and sunflowers, and the join of K2 with stars and wheels. For results on Fibonacci trees see .
In Devakirubanithi and Jeba Jesintha showed that twig diamond graphs without new prime edge coupled with pendent edges, super subdivision of combs, and bistars are odd graceful. The converse skew product G ▽H has vertex set V (G) × V (H) and u = (x1, y1) the electronic journal of combinatorics (2023), #DS6 56 is adjacent to v = (x2, y2) whenever y1 = y2 and x1x2 ∈E(G) or x1x2 ∈E(G) and y1y2 ∈E(H). Hafez, El-Shanawany, and El Atik proved that if G has an odd new graceful labeling f such that G doesn’t contain a path P3 = x1x2x3 where f(x1) < f(x2) < f(x3), then G ▽K(n) and Splm(G) are odd graceful. (Hafez et al. defines K(n) as {Ks(n), s = 1, 2, . . .}, where Ks(n) is the graph obtained from Pn by joining the end vertices of the edge vivi+1 to every vertex in the complete graph Kri, i = 1, 2, . . . , n −1, of order ri, such that (a) if |V (Kri) ∩V (Kri+1)| = t, then t ≤min{ri, ri+1} and (b) Kri and Krj are disjoint when |i −j| > 2).
Wu ( and ) has given a number of methods for constructing larger graceful graphs from graceful graphs. Let G1, G2, . . . , Gp be disjoint connected graphs.
Let wi be in Gi for 1 ≤i ≤p.
Let w be a new vertex not in any Gi.
Form a new graph ⊕w(G1, G2, . . . , Gp) by adjoining to the graph G1 ∪G2 ∪· · · ∪Gp the edges ww1, ww2, . . . , wwp. In the case where each of G1, G2, . . . , Gp is isomorphic to a graph G that has an α-labeling and each wi is the isomorphic image of the same vertex in Gi, Wu shows that the resulting graph is graceful. If f is an α-labeling of a graph, the integer k with the property that for any edge uv either f(u) ≤k < f(v) or f(v) ≤k < f(u) is called the boundary value or critical number of f. Wu has also shown that if G1, G2, . . . , Gp are graphs of the same order and have α-labelings where the labelings for each pair of graphs Gi and Gp−i+1 have the same boundary value for 1 ≤i ≤n/2, then ⊕w(G1, G2, . . . , Gp) is graceful. In Wu proves that if G has n edges and n + 1 vertices and G has an α-labeling with boundary value λ, where |n −2λ −1| ≤1, then G × Pm is graceful for all m.
Given graceful graphs H and G with at least one having an α-labeling Wu and Lu define four graph operations on H and G that when used repeatedly or in turns provide a large number of graceful graphs. In particular, if both H and G have α-labelings, then each of the graphs obtained by the four operations on H and G has an α-labeling.
Ajitha, Arumugan, and Germina use a construction of Koh, Tan, and Rogers to create trees with α-labelings from smaller trees with graceful labelings. These in turn allows them to generate large classes of trees that have a type of called edge-antimagic labelings (see §6.1). Shiue and Lu prove that the graph obtained from K1,k by replacing each edge with a path of length 3 has an α-labeling if and only if k ≤4. In Venkatesh and Bharathi recursively construct new trees starting with caterpillars that admit α-labelings.
Seoud and Helmi have shown that all gear graphs have an α-labeling, all dragons with a cycle of order n ≡0 (mod 4) have an α-labeling, and the graphs obtained by identifying an endpoint of a star Sm (m ≥3) with a vertex of C4n has an α-labeling.
Mavonicolas and Michael say that trees ⟨T1, θ1, w1⟩and ⟨T2, θ2, w2⟩with roots w1 and w2 and |V (T1)| = |V (T2)| are gracefully consistent if either they are identical or they have α-labelings with the same boundary value and θ1(w1) = θ2(w2). They use this concept to show that a number of known constructions of new graceful trees using several identical copies of a given graceful rooted tree can be extended to the case where the copies are replaced by a set of pairwise gracefully consistent trees. In particular, let ⟨T, θ, w⟩and ⟨T0, θ0, w0⟩be gracefully labeled trees rooted at w and w0 respectively.
the electronic journal of combinatorics (2023), #DS6 57 They show that the following four constructions are adaptable to the case when a set of copies of ⟨T, θ, w⟩is replaced by a set of pairwise gracefully consistent trees. When θ(w) = |E(T)| the garland construction due to Koh, Rogers, and Tan gracefully labels the tree consisting of h copies of ⟨T, w⟩with their roots connected to a new vertex r.
In the case when θ(w) = |E(T)| and whenever uw ∈E(T) and θ(u) ̸= 0, then vw ∈E(T) where θ(u) + θ(v) = |E(T)|, the attachment construction of Koh, Tan and Rogers gracefully labels the tree formed by identifying the roots of h copies of ⟨T, w⟩. A construction given by Koh, Tan and Rogers gracefully labels the tree formed by merging each vertex of ⟨T0, w0⟩with the root of a distinct copy of ⟨T, w⟩. When θ0(w0) = |E(T0)|, let N be the set of neighbors of w0 and let x be the vertex of T at even distance from w with θ(x) = 0 or θ(x) = |E(T)|. Then a construction of Burzio and Ferrarese gracefully labels the tree formed by merging each non-root vertex of T0 with the root of a distinct copy of ⟨T, w⟩so that for each v ∈N the edge vw0 is replaced with a new edge xw0 (where x is in the corresponding copy of T).
Snevily says that a graph G eventually has an α-labeling provided that there is a graph H, called a host of G, which has an α-labeling and that the edge set of H can be partitioned into subgraphs isomorphic to G. He defines the α-labeling number of G to be Gα = min{t : there is a host H of G with |E(H)| = t|G|}. Snevily proved that even cycles have α-labeling number at most 2 and he conjectured that every bipartite graph has an α-labeling number. This conjecture was proved by El-Zanati, Fu, and Shiue .
There are no known examples of a graph G with Gα > 2. In Snevily conjectured that the α-labeling number for a tree with n edges is at most n. Shiue and Fu proved that the α-labeling number for a tree with n edges and radius r is at most ⌈r/2⌉n. They also prove that a tree with n edges and radius r decomposes Kt for some t ≤(r+1)n2 +1.
Ahmed and Snevily investigated the claim that for every tree T there exists an α-labeling of T, or else there exists a graph HT with an α-labeling such that HT can be decomposed into two edge-disjoint copies of T. They proved this claim is true for the graphs Cm,k obtained from K1,m by replacing each edge in K1,m with a path of length k.
A graph G with vertex set V and edge set E is called super edge-graceful if there is a bijection f from E to {0, ±1, ±2, . . . , ±(|E| −1)/2} when |E| is odd and from E to {±1, ±2, . . . , ±|E|/2} when |E| is even such that the induced vertex labeling f ∗defined by f ∗(u) = P f(uv) over all edges uv is a bijection from V to {0, ±1, ±2, . . . , ±(|V | −1)/2} when |V | is odd and from V to {±1, ±2, . . . , ±|V |/2} when |V | is even.
Clifton and Khodkar proved that graphs formed by identifying the endpoint of a path Pn and a vertex of a cycle (kites) with n ≥5 vertices, n ̸= 6 are super edge-graceful. Khodkar, Nolen, and Perconti proved that all complete bipartite graphs except for K2,2, K2,3, and K1,n (n odd) are super edge-graceful. Khodkar and proved that all complete tripartite graphs except K1,1,2 are super edge-graceful and that the union of vertex disjoint 3-cycles is super edge-graceful. Lee, Su, and Wei provide a family of trees of odd orders which are super edge-graceful.
For a tree T with m edges, the α-deficit αdef(T) equals m−α(T) where α(T) is defined as the maximum number of distinct edge labels over all bipartite labelings of T. Rosa and Siran showed that for every m ≥1, αdef(Cm,2) = ⌊m/3⌋, which implies that the electronic journal of combinatorics (2023), #DS6 58 (Cm,2)α ≥2 for m ≥3. Ahmed and Snevily define the graph C′ m,j as a comet-like tree with a central vertex of degree m where each neighbor of the central vertex is attached to j pendent vertices for 1 ≤j ≤(m −1). For m ≥3 and 1 ≤j ≤(m −1) they prove: (C′ m,j)α ≤2; (C′ 2k+1,j)α = 2 for 1 ≤j ≤2k and conjecture if ∆T = (2k + 1), then αdef(T) ≤k. Ahmed and Snevily prove that for every comet T (that is, graphs obtained from stars by replacing each edge by a path of some fixed length) there exists an α-labeling of T, or else there exists a graph HT with an α-labeling such that HT can be decomposed into two edge-disjoint copies of T. This is particularly noteworthy since comets are known to have arbitrarily large α-deficits.
Given two bipartite graphs G1 and G2 with partite sets H1 and L1 and H2 and L2, respectively, Snevily defines their weak tensor product G1 NG2 as the bipartite graph with vertex set (H1 × H2, L1 × L2) and with edge (h1, h2)(l1, l2) if h1l1 ∈E(G1) and h2l2 ∈E(G2). He proves that if G1 and G2 have α-labelings then so does G1 NG2.
This result considerably enlarges the class of graphs known to have α-labelings. In López and Muntaner-Batle gave a generalization of Snevily’s weak tensor product that allows them to significantly enlarges the classes of graphs admitting α-labelings, near α-labelings (defined later in this section), and bigraceful graphs.
The sequential join of graphs G1, G2, . . . , Gn is formed from G1 ∪G2 ∪· · · ∪Gn by adding edges joining each vertex of Gi with each vertex of Gi+1 for 1 ≤i ≤n −1. Lee and Wang have shown that for all n ≥2 and any positive integers a1, a2, . . . , an the sequential join of the graphs Ka1, Ka2, . . . , Kan has an α-labeling.
In Gallian and Ropp conjectured that every graph obtained by adding a single pendent edge to one or more vertices of a cycle is graceful.
Qian proved this conjecture and in the case that the cycle is even he shows the graphs have an α-labeling.
He further proves that for n even any graph obtained from an n-cycle by adding one or more pendent edges at some vertices has an α-labeling as long as at least one vertex has degree 3 and one vertex has degree 2.
In Pasotti introduced the following generalization of a graceful labeling. Given a graph G with e = dm edges, an injective function from V (Γ) to the set {0, 1, 2, . . . , d(m+ 1) −1} such that {|f(x) −f(y)| | [x, y] ∈E(Γ)} = {1, 2, 3, . . . , d(m + 1) −1} −{m + 1, 2(m + 1), . . . , (d −1)(m + 1)} is called a d-divisible graceful labeling of G. Note that for d = 1 and of d = e one obtains the classical notion of a graceful labeling and of an odd-graceful labeling (see §3.6 for the definition), respectively. A d-divisible graceful labeling of a bipartite graph G with the property that the maximum value on one of the two bipartite sets is less than the minimum value on the other one is called a d-divisible α-labeling of G. Pasotti proved that these new concepts allow to obtain certain cyclic graph decompositions. In particular, if there exists a d-divisible graceful labeling of a graph G of size e = dm then there exists a cyclic G-decomposition of K e d +1 ×2d and that if there exists a d-divisible α-labeling of a graph Γ of size e then there exists a cyclic G-decomposition of K e d +1 ×2dn for any integer n ≥1. She also it is proved the following: paths and stars admit a d-divisible α-labeling for any admissible d; C4k admits a 2-divisible α-labeling and a 4-divisible α-labeling for any k ≥1; C2k admits a 2-divisible labeling for any odd integer k > 1; and the ladder graph L2k has a 2-divisible α-labeling if and only the electronic journal of combinatorics (2023), #DS6 59 if k is even.
Pasotti generalized the notion of graceful labelings for graphs G with e = d · m edges by defining a d-graceful labeling as an injective function f from V (G) to {0, 1, 2, . . . , d(m + 1) −1} such that {|f(x) −f(y)| | xy ∈E(G)} = {1, 2, . . . , d(m + 1) − 1} −{m + 1, 2(m + 1), . . . , (d −1)(m + 1)}. The case d = 1 is a graceful labeling and the case that d = e is an odd-graceful labeling. A d-graceful α-labeling of a bipartite graph is a d-graceful labeling with the property that the maximum value in one of the two bipartite sets is less than the minimum value on the other bipartite set. Pasotti proved that paths and stars have d-graceful α-labelings for all admissible d, ladders Pn × P2 have a 2-graceful labeling if and only if n is even, and provided partial results about cycles of even length. He showed that the existence of d-graceful labelings can be used to prove that certain complete graphs have cyclic decompositions. Benini and Pasotti used d-divisible α-labelings to construct an infinite class of cyclic Γ-decompositions of the complete multipartite graphs, where Γ is a caterpillar, a hairy cycle or a cycle. Such labelings imply the existence of cyclic Γ-decompositions of certain complete multipartite graphs.
In , Pasotti proved the existence of d-divisible α-labelings for C4k × Pm for any integers k ≥1, m ≥2 for d = 2m −1, 2(2m −1) and 4(2m −1). Benini and Pasotti proved that the generalized Petersen graph P8n,3 admits an α-labeling for any integer n ≥1 confirming that the conjecture posed by A. Vietri in is true.
For any tree T(V, E) whose vertices are properly 2-colored Rosa and Širáň define a bipartite labeling of T as a bijection f : V →{0, 1, 2, . . . , |E|} for which there is a k such that whenever f(u) ≤k ≤f(v), then u and v have different colors. They define the α-size of a tree T as the maximum number of distinct values of the induced edge labels |f(u) −f(v)|, uv ∈E, taken over all bipartite labelings f of T. They prove that the α-size of any tree with n edges is at least 5(n + 1)/7 and that there exist trees whose α-size is at most (5n + 9)/6. They conjectured that minimum of the α-sizes over all trees with n edges is asymptotically 5n/6. This conjecture has been proved for trees of maximum degree 3 by Bonnington and Širáň . For trees with n vertices and maximum degree 3 Brankovic, Rosa, and Širáň have shown that the α-size is at least ⌊6n 7 ⌋−1.
In Brankovic, Murch, Pond, and Rose provide a lower bound for the α-size trees with maximum degree three and a perfect matching as a function of a lower bound for minimum order of such a tree that does not have an α-labeling. Using a computer search they showed that all such trees on less than 30 vertices have an α-labeling. This brought the lower bound for the α-size to 14n/15, for such trees of order n. They conjecture that all trees with maximum degree three and a perfect matching have an α-labeling. Heinrich and Hell defined the gracesize of a graph G with n vertices as the maximum, over all bijections f : V (G) →{1, 2, . . . , n}, of the number of distinct values |f(u) −f(v)| over all edges uv of G. So, from Rosa and Širáň’s result, the gracesize of any tree with n edges is at least 5(n + 1)/7.
In Brinkmann, Crevals, Mélot, Rylands, and Steffan define the parameter αdef which measures how far a tree is from having an α-labeling as it counts the minimum number of errors, that is, the minimum number of edge labels that are missing from the the electronic journal of combinatorics (2023), #DS6 60 set of all possible labels. Trees with an α-labeling have deficit 0. For a tree T = (V, E) with bipartition classes V1 and V2 and a bipartite labeling f : V →{0, . . . , |V | −1} the edge parity of T is (P|E| i=1 i) mod 2 = 1 2(|V | −1)|V | mod 2. So if f is an α-labeling this is the sum of all edge labels modulo 2; it is 0 if |V | ≡0, 1 mod 4 and 1 if |V | ≡2, 3 mod 4.
The vertex parity is the parity of the number of vertices of odd degree with odd label.
Brinkmann et al. proved: in a tree T with α-deficit 0 the edge parity and the vertex parities are equal; and for all non-negative integers k and d and n ≥k2 + k, the number of trees T with n vertices, αdef(T) = d and maximum degree n −k is the same.
Furthermore, they provide computer results on the α-deficit of all trees with up to 26 vertices; with maximum degree 3 and up to 36 vertices, with maximum degree 4 and up to 32 vertices, and with maximum degree 5 and up to 31 vertices.
In Gallian weakened the condition for an α-labeling somewhat by defining a weakly α-labeling as a graceful labeling for which there is an integer k so that for each edge xy either f(x) ≤k ≤f(y) or f(y) ≤k ≤f(x). Unlike α-labelings, this condition allows the graph to have an odd cycle, but still places a severe restriction on the structure of the graph; namely, that the vertex with the label k must be on every odd cycle. Gallian, Prout, and Winters showed that the prisms Cn × P2 with a vertex deleted have α-labelings. The same paper reveals that Cn × P2 with an edge deleted from a cycle has an α-labeling when n is even and a weakly α-labeling when n > 3.
In and Barrientos and Minion focused on the enumeration of graphs with graceful and α-labelings, respectively. They used an extended version of the adjacency matrix of a graph to count the number of labeled graphs. In they count the number of gracefully-labeled graphs of size n and order m, for all possible values of m. In they count the number of α-labeled graphs of size n and order m, for all possible values of m, as well as those α-labeled graphs of size n with boundary value λ. They also count the number of α-labeled graphs of size n, order m, and boundary value for all possible values of m and λ.
A special case of α-labeling called strongly graceful was introduced by Maheo in 1980. A graceful labeling f of a graph G is called strongly graceful if G is bipartite with two partite sets A and B of the same order s, the number of edges is 2t + s, there is an integer k with t −s ≤k ≤t + s −1 such that if a ∈A, f(a) ≤k, and if b ∈B, f(b) > k, and there is an involution π that is an automorphism of G such that: π exchanges A and B and the s edges aπ(a) where a ∈A have as labels the integers between t + 1 and t + s.
Maheo’s main result is that if G is strongly graceful then so is G × Qn. In particular, she proved that (Pn × Qn) × K2, B2n, and B2n × Qn have strongly graceful labelings.
In 1999 Broersma and Hoede conjectured that every tree containing a perfect matching is strongly graceful. Yao, Cheng, Yao, and Zhao proved that this conjec-ture is true for every tree with diameter at most 5 and provided a method for constructing strongly graceful trees.
El-Zanati and Vanden Eynden call a strongly graceful labeling a strong α-labeling.
They show that if G has a strong α-labeling, then G × Pn has an α-labeling. They show that Km,2 × K2 has a strong α-labeling and that Km,2 × Pn has an α-labeling. They also show that if G is a bipartite graph with one more vertex than the number of edges, and if the electronic journal of combinatorics (2023), #DS6 61 G has an α-labeling such that the cardinalities of the sets of the corresponding bipartition of the vertices differ by at most 1, then G × K2 has a strong α-labeling and G × Pn has an α-labeling. El-Zanati and Vanden Eynden also note that K3,3 × K2, K3,4 × K2, K4,4×K2, and C4k×K2 all have strong α-labelings. El-Zanati and Vanden Eynden proved that Km,2 × Qn has a strong α-labeling and that Km,2 × Pn has an α-labeling for all n.
They also prove that if G is a connected bipartite graph with partite sets of odd order such that in each partite set each vertex has the same degree, then G × K2 does not have a strong α-labeling. As a corollary they have that Km,n × K2 does not have a strong α-labeling when m and n are odd.
An α-labeling f of a graph G is called free by El-Zanati and Vanden Eynden in if the critical number k (in the definition of α-labeling) is greater than 2 and if neither 1 nor k −1 is used in the labeling. Their main result is that the union of graphs with free α-labelings has an α-labeling. In particular, they show that Km,n, m > 1, n > 2, has a free α-labeling. They also show that Qn, n ≥3, and Km,2 × Qn, m > 1, n ≥1, have free α-labelings. El-Zanati [personal communication] has shown that the Heawood graph has a free α-labeling.
Wannasit and El-Zanati proved that if G is a cubic bipartite graph each of whose components is either a prism, a Möbius ladder, or has order at most 14, then G admits free α-labeling. They conjecture that every bipartite cubic graph admits a free α-labeling.
In Makadia, Karavadiya, and Kaneria call a vertex v in a graph G with a graceful labeling f a graceful center of G if f(v) = 0 or f(v) = |E(G)|. They say a graph G is a universal graceful graph if for every v ∈V (G), v is a graceful center for G with respect to some graceful labeling of G. They call G a universal α-graceful graph if for every v ∈V (G), v is a graceful center for G with respect to some α-graceful labeling of G. They define the ring sum of two graphs G1 and G2 denoted G1 ⊕G2, as the graph with vertex set (V (G1) ∪V (G2) and edge set E(G1) ∪E(G2) −(E(G1) ∩E(G2)). They proved: any graph G that admits α-labeling has at least four graceful centers; if G is a graceful graph, then G⊕K1,n is graceful; if G is a universal graceful graph, then G⊕K2 is a graceful; if G1 is graceful and G2 has an α-labeling, then the ring sum G1 ⊕G2 with the graceful center of G1 and the graceful center of G2 as a common vertex is a graceful; and if G1 and G2 have α labelings, then the ring sum G1 ⊕G2 with the two graceful centers of G1 and G2 as a common vertex has an α labeling.
For connected bipartite graphs Grannell, Griggs, and Holroyd introduced a labeling that lies between α-labelings and graceful labelings. They call a vertex labeling f of a bipartite graph G with q edges and partite sets D and U gracious if f is a bijection from the vertex set of G to {0, 1, . . . , q} such that the set of edge labels induced by f(u) −f(v) for every edge uv with u ∈U and v ∈D is {1, 2, . . . , q}. Thus a gracious labeling of G with partite sets D and U is a graceful labeling in which every vertex in D has a label lower than every adjacent vertex. They verified by computer that every tree of size up to 20 has a gracious labeling. This led them to conjecture that every tree has a gracious labeling. For any k > 1 and any tree T Grannell et al. say that T has a gracious k-labeling if the vertices of T can be partitioned into sets D and U in such a way the electronic journal of combinatorics (2023), #DS6 62 that there is a function f from the vertices of G to the integers modulo k such that the edge labels induced by f(u) −f(v) where u ∈U and v ∈D have the following properties: the number of edges labeled with 0 is one less than the number of vertices labeled with 0 and for each nonzero integer t the number of edges labeled with t is the same as the number of vertices labeled with t. They prove that every nontrivial tree has a k-gracious labeling for k = 2, 3, 4, and 5 and that caterpillars are k-gracious for all k ≥2. In Bell and Cummins provided new methods for combining certain families of gracefully labeled graphs to produce new gracefully labeled graphs. If the constituent graphs have a gracious labeling, then the methods presented produce a gracious labeling. They also introduce new infinite families of gracious trees and new classes of graceful trees.
The same labeling that is called gracious by Grannell, Griggs, and Holroyd is called a near α-labeling by El-Zanati, Kenig, and Vanden Eynden . The latter prove that if G is a graph with n edges that has a near α-labeling then there exists a cyclic G-decomposition of K2nx+1 for all positive integers x and a cyclic G-decomposition of Kn,n.
They further prove that if G and H have near α-labelings, then so does their weak tensor product (see earlier part of this section) with respect to the corresponding vertex partitions. They conjecture that every tree has a near α-labeling.
In Mahalingam and Rajendram introduced a new labeling called m-bonacci graceful labeling as follows. A graph G on n edges is m-bonacci graceful if the vertices can be labeled with distinct integers from the set of the first n m-bonacci numbers such that the derived edge labels are the first n m-bonacci numbers. They showed that complete graphs, complete bipartite graphs, gear graphs, triangular grid graphs, and wheel graphs are not m-bonacci graceful. They gave m-bonacci graceful labeling for cycles, friendship graphs, polygonal snake graphs, and double polygonal snake graphs and proved that almost all trees are m-bonacci graceful.
For a simple, finite, connected, undirected, non-trivial graph G Sumathi and Raman introduced the notion of arithmetic sequential graceful as an injection f : V (G) → {a, a + d, a + 2d, a + 3d, . . . , 2(a + qd)}, where a ≥0 and d ≥1, with the property that f ∗: E(G) →{d, 2d, 3d, 4d, . . . , qd} defined by f ∗(uv) = |f(u) −f(v)| is a bijection. They proved that stars, double stars, and some star related graphs are arithmetic sequential graceful.
Another kind of labelings for trees was introduced by Ringel, Llado, and Serra in an approach to proving their conjecture Kn,n is edge-decomposable into n copies of any given tree with n edges. If T is a tree with n edges and partite sets A and B, they define a labeling f from the set of vertices to {1, 2, . . . , n} to be a bigraceful labeling of T if f restricted to A is injective, f restricted to B is injective, and the edge labels given by f(y) −f(x) where yx is an edge with y in B and x in A is the set {0, 1, 2, . . . , n −1}.
(Notice that this terminology conflicts with that given in Section 2.7 In particular, the Ringel, Llado, and Serra bigraceful does not imply the usual graceful.) Among the graphs that they show are bigraceful are: lobsters, trees of diameter at most 5, stars Sk,m with k spokes of paths of length m, and complete d-ary trees for d odd. They also prove that if T is a tree then there is a vertex v and a nonnegative integer m such that the addition of m leaves to v results in a bigraceful tree. They conjecture that all trees are bigraceful.
the electronic journal of combinatorics (2023), #DS6 63 A pronic number is one of the form n(n + 1), where n is a positive integer. Porchelvi and Devi defined a pronic graceful labeling of a graph G with n ≥2 vertices as a bijection f : V (G) →{0, 2, . . . , n(n + 1)} such that the upon labeling each edge uv with |f(u) −f(v)| the labels are distinct. A graph G is called a pronic graceful graph if it admits pronic graceful labeling. They proved that paths, cycles, wheels, stars, twigs, and generalized Peterson graphs are pronic graceful. In they proved that the generalized Peterson graphs P(6, 2), P(8, 3), (10, 2), P(10, 3), and P(12, 5) are pronic graceful.
Table 3 summarizes some of the main results about α-labelings; α indicates that the graphs have an α-labeling.
the electronic journal of combinatorics (2023), #DS6 64 Table 3: Summary of Results on α-labelings Graph α-labeling cycles Cn α iff n ≡0 (mod 4) caterpillars α n-cube α books B2n, B4n+1 α , Möbius ladders M2k+1 α Cm ∪Cn α iff m, n are even and m + n ≡0 (mod 4) C4m ∪C4m ∪C4m (m > 1) α C4m ∪C4m ∪C4m ∪C4m α mKs,t (m ≥1, s, t ≥2) iff (m, s, t) ̸= (3, 2, 2) Pn × Qn α B2n × Qn α K1,n × Qn α Km,2 × Qn α Km,2 × Pn α P2 × P2 × · · · × P2 × G α when G = C4m, Pm, K3,3, K4,4 P2 × P2 × · · · × P2 × Pm α P2 × P2 × · · · × P2 × Km,m α when m = 3 or 4 G[Kn] α when G is α the electronic journal of combinatorics (2023), #DS6 65 3.2 γ-Labelings In 2004 Chartrand, Erwin, VanderJagt, and Zhang define a γ-labeling of a graph G of size m as a 1-1 function f from the vertices of G to {0, 1, 2, . . . , m} that induces an edge labeling f ′ defined by f ′(uv) = |f(u)−f(v)| for each edge uv. They define the following pa-rameters of a γ-labeling: val(f) = Σf ′(e) over all edges e of G; valmax(G) = max{ val(f) : f is a γ-labeling of G}, valmin(G) = min{ val(f) : f is a γ −labeling of G}.
Among their results are the following: valmin(Pn) = valmax(Pn) = ⌊(n2 −2)/2⌋; valmin(Cn) = 2(n −1); for even n ≥4, valmax(Cn) = n(n + 2)/2; for odd n ≥3, valmax(Cn) = (n −1)(n + 3)/2; for odd n, valmin(Kn) = n+1 3 ; for odd n, valmax(Kn) = (n2 −1)(3n2 −5n + 6)/24; for even n, valmax(Kn) = n(3n3 −5n2 + 6n −4)/24; for every n ≥3, valmin(K1,n−1) = ⌊n+1 2 ⌋ 2 + ⌈n+1 2 ⌉ 2 ; valmax(K1,n−1) = n 2 for a connected graph of order n and size m, valmin(G) = m if and only if G is isomorphic to Pn; if G is maximal outerplanar of order n ≥2, valmin(G) ≥3n −5 and equality occurs if and only if G = P 2 n; if G is a connected r-regular bipartite graph of order n and size m where r ≥2, then valmax(G) = rn(2m −n + 2)/4.
In another paper on γ-labelings of trees Chartrand, Erwin, VanderJagt, and Zhang prove for p, q ≥2, valmin(Sp,q) (that is, the graph obtained by joining the centers of K1,p and K1,q by an edge)= (⌊p/2⌋+1)2+(⌊q/2⌋+1)2−(np⌊p/2⌋+1)2+(nq⌊(q+2)/2⌋+1)2), where ni is 1 if i is even and ni is 0 if ni is odd; valmin(Sp,q) = (p2+q2+4pq−3p−3q+2)/2; for a connected graph G of order n at least 4, valmin(G) = n if and only if G is a caterpillar with maximum degree 3 and has a unique vertex of degree 3; for a tree T of order n at least 4, maximum degree ∆, and diameter d, valmin(T) ≥(8n + ∆2 −6∆−4d + δ∆)/4 where δ∆is 0 if ∆is even and δ∆is 0 if ∆is odd. They also give a characterization of all trees of order n at least 5 whose minimum value is n + 1.
Saduakdee and Khemmani investigated connected graphs having the unique γ-min labeling. They determined the minimum value of a γ-labeling for some classes of trees and showed that they have no unique γ-min labeling.
In Buratti and Del Fra solved the existence problem for cyclic k-cycle systems of the complete graph Kv with v ≡1 (mod 2k), and the existence problem for cyclic k-cycle systems of the complete m-partite graph Km×k for m and k odd. As a particular consequence, a cyclic p-cycle system of Kv with p a prime exists for all admissible values of v but (p, v) ̸= (3, 9). This was previously known only for p = 3, 5, 7.
In Sanaka determined valmax(Km,n) and valmin(Km,n). In Bunge, Chan-tasartraaamee, El-Zanati, and Vanden Eynden generalized γ-labelings by introducing two labelings for tripartite graphs. Graphs G that admit either of these labelings guarantee the existence of cyclic G-decompositions of K2nx+1 for all positive integers x. They also proved that, except for C3 ∪C3, the disjoint union of two cycles of odd length admits one of these labelings.
the electronic journal of combinatorics (2023), #DS6 66 3.3 Graceful-like Labelings As a means of attacking graph decomposition problems, Rosa invented another analogue of graceful labelings by permitting the vertices of a graph with q edges to assume labels from the set {0, 1, . . . , q +1}, while the edge labels induced by the absolute value of the difference of the vertex labels are {1, 2, . . . , q −1, q} or {1, 2, . . . , q −1, q +1}. He calls these ˆ ρ-labelings. Frucht used the term nearly graceful labeling instead of ˆ ρ-labelings.
Frucht has shown that the following graphs have nearly graceful labelings with edge labels from {1, 2, . . . , q −1, q +1}: Pm ∪Pn; Sm ∪Sn; Sm ∪Pn; G∪K2 where G is graceful; and C3 ∪K2 ∪Sm where m is even or m ≡3 (mod 14). Seoud and Elsakhawi have shown that all cycles are nearly graceful. Barrientos proved that Cn is nearly graceful with edge labels 1, 2, . . . , n −1, n + 1 if and only if n ≡1 or 2 (mod 4). Nurvazly and Sugeng proved that Θ(C3)n graphs (n copies of C3 that share an edge) have ˆ ρ labelings. Gao shows that a variation of banana trees is odd-graceful and in some cases has a nearly graceful labeling. (A graph G with q edges is odd-graceful if there is an injection f from V (G) to {0, 1, 2, . . . , 2q −1} such that, when each edge xy is assigned the label |f(x) −f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q −1}).
For a graph G with p vertices and q directed edges that are assigned distinct vertex labels in {0, . . . , q} and distinct edge labels in {1, . . . , p} so that the label of the directed edge from u to v is (f(v) −f(u)) mod(q + 1) (this generalizes Rosa’s ρ-valuations. Knuth has observed that there is a nice data structure for storing a graph or digraph in a computer. He calls this a graceful data structure labeling.
In 1988 Rosa conjectured that triangular snakes with t ≡0 or 1 (mod 4) blocks are graceful and those with t ≡2 or 3 (mod 4) blocks are nearly graceful (a parity condition ensures that the graphs in the latter case cannot be graceful). Moulton proved Rosa’s conjecture while introducing the slightly stronger concept of almost graceful by permitting the vertex labels to come from {0, 1, 2, . . . , q−1, q+1} while the edge labels are 1, 2, . . . , q −1, q, or 1, 2, . . . , q −1, q + 1. More generally, Rosa conjectured that all triangular cacti are either graceful or near graceful and suggested the use of Skolem sequences to label some types of triangular cacti. Dyer, Payne, Shalaby, and Wicks verified the conjecture for two families of triangular cacti using Langford sequences to obtain Skolem and hooked Skolem sequences with specific subsequences.
Seoud and Elsakhawi and have shown that the following graphs are almost graceful: Cn; Pn + Km; Pn + K1,m; Km,n; K1,m,n; K2,2,m; K1,1,m,n; Pn × P3 (n ≥3); K5 ∪K1,n; K6 ∪K1,n, and ladders.
For a graph G with p vertices, q edges, and 1 ≤k ≤q, Eshghi defines a holey α-labeling with respect to k as an injective vertex labeling f for which f(v) ∈{1, 2, . . . , q+1} for all v, {|f(u) −f(v)| | for all edges uv} = {1, 2, . . . , k −1, k + 1, . . . , q + 1}, and there exist an integer γ with 0 ≤γ ≤q such that min{f(u), f(v)} ≤γ ≤max{f(u), f(v)}. He proves the following: Pn has a holey α-labeling with respect to all k; Cn has a holey α-labeling with respect to k if and only if either n ≡2 (mod 4), k is even, and (n, k) ̸= (10, 6), or n ≡0 (mod 4) and k is odd.
Recall from Section 2.2 that a kCn-snake is a connected graph with k blocks whose the electronic journal of combinatorics (2023), #DS6 67 block-cutpoint graph is a path and each of the k blocks is isomorphic to Cn. In addition to his results on the graceful kCn-snakes given in Section 2.2, Barrientos proved that when k is odd the linear kC6-snake is nearly graceful and that Cm ∪K1,n is nearly graceful when m = 3, 4, 5, and 6.
Yet another kind of labeling introduced by Rosa in his 1967 paper is a ρ-labeling.
(Sometimes called a rosy labeling ). A ρ-labeling (or ρ-valuation) of a graph is an injection from the vertices of the graph with q edges to the set {0, 1, . . . , 2q}, where if the edge labels induced by the absolute value of the difference of the vertex labels are a1, a2, . . . , aq, then ai = i or ai = 2q + 1 −i. Rosa proved that a cyclic decomposition of the edge set of the complete graph K2q+1 into subgraphs isomorphic to a given graph G with q edges exists if and only if G has a ρ-labeling. (A decomposition of Kn into copies of G is called cyclic if the automorphism group of the decomposition itself contains the cyclic group of order n.) It is known that every graph with at most 11 edges has a ρ-labeling and that all lobsters have a ρ-labeling (see ).
In Barrientos and Minion proved that a tree admits a ρ–labeling when the deletion of some of its leaves results in a graceful tree. They use this result to prove the existence of ρ-labeling for several families of trees such as lobsters and those of diameter up to seven. Similarly, they showed that if T is any graceful tree of size n and k is an integer such that 2k ≥n + 1, then any tree of size n + 2k obtained attaching a path of length 2 to k distinct vertices of T has a ρ-labeling.
Donovan, El-Zanati, Vanden Eyden, and Sutinuntopas prove that rCm has a ρ-labeling (or a more restrictive labeling) when r ≤4. They conjecture that every 2-regular graph has a ρ-labeling. Gannon and El-Zanati proved that for any odd n ≥7, rCn admits ρ-labelings. The cases n = 3 and n = 5 were done in and . Aguado, El-Zanati, Hake, Stob, and Yayla give a ρ-labeling of Cr ∪Cs ∪Ct for each of the cases where r ≡0, s ≡1, t ≡1 (mod 4); r ≡0, s ≡3, t ≡3 (mod 4); and r ≡1, s ≡1, t ≡3 (mod 4); (iv) r ≡1, s ≡2, t ≡3 (mod 4); (v) r ≡3, s ≡3, t ≡3 (mod 4).
Caro, Roditty, and Schőnheim provide a construction for the adjacency matrix for every graph that has a ρ-labeling. They ask the following question: If H is a connected graph having a ρ-labeling and q edges and G is a new graph with q edges constructed by breaking H up into disconnected parts, does G also have a ρ-labeling? Kézdy defines a stunted tree as one whose edges can be labeled with e1, e2, . . . , en so that e1 and e2 are incident and, for all j = 3, 4, . . . , n, edge ej is incident to at least one edge ek satisfying 2k ≤j −1. He uses Alon’s “Combinatorial Nullstellensatz” to prove that if 2n + 1 is prime, then every stunted tree with n edges has a ρ-labeling.
Jeba Jesintha and Ezhilarasi Hilda introduced a variation of Rosa’s ρ-labeling as follows. A ρ⋆-labeling of a graph G is an injection from the vertices of the graph with q edges to the set {0, 1, . . . , 2q}, where if the edge labels induced by the absolute value of the difference of the vertex labels are e1, e2, . . . , eq, then ei = i or ei = 2qi. They prove that all paths and shell-butterfly graphs have a ρ⋆-labeling.
In Barrientos and Minion proved the existence of ρ-labelings for some types of forests that considerably reduce the number of trees that need to be studied to prove Kotzig’s Conjecture that states that K2n+1 can be cyclically decomposed into 2n + 1 the electronic journal of combinatorics (2023), #DS6 68 subgraphs isomorphic to a given tree with n edges. Among their results are the following.
If T1 and T2 admit α-labelings such that one of the end-vertices of the edge of weight 1 in T2 is a leaf, then T1 ∪T2 admits a ρ-labeling. If G1, G2, . . . , Gk is a collection of graphs that admit α-labelings, where Gk is a caterpillar of size at least k−2, then Sk i=1 Gi admits a ρ-labeling. Let R denote the family that consists of all trees G such that G has a branch H, (i.e., G −H is a tree) that is a caterpillar, where the excess of G −H is at most the size of H. They prove that G admits a ρ-labeling when G ∈R.
Recall a kayak paddle KP(k, m, l) is the graph obtained by joining Ck and Cm by a path of length l.
Fronček and Tollefeson , proved that KP(r, s, l) has a ρ-labeling for all cases. As a corollary they have that the complete graph K2n+1 is decomposable into kayak paddles with n edges.
In Fronček generalizes the notion of an α-labeling by showing that if a graph G on n edges allows a certain type of ρ-labeling), called α2-labeling, then for any positive integer k the complete graph K2nk+1 can be decomposed into copies of G.
In their investigation of cyclic decompositions of complete graphs El-Zanati, Vanden Eynden, and Punnim introduced two kinds of labelings. They say a bipartite graph G with n edges and partite sets A and B has a θ-labeling h if h is a one-to-one function from V (G) to {0, 1, . . . , 2n} such that {|h(b) −h(a)| ab ∈E(G), a ∈A, b ∈B} = {1, 2, . . . , n}.
They call h a ρ+-labeling of G if h is a one-to-one function from V (G) to {0, 1, . . . , 2n} and the integers h(x) −h(y) are distinct modulo 2n + 1 taken over all ordered pairs (x, y) where xy is an edge in G, and h(b) > h(a) whenever a ∈A, b ∈B and ab is an edge in G. Note that θ-labelings are ρ+-labelings and ρ+-labelings are ρ-labelings. They prove that if G is a bipartite graph with n edges and a ρ+-labeling, then for every positive integer x there is a cyclic G-decomposition of K2nx+1. They prove the following graphs have ρ+-labelings: trees of diameter at most 5, C2n, lobsters, and comets (that is, graphs obtained from stars by replacing each edge by a path of some fixed length). They also prove that the disjoint union of graphs with α-labelings have a θ-labeling and conjecture that all forests have ρ-labelings.
A σ-labeling of G(V, E) is a one-to-one function f from V to {0, 1, . . . , 2|E|} such that {|f(u) −f(v)| | uv ∈E(G)} = {1, 2, . . . , |E|}. Such a labeling of G yields cyclic G-decompositions of K2n+1 and of K2n+2 −F, where F is a 1-factor of K2n+2. El-Zanati and Vanden Eynden (see ) have conjectured that every 2-regular graph with n edges has a ρ-labeling and, if n ≡0 or 3 (mod 4), then every 2-regular graph has a σ-labeling.
Aguado and El-Zanati have proved that the latter conjecture holds when the graph has at most three components.
Given a bipartite graph G with partite sets X and Y and graphs H1 with p vertices and H2 with q vertices, Fronček and Winters define the bicomposition of G and H1 and H2, G[H1, H2], as the graph obtained from G by replacing each vertex of X by a copy of H1, each vertex of Y by a copy of H2, and every edge xy by a graph isomorphic to Kp,q with the partite sets corresponding to the vertices x and y. They prove that if G is a bipartite graph with n edges and G has a θ-labeling that maps the vertex set V = X ∪Y into a subset of {0, 1, 2, . . . , 2n}, then the bicomposition G[Kp, Kq] has a θ-labeling for every p, q ≥1. As corollaries they have: if a bipartite graph G with n edges and at most the electronic journal of combinatorics (2023), #DS6 69 n + 1 vertices has a gracious labeling (see §3.1), then the bicomposition graph G[Kp, Kq] has a gracious labeling for every p, q ≥1, and if a bipartite graph G with n edges has a θ-labeling, then for every p, q ≥1, the bicomposition G[Kp, Kq] decomposes the complete graph K2npq+1.
In a paper published in 2009 El-Zanati and Vanden Eynden survey “Rosa-type” labelings. That is, labelings of a graph G that yield cyclic G-decompositions of K2n+1 or K2nx+1 for all natural numbers x. The 2009 survey by Fronček includes generaliza-tions of ρ- and α-labelings that have been used for finding decompositions of complete graphs that are not covered in .
Blinco, El-Zanati, and Vanden Eynden call a non-bipartite graph almost-bipartite if the removal of some edge results in a bipartite graph. For these kinds of graphs G they call a labeling f a γ-labeling of G if the following conditions are met: f is a ρ-labeling; G is tripartite with vertex tripartition A, B, C with C = {c} and b ∈B such that {b, c} is the unique edge joining an element of B to c; if av is an edge of G with a ∈A, then f(a) < f(v); and f(c)−f(b) = n. (In § 3.2 the term γ-labeling is used for a different kind of labeling.) They prove that if an almost-bipartite graph G with n edges has a γ-labeling then there is a cyclic G-decomposition of K2nx+1 for all x. They prove that all odd cycles with more than 3 vertices have a γ-labeling and that C3 ∪C4m has a γ-labeling if and only if m > 1. In Bunge, El-Zanati, and Vanden Eynden prove that every 2-regular almost bipartite graph other than C3 and C3 ∪C4 have a γ-labeling.
In Blinco, El-Zanati, and Vanden Eynden consider a slightly restricted ρ+-labeling for a bipartite graph with partite sets A and B by requiring that there exists a number λ with the property that ρ+(a) ≤λ for all a ∈A and ρ+(b) > λ for all b ∈B.
They denote such a labeling by ρ++. They use this kind of labeling to show that if G is a 2-regular graph of order n in which each component has even order then there is a cyclic G-decomposition of K2nx+1 for all x. They also conjecture that every bipartite graph has a ρ-labeling and every 2-regular graph has a ρ-labeling.
Dufour and Eldergill have some results on the decomposition of complete graphs using labeling methods. Balakrishnan and Sampathkumar showed that for each positive integer n the graph Kn + 2K2 admits a ρ-labeling. Balakrishnan asks if it is true that Kn +mK2 admits a ρ-labeling for all n and m. Fronček and Fronček and Kubesa have introduced several kinds of labelings for the purpose of proving the existence of special kinds of decompositions of complete graphs into spanning trees.
For positive integers c and d, let Kc×d denote the complete multipartite graph with c parts, each containing d vertices. Let G with n edges be the union of two vertex-disjoint even cycles. In Su et al. use Rosa-type graph labelings to show that there exists a cyclic G- decomposition of K(2n + 1) × t, K(n/2+1)×4t, K5×(n/2)t, and of K2nt for every positive integer t. If n ≡0 (mod 4),then there also exists a cyclic G-decomposition of Kn+1 × 2t, K(n/4)+1 × 8t, K9 × (n/4)t, and of K3×nt for every positive integer t.
For (p, q)-graphs with p = q + 1, Frucht has introduced a stronger version of almost graceful graphs by permitting as vertex labels {0, 1, . . . , q −1, q + 1} and as edge labels {1, 2, . . . , q}.
He calls such a labeling pseudograceful.
Frucht proved that Pn (n ≥3), combs, sparklers (i.e., graphs obtained by joining an end vertex of a path to the electronic journal of combinatorics (2023), #DS6 70 the center of a star), C3 ∪Pn (n ̸= 3), and C4 ∪Pn (n ̸= 1) are pseudograceful whereas K1,n (n ≥3) is not. Kishore proved that Cs ∪Pn is pseudograceful when s ≥5 and n ≥(s + 7)/2 and that Cs ∪Sn is pseudograceful when s = 3, s = 4, and s ≥7. Seoud and Youssef and extended the definition of pseudograceful to all graphs with p ≤q + 1. They proved that Km is pseudograceful if and only if m = 1, 3, or 4 ; Km,n is pseudograceful when n ≥2, and Pm + Kn (m ≥2) is pseudograceful. They also proved that if G is pseudograceful, then G ∪Km,n is graceful for m ≥2 and n ≥2 and G ∪Km,n is pseudograceful for m ≥2, n ≥2 and (m, n) ̸= (2, 2) . They ask if G ∪K2,2 is pseudograceful whenever G is. Seoud and Youssef observed that if G is a pseudograceful Eulerian graph with q edges, then q ≡0 or 3 (mod 4). Youssef has shown that Cn is pseudograceful if and only if n ≡0 or 3 (mod 4), and for n > 8 and n ≡0 or 3 (mod 4), Cn ∪Kp,q is pseudograceful for all p, q ≥2 except (p, q) = (2, 2).
Youssef has shown that if H is pseudograceful and G has an α-labeling with k being the smaller vertex label of the edge labeled with 1 and if either k + 2 or k −1 is not a vertex label of G, then G ∪H is graceful. In Youssef shows that if G is (p, q) pseudograceful graph with p = q + 1, then G ∪Sm is Skolem-graceful (see Section 3.5 for the definition). As a corollary he obtains that for all n ≥2, Pn ∪Sm is Skolem-graceful if and only if n ≥3 or n = 2 and m is even.
In Youssef generalizes his results in and provides new families of discon-nected graphs that have α-labelings and pseudo α-labelings. (A pseudo α-labeling f is an α-labeling for which there is an integer kj with the property that for each edge xy of the graph either f(x) ≤kj < f(y) or f(y) ≤kj < f(x).) For a graph G Ichishima, Muntaner-Batle, and Oshima defined the beta-number of G, β(G), to be either the smallest positive integer n for which there exists an injective function f from the vertices of G to {1, 2, . . . , n} such that when each edge uv is labeled |f(u) −f(v)| the resulting set of edge labels is {c, c + 1, . . . , c + |E(G)| −1} for some positive integer c or +∞if there exists no such integer n.
They defined the strong beta-number of G to be either the smallest positive integer n for which there exists an injective function f from the vertices of G to {1, 2, . . . , n} such that when each edge uv is labeled |f(u) −f(v)| the resulting set of edge labels is {1, 2, . . . , |E(G)|} or +∞if there exists no such integer n. They gave some necessary conditions for a graph to have a finite (strong) beta-number and some sufficient conditions for a graph to have a finite (strong) beta-number. They also determined formulas for the beta-numbers and strong beta-numbers of Cn, 2Cn, Kn (n ≥2), Sm ∪Sn, Pm ∪Sn, and prove that nontrivial trees and forests without isolated vertices have finite strong beta-numbers. In Ichishima, López, Muntaner-Batte, and Oshima proved that if G is a bipartite graph and m is odd, then β(mG) ≤m|E(G)| + m −1. If G has the additional property that G is a graceful nontrivial tree, then β(mG) = m|V (G)| + m −1. They also investigate the (strong) beta-number of forests with components that are isomorphic to either paths or stars. They propose new conjectures on the (strong) beta-number of forests. In Ichishima and Oshima determine a formula for the (strong) beta-number of the linear forests Pm ∪Pn.
As a corollary they provide a partial formula for the beta-number of the disjoint union of multiple copies of the same linear forest. In Ichishima, Muntaner-Batle, Oshima the electronic journal of combinatorics (2023), #DS6 71 provide lower and upper bounds for β(G+nK1) when β(G) = |V (G)|−1 and formulas for β(G + nK1) and βs(G + nK1)) when βs(G) = |V (G)| −1. They also determine formulas for β(G + K1,n) and βs(G + K1,n) when βs(G) = |V (G)| −1. They conclude with two problems.
In Ichishima, Oshima, and Takahashi establish a lower bound for the beta-number of an arbitrary galaxy under certain conditions. They also introduce the notions of odd symmetric and even symmetric galaxies and determine formulas for the beta-number and gamma-number of odd symmetric galaxies.
As corollaries, they provide formulas for the beta-number and gamma-number of the disjoint union of multiple copies of the same galaxy when the number of copies is odd. In addition to these, the present an upper bound for the beta-number of even symmetric galaxies and obtain partial formulas for the beta-number and gamma-number of even symmetric galaxies.
For a graph G of order p and size q and every positive integer n Ichishima, Muntaner-Batle, and Oshima proved if β(G) = p −1, then there exists some positive integer c such that q + np ≤β(G + nK1) ≤c + q + np −1; if βs(G) = p −1, then β(G + nK1) = βs(G + nK1) = q + np and G + nK1 is graceful; and if q = p −1 and βs(G) = p −1, then β(G + Sn) = βs(G + Sn) = (n + 2)p + n −1. In particular, if T is a graceful tree of order p then β(T + nK1) = βs(T + nK1) = (n + 1)p −1. Moreover, T + nK1 and T + Sn are graceful.
In Ichishima, Muntaner-Batle, and Oshima establish a lower bound for the strong beta-number of an arbitrary galaxy (that is, a forest whose components are stars) under certain conditions. They also determine formulas for the (strong) beta-number and gracefulness of galaxies with three and four components. As corollaries, they provide formulas for the beta-number and gracefulness of the disjoint union of multiple copies of the same galaxies if the number of copies is odd. They pose some problems and conjecture.
In Ichishima and Muntaner-Batle determined formulas for the (strong) beta-number and gracefulness of galaxies with five components. In Ichishima, Muntaner-Batle, and Oshima determined formulas for the (strong) beta-number and gamma-number of galaxies with five components. As a corollary of these results, they provide formulas for the beta-number and gamma-number of the disjoint union of multiple copies of the same galaxies if the number of copies is odd.
McTavish has investigated labelings of graphs with q edges where the vertex and edge labels are from {0, . . . , q, q + 1}. She calls these ˜ ρ-labelings. Graphs that have ˜ ρ-labelings include cycles and the disjoint union of Pn or Sn with any graceful graph.
Frucht has made an observation about graceful labelings that yields nearly graceful analogs of α-labelings and weakly α-labelings in a natural way. Suppose G(V, E) is a graceful graph with the vertex labeling f. For each edge xy in E, let [f(x), f(y)] (where f(x) ≤f(y)) denote the interval of real numbers r with f(x) ≤r ≤f(y). Then the intersection ∩[f(x), f(y)] over all edges xy ∈E is a unit interval, a single point, or empty. Indeed, if f is an α-labeling of G then the intersection is a unit interval; if f is a weakly α-labeling, but not an α-labeling, then the intersection is a point; and, if f is a graceful but not a weakly α-labeling, then the intersection is empty. For nearly graceful labelings, the intersection also gives three distinct classes.
the electronic journal of combinatorics (2023), #DS6 72 Let G(V, E) be a graph without isolated vertices and with q edges. The gracefulness grac(G) of G is the smallest positive integer k for which there exists an injective function f : V →{0, 1, 2, . . . , k} such that the edge induced function gf : E →{1, 2, . . . , k} defined by gf(uv) = |f(u) −f(v)| for all edges uv is also injective. Let c(f) = max{i : 1, 2, . . . , i} are edge labels} and let m(G) = maxf{c(f)} where the maximum is taken over all injective functions f from V to the nonnegative integers such that gf is also injective. The measure m(G) is called m-gracefulness of G. It determines how close G is to being graceful. Pereira, Singh, Arumugam prove that there are infinitely many nongraceful graphs with m-gracefulness q−1 and give necessary conditions for an Eulerian graph with q edges and Kp with q edges to have m-gracefulness q −1 and q −2. They prove that K5 is the only complete graph to have m-gracefulness q −1. They also give an upper bound for the highest possible vertex label of Kp if m(Kp) = q −2.
A (p, q)-graph G is said to be a super graceful graph if there is a a bijective function f : V (G) ∪E(G) − →{1, 2, . . . , p + q} such that f(uv) = |f(u) −f(v)| for every edge uv ∈E(G) labeling. Perumal, Navaneethakrishnan, Nagarajan, Arockiaraj and show that the graphs Pn, Cn, Pm ⊙nK1, Km,n, and Pn ⊙K1 minus a pendent edge at an endpoint of Pn are super graceful graphs. Lau, Shiu, and Ng study the super gracefulness of complete graphs, the disjoint union of certain star graphs, the complete tripartite graphs K(1,1,n), and certain families of trees. They also provide four methods of constructing new super graceful graphs. They prove all trees of order at most 7 are super graceful and conjecture that all trees are super graceful. Amutha and Uma Devi proved the following graphs are super graceful: fans, double fans DFn = Pn +K2 (n ≥2), and for (m ≥3, n ≥2) the graphs obtained by identifying a central vertex of the star Sm with an end vertex of path in Pn + K1.
For k ≥1, Lau, Shiu, and Ng say a bijection f : V ∪E →[k, k + p + q −1] is a k-super graceful labeling if f(uv) = |f(u) −f(v)| for every edge uv in G. A graph G is k-super graceful if it admits a k-super graceful labeling. This is a generalization of super graceful labeling defined by Perumal, Navaneethakrishnan, Nagarajan, Arockiaraj in . It was referred to as a k-sequential labeling by Slater in , in which Slater gave necessary and sufficient conditions for a star to be k-sequential.
In Lau, Shiu, and Ng investigated the existence of k-sequential labelings (which they call k-super graceful labelings) of paths, cycles, caterpillars, complete bipartite and complete tripartite graphs.
In Elsonbaty and Daoud introduce a new version of gracefulness called an edge even graceful labeling of graphs. A bijective function f from the edges of a (p, q)-graph G to {2, 4, . . . , 2q} is said to be an edge even graceful labeling of G if the induced function f ∗from the vertices to {0, 2, . . . , 2q} defined by f ∗(e) is the sum of f(e) (mod max(p, q)) is injective. They prove the following graphs have edge even graceful labelings: Pn if and only if n is odd, Cn if and only if n is odd, K1,n if and only if n is even, wheels, fans, friendship graphs, and double wheels Wn,n. The polar grid graph Pm,n consists of n copies of Cm, a new vertex v0, and m copies on Pn+1 that share a endpoint at v0 The graph is drawn as m concentric circles with a center at a new vertex v0 and the m vertices of each cycle lie on a line with one endpoint at v0 and the other endpoint at the outermost the electronic journal of combinatorics (2023), #DS6 73 cycle in such a way that the n vertices of the copies on Pn+1 other the v0 intersect the vertices of cycles. Daoud provided necessary and sufficient conditions for the polar grid graph to be edge even graceful.
Singh and Devaraj call a graph G with p vertices and q edges triangular grace-ful if there is an injection f from V (G) to {0, 1, 2, . . . , Tq} where Tq is the qth triangular number and the labels induced on each edge uv by |f(u) −f(v)| are the first q triangular numbers. They prove the following graphs are triangular graceful: paths, level 2 rooted trees, olive trees (see § 2.1 for the definition), complete n-ary trees, double stars, caterpil-lars, C4n, C4n with pendent edges, the one-point union of C3 and Pn, and unicyclic graphs that have C3 as the unique cycle. They prove that wheels, helms, flowers (see §2.2 for the definition) and Kn with n ≥3 are not triangular graceful. They conjecture that all trees are triangular graceful. In Sethuraman and Venkatesh introduced a new method for combining graceful trees to obtain trees that have α-labelings.
Van Bussel considered two kinds of relaxations of graceful labelings as applied to trees. He called a labeling range-relaxed graceful it is meets the same conditions as a graceful labeling except the range of possible vertex labels and edge labels are not restricted to the number of edges of the graph (the edges are distinctly labeled but not necessarily labeled 1 to q where q is the number of edges). Similarly, he calls a labeling vertex-relaxed graceful if it satisfies the conditions of a graceful labeling while permitting repeated vertex labels. He proves that every tree T with q edges has a range-relaxed graceful labeling with the vertex labels in the range 0, 1, . . . , 2q−d where d is the diameter of T and that every tree on n vertices has a vertex-relaxed graceful labeling such that the number of distinct vertex labels is strictly greater than n/2. In 2017 Sethuraman, Ragukumar, and Slater improved the bound on the range-relaxed graceful labeling given by Van Bussel in in 2002 for a tree T.
The range-relaxed graceful game is a maker-breaker game played in a simple graph G where two players, Alice and Bob, alternately assign an unused label f(v) ∈ {0, . . . , k} (k ≥|E(G)|), to an unlabeled vertex v ∈V (G).
If both ends of an edge vw ∈E(G) are already labeled, then the label of the edge is defined as |f(v) −f(w)|. Al-ice’s goal is to end up with a vertex labeling of G where all edges of G have distinct labels, and Bob’s goal is to prevent this from happening. When it is required that k = |E(G)|, the game is called a graceful game. The range-relaxed graceful game and the graceful game were proposed by Tuza in 2017 . In de Oliveira, Dantas, and Lui, considered a question about the least number of consecutive non-negative integer labels necessary for Alice to win the game on an arbitrary simple graph G and also asked if Alice can win the range-relaxed graceful game on G with the set of labels {0, . . . , k + 1} once it is known that she can win with the set {0, . . . , k}. They investigated the graceful game in Cartesian and corona products of graphs, and determined that Bob has a winning strategy in all investigated families independently of who starts the game. Additionally, they partially answer Tuza’s questions presenting the first results in the range-relaxed graceful game and proving that Alice wins on any simple graph G with order n, size m, and maximum degree ∆, for any set of labels {0, . . . , k} with k ≥(n −1) + 2∆(m −∆) + (∆(∆−1))/2.
In Oliveira, Artigas, Dantas, Frickes, and Luiz studied winning strategies for the new the electronic journal of combinatorics (2023), #DS6 74 graceful game introducted by Tuza in 2017 for the graph classes: paths, complete graphs, cycles, complete bipartite graphs, caterpillars, Wgererat trees, gears, webs, prisms, hypercubes, 2-powers of paths, wheels, and fan graphs.
In , Barrientos and Krop introduce left- and right-layered trees as trees with a specific representation and define the excess of a tree. Applying these ideas, they show a range-relaxed graceful labeling which improves the upper bound for maximum vertex label given by Van Bussel in . They also improve the bounds given by Rosa and Širáň in for the α-size and gracesize of lobsters.
Sekar calls an injective function φ from the vertices of a graph with q edges to {0, 1, 3, 4, 6, 7, . . . , 3(q −1), 3q −2} one modulo three graceful if the edge labels induced by labeling each edge uv with |φ(u) −φ(v)| is {1, 4, 7, . . . , 3q −2}. He proves that the following graphs are one modulo three graceful: Pm; Cn if and only if n ≡0 mod 4; Km,n; C(2) 2n (the one-point union of two copies of C2n); C(t) n for n = 4 or 8 and t > 2; C(t) 6 and t ≥4; caterpillars; stars; lobsters; banana trees; rooted trees of height 2; ladders; the graphs obtained by identifying the endpoints of any number of copies of Pn; the graph obtained by attaching pendent edges to each endpoint of two identical stars and then identifying one endpoint from each of these graphs; the graph obtained by identifying a vertex of C4k+2 with an endpoint of a star; n-polygonal snakes (see §2.2) for n ≡0 (mod 4); n-polygonal snakes for n ≡2 (mod 4) where the number of polygons is even; crowns Cn ⊙K1 for n even; C2n ⊙Pm (C2n with Pm attached at each vertex of the cycle) for m ≥3; chains of cycles (see §2.2) of the form C4,m, C6,2m, and C8,m. He conjectures that every one modulo three graceful graph is graceful.
A subdivided shell graph is obtained by subdividing the edges in the path of the shell graph. Jeba Jesintha and Ezhilarasi Hilda proved that the subdivided uniform shell bow graphs (that is, double shells in which each shell has the same order) are one modulo three graceful. Jeba Jesintha and Ezhilarasi Hilda proved the disjoint union of two subdivided shell graphs are one modulo three graceful.
In Ramachandran and Sekar introduced the notion of one modulo N grace-ful as follows.
For a positive integer N a graph G with q edges is said to be one modulo N graceful if there is an injective function φ from the vertex set of G to {0, 1, N, N + 1, 2N, 2N + 1, . . . , (q −1)N, (q −1)N + 1} such that φ induces a bijection φ∗ from the edge set of G to {1, N + 1, 2N + 1, . . . , (q −1)N + 1} where φ∗(uv) = |φ(u)φ(v)|.
They proved the following graph are one modulo N graceful for all positive integers N: paths, caterpillars, and stars ; n-polygonal snakes, C(t) n , Pa,b ; the splitting graphs S′(P2n), S′(P2n+1), S′(K1,n), all subdivision graphs of double triangular snakes, and all subdivision graphs of 2m-triangular snakes ; the graph Ln ⊗Sm obtained from the ladder Ln (Pn × P2) by identifying one vertex of Ln with any vertex of the star Sm other than the center of Sm ; arbitrary supersubdivisions of paths, disconnected paths, cycles, and stars ; and regular bamboo trees and coconut trees . Ra-machandran and Sekar proved the supersubdivisions of ladders are one modulo N graceful for all positive integers N. In Ramachandran and Sekar proved that the crowns, armed crowns, and chain of even cycles are one modulo N graceful for all positive integers N.
the electronic journal of combinatorics (2023), #DS6 75 In 1983 Bange and Barkauskas z introduced the notion of Fibonacci graceful graphs as follows. A function f : V (G) →{0, 1, 2, . . . , Fq}, where G has q edges and Fq is the qth Fibonacci number, is a Fibonacci graceful labeling if the induced edge labeling f(uv) = |f(u) −f(v)| is a bijection to the set of the first q Fibonacci numbers. Such a graph is called Fibonacci graceful. They derived a number of properties of Fibonacci graceful graphs and provided some forbidden subgraphs of Fibonacci graceful graphs.
Other results include: Cn is Fibonacci graceful if and only if n = 0 or 2 mod 3, trees with at least 7 vertices are Fibonacci graceful, and a maximal outerplanar graph with at least four vertices is Fibonacci graceful if and only if it has exactly two vertices of degree 2. Kathiresan and Amutha in prove the following: Kn is Fibonacci graceful if and only if n ≤3; if an Eulerian graph with q edges is Fibonacci graceful, then q ≡0 (mod 3); paths are Fibonacci graceful; fans Pn ⊙K1 are Fibonacci graceful; squares of paths P 2 n are Fibonacci graceful; and caterpillars are Fibonacci graceful. Dharman and Shanmuga Sundaram proved that balloon trees, barycentric subdivisions of bistars, umbrella graphs, and double comb graphs are Fibonacci graceful graphs.
Kalyan and Kempepatil proved that trees and the graphs obtained by joining a vertex of C3m and a vertex of C3n by P2 or P3 admit Fibonacci graceful labelings. They define a function f : V (G) →{0, F1, F2, . . . , Fq}, where Fi is the ith Fibonacci number, to be a super Fibonacci graceful labeling if the induced labeling f(uv) = |f(u) −f(v)| is onto the set {F1, F2, . . . , Fq}. They show that bistars Bn,n are Fibonacci graceful but not super Fibonacci graceful for n ≥5; cycles Cn are super Fibonacci graceful if and only if n ≡0 (mod 3); if G is Fibonacci or super Fibonacci graceful, then G ⊙K1 is Fibonacci graceful; if G1 and G2 are super Fibonacci graceful graphs in which no two adjacent vertices have the labels 1 and 2, then G1 ∪G2 is Fibonacci graceful; and if G1, G2, . . . , Gn are super Fibonacci graceful graphs in which no two adjacent vertices are labeled with 1 and 2, then the amalgamation of G1, G2, . . . , Gn obtained by identifying the vertices having labels 0 is also a super Fibonacci graceful. Karthikeyan, Arthi, Abinaya, Swathi, Madhumathi proved that friendship graphs C(t) 3 and the graphs obtained by the one-point union of copies of K4 with an edge deleted are super Fibonacci graceful.
Vaidya and Prajapati proved: the graphs obtained joining a vertex of C3m and a vertex of C3n by a path Pk are Fibonacci graceful; the graphs obtained by starting with any number of copies of C3m and joining each copy with a copy of the next by identifying the end points of a path with a vertex of each successive pair of C3m (the paths need not be the same length) are Fibonacci graceful; the one point union of C3m and C3n is Fibonacci graceful; the one point union of k cycles C3m is super Fibonacci graceful; every cycle Cn with n ≡0 (mod 3) or n ≡1 (mod 3) is an induced subgraph of a super Fibonacci graceful graph; and every cycle Cn with n ≡2 (mod 3) can be embedded as a subgraph of a Fibonacci graceful graph. Sridevi, Navaneethakrishnan, and Nagarajan proved the following graphs are super Fibonacci graceful: the graphs obtained by identifying the apex of a fan with the end point of a path, the graphs obtained by identifying the apex of a fan with the vertex of maximum degree of K1,n ⊙P2, the graphs obtained by identifying a vertex of C3n with the end point of a path, the graphs obtained by identifying a vertex of C3n with the center of a star, and the graphs obtained by identifying each endpoint a the electronic journal of combinatorics (2023), #DS6 76 star with the center of K1,2.
For a graph G with q edges an injective function f from the vertices of G to {0, F1, F2, . . . , Fq−1, Fq+1}, where Fi is the ith Fibonacci number (as defined by Kathire-san and Amuth above), is said to be almost super Fibonacci graceful if the induced edge labeling f ∗(uv) = |f(u) −f(v)| is a bijection onto the set {F1, F2, . . . , Fq} or {0, F1, F2, . . . , Fq−1, Fq+1}.
Sridevi, Navaneethakrishnan, and Nagarajan proved that paths, combs, graphs obtained by subdividing each edge of a star, and some special types of extension of cycle related graphs are almost super Fibonacci graceful labeling.
Sridevi, Navaneethakrishnan, and Nagarajan showed that paths, combs, and some special types of extension of cycle-related graphs are almost super Fibonacci graceful.
For a graph G and a vertex v of G, a vertex switching Gv is the graph obtained from G by removing all edges incident to v and adding edges joining v to every vertex not adjacent to v in G. Vaidya and Vihol prove the following: trees are Fibonacci graceful; the graph obtained by switching of a vertex in cycle is Fibonacci graceful; wheels and helms are not Fibonacci graceful; the graph obtained by switching of a vertex in a cycle is super Fibonacci graceful except n ≥6; the graph obtained by switching of a vertex in cycle Cn for n ≥6 can be embedded as an induced subgraph of a super Fibonacci graceful graph; and the graph obtained by joining two copies of a fixed fan with an edge is Fibonacci graceful.
The Perrin sequence of numbers Pn is defined by the linear recurrence relation satisfy-ing the conditions: P1 = 3, P2 = 0, P3 = 2, and Pn = Pn−2 + Pn−3, if n ≥4. Letting Pi be the ith term of the Perrin sequence and P0 = 0, Sugumaran and Rajesh introduced the notion of Perrin graceful labeling as follows: A function f is called a Perrin grace-ful labeling of a graph G, if f : V (G) →{P0, P1, P2, ..., Pq} is injective and the induced function f (∗) : E(G) →{P1, P2, ..., Pq} defined by f ∗(uv) = |f(u) −f(v)| is bijective. A graph that admits Perrin graceful labeling is called a Perrin graceful graph. In Sugumaran and Rajesh proved that the following graphs are Perrin graceful graphs: K1,n, Bn,n, Pn ⊙K1, Cn ⊙K1, and < K1,n; 4 >.
For n ≥1 the Pell numbers are defined as p0 = 0, p1 = 1, and pn+1 = 2pn+pn−1. For a graph G with q edges Muthuramakrishnan and Sutha introduced the concept of Pell graceful labeling as an injective function f from V (G) to the Pell numbers {0, 1, 2, . . . , pq} such that the induced edge labeling f ∗(uv) = |f(u)f(v)| is a bijection onto the Pell numbers {p1, p2, . . . , pq} They prove that paths, combs Pn ⊙K1 (n ≥3), and the graphs obtained by the one point union of paths of lengths 1, 2, . . . , n (n ≥3) are Pell graceful.
In Brešar and Klavžar define a natural extension of graceful labelings of certain tree subgraphs of hypercubes. A subgraph H of a graph G is called isometric if for every two vertices u, v of H, there exists a shortest u-v path that lies in H. The isometric subgraphs of hypercubes are called partial cubes. Two edges xy, uv of G are in Θ-relation if dG(x, u) + dG(y, v) ̸= dG(x, v) + dG(y, u). A Θ-relation is an equivalence relation that partitions E(G) into Θ-classes. A Θ-graceful labeling of a partial cube G on n vertices is a bijection f : V (G) →{0, 1, . . . , n −1} such that, under the induced edge labeling, all edges in each Θ-class of G have the same label and distinct Θ-classes get distinct labels.
They prove that several classes of partial cubes are Θ-graceful and the Cartesian product the electronic journal of combinatorics (2023), #DS6 77 of Θ-graceful partial cubes is Θ-graceful. They also show that if there exists a class of partial cubes that contains all trees and every member of the class admits a Θ-graceful labeling then all trees are graceful.
Cohen and Kovše showed that the graph obtained by merging two vertices of two 4-cycles is not a Θ-graceful partial cube, thereby answering in the negative a question of Brešar and Klavžar who asked whether every partial cube is Θ-graceful.
Table 4 provides a summary results about graceful-like labelings adapted from .
“Y” indicates that all graphs in that class have the labeling; “N” indicates that not all graphs in that class have the labeling; “?” means unknown; “C” means conjectured.
Table 4: Summary of Results on Graceful-like labelings Graph α-labeling β-labeling σ-labeling ρ-labeling Cycle Cn, n ≡0 mod 4 Y Y Y Y Cycle Cn, n ≡3 mod 4 N Y Y Y Wheels N Y , Y Y Trees Yes, if order ≤ 5 35 54 Paths Y Y Y Y Caterpillars Y Y Y Y Firecrackers Y Y Y Y Lobsters N ?C Y Y Bananas ?
Y , Y Y Symmetrical trees N Y Y Y Olive trees ?
Y , Y Y Diameter < 8 N Y Y Y < 5 end vertices N Y Y Y Max degree 3 N C C C Max degree 3 and perfect matching C C C C 3.4 k-graceful Labelings A natural generalization of graceful graphs is the notion of k-graceful graphs introduced independently by Slater in 1982 and by Maheo and Thuillier in 1982. A graph G with q edges is k-graceful if there is labeling f from the vertices of G to {0, 1, 2, . . . , q + k −1} such that the set of edge labels induced by the absolute value of the difference of the labels of adjacent vertices is {k, k + 1, . . . , q + k −1}. Obviously, 1-graceful is graceful and it is readily shown that any graph that has an α-labeling is k-graceful for all k. Graphs that are k-graceful for all k are sometimes called arbitrarily graceful. The result of Barrientos and Minion that all snake polyominoes are α-graphs partially the electronic journal of combinatorics (2023), #DS6 78 answers a question of Acharya and supports his conjecture that if the length of every cycle of a graph is a multiple of 4, then the graph is arbitrarily graceful. In Seoud and Elsakhawi show that P2 ⊕K2 (n ≥2) is arbitrarily graceful. Ng has shown that there are graphs that are k-graceful for all k but do not have an α-labeling.
Results of Maheo and Thuillier together with those of Slater show that: Cn is k-graceful if and only if either n ≡0 or 1 (mod 4) with k even and k ≤(n −1)/2, or n ≡3 (mod 4) with k odd and k ≤(n2 −1)/2. Maheo and Thuillier also proved that the wheel W2k+1 is k-graceful and conjectured that W2k is k-graceful when k ̸= 3 or k ̸= 4. This conjecture was proved by Liang, Sun, and Xu . Kang proved that Pm × C4n is k-graceful for all k. Lee and Wang showed that the graphs obtained from a nontrivial path of even length by joining every other vertex to one isolated vertex (a lotus), the graphs obtained from a nontrivial path of even length by joining every other vertex to two isolated vertices (a diamond), and the graphs obtained by arranging vertices into a finite number of rows with i vertices in the ith row and in every row the jth vertex in that row is joined to the jth vertex and j + 1st vertex of the next row (a pyramid) are k-graceful. Liang and Liu have shown that Km,n is k-graceful. Bu, Gao, and Zhang have proved that Pn × P2 and (Pn × P2) ∪(Pn × P2) are k-graceful for all k. Acharya (see ) has shown that a k-graceful Eulerian graph with q edges must satisfy one of the following conditions: q ≡0 (mod 4), q ≡1 (mod 4) if k is even, or q ≡3 (mod 4) if k is odd. Bu, Zhang, and He have shown that an even cycle with a fixed number of pendent edges adjoined to each vertex is k-graceful. Lu, Pan, and Li have proved that K1,m ∪Kp,q is k-graceful when k > 1, and p and q are at least 2. Jirimutu, Bao, and Kong have shown that the graphs obtained from K2,n (n ≥2) and K3,n (n ≥3) by attaching r ≥2 edges at each vertex is k-graceful for all k ≥2. Seoud and Elsakhawi proved: paths and ladders are arbitrarily graceful; and for n ≥3, Kn is k-graceful if and only if k = 1 and n = 3 or 4. Li, Li, and Yan proved that Km,n is k-graceful graph. Pradhan and Kamesh showed that the hairy cycle Cn ⊙rK1 (n ≡3 (mod 4), the graph obtained by adding a pendent edge to each pendent vertex of hairy cycle Cn ⊙K1; n ≡0 (mod 4), double graphs of path Pn, and double graphs of combs Pn ⊙K1 are k-graceful.
Yao, Cheng, Zhongfu, and Yao have shown: a tree of order p with maximum degree at least p/2 is k-graceful for some k; if a tree T has an edge u1u2 such that the two components T1 and T2 of T −u1u2 have the properties that dT1(u1) ≥|T1|/2 and dT2(u2) ≥|T2|/2, then T is k-graceful for some positive k; if a tree T has two edges u1u2 and u2u3 such that the three components T1, T2, and T3 of T −{u1u2, u2u3} have the properties that dT1(u1) ≥|T1|/2, dT2(u2) ≥|T2|/2, and dT3(u3) ≥|T3|/2, then T is k-graceful for some k > 1; and every Skolem-graceful (see 3.5 for the definition) tree is k-graceful for all k ≥1. They conjecture that every tree is k-graceful for some k > 1.
Several authors have investigated the k-gracefulness of various classes of subgraphs of grid graphs. Acharya proved that all 2-dimensional polyminoes that are convex and Eulerian are k-graceful for all k; Lee showed that Mongolian tents and Mongolian villages are k-graceful for all k (see §2.3 for the definitions); Lee and K. C. Ng proved that all Young tableaus (see §2.3 for the definitions) are k-graceful for all k. (A the electronic journal of combinatorics (2023), #DS6 79 special case of this is Pn × P2.) Lee and H. K. Ng subsequently generalized these results on Young tableaus to a wider class of planar graphs.
In Barrientos and Minion say that two caterpillars Γ and Ωof size n are analogous if the stable sets of Γ have the same cardinalities as the stable sets of Ω. They prove that if Ωis an induced subgraph of a gracefully labeled graph G, such that the induced labeling is a bipartite k-labeling shifted c units, then the graph G′ obtained by replacing Ωwith any other caterpillar Γ analogous to Ω, is a graceful graph. This result is used to generalize several existing results that use k-graceful labelings of paths such as the subdivision of graceful trees , the α-labeling of the ith attachment tree , the α-labelings of path-like trees , the α-labelings of the graphs obtained by identifying the end-vertices of b paths of length a with two new vertices, as well as the graceful labelings of the armed crowns .
Duan and Qi use Gt(m1, n1; m2, n2; . . . ; ms, ns) to denote the graph composed of the s complete bipartite graphs Km1,n1, Km2,n2, . . . , Kms,ns that have only t (1 ≤ t ≤ min{m1, m2, . . . , ms}) common vertices but no common edge and G(m1, n1; m2, n2) to denote the graph composed of the complete bipartite graphs Km1,n1, Km2,n2 with exactly one common edge.
They prove that these graphs are k-graceful graphs for all k.
Let c, m, p1, p2, . . . , pm be positive integers. For i = 1, 2, . . . , m, let Si be a set of pi +1 integers and let Di be the set of positive differences of the pairs of elements of Si. If all these differences are distinct then the system D1, D2, . . . , Dm is called a perfect system of difference sets starting at c if the union of all the sets Di is c, c+1, . . . , c−1+Pm i=1 pi+1 2 .
There is a relationship between k-graceful graphs and perfect systems of difference sets. A perfect system of difference sets starting with c describes a c-graceful labeling of a graph that is decomposable into complete subgraphs. A survey of perfect systems of difference sets is given in .
Acharya and Hegde generalized k-graceful labelings to (k, d)-graceful labelings by permitting the vertex labels to belong to {0, 1, 2, . . . , k + (q −1)d} and requiring the set of edge labels induced by the absolute value of the difference of labels of adjacent vertices to be {k, k + d, k + 2d, . . . , k + (q −1)d}. They also introduce an analog of α-labelings in the obvious way.
Notice that a (1,1)-graceful labeling is a graceful labeling and a (k, 1)-graceful labeling is a k-graceful labeling. Bu and Zhang have shown: Km,n is (k, d)-graceful for all k and d; for n > 2, Kn is (k, d)-graceful if and only if k = d and n ≤4; if mi, ni ≥2 and max{mi, ni} ≥3, then Km1,n1 ∪Km2,n2 ∪· · · ∪Kmr,nr is (k, d)-graceful for all k, d, and r; if G has an α-labeling, then G is (k, d)-graceful for all k and d; a k-graceful graph is a (kd, d)-graceful graph; a (kd, d)-graceful connected graph is k-graceful; and a (k, d)-graceful graph with q edges that is not bipartite must have k ≤(q −2)d.
Let T be a tree with adjacent vertices u0 and v0 and pendent vertices u and v such that the length of the path u0 −u is the same as the length of the path v0 −v. Hegde and Shetty call the graph obtained from T by deleting u0v0 and joining u and v an elementary parallel transformation of T. They say that a tree T is a Tp-tree if it can be transformed into a path by a sequence of elementary parallel transformations. They the electronic journal of combinatorics (2023), #DS6 80 prove that every Tp-tree is (k, d)-graceful for all k and d and every graph obtained from a Tp-tree by subdividing each edge of the tree is (k, d)-graceful for all k and d.
Yao, Cheng, Zhongfu, and Yao have shown: a tree of order p with maximum degree at least p/2 is (k, d)-graceful for some k and d; if a tree T has an edge u1u2 such that the two components T1 and T2 of T −u1u2 have the properties that dT1(u1) ≥|T1|/2 and T2 is a caterpillar, then T is Skolem-graceful (see 3.5 for the definition); if a tree T has an edge u1u2 such that the two components T1 and T2 of T −u1u2 have the properties that dT1(u1) ≥|T1|/2 and dT2(u2) ≥|T2|/2, then T is (k, d)-graceful for some k > 1 and d > 1; if a tree T has two edges u1u2 and u2u3 such that the three components T1, T2, and T3 of T −{u1u2, u2u3} have the properties that dT1(u1) ≥|T1|/2, dT2(u2) ≥|T2|/2, and dT3(u3) ≥|T3|/2, then T is (k, d)-graceful for some k > 1 and d > 1; and every Skolem-graceful tree is (k, d)-graceful for k ≥1 and d > 0. They conjecture that every tree is (k, d)-graceful for some k > 1 and d > 1.
Hegde has proved the following: if a graph is (k, d)-graceful for odd k and even d, then the graph is bipartite; if a graph is (k, d)-graceful and contains C2j+1 as a subgraph, then k ≤jd(q −j −1); Kn is (k, d)-graceful if and only if n ≤4; C4t is (k, d)-graceful for all k and d; C4t+1 is (2t, 1)-graceful; C4t+2 is (2t −1, 2)-graceful; and C4t+3 is (2t + 1, 1)-graceful.
A semismooth graceful graph is a bipartite graph G with the property that for some fixed positive integer t ≤q and all positive integers l there is an injective map g : V − →{0, 1, . . . , t −l, t + l + 1, . . . , q + l} such that the induced edge labeling map g⋆: E − →{1 + l, 2 + l, . . . , q + l} defined by g⋆(e) = |g(u) −g(v)| is a bijection. Kaneria, Gohil, and Makadia prove every semismooth graceful graph is a (k, d)-graceful; graphs obtained by joining two semismooth graceful graphs with an arbitrary path is a semismooth graceful graph; and the notions of graceful labeling and odd-even graceful labelings are equivalent.
(A graph G with q edges is odd-even graceful if there is an injection f from the vertices of G to {1, 3, 5, . . . , 2q + 1} such that, when each edge uv is assigned the label |f(u) −f(v)|, the resulting edge labels are {2, 4, 6, . . . , 2q}). Kaneria, Meghpara and Khoda prove: a smooth graceful labeling for a graph is also an α-labeling for the graph; a graph that has an α-labeling is a semismooth graceful graph; graphs that admit an α-labeling are semismooth graceful graphs; if m is even and H has an α-labeling, then the path union P(m · H) is a smooth graceful graph; and the path union P(m · H) has an α-labeling.
Nurvazly, Chasanah, and Wiranto showed that the corona of the ladder L2 and Kn and of L3 and Kn, which they call millipedes, admits graceful, ˆ ρ, and odd-even graceful labelings. In Sudha and Kanniga proved that tensor product of a star and P2 is odd-even graceful. (The tensor product G ⊗H of graphs G and H, has the vertex set V (G)×V (H) and any two vertices (u, u′) and (v, v′) are adjacent in G⊗H if and only if u′ is adjacent with v′ and u is adjacent with v.) In Venkatesh, Mahalakshmi, and Amirthavahini use Cn,k to denote the dragon obtained by joining an end point of Pk with a vertex of Cn and Ct n,k to denote the graph obtained by taking one-point union of t copies of Cn,k at the common vertex v. They proved that the graph Ct n,k admits a graceful labeling, an odd graceful labeling, and odd-even graceful labeling for all values of t with the electronic journal of combinatorics (2023), #DS6 81 n = 4, k = 1, and that Ct n,1 admits vertex cordial labeling for all values of n and t, except n ≡2 mod 4 (see Section 3.7). Nurvazly and Sugeng proved that Θ(C3)n graphs (n copies of C3 that share an edge) have odd-even graceful labelings. Anitha, Selvam, and Thirusangu provide k-graceful and odd-even graceful labelings for the extended duplicate graph of the kite graph.
For a graph G let G(1), G(2), . . . , G(n) be n ≥2 copies of G. The graph obtained by joining vertices u, v of G(i) with same vertices of the graph G(i+1) by two edges, for all i = 1, 2, . . . , n −1 is called the double path union of n copies of the graph G. Such graphs can obtained in p(p−1) 2 different ways, where p = |V (G)| and are denoted by D(n · G).
Kaneria, Teraiya and Meghpara prove the double path unions of C4m, Km,n, and P2m have α-labelings.
Hegde calls a (k, d)-graceful graph (k, d)-balanced if it has a (k, d)-graceful la-beling f with the property that there is some integer m such that for every edge uv either f(u) ≤m and f(v) > m, or f(u) > m and f(v) ≤m. He proves that if a graph is (1, 1)-balanced then it is (k, d)-graceful for all k and d and that a graph is (1, 1)-balanced graph if and only if it is (k, k)-balanced for all k. He conjectures that all trees are (k, d)-balanced for some values of k and d.
Slater has extended the definition of k-graceful graphs to countable infinite graphs in a natural way. He proved that all countably infinite trees, the complete graph with countably many vertices, and the countably infinite Dutch windmill is k-graceful for all k.
In Hegde and Shivarajkumar extend the idea of k-graceful labeling of undirected graphs to directed graphs as follows. A simple directed graph D with n vertices and e edges is labeled by assigning each vertex a distinct element from the set Ze+k and assigning the edge xy from vertex x to vertex y the label θ(x, y) = θ(y)θ(x) mod(e + k), where θ(y) and θ(x) are the values assigned to the vertices y and x respectively. A labeling is a k-graceful labeling if all θ(x, y) are distinct and belong to {k, k + 1, . . . , k + e −1}. If a digraph D admits a k-graceful labeling then D is called a k-graceful digraph. They provide some values of k for which the unidirectional cycles admit a k-graceful labeling; give a necessary and sufficient condition for the outspoken unicyclic wheel to be k-graceful; and prove that to provide a list of values of k for which the unicyclic wheel is k-graceful is NP-complete.
More specialized results on k-graceful labelings can be found in , , , , , , and .
Graceful-type labelings methods have been used for cryptographical password con-struction for network data , , , and .
3.5 Skolem-Graceful Labelings A number of authors have invented analogues of graceful graphs by modifying the per-missible vertex labels. For instance, Lee (see ) calls a graph G with p vertices and q edges Skolem-graceful if there is an injection from the set of vertices of G to {1, 2, . . . , p} such that the edge labels induced by |f(x)−f(y)| for each edge xy are 1, 2, . . . , q. A neces-the electronic journal of combinatorics (2023), #DS6 82 sary condition for a graph to be Skolem-graceful is that p ≥q+1. Lee and Wui have shown that a connected graph is Skolem-graceful if and only if it is a graceful tree. Yao, Cheng, Zhongfu, and Yao have shown that a tree of order p with maximum degree at least p/2 is Skolem-graceful. Although the disjoint union of trees cannot be graceful, they can be Skolem-graceful. Lee and Wui prove that the disjoint union of 2 or 3 stars is Skolem-graceful if and only if at least one star has even size. In Choudum and Kishore show that the disjoint union of k copies of the star K1,2p is Skolem graceful if k ≤4p + 1 and the disjoint union of any number of copies of K1,2 is Skolem graceful. For k ≥2, let St(n1, n2, . . . , nk) denote the disjoint union of k stars with n1, n2, . . . , nk edges.
Lee, Wang, and Wui showed that the 4-star St(n1, n2, n3, n4) is Skolem-graceful for some special cases and conjectured that all 4-stars are Skolem-graceful. Denham, Leu, and, Liu proved this conjecture. Kishore has shown that a necessary condition for St(n1, n2, . . . , nk) to be Skolem graceful is that some ni is even or (k ≡0) or 1 (mod 4) (see also . He conjectures that each one of these conditions is sufficient. Yue, Yuan-sheng, and Xin-hong show that for k at most 5, a k-star is Skolem-graceful if at one star has even size or k ≡0 or 1 (mod 4). Choudum and Kishore proved that all 5-stars are Skolem graceful.
Lee, Quach, and Wang showed that the disjoint union of the path Pn and the star of size m is Skolem-graceful if and only if n = 2 and m is even or n ≥3 and m ≥1.
It follows from the work of Skolem that nP2, the disjoint union of n copies of P2, is Skolem-graceful if and only if n ≡0 or 1 (mod 4). Harary and Hsu studied Skolem-graceful graphs under the name node-graceful. Frucht has shown that Pm ∪Pn is Skolem-graceful when m + n ≥5. Bhat-Nayak and Deshmukh have shown that Pn1 ∪Pn2 ∪Pn3 is Skolem-graceful when n1 < n2 ≤n3, n2 = t(n1 + 2) + 1 and n1 is even and when n1 < n2 ≤n3, n2 = t(n1 + 3) + 1 and n1 is odd. They also prove that the graphs of the form Pn1 ∪Pn2 ∪· · · ∪Pni where i ≥4 are Skolem-graceful under certain conditions. In Deshmukh states the following results: the sum of all the edges on any cycle in a Skolem graceful graph is even; C5 ∪K1,n if and only if n = 1 or 2; C6 ∪K1,n if and only if n = 2 or 4.
Youssef proved that if G is Skolem-graceful, then G + Kn is graceful. In Youssef shows that that for all n ≥2, Pn ∪Sm is Skolem-graceful if and only if n ≥3 or n = 2 and m is even. Yao, Cheng, Zhongfu, and Yao have shown that if a tree T has an edge u1u2 such that the two components T1 and T2 of T −u1u2 have the properties that dT1(u1) ≥|T1|/2 and T2 is a caterpillar or have the properties that dT1(u1) ≥|T1|/2 and dT2(u2) ≥|T2|/2, then T is Skolem-graceful.
A graph G = (V, E) is said to be (k, d)-Skolem graceful if there exists a bijection f from V (G) to {12, . . . , |V |} such that the induced edge labeling gf defined by gf(uv) = |f(u)−f(v)| is a bijection from E to {k, k+d, . . . , k+(q−1)d} where k and d are positive integers. Such a labeling is called a (k, d)-Skolem graceful labeling of G. In Pereira, Singh, and Arumugam present a few basic results on (k, d)-Skolem graceful graphs and prove that nK2 is (2, 1)-Skolem graceful if and only if n ≡0 or 3 (mod 4), which produces the Langford sequence L(2, n).
Mendelsohn and Shalaby defined a Skolem labeled graph G(V, E) as one for the electronic journal of combinatorics (2023), #DS6 83 which there is a positive integer d and a function L: V →{d, d+1, . . . , d+m}, satisfying (a) there are exactly two vertices in V such that L(v) = d+i, 0 ≤i ≤m; (b) the distance in G between any two vertices with the same label is the value of the label; and (c) if G′ is a proper spanning subgraph of G, then L restricted to G′ is not a Skolem labeled graph.
Note that this definition is different from the Skolem-graceful labeling of Lee, Quach, and Wang. A hooked Skolem sequence of order n is a sequence s1, s2, . . . , s2n+1 such that s2n = 0 and for each j ∈{1, 2, . . . , n}, there exists a unique i ∈{1, 2, . . . , 2n −1, 2n + 1} such that si = si+j = j. Mendelsohn established the following: any tree can be embedded in a Skolem labeled tree with O(v) vertices; any graph can be embedded as an induced subgraph in a Skolem labeled graph on O(v3) vertices; for d = 1, there is a Skolem labeling or the minimum hooked Skolem (with as few unlabeled vertices as possible) labeling for paths and cycles; for d = 1, there is a minimum Skolem labeled graph containing a path or a cycle of length n as induced subgraph. In Mendelsohn and Shalaby prove that the necessary conditions in are sufficient for a Skolem or minimum hooked Skolem labeling of all trees consisting of edge-disjoint paths of the same length from some fixed vertex. Graham, Pike, and Shalaby obtained various Skolem labeling results for grid graphs. Among them are P1×Pn and P2×Pn have Skolem labelings if and only if n ≡0 or 1 mod 4; and Pm × Pn has a Skolem labeling for all m and n at least 3.
In Pike, Sanaei, and Shalaby introduce pseudo-Skolem sequences, which are similar to Skolem-type sequences in their structures and applications. They use known Skolem-type sequences to construct such sequences and discuss applications of these se-quences to Skolem labelings of graphs such that H is bipartite, and give formulas for the gamma-number of rail-siding graphs and caterpillars.
In Clark and Sanaei present (hooked) vertex Skolem labelings for generalized Dutch windmills whenever such labelings exist. They present a novel technique for show-ing that generalized Dutch windmills with more than two cycles cannot be Skolem labelled and that those composed of two cycles of lengths m and n, n ≥m cannot be Skolem labelled if and only if n −m ≡3 or 5 (mod 8) and m is odd.
3.6 Odd-Graceful Labelings Gnanajothi [1143, p. 182] defined a graph G with q edges to be odd-graceful if there is an injection f from V (G) to {0, 1, 2, . . . , 2q −1} such that, when each edge xy is assigned the label |f(x) −f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q −1}. She proved that the class of odd-graceful graphs lies between the class of graphs with α-labelings and the class of bipartite graphs by showing that every graph with an α-labeling has an odd-graceful labeling and every graph with an odd cycle is not odd-graceful.
She also proved the following graphs are odd-graceful: Pn; Cn if and only if n is even; Km,n; combs Pn ⊙K1 (graphs obtained by joining a single pendent edge to each vertex of Pn); books; crowns Cn⊙K1 (graphs obtained by joining a single pendent edge to each vertex of Cn) if and only if n is even; the disjoint union of copies of C4; the one-point union of copies of C4; Cn×K2 if and only if n is even; caterpillars; rooted trees of height 2; the graphs obtained from the electronic journal of combinatorics (2023), #DS6 84 Pn (n ≥3) by adding exactly two leaves at each vertex of degree 2 of Pn; the graphs obtained from Pn × P2 by deleting an edge that joins to end points of the Pn paths; the graphs obtained from a star by adjoining to each end vertex the path P3 or by adjoining to each end vertex the path P4. She conjectures that all trees are odd-graceful and proves the conjecture for all trees with order up to 10. Barrientos has extended this to trees of order up to 12. Eldergill generalized Gnanajothi’s result on stars by showing that the graphs obtained by joining one end point from each of any odd number of paths of equal length is odd-graceful. He also proved that the one-point union of any number of copies of C6 is odd-graceful. Kathiresan has shown that ladders and graphs obtained from them by subdividing each step exactly once are odd-graceful. Barrientos and has proved the following graphs are odd-graceful: every forest whose components are caterpillars; every tree with diameter at most five is odd-graceful; and all disjoint unions of caterpillars. He conjectures that every bipartite graph is odd-graceful. In Neela and Selvaraj partially resolved a Barrientos’s conjecture by showing that the following graphs are odd-graceful: finite unions of paths, stars, and caterpillars; finite unions of ladders; finite unions of paths, bistars and caterpillars; finite unions of graphs obtained by the one end point union of an odd number of paths of uniform length; and the coronas Km;n ⊙rKl. Gao, Zhang, and Xu proved that Pn ×Pm (m = 2, 3 or 4), generalized crown graphs Cn ⊙K1,t, and gears are odd graceful.
Seoud, Diab, and Elsakhawi have shown that a connected complete r-partite graph is odd-graceful if and only if r = 2 and that the join of any two connected graphs is not odd-graceful. Yan proved that Pm × Pn is odd-graceful labeling. Vaidya and Shah prove that the splitting graph and the shadow graph of bistar are odd-graceful.
(The shadow graph D2(G) of a connected graph G is constructed by taking 2 copies G1 and G2 of G and joining each vertex u in G1 to the neighbors of the corresponding vertex v in G2. Li, Li, and Yan proved that Km,n is odd-graceful. Liu, Wang, and Lu that proved that a class of bicyclic graphs with a common edge is odd-graceful. Moussa and Badr proved that ladders and subdivisions of ladders with pendent edges are odd-graceful.
Virk and Riasat showed that the union a specific type of tree new with paths, stars, bistars, or ladders are odd-graceful. Riasat, Kanwal, and Javed give odd-graceful labelings for disjoint unions of graphs consisting of generalized combs, ladders, stars, bistars, caterpillars and paths.
Sekar has shown the following graphs are odd-graceful: the graph obtained by identifying an end point of Pn with every vertex of Cm where n ≥3 and m is even; Pa,b when a ≥2 and b is odd (see §2.7); P2,b and b ≥2; P4,b and b ≥2; Pa,b when a and b are even and a ≥4 and b ≥4; P4r+1,4r+2; P4r−1,4r; all n-polygonal snakes with n even; C(t) n (see §2.2 for the definition); graphs obtained by beginning with C6 and repeatedly forming the one-point union with additional copies of C6 in succession; graphs obtained by beginning with C8 and repeatedly forming the one-point union with additional copies of C8 in succession; graphs obtained from even cycles by identifying a vertex of the cycle with the endpoint of a star; C6,n and C8,n (see §2.7); the splitting graph of Pn (see §2.7) the splitting graph of Cn, n even; lobsters, banana trees, and regular bamboo trees (see §2.1).
the electronic journal of combinatorics (2023), #DS6 85 Jeba Jesinthan, Devakirubanith, and Aathry proved that graphs obtained by new attaching star graphs to the apex and all the path vertices of the subdivided shell graph are odd graceful.
Yao, Cheng, Zhongfu, and Yao have shown the following: if a tree T has an edge u1u2 such that the two components T1 and T2 of T −u1u2 have the properties that dT1(u1) ≥|T1|/2 and T2 is a caterpillar, then T is odd-graceful; and if a tree T has a vertex of degree at least |T|/2, then T is odd-graceful. They conjecture that for trees the properties of being Skolem-graceful and odd-graceful are equivalent. Recall a banana tree is a graph obtained by starting with any number os stars and connecting one end-vertex from each to a new vertex. Zhenbin has shown that graphs obtained by starting with any number of stars, appending an edge to exactly one edge from each star, then joining the vertices at which the appended edges were attached to a new vertex are odd-graceful.
Solairaju and Chithra defined a graph G with q edges to be edge-odd graceful if there is an bijection f from the edges of the graph to {1, 3, 5, . . . , 2q −1} such that, when each vertex is assigned the sum of all the edges incident to it mod 2q, the resulting vertex labels are distinct. They prove they following graphs are odd-graceful: paths with at least 3 vertices; odd cycles; ladders Pn × P2 (n ≥3); stars with an even number of edges; and crowns Cn ⊙K1. In they prove the following graphs have edge-odd graceful labelings: Pn (n > 1) with a pendent edge attached to each vertex (combs); the graph obtained by appending 2n + 1 pendent edges to each endpoint of P2 or P3; and the graph obtained by subdividing each edge of the star K1,2n. In Kumar and new Gayathri showed the existence of edge-odd graceful labeling for Pm × Pn For a graph G, Kulli and Muddebihai define the lict graph of G as the graph whose vertex set is the union of the edges of G and the set of cutpoints of G where two vertices are adjacent if and only if the corresponding members of G are adjacent or the corresponding members of G are incident. They define the litact graph of G as the graph whose vertex set is the union of the edges of G and the set of cutpoints of G where two vertices are adjacent if and only if the corresponding members of G are adjacent or the corresponding members of G are adjacent or incident. Mirajkar and Sthavarmath provided edge-odd graceful labeling for the lict graph of Pn for n > 1 and odd, the lict graph of the one-point of Cn and P2, and the litact graph of Pn for n ≥4.
A subdivided shell graph is obtained by subdividing the edges in the path of the shell graph. Let G1, G2, . . . , Gn be n subdivided shell graphs of any order. The graph SSG(n) is obtained by adding an edge to apexes of Gi and Gi+1, i = 1, 2, . . . , n −1. Jeba Jesintha and Ezhilarasi Hilda that SSG(2) is odd graceful. In and Jeba Jesintha and Ezhilarasi Hilda proved that the subdivided uniform shell bow graphs (that is, double shells in which each shell has the same order) are odd graceful and shell butterfly graphs are edge-odd graceful. Daoud provided necessary and sufficient conditions for Cm × Pn and Cm × Cn to be edge-odd graceful.
Gao has proved the following graphs are odd-graceful: the union of any number of paths; the union of any number of stars; the union of any number of stars and paths; Cm ∪Pn; Cm ∪Cn; and the union of any number of cycles each of which has order divisible the electronic journal of combinatorics (2023), #DS6 86 by 4.
If f is an odd-graceful labeling of a bipartite graph G with bipartition (V1, V2) such that max{f(u) : u ∈V1} < min{f(v) : v ∈V2}, Zhou, Yao, Chen, and Tao say that f is a set-ordered odd-graceful labeling of G. They proved that every lobster is odd-graceful and adding leaves to a connected set-ordered odd-graceful graph is an odd-graceful graph.
In Seoud and Abdel-Aal determined all odd-graceful graphs of order at most 6 and proved that if G is odd-graceful then G ∪Km,n is odd-graceful. In Seoud and Helmi proved: if G has an odd-graceful labeling f with bipartition (V1, V2) such that max{f(x) : f(x) is even, x ∈V1} < min{f(x) : f(x) is odd, x ∈V2}, then G has an α-labeling; if G has an α-labeling, then G ⊙Kn is odd-graceful; and if G1 has an α-labeling and G2 is odd-graceful, then G1 ∪G2 is odd-graceful. They also proved the following graphs have odd-graceful labelings: dragons obtained from an even cycle; graphs obtained from a gear graph by attaching a fixed number of pendent edges to each vertex of degree 2 on rim of the wheel of the graph; C2m ⊙Kn; graphs obtained from an even cycle by attaching a fixed number of pendent edges to every other vertex; graphs obtained by identifying an endpoint of a star Sn (n ≥3) with a vertex of an even cycle; the graphs consisting of two even cycles of the same order that share a common vertex with any number of pendent edges attached at the common vertex; and the graphs obtained by joining two even cycles of the same order by an edge. Seoud, El Sonbaty, and Abd El Rehim proved that the conjunction Pm ∧Pn for all n, m ≥2 and the conjunction K2 ∧Fn for n even are odd-graceful. Jeba Jesintha and Ezhilarasi Hilda proved the disjoint union of two subdivided shell graphs is odd-graceful and the one vertex union of three subdivided shells are odd-graceful.
In and Moussa proved that Cm ∪Pn is odd-graceful in some cases and gave algorithms to prove that for all m ≥2 the graphs P4r−1;m, r = 1, 2, 3 and P4r+1;m, r = 1, 2 are odd-graceful. (Pn;m is the graph obtained by identifying the endpoints of m paths each of length n). He also presented an algorithm that showed that closed spider graphs and the graphs obtained by joining one or two copies of Pm to each vertex of the path Pn are odd-graceful. Moussa and Badr proved that Cm ⊙Pn is odd-graceful if and only if m is even (see also ). Badr, Moussa, and Kathiresan proved ladders are odd graceful.
Moussa defines the tensor product, Pm ∧Pn, of Pm and Pn as the graph with vertices vj i , i = 1, . . . , n; j = 1, . . . , m and edges vj 1vj+1 2 , vj+1 2 vj 3, . . . , vj n−1vj+1 n for j odd and vj 1vj−1 2 , vj−1 2 vj 3, . . . , vj n−1vj−1 n for j even. He proves that Pm ∧Pm is odd-graceful.
In Abdel-Aal generalized the notions of shadow graphs and splitting graphs are follows. The m-shadow graph Dm(G) of a connected graph G is constructed by taking m copies of G1, G2, . . . , Gm of G , and joining each vertex u in Gi to the neighbors of the corresponding vertex v in Gj for 1 ≤i, j ≤m. The m-splitting graph Splm(G) of a graph G is obtained by adding to each vertex v of G m new vertices, v1, v2, . . . , vm, such that vi, 1 ≤i ≤m is adjacent to every vertex that is adjacent to v in Gj. Thus the 2-shadow graph is the shadow graph D2(G) and the 1-splitting graph of G is the splitting graph of G. Abdel-Aal proved the following graphs are odd-graceful: Dm(Pn), Dm(Pn ⊕K2) (the the electronic journal of combinatorics (2023), #DS6 87 symmetric product of Pn and K2), Dm(Kr,s), Splm(Pn), Splm(K1,n), and Splm(Pn ⊕K2).
Vaidya and Bijukumar proved the following are odd-graceful: graphs obtained by joining two copies of Cn by a path; graphs that are two copies of an even cycle that share a common edge; graphs that are the splitting graph of a star; and graphs that are the tensor product of a star and P2. Jeba Jesintha, Jaya Glory, and Elakiya Solai proved that the path unions of caterpillars are odd graceful.
Acharya, Germina, Princy, and Rao proved that every bipartite graph G can be embedded in an odd-graceful graph H. The construction is done in such a way that if G is planar and odd-graceful, then so is H. Varkey and Sunoj investigate some new families of odd graceful graphs generated from various graph operations on the given graph.
In Chawathe and Krishna extend the definition of odd-gracefulness to countably infinite graphs and show that all countably infinite bipartite graphs that are connected and locally finite have odd-graceful labelings.
Solairaju and Chithra defined a graph G with q edges to be edge-odd graceful if there is an bijection f from the edges of the graph to {1, 3, 5, . . . , 2q −1} such that, when each vertex is assigned the sum of all the edges incident to it mod 2q, the resulting vertex labels are distinct. They prove they following graphs are odd-graceful: paths with at least 3 vertices; odd cycles; ladders Pn × P2 (n ≥3); stars with an even number of edges; and crowns Cn ⊙K1. In they prove the following graphs have edge-odd graceful labelings: Pn (n > 1) with a pendent edge attached to each vertex (combs); the graph obtained by appending 2n + 1 pendent edges to each endpoint of P2 or P3; and the graph obtained by subdividing each edge of the star K1,2n.
Singhun proved the following graphs have edge-odd graceful labelings: W2n; Wn⊙K1; and Wn⊙Km, when n is odd, m is even, and n divides m. Seoud and Salim present edge-odd graceful labelings for the following families of graphs: Wn for n ≡1, 2 and 3 (mod 4); Cn ⊙K2m−1; even helms; Pn ⊙K2m; and K2,s. They also provide two theorems about non edge-odd graceful graphs. Susanti, Ernanto1, and Surodjo found edge-odd graceful labelings for some classes of prism related graphs.
In Sridevi, Navaeethakrishnan, Nagarajan, and Nagarajan call a graph G with q edges odd-even graceful if there is an injection f from the vertices of G to {1, 3, 5, . . . , 2q + 1} such that, when each edge uv is assigned the label |f(u) −f(v)|, the resulting edge labels are {2, 4, 6, . . . , 2q}. They proved that Pn, combs Pn ⊙K1, stars K1,n, K1,2,n, Km,n, and bistars Bm,n are odd-even graceful.
Sudha and Babu say a graph G with q edges is even-even graceful if there is an injection f from the edges of G to {2, 4, 6, . . . , 2q} such that, the induced map f + from V (G) to {0, 2, . . . , 2k −2} defined by f ∗(x) = Σ(f(xy) (mod 2k) where k = max(p, q) is injective and each value is f ∗(x) is even. They proved that dumbbells, stars, Cn × P2, and K1 + Cn are even-even graceful.
Behera, Mishra, and Nayak proved the following: bistars Br,r are even-even graceful, combs are even-even graceful, the trees obtained by joining and even number of pendent edges to the endpoint of a path are even-even graceful, the graphs obtained by identifying the center of a star and a vertex of C3 are odd-even graceful, the graphs the electronic journal of combinatorics (2023), #DS6 88 obtained by identifying the center of a star and a vertex of C3 and two pendent edges at the other two vertices are odd-even graceful, and the graphs obtained by identifying the center of a star with a vertex of Cn and the endpoints of the star with the opposite vertices of Cn is odd-even graceful.
In Daoud introduced vertex odd graceful labelings as follows. Let G be a graph with q edges. A function f is called a vertex odd graceful labeling of G if f E(G) → {1, 2, 3, . . . , 2q} is an injection and the induced function f ∗V (G) →{1, 3, . . . , 2q −1} defined as f ∗(u) = P uv∈E(G) f(uv) (mod 2q) is also an injection. A graph that admits a vertex odd graceful labeling is called a vertex odd graceful graph. Necessary and sufficient conditions for prisms, tori, wheels, fans and books to be vertex odd graceful are given.
3.7 Cordial Labelings Cahit has introduced a variation of both graceful and harmonious labelings. Let f be a function from the vertices of G to {0, 1} and for each edge xy assign the label |f(x) −f(y)|. Call f a cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1, and the number of edges labeled 0 and the number of edges labeled 1 differ at most by 1. Cahit proved the following: every tree is cordial; Kn is cordial if and only if n ≤3; Km,n is cordial for all m and n; the friendship graph C(t) 3 (i.e., the one-point union of t 3-cycles) is cordial if and only if t ̸≡2 (mod 4); all fans are cordial; the wheel Wn is cordial if and only if n ̸≡3 (mod 4) (see also ); maximal outerplanar graphs are cordial; and an Eulerian graph is not cordial if its size is congruent to 2 (mod 4). Kuo, Chang, and Kwong determine all m and n for which mKn is cordial. Youssef proved that every Skolem-graceful graph (see 3.5 for the definition) is cordial. Liu and Zhu proved that a 3-regular graph of order n is cordial if and only if n ̸≡4 (mod 8). In Imran, Cancan, Ali, Nadeem, Mushtaq, Aslam, Riaz, proved various comb related graphs are cordial. Kastrati, Myrvold, Panjer, and Williams proved that a forest is cordial if and only if it does not have 4k + 2 components and every vertex has odd-degree.
A k-angular cactus is a connected graph all of whose blocks are cycles with k vertices.
In Cahit proved that a k-angular cactus with t cycles is cordial if and only if kt ̸≡2 (mod 4). This was improved by Kirchherr who showed any cactus whose blocks are cycles is cordial if and only if the size of the graph is not congruent to 2 (mod 4).
Kirchherr also gave a characterization of cordial graphs in terms of their adjacency matrices. Ho, Lee, and Shee proved: Pn × C4m is cordial for all m and all odd n; the composition G and H is cordial if G is cordial and H is cordial and has odd order and even size (see §2.3 for definition of composition); for n ≥4 the composition Cn[K2] is cordial if and only if n ̸≡2 (mod 4); the Cartesian product of two cordial graphs of even size is cordial. Ho, Lee, and Shee showed that a unicyclic graph is cordial unless it is C4k+2 and that the generalized Petersen graph (see §2.7 for the definition) P(n, k) is cordial if and only if n ̸≡2 (mod 4). Khan proved that a graph that consisting of a finite number of cycles of finite length joined at a common cut vertex is cordial if and only if the number of edges is not congruent to 2 mod 4.
the electronic journal of combinatorics (2023), #DS6 89 Du determines the maximal number of edges in a cordial graph of order n and gives a necessary condition for a k-regular graph to be cordial. Riskin proved that Möbius ladders Mn (see §2.3 for the definition) are cordial if and only if n ≥3 and n ̸≡2 (mod 4). (See also .) Diab and Nada show that Pn ⊙Pm is cordial; except for n and m both equal to 2 (mod 4), Cn ⊙Cm is cordial; and when n ≡2 (mod 4) and m is odd, Cn ⊙Cm is not cordial. Nada, Elrokh, Elrayes, and Rabie showed that new Pm ⊙P 4 3 is cordial for all positive m when n ≥7 and n = 3. In Salehi, Mukhin, and Saputro showed that Qn is cordial for all n > 1.
Seoud and Abdel Maqusoud proved that if G is a graph with n vertices and m edges and every vertex has odd degree, then G is not cordial when m + n ≡2 (mod 4). They also prove the following: for m ≥2, Cn × Pm is cordial except for the case C4k+2×P2; P 2 n is cordial for all n; P 3 n is cordial if and only if n ̸= 4; and P 4 n is cordial if and only if n ̸= 4, 5, or 6. Seoud, Diab, and Elsakhawi have proved the following graphs are cordial: Pn +Pm for all m and n except (m, n) = (2, 2); Cm +Cn if m ̸≡0 (mod 4) and n ̸= 2 (mod 4); Cn + K1,m for n ̸≡3 (mod 4) and odd m except (n, m) = (3, 1); Cn + Km when n is odd, and when n is even and m is odd; K1,m,n; K2,2,m; the n-cube; books Bn if and only if n ̸≡3 (mod 4); B(3, 2, m) for all m; B(4, 3, m) if and only if m is even; and B(5, 3, m) if and only if m ̸≡1 (mod 4) (see §2.4 for the notation B(n, r, m)). In Solairaju and Arockiasamy prove that various families of subgraphs of grids Pm × Pn are cordial.
Diab , , and proved the following graphs are cordial: Cm + Pn if and only if (m, n) ̸= (3, 3), (3, 2), or (3,1); Pm + K1,n if and only if (m, n) ̸= (1, 2); Pm ∪K1,n if and only if (m, n) ̸= (1, 2); Cm ∪K1,n; Cm + Kn for all m and n except m ≡3 (mod 4) and n odd, and m ≡2 (mod 4) and n even; Cm ∪Kn for all m and n except m ≡2 (mod 4); Pm + Kn; Pm ∪Kn; P 2 m ∪P 2 n except for (m, n) = (2, 2) or (3,3); P 2 n + Pm except for (m, n) = (3, 1), (3, 2), (2, 2), (3, 3), and (4,2); P 2 n ∪Pm except for (n, m) = (2, 2), (3, 3), and (4,2); P 2 n + Cm if and only if (n, m) ̸= (1, 3), (2, 3), and (3, 3).Pn + Km; Cn + K1,m for all n > 3 and all m except n ≡3 (mod 4); Cn + K1,m for n ≡3 (mod 4) (n ̸= 3) and even m ≥2; and Cm × Cn if and only if 2mn is not congruent to 2 (mod 4).
In Diab proved the graphs Wn+Wm are cordial if and only if one of the following conditions is not satisfied: (i) (n, m) = (3, 3), (ii) n = 3 and m ≡1 (mod 4), (iii) n ≡1 (mod 4) and m ≡3 (mod 4); the graphs Wn ∪Wm are cordial if and only if one of the following conditions is not satisfied: (i) n = 3 and m ≡1 (mod 4), (ii) n ≡1 (mod 4) and m ≡3 (mod 4); the graphs Wn + Pm are cordial if and only if one of the following conditions is not satisfied: (i) (n, m) = (3, 1), (3, 2) and (3, 3), (ii) n ≡3 (mod 4) and m = 1. They also prove that Wn ∪Pm and Wn ∪Cm are cordial for all m and n and Wn + Cm is cordial if and only if (m, n) ̸= (3, 3) and (3, 4). In Diab showed that the second power of Cn is cordial if and only if n = 3 or n is even and greater than 4. He also investigated the cordiality of the join and union of pairs of second power of cycles and graphs consisting of one second power of cycle with one cycle and one path.
In Nada, Diab, Elrokh, and Sabra proved that Pn ⊙Cm is cordial if and only if gcd(n, m) ̸= 1 or 3 (mod 4); in they proved Cn ⊙Pm is cordial for all n ≥3 and m ≥1. Nada, Elrokh, and Elshafey provided necessary and sufficient conditions the electronic journal of combinatorics (2023), #DS6 90 for F 2 n = K1 + P 2 n, F 2 n + F 2 m, and F 2 n + F 2 m to be cordial.
The generalized Jahangir graph Jm,n m > 3, n > 1 is a graph on mn + 1 vertices, consisting of a cycle Cmn with one additional vertex that is adjacent to n vertices of Cmn at distance m to each other on Cmn. Gajjar and Des proved Jm,n is cordial for all m > 3 and n > 1, except for J1,4n−1.
Youssef has proved the following: If G and H are cordial and one has even size, then G ∪H is cordial; if G and H are cordial and both have even size, then G + H is cordial; if G and H are cordial and one has even size and either one has even order, then G + H is cordial; Cm ∪Cn is cordial if and only if m + n ̸≡2 (mod 4); mCn is cordial if and only if mn ̸≡2 (mod 4); Cm + Cn is cordial if and only if (m, n) ̸= (3, 3) and {m (mod 4), n (mod 4)} ̸= {0, 2}; and if P k n is cordial, then n ≥k + 1 + √ k −2.
He conjectures that this latter condition is also sufficient. He confirms the conjecture for k = 5, 6, 7, 8, and 9. Elirokh and Rabie proved P 4 n + P 4 m and P 4 n ∪P 4 m are cordial for all n, m ≥7, and C4 n + C4 m, and C4 n ∪C4 m are cordial for all n, m except (n, m) = (7, 7).
Lee and Liu have shown that the complete n-partite graph is cordial if and only if at most three of its partite sets have odd cardinality (see also ). Lee, Lee, and Chang prove the following graphs are cordial: the Cartesian product of an arbitrary number of paths; the Cartesian product of two cycles if and only if at least one of them is even; and the Cartesian product of an arbitrary number of cycles if at least one of them has length a multiple of 4 or at least two of them are even. Ali Al-Shamiri, Elrokh, El-Mashtawye, and Tallah showed that the Cartesian product of a path and a cycle is cordial under some conditions and that the Cartesian product of two paths is cordial.
Elrokh, Elmshtaye, and Abd El-hay provided necessary and sufficient conditions for cone and lemniscate graphs to be cordial.
Shee and Ho have investigated the cordiality of the one-point union of n copies of various graphs. For C(n) m , the one-point union of n copies of Cm, they prove: (i) If m ≡0 (mod 4), then C(n) m is cordial for all n; (ii) If m ≡1 or 3 (mod 4), then C(n) m is cordial if and only if n ̸≡2 (mod 4); (iii) If m ≡2 (mod 4), then C(n) m is cordial if and only if n is even.
For K(n) m , the one-point union of n copies of Km, Shee and Ho prove: (i) If m ≡0 (mod 8), then K(n) m is not cordial for n ≡3 (mod 4); (ii) If m ≡4 (mod 8), then K(n) m is not cordial for n ≡1 (mod 4); (iii) If m ≡5 (mod 8), then K(n) m is not cordial for all odd n; (iv) K(n) 4 is cordial if and only if n ̸≡1 (mod 4); (v) K(n) 5 is cordial if and only if n is even; (vi) K(n) 6 is cordial if and only if n > 2; (vii) K(n) 7 is cordial if and only if n ̸≡2 (mod 4); (viii) K(2) n is cordial if and only if n has the form p2 or p2 + 1.
For W (n) m , the one-point union of n copies of the wheel Wm with the common vertex being the center, Shee and Ho show: (i) If m ≡0 or 2 (mod 4), then W (n) m is cordial for all n; the electronic journal of combinatorics (2023), #DS6 91 (ii) If m ≡3 (mod 4), then W (n) m is cordial if n ̸≡1 (mod 4); (iii) If m ≡1 (mod 4), then W (n) m is cordial if n ̸≡3 (mod 4). For all n and all m > 1 Shee and Ho prove F (n) m , the one-point union of n copies of the fan Fm = Pm + K1 with the common point of the fans being the center, is cordial (see also ). The flag Flm is obtained by joining one vertex of Cm to an extra vertex called the root. Shee and Ho show all Fl(n) m , the one-point union of n copies of Flm with the common point being the root, are cordial. In his 2001 Ph. D. thesis Selvaraju proves that the one-point union of any number of copies of a complete bipartite graph is cordial. Benson and Lee have investigated the regular windmill graphs K(n) m and determined precisely which ones are cordial for m < 14.
Diab and Mohammedm proved the following: the join of two fans Fn + Fm is cordial if and only if n, m ≥4; Fn ∪Fm is cordial if and only if (n, m) ̸= (1,1) or (2,2); Fn + Pm is cordial if and only if (n, m) ̸= (1,2), (2,1), (2,2), (2,3), or (3,2); Fn ∪Pm is cordial if and only if (n, m) ̸= (1,2); Fn + Cm is cordial if and only if (n, m) ̸= (1,3), (2,3) or (3,3); and Fn ∪Cm is cordial if and only if (n, m) ̸= (2, 3). Hefnawy, Elsid, and Euat Tallah gave necessary and sufficient conditions for a cordial labeling of the sum of the second power of the path P 2 n + K1,m and P 2 n ∪K1,m.
Andar, Boxwala, and Limaye , , and have proved the following graphs are cordial: helms; closed helms; generalized helms obtained by taking a web (see 2.2 for the definitions) and attaching pendent vertices to all the vertices of the outermost cycle in the case that the number cycles is even; flowers (graphs obtained by joining the vertices of degree one of a helm to the central vertex); sunflower graphs (that is, graphs obtained by taking a wheel with the central vertex v0 and the n-cycle v1, v2, . . . , vn and additional vertices w1, w2, . . . , wn where wi is joined by edges to vi, vi+1, where i + 1 is taken modulo n); multiple shells (see §2.2); and the one point unions of helms, closed helms, flowers, gears, and sunflower graphs, where in each case the central vertex is the common vertex.
In , , , , and Prajapati and Gajjar provided results about the existence of cordial labelings of graphs obtained from paths, cycles, flower graphs, sunflower graphs, flower snarks, lotus inside a circle graphs, helms, closed helms, armed helms (Wn ⊕P2), and webs by the duplication of vertices and edges. In Elrokh and Elkom proved that certain classes any four-leaved rose graphs (the one-point union of four cycles of the same length) are cordial.
Du proved that the disjoint union of n ≥2 wheels is cordial if and only if n is even or n is odd and the number of vertices of in each cycle is not 0 (mod 4) or n is odd and the number of vertices of in each cycle is not 3 (mod 4). Prajapati and Gajjar prove Wn is not cordial if n ̸≡4, 7 (mod 8) and Cn is not cordial if n ̸≡4, 7 (mod 8).
Let O be the family of all cordial graphs of odd order and odd size for which there is no cordial labeling g such that eg(0) −eg(1) = 1. Barrientos and Minion proved that if G is a cordial graph such that G ̸∈O, then the corona K1 ⊙G is cordial. They use this result to prove that H ⊙G is cordial when G and H are cordial and G has even order and even size or G ̸∈O. In addition, H ⊙G is cordial when G is a cordial graph of odd order and even size and H is any graph of order m and size n ∈{m −1, m, m + 1}.
If H is bipartite such that the difference of the cardinalities of its partite sets is at most the electronic journal of combinatorics (2023), #DS6 92 one, and G is a cordial graph of even order and odd size that admits a cordial labeling g such that eg(0) −eg(1) = 1, then the corona H ⊙G is cordial. Barrientos and Minion proved the cordiality of certain circulant graphs; they also proved that for every positive integer k, the k-splitting of a cordial graph of even size, results in a cordial graph. They provide sufficient conditions to prove that any super subdivision of a graph G is cordial.
They study the cordiality of the join of two cordial graphs, proving that G + H is cordial when G and H have even order and even size, or both have odd order and even size, or both graphs have odd order, odd size, and the dominating weight in both graphs is not 1, or G has even order, odd size, and the dominating weight on both graphs is not the same, or both G and H have odd order, but only one has odd size, and the dominating weight is 0. They also prove that when G is a cordial graph of odd order and even size, the one-point union of t copies of G is cordial.
In Barrientos and Minion provide necessary conditions for the cordiality of coro-nas of cordial graphs, prove the cordiality of a family of circulant graphs, prove that any splitting graph of a cordial graph of even order and even size is cordial, determine a con-dition that a graph must satisfy in order that any super subdivision of it is cordial, prove the cordiality of the joint of two cordial graphs, and determine when a one-point union of a cordial graph is cordial.
For positive integers m and n divisible by 4 Venkatesh constructs graphs ob-tained by appending a copy of Cn to each vertex of Cm by identifying one vertex of Cn with each vertex of Cm and iterating by appending a copy of Cn to each vertex of degree 2 in the previous step. He proves that the graphs obtained by successive iterations are cordial.
Elumalai and Sethurman proved: cycles with parallel cords are cordial and n-cycles with parallel Pk-chords (see §2.2 for the definition) are cordial for any odd positive integer k at least 3 and any n ̸≡2 (mod 4) of length at least 4. They call a graph H an even-multiple subdivision graph of a graph G if it is obtained from G by replacing every edge uv of G by a pair of paths of even length starting at u and ending at v. They prove that every even-multiple subdivision graph is cordial and that every graph is a subgraph of a cordial graph. In Wen proves that generalized wheels Cn + mK1 are cordial when m is even and n ̸≡2 (mod 4) and when m is odd and n ̸≡3 (mod 4). Kuppusamy and Guruswamy show that the subdivision graph of K2,n is graceful for n ≥1 and the subdivision graph of the shell graph C(n, n −3) is graceful for n ≥4.
Vaidya, Ghodasara, Srivastav, and Kaneria investigated graphs obtained by joining two identical graphs by a path. They prove: graphs obtained by joining two copies of the same cycle by a path are cordial ; graphs obtained by joining two copies of the same cycle that has two chords with a common vertex with opposite ends of the chords joining two consecutive vertices of the cycle by a path are cordial ; graphs obtained by joining two rim vertices of two copies of the same wheel by a path are cordial ; and graphs obtained by joining two copies of the same Petersen graph by a path are cordial . They also prove that graphs obtained by replacing one vertex of a star by a fixed wheel or by replacing each vertex of a star by a fixed Petersen graph are cordial .
In Vaidya, Ghodasara, Srivastav, and Kaneria investigated the electronic journal of combinatorics (2023), #DS6 93 graphs obtained by joining two identical cycles that have a chord are cordial and the graphs obtained by starting with copies G1, G2, . . . , Gn of a fixed cycle with a chord that forms a triangle with two consecutive edges of the cycle and joining each Gi to Gi+1 (i = 1, 2, . . . , n−1) by an edge that is incident with the endpoints of the chords in Gi and Gi+1 are cordial. Vaidya, Dani, Kanani, and Vihol proved that the graphs obtained by starting with copies G1, G2, . . . , Gn of a fixed star and joining each center of Gi to the center of Gi+1 (i = 1, 2, . . . , n −1) by an edge are cordial.
Ghodasara, Rokad, and Jadav prove that the path union of Pn × Pn is cordial.
They also prove that the graph obtained by joining two copies of Pn × Pn by a path is cordial. Ghodasara and Jadav prove: the graph obtained by joining a finite number of copies of Pn × Pn by path is cordial; the star of Pn × Pn is cordial; and the path union of the star of Pn × Pn is cordial. Rokad and Patadiya proved that the shadow graph, splitting graph, and the degree splitting graph of a star are cordial graphs. They also showed that the jewel graph and the jellyfish graph are cordial.
Ghodasara and Rokad prove the star of Kn,n (n ≥2) is cordial, the path union of Kn,n (n ≥2) is cordial, and the graph obtained by joining two copies of Kn,n (n ≥2) by a path is cordial . In the same authors prove that a vertex switching of any non-apex vertex of a wheel graph, a vertex switching of any internal vertex of a flower graph, a vertex switching of any non-apex vertex of a gear graph, and a vertex switching of any non-apex vertex of a shell graph are cordial graphs. In they proved that a barycentric subdivision of a shell graph, a barycentric subdivision of Kn,n, and a barycentric subdivision of a wheel are cordial. Ghodasara and Sonchhatra prove that the graph obtained by joining two copies of the same fan by a path is cordial. They also prove that the star of a fan is cordial and the graph obtained by joining two copies of the star of the same fan by a path is cordial . Elrokh, Nada, and El-Shafey showed that Pk ⊙F 2 m (Fm is the fan graph with m + 1 vertices) is cordial for all k ≥1 and m ≥4.
Vaidya, Kanani, Srivastav, and Ghodasara proved: graphs obtained by subdi-viding every edge of a cycle with exactly two extra edges that are chords with a common endpoint and whose other end points are joined by an edge of the cycle are cordial; graphs obtained by subdividing every edge of the graph obtained by starting with Cn and adding exactly three chords that result in two 3-cycles and a cycle of length n −3 are cordial; graphs obtained by subdividing every edge of a Petersen graph are cordial. Sankar and Sethuramam showed that the subdivision graph S(K2, n) is graceful and cordial and the shell graph S(C(n, n −3)) is graceful and cordial for n ≥4.
Recall the shell C(n, n −3) is the cycle Cn with n −3 cords sharing a common endpoint. Vaidya, Dani, Kanani, and Vihol proved that the graphs obtained by starting with copies G1, G2, . . . , Gn of a fixed shell and joining common endpoint of the chords of Gi to the common endpoint of the chords of Gi+1 (i = 1, 2, . . . , n−1) by an edge are cordial. Vaidya, Dani, Kanani, and Vihol define Cn(Cn) as the graph obtained by subdividing each edge of Cn and connecting the new n vertices to form a copy of Cn inscribed the original Cn. They prove that Cn(Cn) is cordial if n ̸= 2 (mod 4); the graphs obtained by starting with copies G1, G2, . . . , Gk of Cn(Cn) the graph obtained by joining a the electronic journal of combinatorics (2023), #DS6 94 vertex of degree 2 in Gi to a vertex of degree 2 in Gi+1 (i = 1, 2, . . . , n−1) by an edge are cordial; and the graphs obtained by joining vertex of degree 2 from one copy of Cn(Cn) to a vertex of degree 2 to another copy of Cn(Cn) by any finite path are cordial. Vaidya and Shah and proved that following graphs are cordial: the shadow graph of the bistar Bn,n, the splitting graph of Bn,n, the degree splitting graph of Bn,n, alternate triangular snakes, alternate quadrilateral snakes, double alternate triangular snakes, and double alternate quadrilateral snakes. In Vaidya and Shah give cordial labelings of the degree splitting graph of paths, shells, helms, and gears.
A graph C(2n, n −2) is called an alternate shell if C(2n, n −2) is obtained from the cycle C2n (v0, v1, v2, . . . , v2n−1) by adding n −2 chords between the vertex v0 and the vertices v2i+1, for 1 ≤i ≤n −2. Sethuraman and Sankar proved that some graphs obtained by merging alternate shells and joining certain vertices by a path have α-labelings.
Vaidya, Srivastav, Kaneria, and Ghodasara proved that a cycle with two chords that share a common vertex and the opposite ends of which join two consecutive vertices of the cycle is cordial. For a graph G Vaidya, Ghodasara, Srivastav, and Kaneria introduced the graph G∗called the star of G as the graph obtained by replacing each vertex of the star K1,n by a copy of G and prove that C ∗ n admits cordial labeling. Vaidya and Dani proved that the graphs obtained by starting with n copies G1, G2, . . . , Gn of a fixed star and joining each center of Gi to the center of Gi+1 by an edge as well as each of the centers to a new vertex xi (1 ≤i ≤n −1) by an edge admit cordial labelings. An arbitrary supersubdivision H of a graph G is the graph obtained from G by replacing every edge of G by K2,m, where m may vary for each edge arbitrarily. Vaidya and Kanani proved that arbitrary supersubdivisions of paths and stars admit cordial labelings. Vaidya and Dani prove that arbitrary supersubdivisions of trees, Km,n, and Pm × Pn are cordial. They also prove that an arbitrary supersubdivision of the graph obtained by identifying an end vertex of a path with every vertex of a cycle Cn is cordial except when n is odd, mi (1 ≤i ≤n) are odd, and mi (n + 1 ≤i ≤mn) of the K2,mi are even. Recall for a graph G and a vertex v of G Vaidya, Srivastav, Kaneria, and Kanani define a vertex switching Gv as the graph obtained from G by removing all edges incident to v and adding edges joining v to every vertex not adjacent to v in G. They proved that the graphs obtained by the switching of a vertex in Cn admit cordial labelings. They also show that the graphs obtained by the switching of any arbitrary vertex of cycle Cn with one chord that forms a triangle with two consecutive edges of the cycle are cordial.
Moreover they prove that the graphs obtained by the switching of any arbitrary vertex in cycle with two chords that share a common vertex the opposite ends of which join two consecutive vertices of the cycle are cordial.
The middle graph M(G) of a graph G is the graph whose vertex set is V (G) ∪E(G) and in which two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and the other is an edge incident with it. Vaidya and Vihol prove that the middle graph M(G) of an Eulerian graph is Eulerian with |E(M(G))| = Pn i=1(d(vi)2 +2e)/2. They prove that middle graphs of paths, crowns Cn ⊙K1, stars, and tadpoles (that is, graphs obtained by appending a path to a cycle) admit cordial labelings.
the electronic journal of combinatorics (2023), #DS6 95 Vaidya and Dani define the duplication of an edge e = uv of a graph G by a new vertex w as the graph G′ obtained from G by adding a new vertex w and the edges wv and wu. They prove that the graphs obtained by duplication of an arbitrary edge of a cycle and a wheel admit a cordial labeling. Starting with k copies of fixed wheel Wn, W (1) n , W (2) n , . . . , W (k) n , Vaidya, Dani, Kanani, and Vihol define G =< W (1) n : W (2) n : . . . : W (k) n > as the graph obtained by joining the center vertices of each of W (i) n and W (i+1) n to a new vertex xi where 1 ≤i ≤k−1. They prove that < W (1) n : W (2) n : ... : W (k) n > are cordial graphs. Kaneria and Vaidya define the index of cordiality of G as n if the disjoint union of n copies of G is cordial but the disjoint union of fewer than n copies of G is not cordial. They obtain several results on index of cordiality of Kn. In the same paper they investigate cordial labelings of graphs obtained by replacing each vertex of K1,n by a graph G. Kaneria, Jariya, and Karavadiya proved that the index of cordiality for Kn is at most 6 for n at most 105; the index of cordiality for Kn is at most 4, when n can be expressed as sum of square of two integers; and it is at most 8 when a particular different condition on the edge labels are met. See also .
In Andar et al. define a t-ply graph Pt(u, v) as a graph consisting of t internally disjoint paths joining vertices u and v. They prove that Pt(u, v) is cordial except when it is Eulerian and the number of edges is congruent to 2 (mod 4). In Andar, Boxwala, and Limaye prove that the one-point union of any number of plys with an endpoint as the common vertex is cordial if and only if it is not Eulerian and the number of edges is congruent to 2 (mod 4). They further prove that the path union of shells obtained by joining any point of one shell to any point of the next shell is cordial; graphs obtained by attaching a pendent edge to the common vertex of the cords of a shell are cordial; and cycles with one pendent edge are cordial.
For a graph G and a positive integer t, Andar, Boxwala, and Limaye define the t-uniform homeomorph Pt(G) of G as the graph obtained from G by replacing every edge of G by vertex disjoint paths of length t. They prove that if G is cordial and t is odd, then Pt(G) is cordial; if t ≡2 (mod 4) a cordial labeling of G can be extended to a cordial labeling of Pt(G) if and only if the number of edges labeled 0 in G is even; and when t ≡0 (mod 4) a cordial labeling of G can be extended to a cordial labeling of Pt(G) if and only if the number of edges labeled 1 in G is even. In Ander et al. prove that Pt(K2n) is cordial for all t ≥2 and that Pt(K2n+1) is cordial if and only if t ≡0 (mod 4) or t is odd and n ̸≡2 (mod 4), or t ≡2 (mod 4) and n is even.
In Andar, Boxwala, and Limaya show that a cordial labeling of G can be extended to a cordial labeling of the graph obtained from G by attaching 2m pendent edges at each vertex of G. For a binary labeling g of the vertices of a graph G and the induced edge labels given by g(e) = |g(u) −g(v)| let vg(j) denote the number of vertices labeled with j and eg(j) denote the number edges labeled with j. Let i(G) = min{|eg(0) −eg(1)|} taken over all binary labelings g of G with |vg(0) −vg(1)| ≤1. Andar et al. also prove that a cordial labeling g of a graph G with p vertices can be extended to a cordial labeling of the graph obtained from G by attaching 2m + 1 pendent edges at each vertex of G if and only if G does not satisfy either of the conditions: (1) G has an even number of edges and p ≡2 (mod 4); (2) G has an odd number of edges and either p ≡1 (mod 4) the electronic journal of combinatorics (2023), #DS6 96 with eg(1) = eg(0) + i(G) or n ≡3 (mod 4) and eg(0) = eg(1) + i(G). Andar, Boxwala, and Limaye also prove: if g is a binary labeling of the n vertices of graph G with induced edge labels given by g(e) = |g(u) −g(v)| then g can be extended to a cordial labeling of G ⊙K2m if and only if n is odd and i(G) ≡2 (mod 4); Kn ⊙K2m is cordial if and only if n ̸= 4 (mod 8); Kn ⊙K2m+1 is cordial if and only if n ̸= 7 (mod 8); if g is a binary labeling of the n vertices of graph G with induced edge labels given by g(e) = |g(u) −g(v)| then g can be extended to a cordial labeling of G ⊙Ct if t ̸= 3 mod 4, n is odd and eg(0) = eg(1). For any binary labeling g of a graph G with induced edge labels given by g(e) = |g(u) −g(v)| they also characterize in terms of i(G) when g can be extended to graphs of the form G ⊙K2m+1.
For graphs G1, G2, . . . , Gn (n ≥2) that are all copies of a fixed graph G, Shee and Ho call a graph obtained by adding an edge from Gi to Gi+1 for i = 1, . . . , n −1 a path union of G (the resulting graph may depend on how the edges are chosen). Among their results they show the following graphs are cordial: path-unions of cycles; path-unions of any number of copies of Km when m = 4, 6, or 7; path-unions of three or more copies of K5; and path-unions of two copies of Km if and only if m −2, m, or m + 2 is a perfect square. They also show that there exist cordial path-unions of wheels, fans, unicyclic graphs, Petersen graphs, trees, and various compositions.
Lee and Liu give the following general construction for the forming of cordial graphs from smaller cordial graphs. Let H be a graph with an even number of edges and a cordial labeling such that the vertices of H can be divided into t parts H1, H2, . . . , Ht each consisting of an equal number of vertices labeled 0 and vertices labeled 1.
Let G be any graph and G1, G2, . . . , Gt be any t subsets of the vertices of G. Let (G, H) be the graph that is the disjoint union of G and H augmented by edges joining every vertex in Gi to every vertex in Hi for all i.
Then G is cordial if and only if (G, H) is.
From this it follows that: all generalized fans Fm,n = Km + Pn are cordial; the generalized bundle Bm,n is cordial if and only if m is even or n ̸≡2 (mod 4) (Bm,n consists of 2n vertices v1, v2, . . . , vn, u1, u2, . . . , un with an edge from vi to ui and 2m vertices x1, x2, . . . , xm, y1, y2, . . . , ym with xi joined to vi and yi joined to ui); if m is odd the generalized wheel Wm,n = Km + Cn is cordial if and only if n ̸≡3 (mod 4). If m is even, Wm,n is cordial if and only if n ̸≡2 (mod 4); a complete k-partite graph is cordial if and only if the number of parts with an odd number of vertices is at most 3.
Sethuraman and Selvaraju have shown that certain cases of the union of any number of copies of K4 with one or more edges deleted and one edge in common are cordial. Youssef has shown that the kth power of Cn is cordial for all n when k ≡2 (mod 4) and for all even n when k ≡0 (mod 4). Ramanjaneyulu, Venkaiah, and Kothapalli give cordial labelings for a family of planar graphs for which each face is a 3-cycle and a family for which each face is a 4-cycle. Acharya, Germina, Princy, and Rao prove that every graph G can be embedded in a cordial graph H. The construction is done in such a way that if G is planar or connected, then so is H.
Recall from §2.7 that a graph H is a supersubdivision of a graph G, if every edge uv of G is replaced by K2,m (m may vary for each edge) by identifying u and v with the two vertices in K2,m that form the partite set with exactly two members. Vaidya and the electronic journal of combinatorics (2023), #DS6 97 Kanani prove that supersubdivisions of paths and stars are cordial. They also prove that supersubdivisions of Cn are cordial provided that n and the various values for m are odd.
Raj and Koilraj proved that the splitting graphs of Pn, Cn, Km,n, Wn, nK2, and the graphs obtained by starting with k copies of stars K(1) 1,n, K(2) 1,n, . . . , K(k) 1,n and joining the central vertex of K(p−1) 1,n and K(p) 1,n to a new vertex xp−1 for each 2 ≤p ≤k are cordial.
Seoud, El Sonbaty, and Abd El Rehim proved the following graphs are cordial: K1,l,m,n when mn is even; Pm + K1,n if n is even or n is odd and (m ̸= 2); the conjunction graph P4 ∧Cn is cordial if n is even; and the join of the one-point union of two copies of Cn and K1.
Recall < K1,n1, . . . , K1,nt > is the graph obtained by starting with the stars K1,n1, . . . , K1,nt and joining the center vertices of K1,ni and K1,ni+1 to a new vertex vi where 1 ≤i ≤k −1. Kaneria, Jariya, and Meghpara proved that < K1,n1, . . . , K1,nt > is cordial and every graceful graph with |vf(odd) −vf(even)| ≤1 is cordial. Kaneria, Megh-para, and Makadia proved that the cycle of complete graphs C(t · Km,n) and the cycle of wheels C(t·Wn) are cordial. Kaneria, Makadia, and Meghpara proved that the cycle of cycles C(t · Cn) is cordial for t ≥3. Kaneria, Makadia, and Meghpara proved that a star of Kn and a cycle of n copies of Kn are cordial. Kaneria, Viradia, Jariya, and Makadia proved that the cycle of paths C(t · Pn) is cordial, product cordial (see Section 7.5), and total edge product cordial.
Jeba Jesintha and Subashini proved the following graphs are cordial: the cycle of vertex switching of cycles ; the path union of vertex switching of wheels in increasing order ; the path union of jelly fish graphs is cordial and cycle of jelly fish graphs ; the star of fixed trees of diameter three tree ; and the path union of vertex switching of cycles in increasing order . In Jeba Jesintha, Vinodhini, and Lakshmi, proved that a star glued with subdivided shell graph and super subdivision of circular ladders (Cn × K2) admit cordial labelings. In Jeba Jesintha and Devakirubanithi call a new planar graph with n = 3i vertices, for i = 1, 2, 3, . . . that form a sequence of n/3 triangles where pairs of corresponding vertices on consecutive triangles are joined in the sequence nested triangles. They proved nested triangles, the shadow graphs of the nested triangles, and the double graph of the nested triangle graphs admit cordial labelings.
Cordial labelings and variations of them for fractal graphs are given in and .
Cahit calls a graph H-cordial if it is possible to label the edges with the numbers from the set {1, −1} in such a way that, for some k, at each vertex v the sum of the labels on the edges incident with v is either k or −k and the inequalities |v(k) −v(−k)| ≤1 and |e(1) −e(−1)| ≤1 are also satisfied, where v(i) and e(j) are, respectively, the number of vertices labeled with i and the number of edges labeled with j. He calls a graph Hn-cordial if it is possible to label the edges with the numbers from the set {±1, ±2, . . . , ±n} in such a way that, at each vertex v the sum of the labels on the edges incident with v is in the set {±1, ±2, . . . , ±n} and the inequalities |v(i) −v(−i)| ≤1 and |e(i) −e(−i)| ≤1 are also satisfied for each i with 1 ≤i ≤n. Among Cahit’s results are: Kn,n is H-cordial if and only if n > 2 and n is even; and Km,n, m ̸= n, is H-cordial if and only if n ≡0 the electronic journal of combinatorics (2023), #DS6 98 (mod 4), m is even and m > 2, n > 2. Unfortunately, Ghebleh and Khoeilar have shown that other statements in Cahit’s paper are incorrect. In particular, Cahit states that Kn is H-cordial if and only if n ≡0 (mod 4); Wn is H-cordial if and only if n ≡1 (mod 4); and Kn is H2-cordial if and only if n ≡0 (mod 4) whereas Ghebleh and Khoeilar instead prove that Kn is H-cordial if and only if n ≡0 or 3 (mod 4) and n ̸= 3; Wn is H-cordial if and only if n is odd; Kn is H2-cordial if n ≡0 or 3 (mod 4); and Kn is not H2-cordial if n ≡1 (mod 4). Ghebleh and Khoeilar also prove every wheel has an H2-cordial labeling. In Freeda and Chellathurai prove that the following graphs are H2-cordial: the join of two paths, the join of two cycles, ladders, and the tensor product Pn ⊗P2. They also prove that the join of Wn and Wm where n + m ≡0 (mod 4) is H-cordial. Cahit generalizes the notion of H-cordial labelings in .
A graph G(V, E) is called Hk−cordial if it has an H-cordial labeling f such that for each edge e and each vertex v of G have the label 1 ≤|f(e)| ≤k, ≤|f(v)| ≤k and |vf(i)vf(−i))| ≤1, |ef(i))ef(−i)| ≤1 for each i with ≤i ≤k. Ratilal and Parmar investigated Hk-cordial labelings of triangular snakes, double triangular snakes, triple tri-angular snakes, alternate triangular snakes, double alternate triangular snakes, irregular triangular snakes, quadrilateral snakes, double quadrilateral snakes, alternate quadrilat-eral snakes, and irregular quadrilateral snakes. Joshi and Parmar investigated the H-, H2- and H3-cordiality of the following snakes: triangular, double triangular, triple triangular, quadrilateral, double quadrilateral, alternate triangular, double alternate tri-angular, irregular triangular, quadrilateral, double quadrilateral, alternate quadrilateral, and irregular quadrilateral. Joshi and Pamar investigated Hk-cordial labeling of p-triangular, m-polygonal snakes, double m-polygonal snakes, alternate m-polygonal snakes, double alternate m-polygonal snakes, irregular m-polygonal snakes, and double irregular m-polygonal snakes.
Cahit and Yilmaz call a graph Ek-cordial if it is possible to label the edges with the numbers from the set {0, 1, 2, . . . , k −1} in such a way that, at each vertex v, the sum of the labels on the edges incident with v modulo k satisfies the inequalities |v(i)−v(j)| ≤1 and |e(i)−e(j)| ≤1, where v(s) and e(t) are, respectively, the number of vertices labeled with s and the number of edges labeled with t. Cahit and Yilmaz prove the following graphs are E3-cordial: Pn (n ≥3); stars Sn if and only if n ̸≡1 (mod 3); Kn (n ≥3); Cn (n ≥3); friendship graphs; and fans Fn (n ≥3). They also prove that Sn (n ≥2) is Ek-cordial if and only if n ̸≡1 (mod k) when k is odd or n ̸≡1 (mod 2k) when k is even and k ̸= 2. Ni, Liu, and Lu demonstrate the E3-cordiality of Wn, Pm × Pn, Km,n, and trees.
Bapat and Limaye provide E3-cordial labelings for: Kn (n ≥3); snakes whose blocks are all isomorphic to Kn where n ≡0 or 2 (mod 3); the one-point union of any number of copies of Kn where n ≡0 or 2 (mod 3); graphs obtained by attaching a copy of Kn where n ≡0 or 3 (mod 3) at each vertex of a path; and Km ⊙Kn. Sridharan and Umarani proved: for odd n > 1 and k ≥2, Pn ⊙K1 is Ek-cordial; for n even and n ̸= k/2, Pn ⊙K1 is Ek-cordial; and certain cases of fans are Ek-cordial. Youssef gives a necessary condition for a graph to be Ek-cordial for certain k.
He also gives some new families of Ek-cordial graphs and proves Lee’s conjecture about the edge-the electronic journal of combinatorics (2023), #DS6 99 gracefulness of the disjoint union of two cycles. Venkatesh, Salah, and Sethuraman proved that C2n+1 snakes and C2t 2n+1 are E2-cordial. Liu, Liu, and Wu provide two necessary conditions for a graph G to be Ek-cordial and prove that every Pn (n ≥3) is Ep-cordial if p is odd. They also discuss the E2-cordiality of a graph G under the condition that some subgraph of G has a 1-factor. Liu and Liu proved that a graph with no isolated vertex is E2-cordial if and only if it does not have order 4n + 2. Bapat and Limaye prove that helms, one point unions of helms, and path unions of helms are E3-cordial. Jinnah and Beena prove the graphs Pn (n ≥3), Cn where n ̸= 4 mod 8, and Kn (n ≥3) are E4-cordial graphs. They also prove that every graph of order at least 3 is a subgraph of an E4-cordial graph.
Hovey introduced a simultaneous generalization of harmonious and cordial la-belings. For any Abelian group A (under addition) and graph G(V, E) he defines G to be A-cordial if there is a labeling of V with elements of A such that for all a and b in A when the edge ab is labeled with f(a) + f(b), the number of vertices labeled with a and the number of vertices labeled b differ by at most one and the number of edges labeled with a and the number labeled with b differ by at most one. In the case where A is the cyclic group of order k, the labeling is called k-cordial. With this definition we have: if G(V, E) is a graph with |E| ≥|V | −1 then G(V, E) is harmonious if and only if G is |E|-cordial; G is cordial if and only if G is 2-cordial.
Hovey obtained the following: caterpillars are k-cordial for all k; all trees are k-cordial for k = 3, 4, and 5; odd cycles with pendent edges attached are k-cordial for all k; cycles are k-cordial for all odd k; for k even, C2mk+j is k-cordial when 0 ≤j ≤k 2 + 2 and when k < j < 2k; C(2m+1)k is not k-cordial; Km is 3-cordial; and, for k even, Kmk is k-cordial if and only if m = 1.
Hovey advances the following conjectures: all trees are k-cordial for all k; all connected graphs are 3-cordial; and C2mk+j is k-cordial if and only if j ̸= k, where k and j are even and 0 ≤j < 2k. The last conjecture was verified by Tao . Tao’s result combined with those of Hovey show that for all positive integers k the n-cycle is k-cordial with the exception that k is even and n = 2mk + k. Tao also proved that the crown with 2mk + j vertices is k-cordial unless j = k is even, and for 4 ≤n ≤k the wheel Wn is k-cordial unless k ≡5 (mod 8) and n = (k + 1)/2. In Tuczyński, Wenus, and Wesek proved a conjecture of Cichacz, Görlich, and Tuza that all hypertrees are 2-cordial. They also proved that all hypertree are 3-cordial.
In Patrias and Pechenik initiated the study of classes of finite Abelian groups A for which particular graphs are A-cordial. Their results include: P2m and P2m+1 are not Z m 2 -cordial, all paths are A-cordial when A is an Abelian group of odd order, if A is an Abelian group of order n and Pn is A-cordial, then all paths are A-cordial, and P2n is A-cordial when A = Z2 × Zk and n = |A|.
They conjecture that for a finite Abelian group A, all paths are A-cordial if and only if A has an element of order greater than 2. Cichacz proved that all cycle graphs are A-cordial for any Abelian group A of odd order. In and Chidambaram, Athisayanathan, and Ponraj proved that hypercubes, books, Cn × K2, and Pn × K3 and the splitting graphs of paths, cycles, and wheels are {1, −1, i, −i}-cordial.
Since the electronic journal of combinatorics (2023), #DS6 100 the group {1, −1, i, −i} is cyclic, this is same as 4-cordial. In Radha, Venkatesan, Vitaldas, and Perumal prove that triangular ladders, alternate triangular snakes, alternate quadrilateral snakes, and double triangular snakes admit {1, −1, i, −i} cordial labelings.
Erickson et al. showed that the friendship graph Fn is Z3m-cordial and conjectured that Fn is Zm-cordial except when n is even and not divisible by 4 and m = 3n/d, where d is odd.
In Youssef and Al-Kuleab proved the following: if G is a (p1, q1) k-cordial graph and G is a (p2, q2) k-cordial graph with p1 or p2 ≡0 (mod k) and q1 or q2 ≡0 (mod k), then G + H is k-cordial; if G is a (p1, q1) 4-cordial graph and G is a (p2, q2) 4-cordial graph with p1 or p2 ̸≡2 (mod 4) and q1 or q2 ≡0 (mod k), then G+H is 4-cordial; and Km,n,p is 4-cordial if and only if (m, n, p) mod 4 ̸≡(0, 2, 2) or (2, 2, 2).
In ELrokh, Ismail, El-hay, and Elmshtaye define a cubic roots cordial labeling f of the vertices of a graph G with 1, ω, and ω2, where ω3 = 1, with induced edge labeling f ∗: E(G) to {1, ω, ω2} defined by f ∗(uv) = f(u)f(v) if both the number of vertices and the number edges labeled with x and the number of vertices and the number edges labeled with y differ by at most 1. Since {1, ω, ω2} is isomorphic the group Z3, cubic roots cordial is the same as 3-cordial. They prove the all nontrivial cases of paths, cycles, fans, and G ∪H where G and H paths or cycles admit a cubic roots cordial. They also prove that wheels Wn are cubic roots cordial except when n = 2 mod 3 and n is even.
In Youssef obtained the following results: C2k with one pendent edge is not (2k + 1)-cordial for k > 1; Kn is 4-cordial if and only if n ≤6; C2 n is 4-cordial if and only if n ̸≡2 (mod 4); and Km,n is 4-cordial if and only if n ̸≡2 (mod 4); He also provides some necessary conditions for a graph to be k-cordial. Driscol proved that all trees are 7-cordial.
Modha and Kanani prove that following graphs have a 5-cordial labeling: the shadow graph of a path and a cycle, graphs obtained by one point duplication and duplica-tion of an edge by a vertex in cycle, and the graph obtained by the barycentric subdivision of wheel. In Modha and Kanani proved prisms, webs, flowers, and closed helms admit 5-cordial labelings. In they proved that fans are k-cordial for all k and dou-ble fans are k-cordial for all odd k and n = (k + 1)/2. In they proved that the following graphs are k-cordial: Wn for odd k, n = mk + j, m ≥0, 1 ≤j ≤k −1 except for j = (k −1)/2; the total graphs of paths (recall T(Pn) has vertex set V (Pn) ∪E(Pn) with two vertices adjacent whenever they are neighbors in Pn); the square C2 n for odd k ≤n; the path union of n copies of Ck where k is odd; and Cn with one pendent edge for odd k ≤n. Rathod and Kanani proved P 2 n is k-cordial for all k and cycles with a single pendent edge are k-cordial for all even k. In Rathod and Kanani proved the middle graph, total graph, and splitting graph of a path are 4-cordial and P 2 n and triangular snakes are 4-cordial. Modha and Kanani proved: Wn is k-cordial for all odd k and for all n = mk + j, m ≥0, 1 ≤j ≤k −1 except for j = k −1; the path union of copies of Ck is k-cordial for odd k; the total graph of Pn is k-cordial for all k; the square C2 n is k-cordial for odd k odd and n ≥k; and the graphs obtained by appending an edge to Cn is k-cordial for odd k and n ≥k. Modha and Kanani prove the following graphs are k-cordial: Pm × Ck, Pm × Ck+1, Pm × Ck+3 for all odd k and m ≥2, the electronic journal of combinatorics (2023), #DS6 101 and Pm × C2k−1 for all odd k, m ≥2 and m ̸= tk. Rathod and Kanani prove that following graphs are 4-cordial: the splitting graph of K1,n; triangular books; and the one point union any number of copies of the fan f3; braid graphs; triangular ladders; and irregular quadrilateral snakes obtained from the path Pn with consecutive vertices u1, u2, . . . , un and new vertices v1, v2, . . . , vn−2, w1, w2, and edges uivi, wiui+2, viwi for all 1 ≤i ≤n −2. Rathod and Kanani prove wheels, fans, friendship graphs, double fans, and helms are 5-cordial. Driscoll, Krop, and Nguyen proved that all trees are 6-cordial. In , , and Kanani and Modha prove that fans, friendship graphs, ladders, double fans, double wheels, wheels, helms, closed helms, and webs are 7-cordial graphs and wheels, fans and friendship graphs, gears, double fans, and helms are 4-cordial graphs. In Sathish Narayanan and Vijayaragavan obtained 3-divisor cordial labelings for graphs derived from paths.
Cichacz, Görlich and Tuza extended the definition of k-cordial labeling for hy-pergraphs. They presented various sufficient conditions on a hypertree H (a connected hypergraph without cycles) to be k-cordial. From their theorems it follows that every k-uniform hypertree is k-cordial, and every hypertree with odd order or size is 2-cordial.
Modha and Kanani prove the following graphs are k-cordial for all k: bistars, restricted square graphs B2 n,n, the one-point union of C3 and K1,n, and Pn ⊙K1.
In Sethuraman and Selvaraju present an algorithm that permits one to start with any non-trivial connected graph G and successively form supersubdivisions (see §2.7 for the definition) that are cordial in the case that every edge in G is replaced by K2,m where m is even. Sethuraman and Selvaraju also show that the one-vertex union of any number of copies of Km,n is cordial and that the one-edge union of k copies of shell graphs C(n, n −3) (see §2.2) is cordial for all n ≥4 and all k. They conjectured that the one-point union of any number of copies of graphs of the form C(ni, ni −3) for various ni ≥4 is cordial. This was proved by Yue, Yuansheng, and Liping in .
Riskin claimed that Kn is (Z2 × Z2)-cordial if and only if n is at most 3 and Km,n is (Z2 × Z2)-cordial if and only if (m, n) ̸= (2, 2). (Many authors use V4 to denote Z2 × Z2.) However, Pechenik and Wise report that the correct statement for Km,n is Km,n is (Z2 × Z2)-cordial if and only if m and n are not both congruent to 2 mod 4. Seoud and Salim gave an upper bound on the number of edges of a graph that admits a (Z2 ⊕Z2)-cordial labeling in terms the number of vertices. Rathod and Kanani prove the following graphs are (Z2 × Z2)-cordial for all n and m: Cn ⊙mK1, Cn ⊙K2, and graphs obtained by appending a single edge to one vertex of Cn.
In Rathod and Kanani and proved the following graphs are (Z2×Z2)-cordial: alternate triangular snakes, alternate double triangular snakes, alternate triple triangular snakes, quadrilateral snakes, alternate quadrilateral snakes, double quadrilateral snakes, and double alternate quadrilateral snakes.
In Pechenik and Wise investigate Z2×Z2-cordiality of complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. They proved that all complete bipartite graphs are Z2×Z2-cordial except Km,n where m, n ≡2 mod 4; all paths are Z2×Z2-cordial except P4 and P5; all cycles are Z2 ×Z2-cordial except C4, C5, Ck, where k ≡2 mod 4; and all ladders P2 × Pk are Z2 × Z2-cordial except C4. They also introduce a generalization of the electronic journal of combinatorics (2023), #DS6 102 A-cordiality involving digraphs and quasigroups, and show that there are infinitely many Q-cordial digraphs for every quasigroup Q. Jinnah and Nair proved that all trees except P4 and P5 are Z2 ×Z2-cordial and the graphs obtained by subdividing the pendent edges of Cn ⊙K1 are Z2 × Z2 -cordial for all n.
Cairnie and Edwards have determined the computational complexity of cordial and k-cordial labelings.
They prove the conjecture of Kirchherr that deciding whether a graph admits a cordial labeling is NP-complete. As a corollary, this result implies that the same problem for k-cordial labelings is NP-complete. They remark that even the restricted problem of deciding whether connected graphs of diameter 2 have a cordial labeling is also NP-complete.
For a (p, q) graph G and a bijection f from V (G) to {1, 2, . . . , p} Ponraj, Annathurai, and Kala introduced a new graph labeling as follows. For each edge uv assign the remainder when f(u) is divided by f(v) or when f(v) is divided by f(u) depending on whether f(u) ≥f(v) or f(v) ≥f(u). The function f is called a remainder cordial labeling of G if |ηe −ηo| ≤1 where ηe and ηo respectively denote the number of edges labeled with even integers and the number of edges labeled with odd integers. A graph G with a remainder cordial labeling is called a remainder cordial graph. In and they proved that the following graphs are remainder cordial: paths, cycles, stars, bistars, crowns, combs, K2,n, S(K1,n), S(Bn,n), P 2 n, wheels, subdivisions of wheels, K2,2n, and the graph obtained by subdividing the pendent edges of the bistar Bn,n. They also proved the following star related graphs are remainder cordial: K1,n ∪Bn,n, Pn ∪K1,n, Pn ∪Bn,n, K1,n ∪S(K1,n), K1,n ∪S(Bn,n), P 2 n ∪K1,n, Pn 2 ∪Bn,n, and S(K1,n) ∪S(Bn,n).
They conjecture that Kn is remainder cordial if and only if n ≤3. Ponraj, Annathurai, and Kala generalize remainder cordial labelings as follows. Let f be a function from V (G) to {1, 2, . . . , k} where 2 < k ≤|V (G)|. For each edge uv assign the remainder when f(u) is divided by f(v) or when f(v) is divided by f(u) depending on whether f(u) ≥f(v) or f(v) ≥f(u). The function f is called a k-remainder cordial labeling of G if |vf(i) −vf(j)| ≤1, for i, j ∈{1, . . . , k} where vf(x) denote the number of vertices labeled with x and |ηe −ηo| ≤1 where ηe and ηo respectively denote the number of edges labeled with even integers and the number of edges labeled with odd integers. A graph that admits a k-remainder cordial labeling is called a k-remainder cordial graph. In , , , and they proved the following.
Every graph is a subgraph of a connected k-remainder cordial graph for k ≥4. Note that when k = 2, the number of edges with label 0 is q so there does not exists a 2-remainder cordial labeling. They further investigate the 3-remainder cordial labeling behavior of paths, cycles, stars, combs, crowns, wheels, fans, squares of paths, subdivisions of wheels, subdivisions of stars, subdivisions of combs, armed crowns, and K1,n ⊙K2. They further proved that Wn is 3-remainder cordial if and only if n ≡1 (mod 3), K1,n is 3-remainder cordial if and only if n ∈{1, 2, 3, 4, 5, 6, 7, 9}, and Kn is 3-remainder cordial if and only if n ≤3. In , , and Ponraj, Annathurai, and Kala proved the following graphs are 4-remainder cordial: complete graphs, paths, cycles, crowns, stars, bistars, books, subdivisions of stars, subdivisions of bistars, subdivisions of jelly fish, flowers, sunflowers, lotuses inside a circle, friendship graphs, webs, triangular snakes, durer graphs, planar grids, mongolian tents, prisms, the electronic journal of combinatorics (2023), #DS6 103 dragon graphs Cm@Pn (the graph obtained by identifying an endpoint of Pn with one vertex of Cm), crossed prisms CP2n, and K2 + mK1 (m ≡0, 1, 3 (mod 4). They also investigate the 4-remainder cordial labeling of Ln ⊙mK1, Ln ⊙K2, Ln ⊙mK1, Pn ⊙K1, Pn ⊙2K1, Cn ⊙K1, and S(Pn ⊙K1).
In Bapat introduces the following new labeling. A graph G(V, E) has a L-cordial labeling if there is a bijection f from E(G) to {1, 2, . . . , |E|} that assigns 0 to a vertex v if the largest label on the edges incident to v is even and assigns 1 to v otherwise and this assignment satisfies the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1. A graph that admits an L-cordial labeling is called as L-cordial graph. He shows that stars, path, cycles, and triangular snakes are L-cordial graphs In Chartrand, Lee, and Zhang introduced the notion of uniform cordiality as follows. Let f be a labeling from V (G) to {0, 1} and for each edge xy define f ∗(xy) = |f(x) −f(y)|. For i = 0 and 1, let vi(f) denote the number of vertices v with f(v) = i and ei(f) denote the number of edges e with f ∗(e) = i. They call a such a labeling f friendly if |v0(f) −v1(f)| ≤1. A graph G for which every friendly labeling is cordial is called uniformly cordial. They prove that a connected graph of order n ≥2 is uniformly cordial if and only if n = 3 and G = K3, or n is even and G = K1,n−1.
In Riskin introduced two measures of the noncordiality of a graph. He defines the cordial edge deficiency of a graph G as the minimum number of edges, taken over all friendly labelings of G, needed to be added to G such that the resulting graph is cordial.
If a graph G has a vertex labeling f using 0 and 1 such that the edge labeling fe given by fe(xy) = |f(x) −f(y)| has the property that the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1, the cordial vertex deficiency defined as ∞.
Riskin proved: the cordial edge deficiency of Kn (n > 1) is ⌊n 2⌋−1; the cordial vertex deficiency of Kn is j −1 if n = j2 + δ, when δ is −2, 0 or 2, and ∞otherwise. In Riskin determines the cordial edge deficiency and cordial vertex deficiency for the cases when the Möbius ladders and wheels are not cordial. In Riskin determines the cordial edge deficiencies for complete multipartite graphs that are not cordial and obtains a upper bound for their cordial vertex deficiencies.
Recall a graph G the graph G∗, called the star of G, is the graph obtained by replacing each vertex G with the star K1,n. In Kaneria, Patadiya and Teraiya introduced a balanced cordial labeling for a graph by saying that a cordial labeling f is a vertex balanced cordial if it satisfies the conditionvf(0) = vf(1); f is a balanced cordial if it satisfies the conditions ef(0) = ef(1) and vf(0) = vf(1). Kaneria, Teraiya, and Patadiya proved the path union P(t · C4n) is a balanced cordial if t is odd and it is vertex balanced cordial if t is even; C(t · C4n) is a balanced cordial if t ≡0 (mod 4) and it is a vertex balanced cordial if t ≡1, 3 (mod 4); and C⋆ 4n is balanced cordial. They proved Pn × C4t is balanced cordial; C2n × C4t is balanced cordial; and G1 ⊙G2 is cordial when G1 is cordial and G2 is a balanced cordial. Kaneria and Teraiya prove if G is a balanced cordial, then so is G∗; if G is a balanced cordial, then so is P2n+1 × G; and if G is a balanced cordial, then so is G ∗.
An integer cordial labeling of a graph G∗(p, q) is an injective map g : V →[ −p 2 , . . . , p 2]∗ the electronic journal of combinatorics (2023), #DS6 104 or [−⌊p 2⌋, . . . , ⌊p 2⌋] as p is even or odd, which induces an edge labeling g : E →{0, 1} defined by g(uv) = 1 if g(u) + g(v) ≥0 and 0 otherwise such that the number of edges labeled 1 and the number of edges labeled 0 differ by at most 1. If a graph has inte-ger cordial labeling it is called an integer cordial graph. In Gondalia and Rokad investigated the existence of integer cordial labelings of star and bistar related graphs.
For a a planar graph G with an with integer cordial labeling g and the face labeling g ∗∗from the faces of G defined by g ∗∗(f) = 1 if g(v1) + g(v2) + . . . + g(vn) = 0 and g ∗∗(f) = 0 otherwise, where v1, v2, . . . , vn are the vertices of face f. Such a labeling is called a face integer cordial labeling of graph G if the number of faces labeled with 0 and the number of faces labeled with 1 differ by at most 1. Parameswari, Saradha Pritha, and Rajeswari proved that the lilly graph, 2K1,n + 2Pn (n ≥2), admits an integer cordial labeling and the vanessa graph, 2Fn + K1,n (n ≥2), admits an integer cordial labeling and a face integer cordial labeling. Sheriff, Abbas, and Raj proved that wheels, fans, friendship graphs, triangular snakes, double triangular snakes, the star of cycles, the degree splitting graph of bistars are face integer cordial graphs.
For a simple connected graph G(V, E), Sahaya, Maya, and Nicholas introduced the concept of product integer cordial labeling of as an injective mapping f : V → {1, 2, . . . , |V |} such that the induced edge labeling f ∗on E defined f ∗(uv) = 1 or 0 according as f(u)f(v) even or odd respectively, has the property that the number of edges labeled with 1 and the the number of edges labeled with 0 differ by at most 1.
They proved that the following graphs admit product integer cordial labelings: paths, friendship graphs, Cn if and only if n is odd, and Km,n if and only if one of m and n is 1. In Shah and Parmar proved that nontrivial triangular snakes, double triangu-lar snakes, triple triangular snakes, and alternate triangular snakes graph admits integer cordial labelings. In Shah and Parmar proved that m-triangular snakes, quadri-lateral snakes, double quadrilateral snakes, m-quadrilateral snakes, pentagonal snakes, double pentagonal snakes, and m-pentagonal snakes are integer cordial graphs. They also proved that triangular snake graph, double triangular snakes, alternate triangular snakes, m-triangular snakes, quadrilateral snakes, double quadrilateral snakes, m-quadrilateral snakes, pentagonal snakes, double pentagonal snakes, and m-pentagonal snakes graph are integer cordial graphs. In Shah and Parmar proved that alternate m-triangular snakes, quadrilateral snakes, alternate m-quadrilateral snakes, pentagonal snakes, alter-nate m-pentagonal snakes, irregular triangular snakes, irregular quadrilateral snakes, and irregular pentagonal snakes are integer cordial graphs.
If f is a binary vertex labeling of a graph G Lee, Liu, and Tan defined a partial edge labeling of the edges of G by f ∗(uv) = 0 if f(u) = f(v) = 0 and f ∗(uv) = 1 if f(u) = f(v) = 1. They let e0(G) denote the number of edges uv for which f ∗(uv) = 0 and e1(G) denote the number of edges uv for which f ∗(uv) = 1. They say G is balanced if it has a friendly labeling f such that if |e0(f) −e1(f)| ≤1. In the case that the number of vertices labeled 0 and the number of vertices labeled 1 are equal and the number of edges labeled 0 and the number of edges labeled 1 are equal they say the labeling is strongly balanced. They prove: Pn is balanced for all n and is strongly balanced if n is even; Km,n is balanced if and only if m and n are even, m and n are odd and differ by at the electronic journal of combinatorics (2023), #DS6 105 most 2, or exactly one of m or n is even (say n = 2t) and t ≡−1, 0, 1 (mod |m −n|); a k-regular graph with p vertices is strongly balanced if and only if p is even and is balanced if and only if p is odd and k = 2; and if G is any graph and H is strongly balanced, the composition G[H] (see §2.3 for the definition) is strongly balanced. In Kong, Lee, Seah, and Tang show: Cm × Pn is balanced if m and n are odd and is strongly balanced if either m or n is even; and Cm ⊙K1 is balanced for all m ≥3 and strongly balanced if m is even. They also provide necessary and sufficient conditions for a graph to be balanced or strongly balanced. Lee, Lee, and Ng show that stars are balanced if and only if the number of edges of the star is at most 4. Kwong, Lee, Lo, and Wang define a graph G to be uniformly balanced if |e0(f) −e1(f)| ≤1 for every vertex labeling f that satisfies if |v0(f) −v1(f)| ≤1. They present several ways to construct families of uniformly balanced graphs. Kim, Lee, and Ng prove the following: for any graph G, mG is balanced for all m; for any graph G, mG is strongly balanced for all even m; if G is strongly balanced and H is balanced, then G ∪H is balanced; mKn is balanced for all m and strongly balanced if and only if n = 3 or mn is even; if H is balanced and G is any graph, the G × H is strongly balanced; if one of m or n is even, then Pm[Pn] is balanced; if both m and n are even, then Pm[Pn] is balanced; and if G is any graph and H is strongly balanced, then the tensor product G ⊗H is strongly balanced. (The tensor product G ⊗H of graphs G and H, has the vertex set V (G) × V (H) and any two vertices (u, u′) and (v, v′) are adjacent in G ⊗H if and only if u′ is adjacent with v′ and u is adjacent with v.) A graph G is k-balanced if there is a function f from the vertices of G to {0, 1, 2, . . . , k− 1} such that for the induced function f ∗from the edges of G to {0, 1, 2, . . . , k −1} defined by f ∗(uv) = |f(u)−f(v)| the number of vertices labeled i and the number of edges labeled j differ by at most 1 for each i and j. Seoud, El Sonbaty, and Abd El Rehim proved the following: if |E| ≥2k + 1 and |V | ≤k then G(V, E) is not k-balanced; if |E| ≥3k + 1, (k ≥2) and 3k −1 ≥|V | ≥2k + 1 then G(V, E) is not k-balanced; r-regular graphs with 3 ≤r ≤n −1 are not r-balanced; if G1 has m vertices and G2 has n vertices then G1 + G2 is not (m + n)-balanced for m, n ≥5; P3 × Pn with edge set E is 3n-balanced and |E|-balanced; Ln × P2 (Ln = Pn × P2) with vertex set V and edge set E is |V |-balanced and k-balanced for k ≥|E| but not n-balanced for n ≥2; the one-point union of two copies of K2,n is 2n-balanced, |V |-balanced, and |E|-balanced not is 3-balanced when n ≥4. They also proved that the composition graph Pn[P2] is not n-balanced for n ≥3, is not 2n-balanced for n ≥5, and is not |E|-balanced.
A graph whose edges are labeled with 0 and 1 so that the absolute difference in the number of edges labeled 1 and 0 is no more than one is called edge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident to more edges labeled 1 than 0, are labeled 1, and vertices which are incident to more edges labeled 0 than 1, are labeled 0. Vertices that are incident to an equal number of edges of both labels are called unlabeled. Call a procedure on a labeled graph a label switching algorithm if it consists of pairwise switches of labels. Krop, Lee, and Raridan prove that given an edge-friendly labeling of Kn, we show a label switching algorithm producing an edge-friendly relabeling of Kn such that all the vertices are labeled.
the electronic journal of combinatorics (2023), #DS6 106 In 2017 Bapat introduced a new labeling as follows.
A function f from the vertices of a graph G(E, V ) to {0, 1, 2, . . . , |V | −1} is called an extended vertex edge additive cordial labeling if the induced function f ∗from the edges of G to {0, 1} defined by f ∗(uv) = f(u) + f(v) (mod 2) for all edges uv of G has the property that the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1. Bapat proved paths, stars, K2,n, K3,n, K4,n, Pn ⊙C3, and Pn ⊙C4 admit extended vertex edge additive cordial labeling.
Let G(p, q) a simple finite connected graph. Given a bijective function f from E(G) to {0, 1, . . . , q −1} Bapat calls a bijective function f ∗from E(G) to {0, 1, 2, . . . , q −1} an extended edge vertex cordial (eevc) labeling if the induced function f ∗from V (G) to {0, 1} defined by f ∗(u) = Σf(uv) mod 2 where the sum is taken over all edges incident to u has the property that the number of vertices labeled with 0 differs from the number labeled with 1 by at most 1. He shows that Pn (n ̸= 2 mod 4), Cn (n ̸= 2 mod 4), K1,n (n ̸= 1 mod 4), graphs obtained by joining the centers of two copies of K1,2n+1 by an edge, and triangular snakes have eevc labelings.
Murali, Thirusangu, Madura Meenakshi say a graph G = (V, E) is bicondi-tional cordial if there is a function f : V →{0, 1} such that the induced edge function f ∗: E →{0, 1} defined by f ∗(uv) = 1 if (u) = f(v) and 0 if f(u) ̸= f(v) and the number of vertices labeled with 0 and the number labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number labeled with 1 differ by at most 1. Kalaimathi and Balamurugan prove the existence of the biconditional cordial labeling for complete bipartite graphs, books with triangular pages, sunflower graphs and web graphs. Nedumaran, Thirusangu, and Celin Mary proved that the graph con-sisting of k copies of a double star admits a biconditional cordial labeling. Kalaimathi, Balamurugan, and Rao proved the existence of the biconditional cordial labelings for super subdivisions of ladders and grids Pm × Pn.
A total cordial labeling of a graph G is a cordial labeling of vertex set and edge set such that the number of vertices and edges labeled with 0 and the number of vertices and edges labeled with 1 differ by at most 1. Elrokh, Al-Shamiri, Nada, and El-hay provided necessary and sufficient conditions for the existence of cordial and total cordial labelings for the corona product of paths and the one-point union C2 m and C2 n.
3.8 The Friendly Index–Balance Index Recall a function f from V (G) to {0, 1} where for each edge xy, f ∗(xy) = |f(x) − f(y)|, vi(f) is the number of vertices v with f(v) = i, and ei(f) is the num-ber of edges e with f ∗(e) = i is called friendly if |v0(f) −v1(f)| ≤ 1.
Lee and Ng define the friendly index set of a graph G as FI(G)= {|e0(f) − e1(f)| where f runs over all friendly labelings f of G}.
They proved: for any graph G with q edges FI(G) ⊆{0, 2, 4, . . . , q} if q is even and FI(G)⊆{1, 3, . . . , q} if q is odd; for 1 ≤m ≤n, FI(Km,n)= {(m −2i)2| 0 ≤i ≤⌊m/2⌋} if m + n is even; and FI(Km,n)= {i(i + 1)| 0 ≤i ≤m} if m + n is odd.
In Lee and Ng prove the following: FI(C2n) = {0, 4, 8, . . . , 2n} when n is even; FI(C2n) = {2, 6, 10, . . . , 2n} when n the electronic journal of combinatorics (2023), #DS6 107 is odd; and FI(C2n+1) = {1, 3, 5, . . . , 2n −1}. Elumalai defines a cycle with a full set of chords as the graph PCn obtained from Cn = v0, v1, v2, . . . , vn−1 by adding the cords v1vn−1, v2vn−2, . . . , v(n−2)/2, v(n+2)/2 when n is even and v1vn−1, v2vn−2, . . . , v(n−3)/2, v(n+3)/2 when n is odd. Lee and Ng prove: FI(PC2m+1) = {3m −2, 3m −4, 3m −6, . . . , 0} when m is even and FI(PC2m+1) = {3m −2, 3m −4, 3m −6, . . . , 1} when m is odd; FI(PC4) = {1, 3}; for m ≥3, FI(PC2m) = {3m −5, 3m −7, 3m −9, . . . , 1} when m is even; FI(PC2m) = {3m −5, 3m −7, 3m −9, . . . , 0} when m is odd.
Salehi and Lee determined the friendly index for various classes of trees. Among their results are: for a tree with q edges that has a perfect matching, the friendly index is the odd integers from 1 to q and for n ≥2, FI(Pn)= {n −1 −2i| 0 ≤i⌊(n −1)/2⌋.
Law determined the full friendly index sets of spiders and disproved a conjecture by Salehi and Lee that the friendly index set of a tree forms an arithmetic progression.
In Lee, Ng, and Lau determine the friendly index sets of several classes of spiders.
Gao, Sun, and Lee determined the full friendly index of Pm × Pn with the extra mn + 1 −m −n edges uij −u(i+1)(j+1). Sun, Gao, and Lee determined the full friendly index and friendly index for the twisted product of Mőbius ladders. Sinha and Kaur determined the full edge friendly index of stars, wheels, 2-regular graphs, and mPn. In Shiu determined the full edge-friendly index sets of complete bipartite graphs. Salehi and McGinn obtained partial results about the friendly index set of Qn and strengthen a conjecture about the friendly index set of Qn made in . Teffilia1 and Devaraj found the friendly index set of the graphs obtained by identifying the central vertex of a fan with the endpoint of a path (umbrella), the graphs obtained by identifying the central vertex of a star with the endpoint of a path, the graphs obtained by identifying the endpoints of copies of P2 (globe), the splitting graph of a star, and P2 + mK1. Lee, Low, Ng, and Wang determined the friendly index sets for various classes of disjoint unions of stars. Gao, Ruo-Yuan, Lee, Ren, and Sun determined FFI(G), FI(G) and FPCI(G) for a class of cubic graphs G. Ji, Liu, Bai, and Wu new determined the full friendly index sets of mCn.
Lee and Ng define PC(n, p) as the graph obtained from the cycle Cn with consecutive vertices v0, v1, v2, . . . , vn−1 by adding the p cords joining vi to vn−i for 1 ≤ p⌊n/2⌋−1. They prove FI(PC(2m + 1, p)) = {2m + p −1, 2m + p −3, 2m + p −5, . . . , 1} if p is even and FI(PC(2m + 1, p)) = {2m + p −1, 2m + p −3, 2m + p −5, . . . , 0} if p is odd; FI(PC(2m, 1)) = {2m −1, 2m −3, 2m −5, . . . , 1}; for m ≥3, and p ≥2, FI(PC(2m, p)) = {2m + p −4, 2m + p −6, 2m + p −8, . . . , 0} when p is even, and FI(PC(2m, p)) = {2m+p−4, 2m+p−6, 2m+p−8, . . . , 1} when p is odd. More generally, they show that the integers in the friendly index of a cycle with an arbitrary nonempty set of parallel chords form an arithmetic progression with a common difference 2. Shiu and Kwong determine the friendly index of the grids Pn × P2. The maximum and minimum friendly indices for Cm × Pn were given by Shiu and Wong in . In new Liu, Liu, and Ji generalized the concept of the full m index set as follows. For a friendly labeling f of a graph G and an edge uv, f ′(uv) = f(u)+f(v), where ef′(i) is the number of edges labeled with i, and uf(i) is the number of vertices labeled with i. The full friendly m index set of a graph G is the set M(G) = {ef′(1)−ef′(0) | |vf(1)−vf(0)|, 0 ≤m < |V (G)|} the electronic journal of combinatorics (2023), #DS6 108 taken over all friendly labelings f of G. Note that when m = 0 or m = 1, the full m index set of a graph is its full friendly index set. Liu, Liu, and Ji determined the full m index set of P2 × Pn for all m.
In Lee and Ng prove: for n ≥2, FI(C2n×P2) = {0, 4, 8, . . . , 6n−8, 6n} if n is even and FI(C2n ×P2) = {2, 6, 10, . . . , 6n−8, 6n} if n is odd; FI(C3 ×P2) = {1, 3, 5}; for n ≥2, FI(C2m+1×P2) = {6n−1}∪{6n−5−2k| where k ≥0 and 6n−5−2k ≥0}; FI(M4n) (here M4n is the Möbius ladder with 4n steps) = {6n−4−4k| where k ≥0 and 6n−4−4k ≥0}; FI(M4n+2) = {6n+3}∪{6n−5−2k| where k ≥0 and 6n−5−2k > 0}. In Kwong, Lee, and Ng completely determine the friendly index of all 2-regular graphs. As a corollary, they show that Cm ∪Cn is cordial if and only if m + n = 0, 1 or 3 (mod 4). Ho, Lee, and Ng determine the friendly index sets of stars and various regular windmills.
In Wen determines the friendly index of generalized wheels Cn+mK1 for all m > 1.
In Salehi and De determine the friendly index sets of certain caterpillars of diameter 4 and disprove a conjecture of Lee and Ng that the friendly index sets of trees form an arithmetic progression. The maximum and minimum friendly indices for for Cm × Pn were given by Shiu and Wong in . Salehi and Bayot have determined the friendly index set of Pm × Pn. In Lee and Ng determine the friendly index sets for two classes of cubic graphs, prisms and Möbius ladders. Sinha and Kaur investigate the full region index sets of friendly labelings of cycles, wheels fans, and P2 × Pn.
For positive integers a ≤b ≤c, Lee, Ng, and Tong define the broken wheel W(a, b, c) with three spokes as the graph obtained from K4 with vertices u1, u2, u3, c by inserting vertices x1,1, x1,2, . . . , x1,a−1 along the edge u1u2, x2,1, x2,2, . . . , x2,b−1 along the edge u2u3, x3,1, x3,2, . . . , x3,c−1 along the edge u3u1. They determine the friendly index set for broken wheels with three spokes.
Lee and Ng define a parallel chord of Cn as an edge of the form vivn−i (i < n −1) that is not an edge of Cn. For n ≥6, they call the cycle Cn with con-secutive vertices v1, v2, . . . , vn and the edges v1vn−1, v2vn−2, . . . , v(n−2)/2v(n+2)/2 for n even and v2vn−1, v3vn−2, . . . , v(n−1)/2v(n+3)/2 for n odd, Cn with a full set of parallel chords.
They determine the friendly index of these graphs and show that for any cycle with an arbitrary non-empty set of parallel chords the numbers in its friendly index set form an arithmetic progression with common difference 2.
For a graph G(V, E) and a graph H rooted at one of its vertices v, Ho, Lee, and Ng define a root-union of (H, v) by G as the graph obtained from G by replacing each vertex of G with a copy of the root vertex v of H to which is appended the rest of the structure of H. They investigate the friendly index set of the root-union of stars by cycles.
For a graph G(V, E), the total graph T(G) of G, is the graph with vertex set V ∪E and edge set E ∪{(v, uv)| v ∈V, uv ∈E}.
Note that the total graph of the n-star is the friendship graph and the total graph of Pn is a triangular snake.
Lee and Ng use SP(1n, m) to denote the spider with one central vertex joining n isolated vertices and a path of length m. They show: FI(K1 + 2nK2) (friendship graph with 2n triangles) = {2n, 2n −4, 2n −8, . . . , 0} if n is even; {2n, 2n −4, 2n −8, . . . , 2} if n is odd; FI(K1+(2n+1)K2) = {2n+1, 2n−1, 2n−3, . . . , 1}; for n odd, FI(T(Pn)) = {3n−7, 3n− the electronic journal of combinatorics (2023), #DS6 109 11, 3n −15, . . . , z} where z = 0 if n ≡1 (mod 4) and z = 2 if n ≡3 (mod 4); for n even, FI(T(Pn)) = {3n−7, 3n−11, 3n−15, . . . , n+1}∪{n−1, n−3, n−5, . . . , 1}; for m ≤n−1 and m+n even, FI(T(SP(1n, m))) = {3(m+n)−4, 3(m+n)−8, 3(m+n)−12, . . . , (m+n) (mod 4)}; for m + n odd, FI(T(SP(1n, m))) = {3(m + n) −4, 3(m + n) −8, 3(m + n) − 12, . . . , m + n + 2} ∪{m + n, m + n −2, m + n −4, . . . , 1}; for n ≥m and m + n even, FI(T(SP(1n, m))) = {|4k −3(m+n)| |(n−m+2)/2 ≤k ≤m+n}; for n ≥m and m+n odd, FI(T(SP(1n, m))) = {|4k −3(m + n)| |(n −m + 3)/2 ≤k ≤m + n}.
Kwong and Lee determine the friendly index any number of copies of C3 that share an edge in common and the friendly index any number of copies of C4 that share an edge in common. Lau, Gao, Lee, and Sun determine the friendly index sets and the cordiality of the edge-gluing of a complete graph Kn and n copies of cycles C3.
For a planar graph G(V, E) Sinha and Kaur extended the notion of an index set of a friendly labeling to regions of a planar graph and determined the full region index sets of friendly labeling of cycles, wheels fans, and grids Pn × P2.
An edge-friendly labeling f of a graph G induces a function f ∗from V (G) to {0, 1} defined as the sum of all edge labels mod 2. The edge-friendly index set, If(G), of f is the number of vertices of f labeled 1 minus the number of vertices labeled 0. The edge-friendly index set of a graph G, EFI(G), is {|If(G)|} taken over all edge-friendly labelings f of G. The full edge-friendly index set of a graph G, FEFI(G), is {If(G)} taken over all edge-friendly labelings f of G. Sinha and Kaur determined the full edge-friendly index sets of stars, 2-regular graphs, wheels, and mPn. In Sinha and Kaur extended the notion of index set of an edge-friendly labeling to regions of a planar graph and determined the full region index set of edge-friendly labelings of cycles, wheels, fans Pn + K1, double fans Pn + K2, and grids Pm × Pn (m ≥2, n ≥3).
Sinha and Kaur investigate the full edge-friendly index sets of double stars, fans generalized fans, and Pn × P2. In Shiu determined the extreme values of edge-friendly indices of complete bipartite graphs.
In Kim, Lee, and Ng define the balance index set of a graph G as {|e0(f)−e1(f)|} where f runs over all friendly labelings f of G. Zhang, Lee, and Wen investigate the balance index sets for the disjoint union of up to four stars and Zhang, Ho, Lee, and Wen investigate the balance index sets for trees with diameter at most four.
Kwong, Lee, and Sarvate determine the balance index sets for cycles with one pendent edge, flowers, and regular windmills. Lee, Ng, and Tong determine the balance index set of certain graphs obtained by starting with copies of a given cycle and successively identifying one particular vertex of one copy with a particular vertex of the next. For graphs G and H and a bijection π from G to H, Lee and Su define Perm(G, π, H) as the graph obtaining from the disjoint union of G and H by joining each v in G to π(v) with an edge. They determine the balanced index sets of the disjoint union of cycles and the balanced index sets for graphs of the form Perm(G, π, H) where G and H are regular graphs, stars, paths, and cycles with a chord. They conjecture that the balanced index set for every graph of the form Perm(G, π, H) is an arithmetic progression.
Lee, Ho, and Su investigated the balance index sets of k-level wheel graphs.
Wen determines the balance index set of the graph that is constructed by the electronic journal of combinatorics (2023), #DS6 110 identifying the center of a star with one vertex from each of two copies of Cn and provides a necessary and sufficient for such graphs to be balanced. In Lee, Su, and Wang determine the balance index sets of the disjoint union of a variety of regular graphs of the same order. Kwong determines the balanced index sets of rooted trees of height at most 2, thereby settling the problem for trees with diameter at most 4. His method can be used to determine the balance index set of any tree. The homeomorph Hom(G, p) of a graph G is the collection of graphs obtained from G by adding p (p ≥0) additional degree 2 vertices to its edges. For any regular graph G, Kong, Lee, and Lee studied the changes of the balance index sets of Hom(G, p) with respect to the parameter p. They derived explicit formulas for their balance index sets provided new examples of uniformly balanced graphs. In Bouchard, Clark, Lee, Lo, and Su investigate the balance index sets of generalized books and ear expansion graphs.
In Rose and Su provided an algorithm to calculate the balance index sets of a graph. Hua and Raridan determine the balanced index sets of all complete bipartite graphs with a larger part of odd cardinality and a smaller part of even cardinality.
In Shiu and Kwong made a major advance by introducing an easier approach to find the balance index sets of a large number of families of graphs in a unified and uniform manner. They use this method to determine the balance index sets for r-regular graphs, amalgamations of r-regular graphs, complete bipartite graphs, wheels, one point unions of regular graphs, sun graphs, generalized theta graphs, m-ary trees, spiders, grids Pm × Pn, and cylinders Cm × Pn. They provide a formula that enables one to determine the balance index sets of many biregular graphs (that is, graphs with the property that there exist two distinct positive integers r and s such that every vertex has degree r or s).
A labeling f from the vertices of a graph G to {0, 1} is said to be vertex-friendly if the number of vertices labeled with 0 and the number labeled with 1 differ by at most 1. The vertex balance index set of G is |e0(f) −e1(f)| taken over all vertex-friendly labelings f.
Adiga, Subbaraya, Shrikanth and Sriraj completely determined the vertex balance index set of Kn, Km,n, Cn × P2, and complete binary trees.
Manico and Pedrano prove that if the number of edges in a vertex-friendly of a graph G is even, then the vertex balance index of G contains only even numbers, and if the number of edges in a vertex-friendly graph G is odd, then the the vertex balance index of G contains only odd numbers. Furthermore, they provide the vertex balance index set of triangular snakes, quadrilateral snakes, double triangular snakes), and double quadrilateral snakes.
In Shiu and Kwong define the full friendly index set of a graph G as {e0(f) − e1(f)} where f runs over all friendly labelings of G. The full friendly index for P2 × Pn is given by Shiu and Kwong in . The full friendly index of Cm × Cn is given by Shiu and Ling in . In and Sinha and Kaur investigated the full friendly index sets complete graphs, cycles, fans, double fans, wheels, double stars, P3 × Pn, and the tensor product of P2 and Pn. Shiu and Ho investigated the full friendly index sets of cylinder graphs Cm × P2 (m ≥3), Cm × P3 (m ≥4), and C3 × Pn (n ≥4). These results, together with previously proven ones, completely determine the full friendly index the electronic journal of combinatorics (2023), #DS6 111 of all cylinder graphs. Shiu and Ho study the full friendly index set and the full product-cordial index set of odd twisted cylinders and two permutation Petersen graphs.
Gao determined the full friendly index set of Pm × Pn, but he used the terms “edge difference set” instead of “full friendly index set” and “direct product” instead of “Cartesian product.” The twisted cylinder graph is the permutation graph on 4n (n ≥2) vertices, P(2n; σ), where σ = (1, 2)(3, 4) · · · (2n−1, 2n) (the product of n transpositions).
Shiu and Lee determined the full friendly index sets of twisted cylinders.
In and Chopra, Lee, and Su and Kwong and Lee introduce a dual of balance index sets as follows. For an edge labeling f using 0 and 1 they define a partial vertex labeling f ∗by assigning 0 or 1 to f ∗(v) depending on whether there are more 0-edges or 1-edges incident to v and leaving f ∗(v) undefined otherwise. For i = 0 or 1 and a graph G(V, E), let ef(i) = |{uv ∈E : f(uv) = i}| and vf(i) = |{v ∈V : f ∗(v) = i}|. They define the edge-balance index of G as EBI(G) = {|vf(0) −vf(1)| : the edge labeling f satisfies |ef(0) −ef(1)| ≤1}. Among the graphs whose edge-balance index sets have been investigated by Lee and his colleagues are: fans and wheels ; generalized theta graphs ; flower graphs and ; stars, paths, spiders, and double stars ; (p, p + 1)-graphs ; prisms and Möbius ladders ; 2-regular graphs, complete graphs ; and the envelope graphs of stars, paths, and cycles . (The envelope graph of G(V, E) is the graph with vertex set V (G) ∪E(G) and set E(G) ∪{(u, (u, v)) : U ∈V, (u, v) ∈E)}).
Lee, Kong, Wang, and Lee found the EBI(Km,n) for m = 1, 2, 3, 4, 5 and m = n.
Krop, Minion, Patel, and Raridan did the case for complete bipartite graphs with both parts of odd cardinality. Dao, Hua, Ngo, and Raridan determined the edge-balanced index sets for complete even bipartite graphs. Krop and Sikes determined EBI(Km,m−2a) for 1 ≤a ≤(m −3)/4 and m odd.
For a graph G and a connected graph H with a distinguished vertex s, the L-product of G and (H, s), G ×L (H, s), is the graph obtained by taking |V (G)| copies of (H, s) and identifying each vertex of G with s of a single copy of H. In and Chou, Galiardi, Kong, Lee, Perry, Bouchard, Clark, and Su investigated the edge-balance index sets of L-product of cycles with stars. Bouchard, Clark, and Su gave the exact values of the edge-balance index sets of L-product of cycles with cycles.
Chopra, Lee, and Su prove that the edge-balance index of the fan P3 + K1 is {0, 1, 2} and edge-balance index of the fan Pn + K1, n ≥4, is {0, 1, 2, . . . , n −2}.
They define the broken fan graphs BF(a, b) as the graph with V (BF(a, b)) = {c} ∪ {v1, . . . , va} ∪{u1, . . . , ub} and E(BF(a, b)) = {(c, vi)| i = 1, . . . , a} ∪{(c, ui)| 1, . . . , b} ∪ E(Pa) ∪E(Pb) (a ≥2 and b ≥2). They prove the edge-balance index set of BF(a, b) is {0, 1, 2, . . . , a + b −4}. In Lee, Su, and Todt give the edge-balance index sets of broken wheels. See also and . In Lee, Lee, and Su present a technique that determines the balance index sets of a graph from its degree sequence. In addition, they give an explicit formula giving the exact values of the balance indices of generalized friendship graphs, envelope graphs of cycles, and envelope graphs of cubic trees.
the electronic journal of combinatorics (2023), #DS6 112 3.9 k-equitable Labelings In 1990 Cahit proposed the idea of distributing the vertex and edge labels among {0, 1, . . . , k −1} as evenly as possible to obtain a generalization of graceful labelings as follows. For any graph G(V, E) and any positive integer k, assign vertex labels from {0, 1, . . . , k−1} so that when the edge labels induced by the absolute value of the difference of the vertex labels, the number of vertices labeled with i and the number of vertices labeled with j differ by at most one and the number of edges labeled with i and the number of edges labeled with j differ by at most one. Cahit has called a graph with such an assignment of labels k-equitable. Note that G(V, E) is graceful if and only if it is |E| + 1-equitable and G(V, E) is cordial if and only if it is 2-equitable. Cahit has shown the following: Cn is 3-equitable if and only if n ̸≡3 (mod 6); the triangular snake with n blocks is 3-equitable if and only if n is even; the friendship graph C(n) 3 is 3-equitable if and only if n is even; an Eulerian graph with q ≡3 (mod 6) edges is not 3-equitable; and all caterpillars are 3-equitable . Cahit claimed to prove that Wn is 3-equitable if and only if n ̸≡3 (mod 6) but Youssef proved that Wn is 3-equitable for all n ≥4. Youssef also proved that if G is a k-equitable Eulerian graph with q edges and k ≡2 or 3 (mod 4) then q ̸≡k (mod 2k). Cahit conjectures that a triangular cactus with n blocks is 3-equitable if and only if n is even. In Cahit proves that every tree with fewer than five end vertices has a 3-equitable labeling.
He conjectures that all trees are k-equitable . In 1999 Speyer and Szaniszló proved Cahit’s conjecture for k = 3. Coles, Huszar, Miller, and Szaniszlo proved caterpillars, symmetric generalized n-stars (or symmetric spiders), and complete n-ary trees are 4-equitable. Vaidya and Shah proved that the splitting graphs of K1,n and the bistar Bn,n and the shadow graph of Bn,n are 3-equitable. Rokad found 3-equitable labelings of the ring sum of different graphs.
Vaidya, Dani, Kanani, and Vihol proved that the graphs obtained by starting with copies G1, G2, . . . , Gn of a fixed star and joining each center of Gi to the center of Gi+1 (i = 1, 2, . . . , n−1) by an edge are 3-equitable. Recall the shell C(n, n−3) is the cycle Cn with n −3 cords sharing a common endpoint called the apex. Vaidya, Dani, Kanani, and Vihol proved that the graphs obtained by starting with copies G1, G2, . . . , Gn of a fixed shell and joining each apex of Gi to the apex of Gi+1 (i = 1, 2, . . . , n −1) by an edge are 3-equitable. For a graph G and vertex v of G, Vaidya, Dani, Kanani, and Vihol prove that the graphs obtained from the wheel Wn, n ≥5, by duplicating (see 3.7 for the definition) any rim vertex is 3-equitable and the graphs obtained from the wheel Wn by duplicating the center is 3-equitable when n is even and not 3-equitable when n is odd and at least 5. They also show that the graphs obtained from the wheel Wn, n ̸= 5, by duplicating every vertex is 3-equitable.
Vaidya, Srivastav, Kaneria, and Ghodasara prove that cycle with two chords that share a common vertex with opposite ends that are incident to two consecutive vertices of the cycle is 3-equitable. Vaidya, Ghodasara, Srivastav, and Kaneria prove that star of cycle C ∗ n is 3-equitable for all n.
Vaidya and Dani proved that the graphs obtained by starting with n copies G1, G2, . . . , Gn of a fixed star and the electronic journal of combinatorics (2023), #DS6 113 joining the center of Gi to the center of Gi+1 by an edge and each center to a new vertex xi (1 ≤i ≤n −1) by an edge have 3-equitable labeling. Vaidya and Dani prove that the graphs obtained by duplication of an arbitrary edge of a cycle or a wheel have 3-equitable labelings.
The Mycielski graph of a graph G is obtained from G by adding to each vertex v a new vertex u that is adjacent to the neighbors of v and a adding a new vertex w that is adjacent to every u. In Sangeeta, Parthiban, Selvaraju proved the non-existence of 3-equitable labelings for non-trivial cases of the total graphs of fans, the middle graph of ladders, the degree splitting graphs of friendship graphs, and the Mycielskian graphs of paths.
Recall G =< W (1) n : W (2) n : . . . : W (k) n > 1s the graph obtained by joining the center vertices of each of W (i) n and W (i+1) n to a new vertex xi where 1 ≤i ≤k −1. Vaidya, Dani, Kanani, and Vihol prove that < W (1) n : W (2) n : ... : W (k) n > is 3-equitable. Vaidya and Vihol prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph thereby ruling out any possibility of obtaining any forbidden subgraph characterization for 3-equitable graphs.
The shadow graph D2(G) of a connected graph G is constructed by taking two copies of G, G′ and G′′ and joining each vertex u′ in G′ to the neighbors of the corresponding vertex u′′ in G′′. Vaidya, Vihol, and Barasara prove that the shadow graph of Cn is 3-equitable except for n = 3 and 5 while the shadow graph of Pn is 3-equitable except for n = 3. They also prove that the middle graph of Pn is 3-equitable and the middle graph of Cn is 3-equitable for n even and not 3-equitable for n odd.
Bhut-Nayak and Telang have shown that crowns Cn ⊙K1, are k-equitable for k = n, . . . , 2n −1 and Cn ⊙K1 is k-equitable for all n when k = 2, 3, 4, 5, and 6 .
In Seoud and Abdel Maqsoud prove: a graph with n vertices and q edges in which every vertex has odd degree is not 3-equitable if n ≡0 (mod 3) and q ≡3 (mod 6); all fans except P2 + K1 are 3-equitable; all double fans Pn + K2 except P4 + K2 are 3-equitable; P 2 n is 3-equitable for all n except 3; K1,1,n is 3-equitable if and only if n ≡0 or 2 (mod 3); K1,2,n, n ≥2, is 3-equitable if and only if n ≡2 (mod 3); Km,n, 3 ≤m ≤n, is 3-equitable if and only if (m, n) = (4, 4); and K1,m,n, 3 ≤m ≤n, is 3-equitable if and only if (m, n) = (3, 4). They conjectured that C2 n is not 3-equitable for all n ≥3.
However, Youssef proved that C2 n is 3-equitable if and only if n is at least 8.
Youssef also proved that Cn + K2 is 3-equitable if and only if n is even and at least 6 and determined the maximum number of edges in a 3-equitable graph as a function of the number of its vertices. For a graph with n vertices to admit a k-equitable labeling, Seoud and Salim proved that the number of edges is at most k⌈(n/k)⌋2 + k −1.
Bapat and Limaye have shown the following graphs are 3-equitable: helms Hn, n ≥4; flowers (see §2.2 for the definition); the one-point union of any number of helms; the one-point union of any number of copies of K4; K4-snakes (see §2.2 for the definition); Ct-snakes where t = 4 or 6; C5-snakes where the number of blocks is not congruent to 3 modulo 6. A multiple shell MS{nt1 1 , . . . , ntr r } is a graph formed by ti shells each of order ni, 1 ≤i ≤r, that have a common apex. Bapat and Limaye show that every multiple shell is 3-equitable and Chitre and Limaye show that every multiple the electronic journal of combinatorics (2023), #DS6 114 shell is 5-equitable. In Chitre and Limaye define the H-union of a family of graphs G1, G2, . . . , Gt, each having a graph H as an induced subgraph, as the graph obtained by starting with G1 ∪G2 ∪· · · ∪Gt and identifying all the corresponding vertices and edges of H in each of G1, . . . , Gt. In and they proved that the Kn-union of gears and helms Hn (n ≥6) are edge-3-equitable.
Szaniszló has proved the following: Pn is k-equitable for all k; Kn is 2-equitable if and only if n = 1, 2, or 3; Kn is not k-equitable for 3 ≤k < n; Sn is k-equitable for all k; K2,n is k-equitable if and only if n ≡k −1 (mod k), or n ≡0, 1, 2, . . . , ⌊k/2⌋−1 (mod k), or n = ⌊k/2⌋and k is odd. She also proves that Cn is k-equitable if and only if k meets all of the following conditions: n ̸= k; if k ≡2, 3 (mod 4), then n ̸= k −1 and n ̸≡k (mod 2k). Coles, Huszar, Miller, and Szaniszló proved that all caterpillars, symmetric generalized n-stars (or symmetric spiders), and complete n-ary trees for all are 4-equitable.
Vickrey has determined the k-equitability of complete multipartite graphs. He shows that for m ≥3 and k ≥3, Km,n is k-equitable if and only if Km,n is one of the following graphs: K4,4 for k = 3; K3,k−1 for all k; or Km,n for k > mn. He also shows that when k is less than or equal to the number of edges in the graph and at least 3, the only complete multipartite graphs that are k-equitable are Kkn+k−1,2,1 and Kkn+k−1,1,1. Partial results on the k-equitability of Km,n were obtained by Krussel .
In Youssef and Al-Kuleab proved the following: C3 n is 3-equitable if and only if n is even and n ≥12; gear graphs are k-equitable for k = 3, 4, 5, 6; ladders Pn × P2 are 3-equitable for all n ≥2; Cn × P2 is 3-equitable if and only if n ̸≡(mod 6); Möbius ladders Mn are 3-equitable if and only if n ̸≡(mod6); and the graphs obtained from Pn × P2 (n ≥2) where by adding the edges uivi+1 (1 ≤i ≤n −1) to the path vertices u1, u2, . . . , un and v1, v2, . . . , vn.
In López, Muntaner-Batle, and Rius-Font prove that if n is an odd integer and F is optimal k-equitable for all proper divisors k of |E(F)|, then nF is optimal k-equitable for all proper divisors k of |E(F)|. They also prove that if m −1 and n are odd, then then nCm is optimal k-equitable for all proper divisors k of |E(F)|.
As a corollary of the result of Cairnie and Edwards on the computational com-plexity of cordially labeling graphs it follows that the problem of finding k-equitable labelings of graphs is NP-complete as well.
Seoud and Abdel Maqsoud call a graph k-balanced if the vertices can be labeled from {0, 1, . . . , k −1} so that the number of edges labeled i and the number of edges labeled j induced by the absolute value of the differences of the vertex labels differ by at most 1. They prove that P 2 n is 3-balanced if and only if n = 2, 3, 4, or 6; for k ≥4, P 2 n is not k-balanced if k ≤n −2 or n + 1 ≤k ≤2n −3; for k ≥4, P 2 n is k-balanced if k ≥2n −2; for k, m, n ≥3, Km,n is k-balanced if and only if k ≥mn; for m ≤n, K1,m,n is k-balanced if and only if (i) m = 1, n = 1 or 2, and k = 3; (ii) m = 1 and k = n + 1 or n + 2; or (iii) k ≥(m + 1)(n + 1).
In Youssef gave some necessary conditions for a graph to be k-balanced and some relations between k-equitable labelings and k-balanced labelings. Among his results are: Cn is 3-balanced for all n ≥3; Kn is 3-balanced if and only if n ≤3; and all trees the electronic journal of combinatorics (2023), #DS6 115 are 2-balanced and 3-balanced. He conjectures that all trees are k-balanced (k ≥2).
Bloom has used the term k-equitable to describe another kind of labeling (see and ). He calls a graph k-equitable if the edge labels induced by the absolute value of the difference of the vertex labels have the property that every edge label occurs exactly k times. Bloom calls a graph of order n minimally k-equitable if the vertex labels are 1, 2,. . ., n and it is k-equitable. Both Bloom and Wojciechowski , proved that Cn is minimally k-equitable if and only if k is a proper divisor of n. Barrientos and Hevia proved that if G is k-equitable of size q = kw (in the sense of Bloom), then δ(G) ≤w and ∆(G) ≤2w. Barrientos, Dejter, and Hevia have shown that forests of even size are 2-equitable. They also prove that for k = 3 or k = 4 a forest of size kw is k-equitable if and only if its maximum degree is at most 2w and that if 3 divides mn + 1, then the double star Sm,n is 3-equitable if and only if q/3 ≤m ≤⌊(q −1)/2⌋. (Sm,n is P2 with m pendent edges attached at one end and n pendent edges attached at the other end.) They discuss the k-equitability of forests for k ≥5 and characterize all caterpillars of diameter 2 that are k-equitable for all possible values of k. Acharya and Bhat-Nayak have shown that coronas of the form C2n ⊙K1 are minimally 4-equitable. In Barrientos proves that the one-point union of a cycle and a path (dragon) and the disjoint union of a cycle and a path are k-equitable for all k that divide the size of the graph. Barrientos and Havia have shown the following: Cn × K2 is 2-equitable when n is even; books Bn (n ≥3) are 2-equitable when n is odd; the vertex union of k-equitable graphs is k-equitable; and wheels Wn are 2-equitable when n ̸≡3 (mod 4). They conjecture that Wn is 2-equitable when n ≡3 (mod 4) except when n = 3. Their 2-equitable labelings of Cn × K2 and the n-cube utilized graceful labelings of those graphs.
M. Acharya and Bhat-Nayak have proved the following: the crowns C2n ⊙K1 are minimally 2-equitable, minimally 2n-equitable, minimally 4-equitable, and minimally n-equitable; the crowns C3n ⊙K1 are minimally 3-equitable, minimally 3n-equitable, minimally n-equitable, and minimally 6-equitable; the crowns C5n ⊙K1 are minimally 5-equitable, minimally 5n-equitable, minimally n-equitable, and minimally 10-equitable; the crowns C2n+1 ⊙K1 are minimally (2n + 1)-equitable; and the graphs Pkn+1 are k-equitable.
In Barrientos calls a k-equitable labeling optimal if the vertex labels are con-secutive integers and complete if the induced edge labels are 1, 2, . . . , w where w is the number of distinct edge labels. Note that a graceful labeling is a complete 1-equitable labeling. Barrientos proves that Cm ⊙nK1 (that is, an m-cycle with n pendent edges attached at each vertex) is optimal 2-equitable when m is even; C3 ⊙nK1 is complete 2-equitable when n is odd; and that C3 ⊙nK1 is complete 3-equitable for all n. He also shows that Cn ⊙K1 is k-equitable for every proper divisor k of the size 2n. Barrientos and Havia have shown that the n-cube (n ≥2) has a complete 2-equitable labeling and that Km,n has a complete 2-equitable labeling when m or n is even. They conjecture that every tree of even size has an optimal 2-equitable labeling.
the electronic journal of combinatorics (2023), #DS6 116 3.10 Hamming-graceful Labelings Mollard, Payan, and Shixin introduced a generalization of graceful graphs called Hamming-graceful. A graph G = (V, E) is called Hamming-graceful if there exists an injective labeling g from V to the set of binary |E|-tuples such that {d(g(v), g(u))| uv ∈ E} = {1, 2, . . . , |E|} where d is the Hamming distance. Shixin and Yu have shown that all graceful graphs are Hamming-graceful; all trees are Hamming-graceful; Cn is Hamming-graceful if and only if n ≡0 or 3 (mod 4); if Kn is Hamming-graceful, then n has the form k2 or k2 + 2; and Kn is Hamming-graceful for n = 2, 3, 4, 6, 9, 11, 16, and 18.
They conjecture that Kn is Hamming-graceful for n of the forms k2 and k2 + 2 for k ≥5.
the electronic journal of combinatorics (2023), #DS6 117 4 Variations of Harmonious Labelings 4.1 Sequential and Strongly c-harmonious Labelings Chang, Hsu, and Rogers and Grace , have investigated subclasses of harmonious graphs. Chang et al. define an injective labeling f of a graph G with q vertices to be strongly c-harmonious if the vertex labels are from {0, 1, . . . , q −1} and the edge labels induced by f(x) + f(y) for each edge xy are c, . . . , c + q −1. Strongly 1-harmonious labelings are more simply called strongly harmonious. Grace called such a labeling sequential. In the case of a tree, Chang et al. modify the definition to permit exactly one vertex label to be assigned to two vertices whereas Grace allows the vertex labels to range from 0 to q with no vertex label being used twice. For graphs other than trees, we use the term c-sequential labelings interchangeably with strongly c-harmonious labelings. By taking the edge labels of a sequentially labeled graph with q edges modulo q, we obviously obtain a harmoniously labeled graph. It is not known if there is a graph that can be harmoniously labeled but not sequentially labeled. Grace proved that caterpillars, caterpillars with a pendent edge, odd cycles with zero or more pendent edges, trees with α-labelings, wheels W2n+1, and P 2 n are sequential. Liu and Zhang finished off the crowns C2n ⊙K1. (The case C2n+1 ⊙K1 was a special case of Grace’s results.
Liu proved crowns are harmonious.) Bača and Youssef investigated the existence of harmonious labelings for the corona graphs of a cycle and a graph G. They proved that if G+K1 is strongly harmonious with the 0 label on the vertex of K1, then Cn ⊙G is harmonious for all odd n ≥3. By combining this with existing results they have as corollaries that the following graphs are harmonious: Cn ⊙Cm for odd n ≥3 and m ̸≡2 (mod 3); Cn ⊙Ks,t for odd n ≥3; and Cn ⊙K1,s,t for odd n ≥3.
Bu also proved that crowns are sequential as are all even cycles with m pendent edges attached at each vertex. Figueroa-Centeno, Ichishima, and Muntaner-Batle proved that all cycles with m pendent edges attached at each vertex are sequential. Wu has shown that caterpillars with m pendent edges attached at each vertex are sequential. exactly one path of fixed length to each vertex of some path is sequential.
Singh has proved the following: Cn ⊙K2 is sequential for all odd n > 1 ; Cn ⊙P3 is sequential for all odd n ; K2 ⊙Cn (each vertex of the cycle is joined by edges to the end points of a copy of K2) is sequential for all odd n ; helms Hn are sequential when n is even ; and K1,n + K2, K1,n + K2, and ladders are sequential .
Santhosh has shown that Cn ⊙P4 is sequential for all odd n ≥3. Both Grace and Reid (see ) have found sequential labelings for the books B2n. Jungreis and Reid have shown the following graphs are sequential: Pm × Pn (m, n) ̸= (2, 2); C4m × Pn (m, n) ̸= (1, 2); C4m+2 × P2n; C2m+1 × Pn; and C4 × C2n (n > 1). The graphs C4m+2 × C2n+1 and C2m+1 × C2n+1 fail to satisfy a necessary parity condition given by Graham and Sloane . The remaining cases of Cm × Pn and Cm × Cn are open.
Gallian, Prout, and Winters proved that all graphs Cn × P2 with a vertex or an edge deleted are sequential. Zhu and Liu give necessary and sufficient conditions the electronic journal of combinatorics (2023), #DS6 118 for sequential graphs, provide a characterization of non-tree sequential graphs by way of by vertex closure, and obtain characterizations of sequential trees.
Gnanajothi [pp. 68-78] has shown the following graphs are sequential: K1,m,n; mCn, the disjoint union of m copies of Cn if and only if m and n are odd; books with triangular pages or pentagonal pages; and books of the form B4n+1, thereby answering a question and proving a conjecture of Gallian and Jungreis . Sun has also proved that Bn is sequential if and only if n ̸≡3 (mod 4). Ichishima and Oshima pose determining whether or not mKs,t is sequential as a problem.
Yuan and Zhu have shown that mCn is sequential when m and n are odd.
Although Graham and Sloane proved that the Möbius ladder M3 is not harmonious, Gallian established that all other Möbius ladders are sequential (see §2.3 for the definition of Möbius ladder). Chung, Hsu, and Rogers have shown that Km,n + K1, which includes Sm + K1, is sequential.
Seoud and Youssef proved that if G is sequential and has the same number of edges as vertices, then G + Kn is sequential for all n. Recall that Θ(Cm)n denotes the book with n m-polygonal pages. Lu proved that Θ(C2m+1)2n is 2mn-sequential for all n and m = 1, 2, 3, 4, and Θ(Cm)2 is (m−2)-sequential if m ≥3 and m ≡2, 3, 4, 7 (mod 8).
Zhou and Yuan have shown that for every c-sequential graph G with p vertices and q edges and any positive integer m the graph (G + Km) + Kn is also c-sequential when q −p + 1 ≤m ≤q −p + c. Zhou has shown that the analogous results hold for strongly c-harmonious graphs. Zhou and Yuan have shown that for every c-sequential graph G with p vertices and q edges and any positive integer m the graph (G + Km) + Kn is c-sequential when q −p + 1 ≤m ≤q −p + c.
Shee proved that every graph is a subgraph of a sequential graph. Acharya, Germina, Princy, and Rao prove that every connected graph can be embedded in a strongly c-harmonious graph for some c. Miao and Liang use Cn(d; i, j; Pk) to denote a cycle Cn with path Pk joining two nonconsecutive vertices xi and xj of the cycle, where d is the distance between xi and xj on Cn. They proved that the graph Cn(d; i, j; Pk) is strongly c-harmonious when k = 2, 3 and integer n ≥6.
Lu provides three techniques for constructing larger sequential graphs from some smaller one: an attaching construction, an adjoining construction, and the join of two graphs.
Using these, he obtains various families of sequential or strongly c-indexable graphs.
For 1 ≤s ≤n3, let Cn(i : i1, i2, . . . , is) denote an n-cycle with consecutive vertices x1, x2, . . . , xn to which the s chords xixi1, xixi2, . . . , xixis have been added. Liang proved a variety of graphs of the form Cn(i : i1, i2, . . . , is) are strongly c-harmonious.
Youssef observed that a strongly c-harmonious graph with q edges is c-cordial for all c ≥q and a strongly k-indexable graph is k-cordial for every k. The converse of this latter result is not true.
In Ichishima and Oshima show that the hypercube Qn (n ≥2) is sequential if and only if n ≥4. They also introduce a special kind of sequential labeling of a graph G with size 2t + s by defining a sequential labeling f to be a partitional labeling if G is bipartite with partite sets X and Y of the same cardinality s such that f(x) ≤t + s −1 for all x ∈X and f(y) ≥t −s for all y ∈Y , and there is a positive integer m such that the electronic journal of combinatorics (2023), #DS6 119 the induced edge labels are partitioned into three sets [m, m+t−1], [m+t, m+t+s−1], and [m + t + s, m + 2t + s −1] with the properties that there is an involution π, which is an automorphism of G such that π exchanges X and Y , xπ(x) ∈E(G) for all x ∈X, and {f(x) + f(π(x))| x ∈X} = [m + t, m + t + s −1]. They prove if G has a partitional labeling, then G × Qn has a partitional labeling for every nonnegative integer n. Using this together with existing results and the fact that every graph that has a partitional labeling is sequential, harmonious, and felicitous (see §4.5) they show that the following graphs are partitional, sequential, harmonious, and felicitous: for n ≥4, hypercubes Qn; generalized books S2m × Qn; and generalized ladders P2m+1 × Qn.
In Ichishma and Oshima proved the following: if G is a partitional graph, then G × K2 is partitional, sequential, harmonious and felicitous; if G is a connected bipartite graph with partite sets of distinct odd order such that in each partite set each vertex has the same degree, then G × K2 is not partitional; for every positive integer m, the book Bm is partitional if and only if m is even; the graph B2m × Qn is partitional if and only if (m, n) ̸= (1, 1); the graph Km,2 × Qn is partitional if and only if (m, n) ̸= (2, 1); for every positive integer n, the graph Km,3 × Qn is partitional when m = 4, 8, 12, or 16. As open problems they ask which m and n is Km,n × K2 partitional and for which l, m and n is Kl,m × Qn partitional?
Ichishma and Oshima also investigated the relationship between partitional graphs and strongly graceful graphs (see §3.1 for the definition) and partitional graphs and strongly felicitous graphs (see §4.5) for the definition). They proved the following. If G is a partitional graph, then G×K2 is partitional, sequential, harmonious and felicitous.
Assume that G is a partitional graph of size 2t + s with partite sets X and Y of the same cardinality s, and let f be a partitional labeling of G such that λ1 = max{f(x) : x ∈X} and λ2 = max{f(y) : y ∈Y }. If λ1 + 1 = m + 2t + s −λ2, where m = min{f(x) + f(y) : xy ∈E(G)} = min{f(y) : y ∈Y }, then G has a strong α-valuation. Assume that G is a partitional graph of size 2t + s with partite sets X and Y of the same cardinality s, and let f be a partitional labeling of G such that λ1 = max{f(x) : x ∈X} and λ2 = max{f(y) : y ∈Y }. If λ1 + 1 = m + 2t + s −λ2, where m = min{f(x) + f(y) : xy ∈E(G)} = min{f(y) : y ∈Y }, then G is strongly felicitous. Assume that G is a partitional graph of size 2t + s with partite sets X and Y of the same cardinality s, and let f be a partitional labeling of G such that µ1 = f(x1) = min{f(x) : x ∈X} and µ2 = f(y1) = min{f(y) : y ∈Y }. If t + s = m + 1 and µ1 + µ2 = m, where m = min{f(x) + f(y) : xy ∈E(G)} and x1y1 ∈E(G), then G has a strong α-valuation and strongly felicitous labeling.
Vaidya and Lekha proved the following graphs are odd sequential: Pn, Cn for n ≡0 (mod 4), crowns Cn J K1 for even n, the graph obtained by duplication of arbitrary vertex in even cycles, path unions of stars, arbitrary super subdivisions in Pn, and shadows of stars. They also introduced the concept of a bi-odd sequential labeling of a graph G as one for which both G and its line graph L(G) admit odd sequential labeling. They proved Pn and Cn for n ≡(mod 4) are bi-odd sequential graphs and trees are bi-odd sequential if and only if they are paths. They also prove that P4 is the only graph with the property that it and its complement are odd sequential.
the electronic journal of combinatorics (2023), #DS6 120 Arockiaraj, Mahalakshmi, and Namasivayam proved that the subdivision graphs of the following graphs have odd sequential labelings (they call them odd sum labelings): triangular snakes; quadrilateral snakes; slanting ladders SLn (n > 1) (the graphs obtained from two paths u1u2 . . . un and v1v2 . . . vn by joining each ui with vi+1); Cp ⊙K1, Hn ⊙ K1, Cm@Cn (the graph obtained by attaching paths Pn to Cm by identifying the endpoints of the paths with each successive pairs of vertices of Cm); Pm×Pn; and graphs obtained by the duplication of a vertex of a path and the duplication of a vertex of a cycle. Arockiaraj, Mahalakshmi, and Namasivayam investigate the odd sum labeling behavior of paths, combs, cycles, crowns, and ladders under duplication of an edge. In they investigated the odd sum property of shadow graphs, edge duplication graphs and vertex identification graphs.
In Gopi proved the following graphs are odd sum graphs: graphs Hn obtained from two copies of Pn (n ≥3) with vertices v1, v2, . . . , vn and u1, u2, . . . , un by joining v(n+1)/2 and u(n+1)/2 if n is odd and vn/2 and u(n+2)/2 if n is even; graphs obtained from Hn by attaching a fixed number of pendent edges at each vertex, graphs obtained from Pn (n ≥4) by attaching a two pendent edges at each interior vertex; and graphs obtained from Pm (m ≥4) by identifying an endpoint of the star Sn (n ≥2) with each vertex of Pm. In Gopi and Irudaya Mary proved that slanting ladders, shadow graphs of stars and bistars and mirror graphs and duplicate vertex graphs of paths with at least four vertices are odd sum graphs. In Gopi proved that alternative quadrilateral snakes A(D(Qn)) (n ≥4) are odd sum graphs.
Arockiaraj and Mahalakshmi proved the following graphs have odd sequential labelings (odd sum labelings): Pn (n > 1), Cn if and only if n ≡0 (mod 4); C2n ⊙ K1; Pn × P2 (n > 1); Pm ⊙K1 if m is even or m is odd and n = 1 or 2; the balloon graph Pm(Cn) obtained by identifying an end point of Pm with a vertex of Cn if either n ≡0 (mod 4) or n ≡2 (mod 4) and m ̸≡1 (mod 3); quadrilateral snakes Qn; Pm ⊙Cn if m > 1 and n ≡0 (mod 4); Pm ⊙Q3; bistars; C2n × P2; the trees T n p obtained from n copies of Tp by joining an edge uu′ between every pair of consecutive paths where u is a vertex in ith copy of the path and u′ is the corresponding vertex in the (i + 1)th copy of the path; Hn-graphs obtained by starting with two copies of Pn with vertices v1, v2, . . . , vn and u1, u2, . . . , un and joining the vertices v(n+1)/2 and u(n+1)/2 if n is odd and the vertices vn/2+1 and un/2 if n; and Hn ⊙mK1.
Arockiaraj and Mahalakshmi proved the splitting graphs of following graphs have odd sequential labelings (odd sum labelings): Pn; Cn if and only if n ≡0 (mod 4); Pn ⊙ K1; C2n ⊙K1; K1,n if and only if n ≤2; Pn × P2 (n > 1); slanting ladders SLn (n > 1); the quadrilateral snake Qn; and Hn-graphs.
Among the strongly 1-harmonious (also called strongly harmonious) graphs are: fans Fn with n ≥2 ; wheels Wn with n ̸≡2 (mod 3) ; Km,n + K1 ; French windmills K(t) 4 , ; the friendship graphs C(n) 3 if and only if n ≡0 or 1 (mod 4) , , ; C(t) 4k ; and helms .
Seoud, Diab, and Elsakhawi have shown that the following graphs are strongly harmonious: Km,n with an edge joining two vertices in the same partite set; K1,m,n; the composition Pn[P2] (see §2.3 for the definition); B(3, 2, m) and B(4, 3, m) for all m (see §2.4 for the notation); P 2 n (n ≥3); and P 3 n (n ≥3).
Seoud et al. have also the electronic journal of combinatorics (2023), #DS6 121 proved: B2n is strongly 2n-harmonious; Pn is strongly ⌊n/2⌋-harmonious; ladders L2k+1 are strongly (k + 1)-harmonious; and that if G is strongly c-harmonious and has an equal number of vertices and edges, then G + Kn is also strongly c-harmonious.
Bača and Youssef investigated the existence of harmonious labelings for the corona graphs of a cycle and a graph G, and for the corona graph of K2 and a tree. They prove: if join of a graph G of order p and K1, G + K1, is strongly harmonious with the 0 label on the vertex of K1, then the corona of Cn with G, Cn ⊙G, is harmonious for all odd n ≥3; if T is a strongly c-harmonious tree of odd size q and c = q+1 2 then the corona of K2 with T, K2 ⊙T, is also strongly c-harmonious; if a unicyclic graph G of odd size q is a strongly c-harmonious and c = q−1 2 then the corona of K2 with G, K2 ⊙G, is also strongly c-harmonious.
Seenivasan and Lourdusamy define an absolutely harmonious labeling f as an injection from the vertex set of a graph G with q edges to the set {0, 1, 2, . . . , q −1}, if when each edge uv is assigned f(u) + f(v), the resulting edge labels can be arranged as a0, a1, a2, . . . , aq−1 where ai = q −i or q + i for 0 ≤i ≤q −1. When G is a tree one of the vertex labels may be assigned to exactly two vertices. A graph that admits absolutely harmonious labeling is called an absolutely harmonious graph. Observe that a strongly harmonious graph is an absolutely harmonious graph. They prove the following graphs are absolutely harmnious: Pn (n ≥3), Pn ⊙Km, Cn ⊙Km, the banana tree obtained by joining a vertex of degree 1 of each of any number of copies of K1,n to an isolated vertex, ladders, triangular snakes, quadrilateral snakes, mK4, Kn if and only if n = 3 or 4. They also prove that if G is an absolutely harmonious graph, then there exists a partition (V1, V2) of the vertex set V (G), such that the number of edges connecting the vertices of V1 to the vertices of V2 is exactly ⌈q/2⌉and that if every vertex of an absolutely harmonious graph with q edges is even then q ≡1 or 2. As corollaries of the latter condition, they have that Cn when n ≡1 or 2 (mod 4), Cm × Cn when m and n are odd, and mK3, m ≥2 are not absolutely harmonious.
Sethuraman and Selvaraju have proved that the graph obtained by joining two complete bipartite graphs at one edge is graceful and strongly harmonious. They ask whether these results extend to any number of complete bipartite graphs.
For a graph G(V, E) Gayathri and Hemalatha define an even sequential harmo-nious labeling f of G as an injection from V to {0, 1, 2, . . . , 2|E|} with the property that the induced mapping f + from E to {2, 4, 6, . . . , 2|E|} defined by f +(uv) = f(u) + f(v) when f(u)+f(v) is even, and f +(uv) = f(u)+f(v)+1 when f(u)+f(v) is odd, is an in-jection. They prove the following have even sequential harmonious labelings (all cases are the nontrivial ones): Pn, P + n , Cn( n ≥3), triangular snakes, quadrilateral snakes, Möbius ladders, Pm × Pn (m ≥2, n ≥2), Km,n; crowns Cm ⊙K1, graphs obtained by joining the centers of two copies of K1,n by a path; banana trees (see §2.1), P 2 n, closed helms (see §2.2), C3 ⊙nK1 (n ≥2); D ⊙K1,n where D is a dragon (see §2.2); ⟨K1,n : m⟩(m, n ≥2) (see §4.5); the wreath product Pn ∗K2 (n ≥2) (see §4.5); combs Pn ⊙K1; the one-point union of the end point of a path to a vertex of a cycle (tadpole); the one-point union of the end point of a tadpole and the center of a star; the graphs PCn obtained from Cn = v0, v1, v2, . . . , vn−1 by adding the cords v1vn−1, v2vn−2, . . . , v(n−2)/2, v(n+2)/2 when n the electronic journal of combinatorics (2023), #DS6 122 is even and v1vn−1, v2vn−2, . . . , v(n−3)/2, v(n+3)/2 when n is odd (that is, cycles with a full set of cords); Pm ⊙nK1; the one-point union of a vertex of a cycle and the center of a star; graphs obtained by joining the centers of two stars with an edge; graphs obtained by joining two disjoint cycles with an edge (dumbbells); graphs consisting of two even cycles of the same order sharing a common vertex with an arbitrary number of pendent edges attached at the common vertex (butterflies).
In Ichishima, Muntaner-Batle, and Oshima define the harmonious number, η(G), of a graph G with q edges as the smallest positive integer n for which there exists an injective function f from V (G) to Zn+1 such that each uv of G is labeled f(u) + f(v) (mod q) and the resulting edge labels are distinct, or +∞if there exists no such integer n.
If such functions exist, they are called harmonious numberings. The strong harmonious number, ηs(G), of a graph G is defined to be either the smallest positive integer n such that n = η(G) with the additional property that there exists an integer λ such that min{f(u), f(v)} ≤λ ≤max{f(u), f(v)} for each edge in G or +∞if there exists no such integer n. They provide a necessary condition for a graph to have a finite harmonious number and sufficient conditions for a graph to have an infinite (strong) harmonious number. In addition, they examine the relations between harmonious numbers, gamma-numbers, alpha-numbers, and super edge magic deficiencies (see §5.2). They determine the formulas for the (strong) harmonious numbers of some 2-regular graphs and all complete bipartite graphs.
In her PhD thesis (see also ) Muthuramakrishnan defined a labeling f of a graph G(V, E) to be k-even sequential harmonious if f is an injection from V to {k −1, k, k + 1, . . . , k + 2q −1} such that the induced mapping f + from E to {2k, 2k + 2, 2k + 4, . . . , 2k + 2q −2} defined by f +(uv) = f(u) + f(v) if f(u) + f(v) is even and f +(uv) = f(u) + f(v) + 1 if f(u) + f(v) is odd are distinct. A graph G is called a k-even sequential harmonious graph if it admits a k-even sequential harmonious labeling. Among the numerous graphs that she proved to be k-even sequential harmonious are: paths, cycles, Km,n, P 2 n (n ≥3), crowns Cm ⊙K1, Cm@Pn (the graph obtained by identifying an endpoint of Pn with one vertex of Cm), double triangular snakes, double quadrilateral snakes, bistars, grids Pm×Pn (m, n ≥2), Pn[P2], C3⊙nK1 (n ≥2), flags Flm (the cycle Cm with one pendent edge), dumbbell graphs (two disjoint cycles joined by an edge) butterfly graphs Bn (two even cycles of the same order sharing a common vertex with an arbitrary number of pendent edges attached at the common vertex), K2 + nK1, Kn + 2K2, banana trees, sparklers Pm@K1,n (m, n ≥2), (graphs obtained by identifying an endpoint of Pm with the center of a star), twigs (graphs obtained from Pn (n ≥3) by attaching exactly two pendent edges at each internal vertex of Pn), festoon graphs Pm ⊙nK1 (m ≥2), the graphs Tm,n,t obtained from a path Pt by appending m edges at one endpoint of Pt and n edges at the other endpoint of Pt, Ln ⊙K1 (Ln is the ladder Pn × P2), shadow graphs of paths, stars and bistars, and split graphs of paths and stars. Muthuramakrishnan also defines k-odd sequential harmonious labeling of graphs in the natural way and obtains a handful of results.
In Beatress and Sarasija introduced a new harmonious-like labeling as follows.
A graph G(V, E) with n vertices and m edges is said to be a square harmonious graph if the electronic journal of combinatorics (2023), #DS6 123 there exists an injection f from V (G) to {1, 2, . . . , m2 +1} such that the induced mapping f ∗from E(G) to {1, 4, 9, . . . , m2} defined by f ∗(uv) = (f(u) + f(v)) mod(m2 + 1) is a bijection. Such a function f is called a square harmonious labeling of G. They prove that Pn (n ≥3), K1,n (n ≥2), bistars, combs, Pn ⊙pK1 (n ≥2), and C3@pK1 (p ≥2) are square harmonious graphs. Lawas and Lim proved that stars have a square harmonious labelings.
In Barrientos and Youssef generalize the concepts of harmonious and (k, d)-arithmetic graphs by relaxing the injectivity constraint of the corresponding labelings.
They call these labelings semi harmonious and semi (k, d)-arithmetic.
They showed the existence of a semi harmonious labeling for several types of cycle-related graphs and characterized the cycles that admit semi harmonious labelings. They proved that if G is semi (k, d)-arithmetic, then it is also semi (rk, rd)-arithmetic for every r ≥1 and that nG is both semi (k, d)-arithmetic and semi (k + d, d)-arithmetic. They further showed that any graph whose components are complete bipartite graphs is semi (k, d)-arithmetic for any ordered pair (k, d) of positive integers, and if Gi is a semi (ki, d)-arithmetic graph of size qi for each i = 1, 2, then G1 ∪G2 is semi (ki, d)-arithmetic.
4.2 (k, d)-arithmetic Labelings Acharya and Hegde have generalized sequential labelings as follows.
Let G be a graph with q edges and let k and d be positive integers. A labeling f of G is said to be (k, d)-arithmetic if the vertex labels are distinct nonnegative integers and the edge labels induced by f(x)+f(y) for each edge xy are k, k+d, k+2d, . . . , k+(q−1)d. They obtained a number of necessary conditions for various kinds of graphs to have a (k, d)-arithmetic labeling. The case where k = 1 and d = 1 was called additively graceful by Hegde .
Hegde showed: Kn is additively graceful if and only if n = 2, 3, or 4; every additively graceful graph except K2 or K1,2 contains a triangle; and a unicyclic graph is additively graceful if and only if it is a 3-cycle or a 3-cycle with a single pendent edge attached.
Jinnah and Singh noted that P 2 n is additively graceful. Hegde proved that if G is strongly k-indexable, then G and G + Kn are (kd, d)-arithmetic. Acharya and Hegde proved that Kn is (k, d)-arithmetic if and only if n ≥5 (see also ).
They also proved that a graph with an α-labeling is a (k, d)-arithmetic for all k and d. Bu and Shi proved that Km,n is (k, d)-arithmetic when k is not of the form id for 1 ≤i ≤n −1. For all d ≥1 and all r ≥0, Acharya and Hegde showed the following: Km,n,1 is (d + 2r, d)-arithmetic; C4t+1 is (2dt + 2r, d)-arithmetic; C4t+2 is not (k, d)-arithmetic for any values of k and d; C4t+3 is ((2t + 1)d + 2r, d)-arithmetic; W4t+2 is (2dt + 2r, d)-arithmetic; and W4t is ((2t + 1)d + 2r, d)-arithmetic. They conjecture that C4t+1 is (2dt + 2r, d)-arithmetic for some r and that C4t+3 is (2dt + d + 2r, d)-arithmetic for some r. Hegde and Shetty proved the following: the generalized web W(t, n) (see §2.2 for the definition) is ((n −1)d/2, d)-arithmetic and ((3n −1)d/2, d)-arithmetic for odd n; the join of the generalized web W(t, n) with the center removed and Kp where n is odd is ((n −1)d/2, d)-arithmetic; every Tp-tree (see §3.2 for the definition) with q edges and every tree obtained by subdividing every edge of a Tp-tree exactly once is the electronic journal of combinatorics (2023), #DS6 124 (k +(q −1)d, d)-arithmetic for all k and d. Lu, Pan, and Li proved that K1,m ∪Kp,q is (k, d)-arithmetic when k > (q −1)d + 1 and d > 1.
Yu proved that a necessary condition for C4t+1 to be (k, d)-arithmetic is that k = 2dt + r for some r ≥0 and a necessary condition for C4t+3 to be (k, d)-arithmetic is that k = (2t + 1)d + 2r for some r ≥0. These conditions were conjectured by Acharya and Hegde . Singh proved that the graph obtained by subdividing every edge of the ladder Ln is (5, 2)-arithmetic and that the ladder Ln is (n, 1)-arithmetic .
He also proves that Pm × Cn is ((n −1)/2, 1)-arithmetic when n is odd . Acharya, Germina, and Anandavally proved that the subdivision graph of the ladder Ln is (k, d)-arithmetic if either d does not divide k or k = rd for some r ≥2n and that Pm ×Pn and the subdivision graph of the ladder Ln are (k, k)-arithmetic if and only if k is at least 3. Lu, Pan, and Li proved that Sm ∪Kp,q is (k, d)-arithmetic when k > (q −1)d+1 and d > 1.
A graph is called arithmetic if it is (k, d)-arithmetic for some k and d. Singh and Vilfred showed that various classes of trees are arithmetic. Singh has proved that the union of an arithmetic graph and an arithmetic bipartite graph is arithmetic. He conjectures that the union of arithmetic graphs is arithmetic. He provides an example to show that the converse is not true.
Germina and Anandavally investigated embedding of graphs in arithmetic graphs.
They proved: every graph can be embedded as an induced subgraph of an arithmetic graph; every bipartite graph can be embedded in a (k, d)-arithmetic graph for all k and d such that d does not divide k; and any graph containing an odd cycle cannot be embedded as an induced subgraph of a connected (k, d)-arithmetic with k < d.
In Yao, Liu, and Yao give necessary and sufficient conditions for a tree to have the following mutually equivalent labelings: set-ordered odd-graceful, (k, d)-graceful, super edge-magic total, odd-elegant (see §4.4), harmonious, (k, d)-arithmetic, and edge-antimagic (see §6.1).
4.3 (k, d)-indexable Labelings Acharya and Hegde call a graph with p vertices and q edges (k, d)-indexable if there is an injective function from V to {0, 1, 2, . . . , p−1} such that the set of edge labels induced by adding the vertex labels is a subset of {k, k+d, k+2d, . . . , k+q(d−1)}. When the set of edges is {k, k+d, k+2d, . . . , k+q(d−1)} the graph is said to be strongly (k, d)-indexable. A (k, 1)-graph is more simply called k-indexable and strongly 1-indexable graphs are simply called strongly indexable. Notice that strongly indexable graphs are a stronger form of sequential graphs and for trees and unicyclic graphs the notions of sequential labelings and strongly k-indexable labelings coincide. Hegde and Shetty have shown that the notions of (1, 1)-strongly indexable graphs and super edge-magic total labelings (see §5.2) are equivalent.
Zhou has shown that for every k-indexable graph G with p vertices and q edges the graph (G+Kq−p+k)+K1 is strongly k-indexable. Acharaya and Hegde prove that the only nontrivial regular graphs that are strongly indexable are K2, K3, and K2 × K3, and the electronic journal of combinatorics (2023), #DS6 125 that every strongly indexable graph has exactly one nontrivial component that is either a star or has a triangle. Acharya and Hegde call a graph with p vertices indexable if there is an injective labeling of the vertices with labels from {0, 1, 2, . . . , p −1} such that the edge labels induced by addition of the vertex labels are distinct. They conjecture that all unicyclic graphs are indexable. This conjecture was proved by Arumugam and Germina who also proved that all trees are indexable. Bu and Shi also proved that all trees are indexable and that all unicyclic graphs with the cycle C3 are indexable.
Hegde has shown the following: every graph can be embedded as an induced subgraph of an indexable graph; if a connected graph with p vertices and q edges (q ≥2) is (k, d)-indexable, then d ≤2; Pm × Pn is indexable for all m and n; if G is a connected (1, 2)-indexable graph, then G is a tree; the minimum degree of any (k, 1)-indexable graph with at least two vertices is at most 3; a caterpillar with partite sets of orders a and b is strongly (1, 2)-indexable if and only if |a −b| ≤1; in a connected strongly k-indexable graph with p vertices and q edges, k ≤p −1; and if a graph with p vertices and q edges is (k, d)-indexable, then q ≤(2p −3 −k + d)/d. As a corollary of the latter, it follows that Kn (n ≥4) and wheels are not (k, d)-indexable.
Lee and Lee provide a way to construct a (k, d)-strongly indexable graph from two given (k, d)-strongly indexable graphs. Lee and Lo show that every given (1,2)-strongly indexable spider can extend to an (1,2)-strongly indexable spider with arbitrarily many legs.
Seoud, Abd El Hamid, and Abo Shady proved the following graphs are in-dexable: Pm × Pn (m, n ≥2); the graphs obtained from Pn + K1 by inserting one vertex between every two consecutive vertices of Pn; the one-point union of any number of copies of K2,n; and the graphs obtained by identifying a vertex of a cycle with the center of a star.
They showed Pn is strongly ⌈n/2⌉-indexable; odd cycles Cn are strongly ⌈n/2⌉-indexable; K(m,n) (m, n > 2) is indexable if and only if m or n is at most 2. For a simple indexable graph G(V, E) they proved |E| ≤2|V | −3. Also, they determine all indexable graphs of order at most 6.
Hegde and Shetty also prove that if G is strongly k-indexable Eulerian graph with q edges then q ≡0, 3 (mod 4) if k is even and q ≡0, 1 (mod 4) if k is odd. They further showed how strongly k-indexable graphs can be used to construct polygons of equal internal angles with sides of different lengths.
Germina has proved the following: fans Pn + K1 are strongly indexable if and only if n = 1, 2, 3, 4, 5, 6; Pn + K2 is strongly indexable if and only if n ≤2; the only strongly indexable complete m-partite graphs are K1,n and K1,1,n; ladders Pn×P2 are ⌈n 2⌉-strongly indexable, if n is odd; Kn×Pk is a strongly indexable if and only if n = 3; Cm×Pn is 2-strongly indexable if m is odd and n ≥2; K1,n+Ki is not strongly indexable for n ≥2; for Gi ∼ = K1,n, 1 ≤i ≤n, the sequential join G ∼ = (G1+G2)∪(G2+G3)∪· · ·∪(Gn−1+Gn) is strongly indexable if and only if, either i = n = 1 or i = 2 and n = 1 or i = 1, n = 3; P1∪Pn is strongly indexable if and only if n ≤3; P2∪Pn is not strongly indexable; P2∪Pn is ⌈n+3 2 ⌉-strongly indexable; mCn is k-strongly indexable if and only if m and n are odd; K1,n ∪K1,n+1 is strongly indexable; and mK1,n is ⌈3m−1 2 ⌉-strongly indexable when m is odd.
the electronic journal of combinatorics (2023), #DS6 126 Acharya and Germina proved that every graph can be embedded in a strongly indexable graph and gave an algorithmic characterization of strongly indexable unicyclic graphs. In they provide necessary conditions for an Eulerian graph to be strongly k-indexable and investigate strongly indexable (p, q)-graphs for which q = 2p −3.
Hegde and Shetty proved that for n odd the generalized web graph W(t, n) with the center removed is strongly (n−1)/2-indexable. Hegde and Shetty define a level joined planar grid as follows. Let u be a vertex of Pm × Pn of degree 2. For every pair of distinct vertices v and w that do not have degree 4, introduce an edge between v and w provided that the distance from u to v equals the distance from u to w. They prove that every level joined planar grid is strongly indexable. For any sequence of positive integers (a1, a2, . . . , an) Lee and Lee show how to associate a strongly indexable (1, 1)-graph. As a corollary, they obtain the aforementioned result Hegde and Shetty on level joined planar grids.
Section 5.2 of this survey includes a discussion of a labeling method called super edge-magic. In 2002 Hegde and Shetty showed that a graph has a strongly k-indexable labeling if and only if it has a super edge-magic labeling.
4.4 Elegant Labelings In 1981 Chang, Hsu, and Rogers defined an elegant labeling f of a graph G with q edges as an injective function from the vertices of G to the set {0, 1, . . . , q} such that when each edge xy is assigned the label f(x)+f(y) (mod (q+1)) the resulting edge labels are distinct and nonzero. An injective labeling f of a graph G with q vertices is called strongly k-elegant if the vertex labels are from {0, 1, . . . , q} and the edge labels induced by f(x) + f(y) (mod (q + 1)) for each edge xy are k, . . . , k + q −1. Note that in contrast to the definition of a harmonious labeling, for an elegant labeling it is not necessary to make an exception for trees.
Whereas the cycle Cn is harmonious if and only if n is odd, Chang et al. proved that Cn is elegant when n ≡0 or 3 (mod 4) and not elegant when n ≡1 (mod 4). Chang et al. further showed that all fans are elegant and the paths Pn are elegant for n ̸≡0 (mod 4). Cahit then showed that P4 is the only path that is not elegant. Balakrishnan, Selvam, and Yegnanarayanan have proved numerous graphs are elegant. Among them are Km,n and the mth-subdivision graph of K1,2n for all m. They prove that the bistar Bn,n (K2 with n pendent edges at each endpoint) is elegant if and only if n is even. They also prove that every simple graph is a subgraph of an elegant graph and that several families of graphs are not elegant. Deb and Limaye have shown that triangular snakes (see §2.2 for the definition) are elegant if and only if the number of triangles is not equal to 3 (mod 4).
In the case where the number of triangles is 3 (mod 4) they show the triangular snakes satisfy a weaker condition they call semi-elegant whereby the edge label 0 is permitted. In Deb and Limaye define a graph G with q edges to be near-elegant if there is an injective function f from the vertices of G to the set {0, 1, . . . , q} such that when each edge xy is assigned the label f(x) + f(y) (mod (q + 1)) the resulting edge labels are distinct and not equal to q. Thus, in a near-elegant the electronic journal of combinatorics (2023), #DS6 127 labeling, instead of 0 being the missing value in the edge labels, q is the missing value.
Deb and Limaye show that triangular snakes where the number of triangles is 3 (mod 4) are near-elegant.
For any positive integers α ≤β ≤γ where β is at least 2, the theta graph θα,β,γ consists of three edge disjoint paths of lengths α, β, and γ having the same end points. Deb and Limaye provide elegant and near-elegant labelings for some theta graphs where α = 1, 2, or 3. Seoud and Elsakhawi have proved that the following graphs are elegant: K1,m,n; K1,1,m,n; K2 + Km; K3 + Km; and Km,n with an edge joining two vertices of the same partite set. Elumalai and Sethuraman proved P n 2 , P 2 m + Kn, Sm + Sn, Sm + Km, C3 × Pm, and even cycles C2n with vertices a0, a1, . . . , a2n−1, a0 and 2n −3 chords a0a2, a0a3, . . . , a0a2n−2 (n ≥2) are elegant. Zhou has shown that for every strongly k-elegant graph G with p vertices and q edges and any positive integer m the graph (G + Km) + Kn is also strongly k-elegant when q −p + 1 ≤m ≤q −p + k.
If f is a strongly k-elegant labeling of a bipartite graph G with partite sets V1 and V2 and max f(u) < min f(v) for all u in V1 and v in V2, f is said to be a set-ordered strongly k-elegant labeling of G. Su, Wang, and Yao proved that if a connected (p, q)-graph admits a strongly k-elegant labeling, then q ≤2p−3 and if a graph is a set-ordered strongly k-elegant, then q = p −1. They constructed several classes of large-scale trees that have strongly k-elegant labelings through graph operations that connect edges between two vertices or identify two vertices to form a new graph and proved that caterpillars with p vertices admits a set-ordered strongly k-elegant labelings. They further showed that a graph admits a strongly k-elegant labeling if and only if it has a super edge-magic total labeling (SEMT)–see Section 5.2.
Sethuraman and Elumalai proved that every graph is a vertex induced subgraph of a elegant graph and present an algorithm that permits one to start with any non-trivial connected graph and successively form supersubdivisions (see §2.7) that have a strong form of elegant labeling. Acharya, Germina, Princy, and Rao prove that every (p, q)-graph G can be embedded in a connected elegant graph H. The construction is done in such a way that if G is planar and elegant (harmonious), then so is H.
In Sethuraman and Elumalai define a graph H to be a K1,m-star extension of a graph G with p vertices and q edges at a vertex v of G where m > p −1 −deg(v) if H is obtained from G by merging the center of the star K1,m with v and merging p−1−deg(v) pendent vertices of K1,m with the p −1 −deg(v) nonadjacent vertices of v in G. They prove that for every graph G with p vertices and q edges and for every vertex v of G and every m ≥2p−1 −1 −q, there is a K1,m-star extension of G that is both graceful and harmonious. In the case where m ≥2p−1 −q, they show that G has a K1,m-star extension that is elegant. Sethuraman and Selvaraju have shown that certain cases of the union of any number of copies of K4 with one or more edges deleted and one edge in common are elegant.
In Ephremnath and Elumlai say a graph G is a cycle with a chord Hamiltonian path if G is obtained from the cycle v0, v1, . . . , vn−1, v0 (n ≥6) by adding the chords v1vn−1, vvvn−2, . . . , vαvβ where α = β = (n −2)/2 if n is even and α = (n + 3)/2, β = (n −1)/2 if n is odd. They proved that Cn (n ≥6) with a chord Hamiltonian path is the electronic journal of combinatorics (2023), #DS6 128 harmonious and elegant.
Gallian extended the notion of harmoniousness to arbitrary finite Abelian groups as follows. Let G be a graph with q edges and H a finite Abelian group (under addition) of order q. Define G to be H-harmonious if there is an injection f from the vertices of G to H such that when each edge xy is assigned the label f(x) + f(y) the resulting edge labels are distinct. When G is a tree, one label may be used on exactly two vertices. Beals, Gallian, Headley, and Jungreis have shown that if H is a finite Abelian group of order n > 1 then Cn is H-harmonious if and only if H has a non-cyclic or trivial Sylow 2-subgroup and H is not of the form Z2 × Z2 × · · · × Z2. Thus, for example, C12 is not Z12-harmonious but is (Z2 ×Z2 ×Z3)-harmonious. In Ehard, Glock, and Joos apply rainbow colorings to graph decompositions and harmonious labeling of graphs.
Analogously, the notion of an elegant graph can be extended to arbitrary finite Abelian groups. Let G be a graph with q edges and H a finite Abelian group (under addition) with q + 1 elements. We say G is H-elegant if there is an injection f from the vertices of G to H such that when each edge xy is assigned the label f(x) + f(y) the resulting set of edge labels is the non-identity elements of H. Beals et al. proved that if H is a finite Abelian group of order n with n ̸= 1 and n ̸= 3, then Cn−1 is H-elegant using only the non-identity elements of H as vertex labels if and only if H has either a non-cyclic or trivial Sylow 2-subgroup. This result completed a partial characterization of elegant cycles given by Chang, Hsu, and Rogers by showing that Cn is elegant when n ≡2 (mod 4). Mollard and Payan also proved that Cn is elegant when n ≡2 (mod 4) and gave another proof that Pn is elegant when n ̸= 4. In 2014 Ollis used harmonious labelings for Zm given by Beals, Gallian, Headley, and Jungreis in to construct new Latin squares of odd order.
A function f is said to be an odd-elegant labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q −1 such that the induced mapping f ∗(uv) = f(u) + f(v) (mod 2q) from the edges of G to the odd integers between 1 to 2q −1 is a bijection. Zhou, Yao, and Chen proved that every lobster is odd-elegant. In Wang, Xu, Ma, and Zhang gave a new type of graphical passwords based on odd-elegant labeled graphs. See also and .
For a graph G(V, E) and an Abelian group H Valentin defines a polychrome labeling of G by H to be a bijection f from V to H such that the edge labels induced by f(uv) = f(v) + f(u) are distinct. Valentin investigates the existence of polychrome labelings for paths and cycles for various Abelian groups.
4.5 Felicitous Labelings Another generalization of harmonious labelings are felicitous labelings. An injective func-tion f from the vertices of a graph G with q edges to the set {0, 1, . . . , q} is called felicitous if the edge labels induced by f(x) + f(y) (mod q) for each edge xy are distinct. (Recall a harmonious labeling only allows the vertex labels 0, 1, . . . , q −1.) This definition first appeared in a paper by Lee, Schmeichel, and Shee in and is attributed to E.
Choo. labeling of the graph. Balakrishnan and Kumar proved the conjecture of the electronic journal of combinatorics (2023), #DS6 129 Lee, Schmeichel, and Shee that every graph is a subgraph of a felicitous graph by showing the stronger result that every graph is a subgraph of a sequential graph. Among the graphs known to be felicitous are: Cn except when n ≡2 (mod 4) ; Km,n when m, n > 1 ; P2 ∪C2n+1 ; P2 ∪C2n ; P3 ∪C2n+1 ; Sm ∪C2n+1 ; Kn if and only if n ≤4 ; Pn+Km ; the friendship graph C(n) 3 for n odd ; Pn ∪C3 ; Pn ∪Cn+3 ; and the one-point union of an odd cycle and a caterpil-lar . Shee conjectured that Pm ∪Cn is felicitous when n > 2 and m > 3. Lee, Schmeichel, and Shee ask for which m and n is the one-point union of n copies of Cm felicitous. They showed that in the case where mn is twice an odd integer the graph is not felicitous. In contrast to the situation for felicitous labelings, we remark that C4k and Km,n where m, n > 1 are not harmonious and the one-point union of an odd cycle and a caterpillar is not always harmonious. Lee, Schmeichel, and Shee conjectured that the n-cube is felicitous. This conjecture was proved by Figueroa-Centeno and Ichishima in 2001 .
Balakrishnan, Selvam, and Yegnanarayanan obtained numerous results on fe-licitous labelings.
The wreath product, G ∗H, of graphs G and H has vertex set V (G) × V (H) and (g1, h1) is adjacent to (g2, h2) whenever g1g2 ∈E(G) or g1 = g2 and h1h2 ∈E(H). They define Hn,n as the graph with vertex set {u1, . . . , un; v1, . . . , vn} and edge set {uivj| 1 ≤i ≤j ≤n}. They let ⟨K1,n : m⟩denote the graph obtained by taking m disjoint copies of K1,n, and joining a new vertex to the centers of the m copies of K1,n. They prove the following are felicitous: Hn,n; Pn ∗K2; ⟨K1,m : m⟩; ⟨K1,2 : m⟩ when m ̸≡0 (mod 3), or m ≡3 (mod 6), or m ≡6 (mod 12); ⟨K1,2n : m⟩for all m and n ≥2; ⟨K1,2t+1 : 2n+1⟩when n ≥t; P k n when k = n−1 and n ̸≡2 (mod 4), or k = 2t and n ≥3 and k < n −1; the join of a star and Kn; and graphs obtained by joining two end vertices or two central vertices of stars with an edge. Yegnanarayanan conjectures that the graphs obtained from an even cycle by attaching n new vertices to each vertex of the cycle is felicitous. This conjecture was verified by Figueroa-Centeno, Ichishima, and Muntaner-Batle in . In Sethuraman and Selvaraju have shown that certain cases of the union of any number of copies of K4 with 3 edges deleted and one edge in common are felicitous. Sethuraman and Selvaraju present an algorithm that permits one to start with any non-trivial connected graph and successively form supersubdivisions (see §2.7) that have a felicitous labeling. Krisha and Dulawat give algorithms for finding graceful, harmonious, sequential, felicitous, and antimagic (see §5.7) labelings of paths. A linear cactus Pm(Kn) is a connected graph in which all the blocks are isomorphic to a complete graph Kn and block-cutpoint is a path P2m−1. Go-mathi proved the follow graphs are felicitous: Pm(K4), splitting graphs of (Bn,n), planar graphs Plm,n, and C2k+1⊙Sm. Gomathi and Nagarajan proved the following graphs are felicitous: a vertex switching of Cn (n ≥4), a vertex switching of Cn (n ≥4) with one chord, a vertex duplication of Cn, and the square of the book Bn,n (n ≥2).
Ezhilarasi Hilda and Jeba Jesintha proved that all shell flower graphs are felicitous.
Devakirubanithi, Jeba Jesintha, and Abigail proved that the duplication of the apex new vertex of bistars, braid graphs, and globe graphs are felicitous.
In Sudhakar, Ranjani, Swathy, and Balaji provided a technique for coding a the electronic journal of combinatorics (2023), #DS6 130 secret messages by applying an even felicitous labeling for the union of two star graphs using a GMJ (Graph Message Jumbled) code. They include two illustrations for convert-ing plain text into cipher text (picture coding) and a method for a felicitous labeling of a graph. In Manshath, Hariprabakaran, Veerasamy, and Balaji used a GMJ code to create a confidential message by applying an even felicitous labeling to a bistar.
In Elumalai calls the graph obtained from a shell graph C(2n+3, 2n) by adding new a vertex in between each pair of adjacent vertices on the cycle C2n, adding an edge in apex, and adding two more chords a stem-lotus graph . He proved that stem-lotus graphs are graceful and felicitous.
Figueroa-Centeno, Ichishima, and Muntaner-Batle define a felicitous graph to be strongly felicitous if there exists an integer k so that for every edge uv, min{f(u), f(v)} ≤k < max{f(u), f(v)}. For a graph with p vertices and q edges with q ≥p −1 they show that G is strongly felicitous if and only if G has an α-labeling (see §3.1). They also show that for graphs G1 and G2 with strongly felicitous labelings f1 and f2 the graph obtained from G1 and G2 by identifying the vertices u and v such that f1(u) = 0 = f2(v) is strongly felicitous and that the one-point union of two copies of Cm where m ≥4 and m is even is strongly felicitous. As a corollary they have that the one-point union of n copies of Cm where m is even and at least 4 and n ≡2 (mod 4) is felicitous. They conjecture that the one-point union of n copies of Cm is felicitous if and only if mn ≡0, 1, or 3 (mod 4). In Figueroa-Centeno, Ichishima, and Muntaner-Batle prove that 2Cn is strongly felicitous if and only if n is even and at least 4. They conjecture that mCn is felicitous if and only if mn ̸≡2 (mod 4) and that Cm ∪Cn is felicitous if and only if m + n ̸≡2 (mod 4).
As consequences of their results about super edge-magic labelings (see §5.2) Figueroa-Centeno, Ichishima, Muntaner-Batle, and Oshima have the following corollaries: if m and n are odd with m ≥1 and n ≥3, then mCn is felicitous; 3Cn is felicitous if and only if n ̸≡2 (mod 4); and C5 ∪Pn is felicitous for all n.
For a graph G with q edges Shainy and Balaji call a one-to-one function f from V (G) to {0, 1, 2, . . . , 2q −1} a even felicitous if the edge labels generated by (f(r) + f(s)) mod(2q −1) for each edge are even and distinct. They proved that stars, bistars, the union two stars, and the union of three stars are even felicitous graph.
In Manickam, Marudai, and Kala prove the following graphs are felicitous: the one-point union of m copies of Cn if mn ≡1, 3 mod 4; the one-point union of m copies of C4; mCn if mn ≡1, 3 (mod 4); and mC4. These results partially answer questions raised by Figueroa-Centeno, Ichishima, Muntaner-Batle, and Oshima in and .
Chang, Hsu, and Rogers have given a sequential counterpart to felicitous la-belings. They call a graph with q edges strongly c-elegant if the vertex labels are from {0, 1, . . . , q} and the edge labels induced by addition are {c, c+1, . . . , c+q−1}. (A strongly 1-elegant labeling has also been called a consecutive labeling.) Notice that every strongly c-elegant graph is felicitous and that strongly c-elegant is the same as (c, 1)-arithmetic in the case where the vertex labels are from {0, 1, . . . , q}. Chang et al. have shown: Kn is strongly 1-elegant if and only if n = 2, 3, 4; Cn is strongly 1-elegant if and only if n = 3; and a bipartite graph is strongly 1-elegant if and only if it is a star. Shee the electronic journal of combinatorics (2023), #DS6 131 has proved that Km,n is strongly c-elegant for a particular value of c and obtained several more specialized results pertaining to graphs formed from complete bipartite graphs.
Seoud and Elsakhawi have shown: Km,n (m ≤n) with an edge joining two vertices of the same partite set is strongly c-elegant for c = 1, 3, 5, . . . , 2n + 2; K1,m,n is strongly c-elegant for c = 1, 3, 5, . . . , 2m when m = n, and for c = 1, 3, 5, . . . , m + n + 1 when m ̸= n; K1,1,m,m is strongly c-elegant for c = 1, 3, 5, . . . , 2m+1; Pn +Km is strongly ⌊n/2⌋-elegant; Cm + Kn is strongly c-elegant for odd m and all n for c = (m −1)/2, (m − 1)/2 + 2, . . . , 2m when (m −1)/2 is even and for c = (m −1)/2, (m −1)/2 + 2, . . . , 2m − (m −1)/2 when (m −1)/2 is odd; ladders L2k+1 (k > 1) are strongly (k + 1)-elegant; and B(3, 2, m) and B(4, 3, m) (see §2.4 for notation) are strongly 1-elegant and strongly 3-elegant for all m; the composition Pn[P2] (see §2.3 for the definition) is strongly c-elegant for c = 1, 3, 5, . . . , 5n −6 when n is odd and for c = 1, 3, 5, . . . , 5n −5 when n is even; Pn is strongly ⌊n/2⌋-elegant; P 2 n is strongly c-elegant for c = 1, 3, 5, . . . , q where q is the number of edges of P 2 n; and P 3 n (n > 3) is strongly c-elegant for c = 1, 3, 5, . . . , 6k−1 when n = 4k; c = 1, 3, 5, . . . , 6k + 1 when n = 4k + 1; c = 1, 3, 5, . . . , 6k + 3 when n = 4k + 2; c = 1, 3, 5, . . . , 6k + 5 when n = 4k + 3.
In Barrientos and Minion study a technique to transform an α-labeling of some snakes whose cells are squares into a felicitous labeling and the felicitous labeling into a harmonious labeling. They prove that all quadrilateral snakes, all snake polyominoes, and all hybrid quadrilateral snakes are both, felicitous (see §4.5) and harmonious. A hybrid quadrilateral snake is a snake obtained with n copies of C4 where the ith copy of C4 is attached to the (i+1)th copy via vertex amalgamation or edge amalgamation. Barrientos and Minion prove that all hybrid quadrilateral snakes admit α-labelings.
4.6 Odd Harmonious and Even Harmonious Labelings Liang and Bai introduced odd harmonious labelings by defining a function f to be an odd harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q −1 such that the induced mapping f ∗(uv) = f(u)+f(v) from the edges of G to the odd integers between 1 to 2q −1 is a bijection. A function f is said to be a strongly odd harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to q such that the induced mapping f ∗(uv) = f(u) + f(v) from the edges of G to the odd integers between 1 to 2q −1 is a bijection. Liang and Bai have shown the following: odd harmonious graphs are bipartite; if a (p, q)-graph is odd harmonious, then 2√q ≤p ≤2q −1; if a (p, q)-graph with degree sequence (d1, d2, . . . , dp) is odd harmonious, then gcd(d1, d2, . . . , dp) divides q2; Pn (n > 1) is odd harmonious and strongly odd harmonious; Cn is odd harmonious if and only if n ≡0 mod 4; Kn is odd harmonious if and only if n = 2; Kn1,n2,...,nk is odd harmonious if and only if k = 2; Kt n is odd harmonious if and only if n = 2; Pm × Pn is odd harmonious; the tadpole graph obtained by identifying the endpoint of a path with a vertex of an n-cycle is odd harmonious if n ≡0 mod 4; the graph obtained by appending two or more pendent edges to each vertex of C4n is odd harmonious; the graph obtained by subdividing every edge of the cycle of a wheel (gear graphs) is odd harmonious; the the electronic journal of combinatorics (2023), #DS6 132 graph obtained by appending an edge to each vertex of a strongly odd harmonious graph is odd harmonious; and caterpillars and lobsters are odd harmonious. They conjecture that every tree is odd harmonious.
In , Seoud and Hafez z proved that if G is a strongly odd harmonious graph, new then the graph obtained by identifying a distinguished vertex of G to each vertex of C4m is strongly odd harmoniou. They further proved that the Cartesian product of strongly odd harmonious trees is strongly odd harmonious.
Liang and Bai also showed that the kC4-snake graph is an odd harmonious graph. Abdel-Aal generalize this result by showing that the kCn-snake with string 1, 1, . . . , 1 for n ≡0 (mod 4) are odd harmonious. He also showed that the kC4 snake with m pendent edges is odd harmonious and that all subdivisions of 2m-triangular snakes are odd harmonious.
Alyani, Firmansah, Giyarti, and Sugeng constructed odd harmonious labelings for kCn-snakes for n = 4 and n = 8 and gave odd harmonious labelings for a variation of kCn-snakes.
An n hair-kC4 snake is a graph obtained by attaching n leaves to vertices of degree two in a kC4-snake graph. Mumtaz and Silaban proved that n hair-kC4-snakes are odd harmonious.
A k(G) snake graph is a graph obtained from a path on k edges by replacing each edge by a graph isomorphic to G. Asumpta, Purwanto, and Chandra showed that the snake graph k(G) is odd harmonious when G is a gear graph based on W3 and when G is K2,m (m ≥2).
Renuka and Palanivelu proved that the one vertex union of some classes of complete bipartite graphs, the one vertex union of paths, and extended bistars and their subdivisions are odd harmonious. They further proved that the collection of paths passing through another path at a selected common mid vertex and each vertex of ladder appended by an edge are strongly odd harmonious and that the subdivision of the Cartesian product of two paths is strongly odd harmonious.
In , Seoud and Hafez proved that a graph has an odd harmonious labeling if new and only if it has a bigraceful labeling. They also proved that the number of distinct odd harmonious labeled graphs on m edges is m! and the number of distinct strongly odd harmonious labeled graphs on m edges is ⌊m/2⌋!⌊m/2⌋!
Abdel-Aal also proved that a necessary condition for odd harmonious Eulerian graphs with q edges is q ≡0 (mod 4) and that the following graphs are odd harmonious: Cm × Pn (n ≥2, m ≡0 (mod 4); C4m ⊙C4; Sn ⊙Km; two copies of an even n-cycle sharing a common edge is an odd harmonious graph when n ≡0 (mod 4); two copies of an even n-cycle sharing a common vertex is odd harmonious when n ≡0 (mod 4); and graphs obtained from K2,n (n ≥2) by adding r pendent edges to one of the two vertices of degree n and s pendent edges to the other vertex of degree n. In Mumtaz, John, and Silaban proved that the grid-like graph of order 2mn+mm+n and size 4mn obtained by arranging m-rows connected rows of nC4-snake graphs is odd harmonious.
Vaidya and Shah prove that the shadow graphs (see §3.8 for the definition) of path Pn and star K1n are odd harmonious. They also show that the splitting graphs (see §2.7 for the definition) ) of path Pn and star K1,n are odd harmonious. In Vaidya and Shah proved the following graphs are odd harmonious: the shadow graph and the splitting graph of bistar Bn,n; the arbitrary supersubdivision of paths; graphs the electronic journal of combinatorics (2023), #DS6 133 obtained by joining two copies of cycle Cn for n ≡0(mod 4) by an edge; and the graphs Hn,n, where V (Hn,n) = {v1, v2, . . . , vn, u1, u2, . . . , vn} and E(Hn,n) = {viuj : 1 ≤i ≤ n, n −i + 1 ≤j ≤n}. In Yan proves that Pm × Pn is strongly odd harmonious.
Koppendrayer has proved that every graph with an α-labeling is odd harmonious.
Li, Li, and Yan proved that Km,n is odd strongly harmonious. Saputri, Sugeng, and Fronček proved that the graph obtained by joining Cn to Ck by an edge (dumbbell graph Dn,k,2) is odd harmonious for n ≡k ≡0 (mod 4) and n ≡k ≡2 (mod 4), and Cn × Pm is odd harmonious if and only if n ≡0 (mod 4). They also observe that Cn ⊙K1 with n ≡0 (mod 4) is odd harmonious. Pujiwatia, Halikinb, and Wijayac proved double stars and even cycles odd harmonious. Sugeng, Surip, and Rismayati proved that the m-shadow of C4n and nontivial gears are odd harmonious graphs.
Jeyanthi proved that the shadow and splitting graphs of K2,n, C4n, the double quadrilateral snakes DQ(n) (n ≥2), and the graph Hn,n with vertex set V (Hn,n) = {v1, v2, . . . , vn, u1, u2, . . . , un} and the edge set E(Hn,n) = {viuj : 1 ≤i ≤n, n −i + 1 ≤ j ≤n} are odd harmonious. Jeyanthi and Philo proved that the shadow graphs D2(K2,n) and D2(Hn,n) are odd harmonious and the splitting of graphs of K2,n and Hn,n are odd harmonious. They also showed that the shadow graph D2(Cn) is odd harmonious if n ≡0 (mod 4), the splitting of Cn is odd harmonious if n ≡0 (mod 4), and the double quadrilateral snake DQ(n) is odd harmonious for n ≥2.
In Jeyanthi and Philo prove that super subdivision of cycles, ladders, C4n ⊕K1,m, and uniform fire crackers are odd harmonious graphs. Jeyanthi and Philo proved that the graph Pn−1(1, 2, 3, . . . , n) obtained from a path of n vertices v1, v2, . . . , vn−1 by appending a path of length n −i at each vi and certain one point unions of the end points of paths are odd harmonious.
Philo, Jeyanthi, and Davazz proved that the following graphs are odd harmo-nious: the path union of r copies of Km,n; the path union of copies of complete bipartite graphs; the graphs obtained from t copies of Km,n by joining each one to the next one with an isolated vertex (the join sum) ; the one point union of t copies of complete bipartite graphs (not necessarily the same); the graphs obtained by replacing each vertex of K1,t, except the apex vertex, with the graph Km,n; and the one point union of isomorphic path graphs of Km,n. Let ui,1, ui,2, . . . , ui,m and vi,1, vi,2, . . . , vi,n be the vertices of r copies of Km,n. The circle formation of r copies of Km,n is the graph obtained by joining the ith copy of Km,n (1 ≤i ≤r) by joining the vertices ui,m to ui+1,1 for 1 ≤i ≤r−1 and joining the vertex ur,m to u1,1. Philo, Jeyanthi, and Davazz proved that he circle formation of r copies of Km,n when r = 0(mod 4) is odd harmonious.
Suptri, Sugeng, and Fronček proved that that dumbbell graphs Dn,k,2 are odd harmonious if and only if n, k = 2(mod 4). In Jeyanthi and Philo proved that the graphs obtained by attaching m pendant vertices to each vertex of paths of odd length, the splitting graph of combs, slanting ladders, graphs obtained from m copies of ⟨K1,n : K1,n⟩by joining one leaf of ith copy of ⟨K1,n : K1,n⟩with the center of (i + 1)th copy of ⟨K1,n : K1,n⟩where 1 ≤i ≤m −1, and H-super subdivisions of Pn and C4n admit odd harmonious labeling. Moreover, they observe that all strongly odd harmonious graphs admit a mean labeling, an odd mean labeling, an odd sequential labeling, all the electronic journal of combinatorics (2023), #DS6 134 odd sequential graphs are odd harmonious, and all odd harmonious graphs are even sequential harmonious.
They also proved the n-splitting graphs for paths, stars, and symmetric product between paths and null graphs are odd harmonious graphs for all n.
Selvaraju, Balaganesan, and Renuka proved that quadrilateral snakes and k-regular caterpillars are odd harmonious. Abdel-Aal1 and Seoud proved that the m-shadow graphs for paths and complete bipartite graphs are odd harmonious graphs for all m.
For a Tp-tree T with vertices v1, v2, . . . , vn, the graph Tˆ ◦Pm is obtained from T and n copies of Pm by identifying a pendant vertex of ith copy of Pm with vertex vi of T. For Cn with consecutive vertices v1, v2, . . . , vn, the graph Cnˆ ◦Pm is obtained from Cn and n copies of Pm by identifying a pendant vertex of ith copy of Pm with vertex vi of Cn. Jeyanthi and Philo proved that Tp-trees, Tˆ ◦Pm, Tˆ ◦2Pm, regular bamboo trees, Cnˆ ◦Pm, Cnˆ ◦2Pm, and subdivided grid graphs are odd harmonious.
Recall a subdivided shell graph is obtained by subdividing the edges in the path of the shell graph. Let G1, G2, . . . , Gn be n subdivided shell graphs of any order. The graph SSG(n) is obtained by adding an edge to apexes of Gi and Gi+1, i = 1, 2, . . . , (n−1). Jeba Jesintha and Ezhilarasi Hilda proved that the subdivided shell graph and SSG(2) are odd harmonious.
The following definitions are taken from . The m-shadow graph Dm(G) of a connected graph G is constructed by taking m-copies of G, G1,G2,G3,…,Gm, and joining each vertex u in Gi to the neighbors of the corresponding vertex v in Gj, 1 ≤j ≤m.
The m-splitting graph Splm(G) of a graph G is obtained by adding to each vertex v of G m new vertices, v1, v2, . . . , vm such that vi, 1 ≤i ≤m, is adjacent to every vertex that is adjacent to v in G. Note that the 2-shadow graph is the shadow graph D2(G) and the 1-splitting graph is splitting graph. The m-mirror graph Mm(G) is defined as the disjoint union of m copies of G, G1, G2, . . . , Gm, together with additional edges joining each vertex of Gi to its corresponding vertex in Gi+1, 1 ≤i ≤m −1. The graph Wm,n is obtained from the gear graph arising from the wheel Wn as follows: Join the vertices vi and vi+2 with the new vertices vj i+1 for 1 ≤j ≤m and 2 ≤i ≤n −2 and join vn and v2 with v2i−1. The graph K2,n(r, s) is obtained from K2,n (n ≥2) by adding r and s pendent edges to the two vertices of degree n. The graph G = ⟨Cn : K2,m : Cr⟩is obtained from K2,m with the partite set {u, v} by identifying the vertex u with a vertex of Cn and the vertex v with a vertex of Cr. Let Pn be a path on n vertices denoted by (1, 1), (1, 2), . . . , (1, n) and with n−1 edges denoted by e1, e2, . . . , en−1 where ei is the edge joining the vertices (1, i) and (1, i + 1). The step ladder graph S(Tn) has (n2 + 3n −2)/2 vertices denoted by (1, 1), (1, 2), . . . , (1, n), (2, 1), (2, 2), . . . , (2, n), (3, 1), (3, 2), . . . , (3, n − 1), (4, 1), . . . , (4, n −2), . . . , (n, 1), (n, 2) and n2 + n + 2 edges. In any ordered pair (i, j), i denotes the row (counted from bottom to top) and j denotes the column(from left to right) in which the vertex occurs.
The cocktail party graph, Hm,n (m, n ≥2), is the graph with a vertex set V = {v1, v2, . . . , vmn} partitioned into n independent sets V = {I1, I2, . . . , In} each of size m such that vivj ∈E for all i, j ∈{1, 2, . . . , mn} where i ∈Ip, j ∈Iq, p ̸= q.
Jeyanthi and Philo proved that following graphs are odd harmonious: Dm(Pn) for all m, n ≥2; Splm(Pn) for m, n ≥2; Dm(Hn,n) for all m ≥2 and n ≥1; Splm(Hn,n) the electronic journal of combinatorics (2023), #DS6 135 for all m ≥2 and n ≥1; Dm(Kr,s) for all r, s ≥1; Splm(Kr,s) for all m ≥2 and r, s ≥1; Dm(Pn ⊕K2) for all m, n ≥2; Splm(Pn ⊕K2), m, n ≥2; and Splm(Cn) if and only if n ≡0 (mod 4).
Jeyanthi and Philo proved that following graphs are odd harmonious: Wm,n for n ≡0 (mod 4), m ≥1; Dm(Pn ⊙K1) (the authors use the notion Cbn for the comb Pn ⊙K1) for all m ≥2 and n ≥1; Splm(K2,n(r, s)); ⟨Cn : K2,m : Cr⟩for n, r ≡0 (mod 4) and m ≥2; and the graphs obtained by arranging vertices into a finite number of rows (at least 2) with i vertices in the ith row and in every row the jth vertex in that row is joined to the jth vertex and j + 1st vertex of the next row (a pyramid) for n ≥2. They also prove that if G is a strongly odd harmonious tree, then Mm(G) is odd harmonious.
Let P2n be a path of length 2n−1 with 2n vertices, denoted by (1, 1), (1, 2), . . . , (1, 2n) and with 2n−1 edges, denoted by e1, e2, . . . , e2n−1 where ei is the edge joining the vertices (1, i) and (1, i+1). On each edge ei for i = n+1, n+2, . . . , 2n−1, we erect a ladder with 2n+1−i steps including the edge ei. The double sided step ladder graph 2S(T2×n) has ver-tices denoted by (1, 1), (1, 2), . . . , (1, 2n),(2, 1), (2, 2), . . . , (2, 2n),(3, 2), (3, 3), . . . , (3, 2n − 1),(4, 3), (4, 4), . . . , (4, 2n −2), . . . , (n + 1, n), (n + 1, n + 1). In any ordered pair (i, j), i denotes the row (counted from bottom to top) and j denotes the column (from left to right) in which the vertex occurs. Jeyanthi and Philo proved that the path union of t copies of S(Tn), the double sided step ladder 2S(T2×n), the path union of t copies of 2S(T2×n), S(t.Cbn), S(t.C4), C4 t, C6 t, and C8 t are odd harmonious graphs.
Jeyanthi and Philo proved that path union of r copies of Km,n, the path union of r copies of Kmi,ni, 1 ≤i ≤r, Kt m,n, Kt (m1,n1),(m2,n2),...,(mt,nt), the join sum of graph ⟨Km,n; Km,n; . . . , Km,n (t copies ⟩, ⟨Km1,n1; Km2,n2; . . . , Kmt,nt⟩, the circle formation of r copies of Km,n when r ≡0 (mod 4), S(t.Km,n) and P t n(t.n.Kp,q) are odd harmonious graphs. Jeyanthi and Philo proved that the subdivided shell graphs, disjoint union of two subdivided shell graphs, subdivided shell flower graphs, and subdivided uniform shell bow graphs are odd harmonious. Jeyanthi, Philo, and Youssef proved that the path union of t copies of Pm × Pn, the path union of t copies of Pmi × Pni where 1 ≤i ≤t, the vertex union of t copies of Pm × Pn, the vertex union of t different copies of Pmi × Pni where 1 ≤i ≤t, the one point union of path of P t n(t.n.Pm × Pm), and the super subdivision of grid graph Pm × Pn are odd harmonious graphs.
Recall from Section 2.7 that for even n > 2 a plus graph of size n (denoted by Pln) is the graph obtained by starting with paths P2, P4, . . . , Pn−2, Pn, Pn, Pn−2, . . . , P4, P2 arranged vertically parallel with the vertices in the paths forming horizontal rows and edges joining the vertices of the rows. Jeyanthi proved that following graphs are odd harmonious: Pln where n ≡0 (mod 2), n ̸= 2; path unions of finitely many copies of Pln where n ≡0 (mod 2), n ̸= 2; open stars of plus graphs S(t.Pln) where n ≡0 (mod 2), n ̸= 2 and t odd; graphs obtained by joining Cm, m ≡0 (mod 4) and a plus graph Pln, n ≡0 (mod 2), n ̸= 2 with a path of arbitrary length; the graph obtained by replacing all vertices of K1,t, except the apex vertex, by the path union of n copies of the graph Plm.
In Jeyanthi and Philo prove that super subdivision of cycles, ladders, C4n⊕K1,m, and uniform fire crackers are odd harmonious graphs.
They also proved the (m, n)-firecracker graph obtained by the concatenation of m n-stars by linking one leaf from each the electronic journal of combinatorics (2023), #DS6 136 is odd harmonious; the arbitrary super subdivision of cycles Cm are odd harmonious; and the super subdivision of ladders are odd harmonious. Jeyanthi and Philo proved that the m-mirror graph Mm(G) (m ≥2), m-splitting graph of K2,n(r, s) (obtained from K2,n, (n > 2) by adding r and s (r, s > 1) pendent edges to the two vertices of degree n), W(m,4n) obtained from the gear graph of Wn by joining the vertices vi and vi+2 with the new vertices vj i+1 for 1 ≤j ≤m and 2 ≤i ≤n2 and joining vn and v2 with vj 1 for 1 ≤j ≤m, ⟨C4n : K2,m : C4r⟩obtained from K2,m with one partite set V1 = {u, v} and Cr by identifying the vertex u of V1 with a vertex of Cn and the other vertex v of V1 with a vertex of Cr, and the pyramid graph PYn(n ≥2) are odd harmonious graphs. They also proved that G is a strongly odd harmonious tree, then Mm(G) is an odd harmonious.
The edge comb product of two graphs G and H is the graph formed by taking one copy of G and |E(G)| copies of H, then attaching the i-th copy of H at the edge e to the i-th edge of G. In Sarasvati, Halikin, and Wijaya showed that graphs constructed by the edge comb product of Pn and C4 and the shadow of C4 are odd harmonious.
Firmansah and Tasari gave odd harmonious labelings for a line amalgamation of double quadrilateral graphs and the graphs obtained by connecting two copies of double quadrilateral graph by an edge. Firmansah and Giyarti proved that graphs obtained by the edge amalgamation of double quadrilateral graphs are odd harmonious. Febriana and Sugeng proved that squid graphs (obtained from Cn by add pendant edges to one vertex of Cn) are odd harmonious if and only if n is even and double squid graphs (obtained from two copies of Cn that share one common vertex and adding pendant edges to the common vertex) are odd harmonious if and only n ≥4 is even. In , , Firmansah proved that multiple net snake graphs and a variation of the double quadrilateral windmill graphs, layered graphs, and some classes of string graphs are odd harmonious graphs. Firmansah, Tasari, and Yuwono proved that the zinnia flower graph and its variations are odd harmonious graphs.
In Devakirubanithi, Jeba Jesintha, and Benita define a modified arrow graph, new MAnm, with dimensions m (width) and n (length) as one obtained by adding a vertex v and connecting it with the first and last superior vertices of Pm × Pn with two additional edges coming from one end. They define a modified double arrow graph, MD(Anm, with dimensions m (width) and n (length) as one obtained by connecting two vertices v and w with first and last superior vertices of Pm × Pn with two additional edges coming from the two ends. They proved that modified double arrow graphs, modified arrow graphs, and the corona product of a Tp tree and a star are odd harmonious..
In Jeyanthi and Philo modified the notion of odd harmonious by defining an odd harmonious labelings as a function f to be an odd harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q −1 such that the induced mapping f ∗(uv) = f(u) + f(v) mod (2q) from the edges of G to the odd integers between 1 to 2q −1 is a bijection. Using this definition they proved that an m-cycle and an n-cycle sharing a common vertex is an odd harmonious if and only if either both m, n ≡0 (mod 4) or both m, n ≡2 (mod 4) and the same holds for an m-cycle and an n-cycle sharing a common edge. They also proved that any two even cycles sharing a common vertex and a common edge are odd harmonious graphs.
the electronic journal of combinatorics (2023), #DS6 137 Sarasija and Binthiya say a function f is an even harmonious labeling of a graph G with q edges if f : V →{0, 1, . . . , 2q} is injective and the induced function f ∗: E →{0, 2, . . . , 2(q−1)} defined as f ∗(uv) = f(u)+f(v) (mod 2q) is bijective. Notice that for an even harmonious labeling of a connected graph all the vertex labels must have the same parity. Moreover, in the case of even harmonious labelings for connected graphs there is no loss of generality to assume that all the vertex labels are even integers and the duplicate vertex is 0. They proved the following graphs are even harmonious: non-trivial paths; complete bipartite graphs; odd cycles; bistars Bm,n; K2 + Kn; P 2 n; and the friendship graphs F2n+1. López, Muntaner-Batle and Rious-Font proved that every super edge-magic graph (see Section 5.2 for the definition of super edge-magic) with p vertices and q edges where q ≥p −1 has an even harmonious labeling. In Youssef provided a necessary condition for some regular graphs to be even harmonious, showed that the disjoint union of two k-sequential graphs is even 2k-sequential under some conditions, and showed that in some cases G is k-sequential implies mG is even 2k-sequential for all positive integer m.
Because 0 and 2q are equal modulo 2q the following restricted form of even harmonious labelings is of interest. A function f is said to be a properly even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q −1 and the induced function f ∗from the edges of G to {0, 2, . . . , 2q −2} defined by f ∗(xy) = f(x)+f(y)(mod 2q) is bijective. In their definition of properly even harmonious in Gallian and Schoenhard incorrectly required that the vertex labels should be the even integers from 0 to 2q −2. For connected graphs the two definitions are equivalent but for disconnected graph they are not. They used vertex labels from 0 to 2q −1 for their results on disconnected graphs.
A graph with a properly even harmonious labeling is said to be properly even harmo-nious. Gallian and Schoenhard say a properly even harmonious labeling of a graph with q edges is strongly even harmonious if it satisfies the additional condition that for any two adjacent vertices with labels u and v, 0 < u + v ≤2q.
Jared Bass has observed that for connected graphs any harmonious labeling of a graph with q edges yields an even harmonious labeling by simply multiplying each vertex label by 2 and adding the vertex labels modulo 2q. Thus we know that every connected harmonious graph is an even harmonious graph and every connected graph that is not a tree that has a harmonious labeling also has a properly even harmonious labeling. Conversely, a properly even harmonious labeling of a connected graph with q edges (assuming that the vertex labels are even) yields a harmonious labeling of the graph by dividing each vertex label by 2 and adding the vertex labels modulo q.
Gallian and Schoenhard proved the following: wheels Wn and helms Hn are properly even harmonious when n is odd; nP2 is even harmonious for n odd; nP2 is properly even harmonious if and only if n is even; Kn is even harmonious if and only if n ≤4; C2n is not even harmonious when n is odd; Cn ∪P3 is properly even harmonious when odd n ≥3; C4∪Pn is even harmonious when n ≥2; C4∪Fn is even harmonious when n ≥2; Sm ∪Pn is even harmonious when n ≥2; K4 ∪Sn is properly even harmonious; Pm ∪Pn is properly even harmonious for all m ≥2 and n ≥2; C3 ∪P 2 n is even harmonious the electronic journal of combinatorics (2023), #DS6 138 when n ≥2; C4 ∪P 2 n is even harmonious when n ≥2; the disjoint union of two or three stars where each star has at least two edges and one has at least three edges is properly even harmonious; P 2 m∪Pn is even harmonious for m ≥2 and 2 ≤n < 4m−5; the one-point union of two complete graphs each with at least 3 vertices is not even harmonious; Sm∪Pn is strongly even harmonious if n ≥2; and Sn1 ∪Sn2 ∪· · ·∪Snt is strongly even harmonious for n1 ≥n2 ≥· · · ≥nt and t < n1 2 + 2. They conjecture that Sn1 ∪Sn2 ∪· · · ∪Snt is strongly even harmonious if at least one star has more than 2 edges. They also note that C4, C8, C12, C16, C20, C24 are even harmonious and conjecture that C4n is even harmonious for all n. This conjecture was proved by Youssef who also proved that if a connected even harmonious graph with q edges where q is even and each vertex has degree divisible by 2k (k ≥1), then q is divisible by 2k+1. As a corollary of the latter he gets that C2 4n+2 is not even harmonious. Hall, Hillesheim, Kocina, and Schmit proved that nC2m+1 is properly even harmonious for all n and m.
In and Gallian and Stewart investigated properly even harmonious la-belings of unions of graphs.
They use Pm +t to denote the graph obtained from the path Pm by appending t edges to an endpoint; Catm +t to denote a caterpillar of path length m with t pendent edges; and Cm +t to denote an m-cycle with t pendent edges.
They proved the following graphs are properly even harmonious: nPm if n is even and m ≥2; Pn ∪Km,2 for n odd and n > 1, m > 1; Pn ∪Sm1 ∪Sm2 for n > 2 and m1 + m2 is odd; Cn ∪Sm1 ∪Sm2 for n odd and m1, m2 > 3; Pm +t ∪Pn +s; the union of any number of caterpillars; Cm ∪Catn +t for m > 1 odd, n > 1; C4 ∪Catm +t; the union of C4 and a hairy cycle; K4 ∪Cm +n for some cases; W4 ∪Cm +n for some cases; C4 ∪(Pn + K2) for n > 1; K4 ∪(Pn + Km) for n ≡1, 2 (mod 4); C3 ∪(Pn + Km) for n ≡1, 2 (mod 4); W4∪(Pn+Km) for n ≡1, 2 ( mod 4); W4∪Pn for n ≡1, 2 ( mod 4); K4∪Pn for n > 1 and n ≡1, 2 ( mod 4); K4∪P 2 m1 ∪P 2 m2 ∪· · ·∪P 2 mn for mi > 2, n ≥1; W4∪P 2 m1 ∪P 2 m2 ∪· · ·∪P 2 mn for mi > 2, n ≥1; Cm ∪P 2 n for m ≡3 (mod 4) and n > 1; and 2Pm ∪2Pn. They also prove that nP3 is even harmonious if n > 1 is odd and P 2 m1 ∪P 2 m2 ∪· · · ∪P 2 mn is strongly even harmonious for m > 2, n ≥1.
Gallian and Stewart call an injective labeling f of a graph G with q edges even 2a-sequential if the vertex labels are from {0, 1, . . . , 2q −1} and the edge labels induced by f(u) + f(v) for each edge uv are 2a, 2a + 2, . . . , 2a + 2q −2. When G is a tree, the allowable vertex labels are 0, 1, . . . , 2q. For connected a-sequential graphs, a connected 2a-sequential graph can be obtained by multiplying all the vertex labels by 2. Notice that the vertex labels in resulting graph belong to {0, 2, . . . , 2q −2} (or {0, 2, . . . , 2q} for trees) and the edges labels are from 2a to 2a + 2q −2. Moreover, a connected a-sequential graph can be obtained from a connected even 2a-sequential graph with even vertex labels by dividing all the vertex labels by 2. Likewise, a 2a-sequential labeling of a connected graph with odd vertex labels induces an a-sequential labeling of the graph by subtracting 1 from each vertex label and dividing by 2. Thus for connected graphs, a-sequential is equivalent to 2a-sequential. They prove that if G is even 2a-sequential the following graphs are properly even harmonious: G ∪P 2 m for m > 2, G ∪Pn for n > 1, n ≡1, 2 (mod 4), G ∪Cm +t for some cases, G ∪Catm +n for m > 1, and G ∪W2n+1.
For n and k odd and m, n, k, t > 1, Mbianda and Gallian (see ) proved the the electronic journal of combinatorics (2023), #DS6 139 following graphs have properly even harmonious labelings: mP3 for even m; 2Pm∪2Pn∪St; 2Pm ∪2Pn ∪Pk; 2Pm ∪2Pn ∪Ck; 2Pm ∪2Pn ∪C4; 2Pm ∪2Pn ∪2K4; 2Pm ∪2Pn ∪2W4; 2Pm ∪2Pn ∪2Ck; Fn ∪K4 (Fn = Pn + K1 is the fan); Fn ∪2K4; Fn ∪W4; Fn ∪2W4; Wn∪K4; Wn∪2K4; Wn∪W4; Wn∪2W4; (Cn+K1)∪K4; (Cn+K1)∪W4; (Cn+K1)∪2K4; (Cn+K1)∪2W4; and (Cn+K2)∪K4 ((Cn+K2) is the double cone). Gallian proved the following graphs have properly even harmonious labelings (in all cases m, n > 1): mPn for m even; 2Pm ∪2Pn ∪2C3; 2Pm ∪2Pn ∪2C4; 2Pm ∪2Pn ∪C3 ∪C4; Fn ∪P4; Fn ∪2P4; Fn ∪C4; and Fn ∪2C4.
Binthiya and Sarasija prove the following graphs are even harmonious: Cn ⊙ mK1 (n odd), Pn ⊙mK1 (n > 1 odd), C2n@K2, Pn (n even) with n−1 copies of mK1, the shadow graph D2(K1,n), the splitting graph spl(K1,n), and the graph obtained from the Pn (n even) with n−1 copies of Km incident with first n−1 vertices of Pn. Vargheese and Arun prove that the triangular books, the disjoint union of two triangular book graphs, total graphs T(Pn), the disjoint union of T(Pn) and a triangular book, and the graph obtained by joining the centers of two disjoint copies of K1,n to an isolated vertex are even harmonious.
Lasim, Halikin, and Wijaya showed how to build new harmonious, odd harmo-nious, even harmonious labelings based on the existing such labelings.
In Beatress and Sarasija introduced the notion of even-odd harmonious graphs as follows. Let G be a graph with m vertices and n edges. An injective mapping from the vertices of G to {1, 3, 5, . . . , 2m−1} is called an even-odd harmonious labeling of G if the induced edge mapping f ∗from the edges of G to {0, 2, 4, . . . , 2(n −1)} is a bijection and f ∗(uv) = (f(u)∗+ f ∗(v)) mod 2n for all edges uv. They proves that the bistars, cycles with one pendent edge, crowns, K1,m,n, P 2 n (n ≥4), and nP2 are even-odd harmonious graphs. Kalaimathi and Balamurugan proved caterpillars, lobsters, coconut trees, spider trees, and star graphs admit even-odd harmonious labelings. Zala, Chotaliya, and Chaurasiya proved H-graphs, combs, and bistars graph have even-odd harmonious labelings. Ganeshwari and Sudhana proved that paths, cycles, stars, bistars, combs, P 2 n , C3@pK1 and C2r−1@K1 are even-odd average harmonious graphs.
the electronic journal of combinatorics (2023), #DS6 140 5 Magic-type Labelings 5.1 Magic Labelings Motivated by the notion of magic squares in number theory, magic labelings were in-troduced by Sedláček in 1963.2 Responding to a problem raised by Sedláček, Stewart and studied various ways to label the edges of a graph in the mid 1960s. Stewart calls a connected graph semi-magic if there is a labeling of the edges with integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v. (Berge used the term “regularisable” for this notion.) A semi-magic labeling where the edges are labeled with distinct positive integers is called a magic labeling. Stewart calls a magic labeling supermagic if the set of edge labels consists of consecutive positive integers. The classic concept of an n × n magic square in number theory corresponds to a supermagic labeling of Kn,n. Stewart proved the following: Kn is magic for n = 2 and all n ≥5; Kn,n is magic for all n ≥3; fans Fn are magic if and only if n is odd and n ≥3; wheels Wn are magic for n ≥4; and Wn with one spoke deleted is magic for n = 4 and for n ≥6. Stewart also proved that Km,n is semi-magic if and only if m = n. In Stewart proved that Kn is supermagic for n ≥5 if and only if n > 5 and n ̸≡0 (mod 4). Sedláček showed that Möbius ladders Mn (see §2.3 for the definition) are supermagic when n ≥3 and n is odd and that Cn × P2 is magic, but not supermagic, when n ≥4 and n is even. Shiu, Lam, and Lee have proved: the composition of Cm and Kn (see §2.3 for the definition) is supermagic when m ≥3 and n ≥2; the complete m-partite graph Kn,n,...,n is supermagic when n ≥3, m > 5 and m ̸≡0 (mod 4); and if G is an r-regular supermagic graph, then so is the composition of G and Kn for n ≥3. Ho and Lee showed that the composition of Km and Kn is supermagic for m = 3 or 5 and n = 2 or n odd. Bača, Holländer, and Lih have found two families of 4-regular supermagic graphs. Shiu, Lam, and Cheng proved that for n ≥2, mKn,n is supermagic if and only if n is even or both m and n are odd. Ivančo gave a characterization of all supermagic regular complete multipartite graphs. He proved that Qn is supermagic if and only if n = 1 or n is even and greater than 2 and that Cn × Cn and C2m × C2n are supermagic. He conjectures that Cm × Cn is supermagic for all m and n. Trenklér has proved that a connected magic graph with p vertices and q edges other than P2 exits if and only if 5p/4 < q ≤p(p −1)/2.
In Sun, Guan, and Lee give an efficient algorithm for finding a magic labeling of a graph. In Wen, Lee, and Sun show how to construct a supermagic multigraph from a given graph G by adding extra edges to G.
Sudarsana, Suryanto, Lusianti, and Putri show how magic graph labelings can be used to increase the security level of encrypted text on social media. Angel Sherin and Maheswari and Ali Ahmed and Baskar Babujee used magic labeling of graph to devise encryption and decryption schemes. In Lakshmi, Sudhakar, and Sudhakar provided a cryptographic technique for data encryption and decryption using 2A comprehensive expository treatment of magic labelings is given by Bača, Miller, Ryan, and Semaničová-Feňovčíkováin .
the electronic journal of combinatorics (2023), #DS6 141 magic labelings of graphs.
In Kovář provides a general technique for constructing supermagic labelings of copies of certain kinds of regular supermagic graphs. In particular, he proves: if G is a supermagic r-regular graph (r ≥3) with a proper edge r coloring, then nG is supermagic when r is even and supermagic when r and n are odd; if G is a supermagic r-regular graph with m vertices and has a proper edge r coloring and H is a supermagic s-regular graph with n vertices and has a proper edge s coloring, then G × H is supermagic when r is even or n is odd and is supermagic when s or m is odd.
Kovář, Kravčenko, Krbeček, and Silber affirmatively answered a question by Madaras about existence of supermagic graphs with arbitrarily many different degrees.
Their construction provided graphs with all degrees even. They asked if there exists a supermagic graph with d different odd degrees for any positive integer d. This question was answered affirmatively by Fronček and Qiu with a construction based on the use of 3-dimensional magic rectangles.
In Drajnová, Ivančo, and Semaničová proved that the maximal number of edges in a supermagic graph of order n is 8 for n = 5 and n(n−1) 2 for 6 ≤n ̸≡0 (mod 4), and n(n−1) 2 −1 for 8 ≤n ≡0 (mod 4). They also establish some bounds for the minimal number of edges in a supermagic graph of order n. Ivančo, and Semaničová proved that every 3-regular triangle-free supermagic graph has an edge such that the graph obtained by contracting that edge is also supermagic and the graph obtained by contracting one of the edges joining the two n-cycles of Cn × K2 (n ≥3) is supermagic.
Ivančo proved: the complement of a d-regular bipartite graph of order 8k is supermagic if and only if d is odd; the complement of a d-regular bipartite graph of order 2n where n is odd and d is even is supermagic if and only if (n, d) ̸= (3, 2); if G1 and G2 are disjoint d-regular Hamiltonian graphs of odd order and d ≥4 and even, then the join G1 ⊕G2 is supermagic; and if G1 is d-regular Hamiltonian graph of odd order n, G2 is d−2-regular Hamiltonian graph of order n and 4 ≤d ≡0 (mod 4), then the join G1 ⊕G2 is supermagic.
For k ≥2 and graphs G and H, the graph G ⊙k H defined as (G ⊙k−1 H) ⊙H (where G ⊙1 H = G ⊙H) is called the k-multilevel corona of G with H. Marbun and Salman proved (Wn⊙k−1) ⊙Cn is Wn-edge magic.
In Bezegová and Ivančo extended the notion of supermagic regular graphs by defining a graph to be degree-magic if the edges can be labeled with {1, 2, . . . , |E(G)|} such that the sum of the labels of the edges incident with any vertex v is equal to (1 + |E(G))/deg(v).
They used this notion to give some constructions of supermagic graphs and proved that for any graph G there is a supermagic regular graph which con-tains an induced subgraph isomorphic to G.
In they gave a characterization of complete tripartite degree-magic graphs and in they provided some bounds on the number of edges in degree-magic graphs. They say a graph G is conservative if it admits an orientation and a labeling of the edges by {1, 2, . . . , |E(G)|} such that at each vertex the sum of the labels on the incoming edges is equal to the sum of the labels on the out-going edges. In Bezegová and Ivančo introduced some constructions of degree-magic labelings for a large family of graphs using conservative graphs. Using a connection be-the electronic journal of combinatorics (2023), #DS6 142 tween degree-magic labelings and supermagic labelings they also constructed supermagic labelings for the disjoint union of some regular non-isomorphic graphs. Among their re-sults are: If G is a δ-regular graph where δ is even and at least 6, and each component of G is a complete multipartite graph of even size, then G is a supermagic graph; for any δ-regular supermagic graph H, the union of disjoint graphs H and G is supermagic; if G is a δ-regular graph with δ ≡0 (mod 8) and each component is a circulant graph, then G is a supermagic graph; for any δ-regular supermagic graph H, the union of dis-joint graphs H and G is a supermagic graph; and that the complement of the union of disjoint cycles Cn1, . . . , Cnk is supermagic when k ≡1 (mod 4) and 11 ≤ni ≡3 (mod 8) for all i = 1, . . . , k. In Inpoonjal gave necessary and sufficient conditions for the existence of degree-magic labelings of graphs obtained by taking the join and composition of complete tripartite graphs.
A graph G is said to be (F, H)-sim-(super)magic if there exists a bijection f that is simultaneously F-(super) magic and H-(super) magic. In , Ashari, Salman, and Simanjuntak consider (K2, H)-sim-(super) magic graphs where H is isomorphic to three classes of graphs with varied symmetry: a cycle which is symmetric (both vertex-transitive and edge-transitive), a star which is edge-transitive but not vertex-transitive, and a path which is neither vertex-transitive nor edge-transitive. They provide forbidden subgraphs for the existence of (K2, H)-sim-(super) magic graphs and classify classes of (K2, H)-sim-(super) magic graphs. They also derive sufficient conditions for edge-(super) magic graphs to be (K2, H)-sim-(super) magic and utilize such conditions to characterize some (K2, H)-sim-(super) magic graphs.
Let G be a copy of a simple graph G and for each vertex vi of G let ui be the vertex of G corresponding with vi. The double graph has vertex set V (G) ∪V (G′) and edge set E(G) ∪E(G′) ∪{uivj | ui ∈V (G); vj ∈V (G′) and uiuj ∈E(G)}. Ivančo establishes sufficient conditions for generalized double graphs to be degree-magic and constructs supermagic labelings of some graphs generalizing double graphs.
Sedláček proved that graphs obtained from an odd cycle with consecutive ver-tices u1, u2, . . . , um, um+1, vm, . . . , v1 (m ≥2) by joining each ui to vi and vi+1 and u1 to vm+1, um to v1 and v1 to vm+1 are magic. Trenklér and Vetchý have shown that if G has order at least 5, then Gn is magic for all n ≥3 and G2 is magic if and only if G is not P5 and G does not have a 1-factor whose every edge is incident with an end-vertex of G. Avadayappan, Jeyanthi, and Vasuki have shown that k-sequential trees are magic (see §4.1 for the definition).
Seoud and Abdel Maqsoud proved that K1,m,n is magic for all m and n and that P 2 n is magic for all n. However, Serverino has reported that P 2 n is not magic for n = 2, 3, and 5 . Jeurissan characterized magic connected bipartite graphs.
Ivančo proved that bipartite graphs with p ≥8 vertices, equal sized partite sets, and minimum degree greater than p are magic. Bača characterizes the structure of magic graphs that are formed by adding edges to a bipartite graph and proves that a regular connected magic graph of degree at least 3 remains magic if an arbitrary edge is deleted. In Solairaju and Arockiasamy prove that various families of subgraphs of grids Pm × Pn are magic. Dayanand and Ahmed investigate super magic properties the electronic journal of combinatorics (2023), #DS6 143 of several classes of connected and disconnected graphs. They show that there can be arbitrarily large gaps among the possible valences for certain super magic graphs. They also prove that the disjoint union of multiple copies of a super magic linear forest is super magic if the number of copies is odd and that the super magic labeling is complementary edge antimagic as well. The broom Bn,t is a graph obtained by attaching n −t pendent edges to an end point vertex of the path Pt. Marimuthu, Raja, and Raja Durga prove that Bn,n−1 is E-super vertex magic if and only if n ≥3 is odd and Bn,t is not E-super vertex magic for n ≥4 and t ≥3. In Nemani and Joshi defined a new class of graph called the cartoon flower and showed that a E-super vertex magic labeling does not exist for the class of cartoon flower graphs.
They also define the wounded cartoon flower graphs and establish some sufficient conditions for the graph not to be E-super vertex magic. They give examples of some wounded cartoon flowers that admit an E-super vertex magic labeling and some others that do not. In Kumar and Marimuthu proved that semi-regular bipartite graphs are not E-super vertex-magic and gave an upper bound for the maximum degree of an E-super vertex-magic graph. They also ontained upper and lower bounds of any vertex degree d of a E-super vertex-magic graph.
A Γ-supermagic labeling of a graph G(V, E) with |E| = k is a bijection from E to an Abelian group Γ of order k such that the sum of labels of all incident edges of every vertex x ∈V is equal to the same element µ ∈Γ. An existence of a Z2nm-supermagic labeling of Cartesian product of two cycles, Cn × Cm for n odd was proved recently. This along with an earlier result by Ivanč proved the existence of a Z2nm-supermagic labeling of Cm × Cn for every m, n ≥3 and conjectured that such labeling is possible for all Cm × Cn. In Fronček and McKeown present a simple unified labeling method for Fronček proved this conjecture for all m, n odd that not relatively prime. In Froncek, Paananen, Sorensen proved that the Cartesian product of any two odd new cycles admits a Γ-supermagic labeling for all Abelian groups Γ of order 2mn. In new they proved an analogous result for the product of two even cycles, and showed that there is a Γ-supermagic labeling for all Abelian groups Γ of order 2mn for all even m, n greater then 2. The case of a product of an odd and even cycle, remains widely open with three exceptions.
Ponnappan, Nagaraj, and Prabakaran say a vertex magic labeling f of a graph G(V, E) is an odd vertex magic if f maps V to {1, 3, 5, . . . , 2|V | −1} and E to {1, 2, 3, 4, . . . , |V |+|E|}−{1, 3, 5, . . . , 2|V |−1} if |E| ≥|V |−1) and otherwise f maps E to {2, 4, 6, . . . , 2|E|} and V to {1, 2, 3, 4, . . . , |V | + |E|} −{2, 4, 6, . . . , 2|E|}. They prove that Pn (n ≥3), Cn and mC3 are odd vertex magic if and only if n is odd, (3, t)-kites are vertex magic if and only if t is even, and Cn ⊙K1 are not odd vertex for all n.
A triplet [H, φ, t] is called a supermagic frame of G if φ is a homomorphism of H onto G and t : E(H) →{1, 2, . . . , |E(H)|} is an injective mapping such that the sum of t(uw) over all u ∈φ−1(v) is independent of the vertex v ∈V (G). In 2000, Ivančo proved that if there is a supermagic frame of a graph G, then G is supermagic. Singhun, Boonklurb, and Charnsamorn construct a supermagic frame of m ≥2 copies of the Cartesian product of cycles and show that m copies of the Cartesian product of cycles is supermagic.
the electronic journal of combinatorics (2023), #DS6 144 A prime-magic labeling is a magic labeling for which every label is a prime. Sedláček proved that the smallest magic constant for prime-magic labeling of K3,3 is 53 while Bača and Holländer showed that the smallest magic constant for a prime-magic labeling of K4,4 is 114. Letting σn be the smallest natural number such that nσn is equal to the sum of n2 distinct prime numbers we have that the smallest magic constant for a prime-magic labeling of Kn,n is σn. Bača and Hollaänder conjecture that for n ≥5, Kn,n has a prime-magic labeling with magic constant σn.
They proved the conjecture for 5 ≤n ≤17 and confirmed the conjecture for n = 5, 6 and 7.
Characterizations of regular magic graphs were given by Doob and necessary and sufficient conditions for a graph to be magic were given in , , and .
Some sufficient conditions for a graph to be magic are given in , , and .
Bertault, Miller, Pé-Rosés, Feria-Puron, and Vaezpour provided a heuristic algorithm for finding magic labelings for specific families of graphs. The notion of magic graphs was generalized in and .
Let m, n, a1, a2, . . . , am be positive integers where 1 ≤ai ≤⌊n/2⌋and the ai are dis-tinct. The circulant graph Cn(a1, a2, . . . , am) is the graph with vertex set {v1, v2, . . . , vm} and edge set {vivi+aj | 1 ≤i ≤n, 1 ≤j ≤m} where addition of indices is done modulo n. In Semaničová characterizes magic circulant graphs and 3-regular supermagic circulant graphs. In particular, if G = Cn(a1, a2, . . . , am) has degree r at least 3 and d = gcd(a1, n/2) then G is magic if and only if r = 3 and n/d ≡2 (mod 4), a1/d ≡1 (mod 2), or r ≥4 (a necessary condition for Cn(a1, a2, . . . , am) to be 3-regular is that n is even). In the 3-regular case, Cn(a1, n/2) is supermagic if and only n/d ≡2 (mod 4), a1/d ≡1 (mod 2) and d ≡1 (mod 2). Semaničová also notes that a bipartite graph that is decomposable into an even number of Hamilton cycles is supermagic. As a corollary she obtains that Cn(a1, a2, . . . , a2k) is supermagic in the case that n is even, every ai is odd, and gcd(a2j−1, a2j, n) = 1 for i = 1, 2, . . . , 2k and j = 1, 2, . . . , k.
Ivančo, Kovář, and Semaničová-Feňovčková characterize all pairs n and r for which an r-regular supermagic graph of order n exists.
They prove that for positive integers r and n with n ≥r + 1 there exists an r-regular supermagic graph of order n if and only if one of the following statements holds: r = 1 and n = 2; 3 ≤r ≡1 (mod 2) and n ≡2 (mod 4); and 4 ≤r ≡0 (mod 2) and n > 5. The proof of the main result is based on finding supermagic labelings of circulant graphs. The authors construct supermagic labelings of several circulant graphs.
In Ivančo completely determines the supermagic graphs that are the disjoint unions of complete k-partite graphs where every partite set has the same order.
Trenklér extended the definition of supermagic graphs to include hypergraphs and proved that the complete k-uniform n-partite hypergraph is supermagic if n ̸= 2 or 6 and k ≥2 (see also ). In Sugiyama gave a generalized definition of magic graphs, for which any number of digits can be used to label a vertex and edge, and de-scribed the construction of such magic graphs and their properties. He determined the minimum and maximum magic sums for regular graphs, including polygons and polyhe-drons, and provided techniques for transforming and synthesizing magic graphs using an affine transform.
the electronic journal of combinatorics (2023), #DS6 145 For connected graphs of size at least 5, Ivančo, Lastivkova, and Semaničová provide a forbidden subgraph characterization of the line graphs that can be magic. As a corollary they obtain that the line graph of every connected graph with minimum degree at least 3 is magic. They also prove that the line graph of every bipartite regular graph of degree at least 3 is supermagic.
For any non-trivial Abelian group A under addition, a graph G is said to be strong A-magic if there exists a labeling f of the edges of G with non-zero elements of A such that the vertex labeling f + defined as f +(v) = P f(uv) taken over all edges uv incident at v is a constant, and the constant is same for all possible values of |V (G)|. Stella Arputha Mary, Navaneethakrishnan, and Nagarajan provide strong Z4-magic labelings for various graphs and strong Z4p-magic labelings for those graphs.
In Razzaq, Rizvi, and Ali introduce the concept of an H-group magic total labeling of a graph G over a finite Abelian group A as a bijection λ : V (G) ∪E(G) →A such that for any subgraph H′(V ′, E′) of G isomorphic to H, the sum P v∈V ′ λ(v) + P e∈E′ λ(e) is equal to magic constant k′. A graph is called H-group magic if it admits an H-group magic total labeling. They determine the H-group magic total labelings of fan graphs over the finite Abelian group A ∼ = Z3 × Zt, where t ≥3 and show that disjoint union of isomorphic as well as non-isomorphic copies of fan graphs are H-group magic over A ∼ = Z3 × Zt.
For a graph G(V, E) and a set of positive integers S with |S| = |V |, Godinho, Singh, and Arumugam say G is S-magic if there exists a bijection φ : V →S such that P φ(u) over all u ∈N(v) is a constant k for all v ∈V . They proved that if G is S-magic, then the corresponding magic constant is unique. They proved that several families of graphs are S-magic and that several families are not S-magic. They also determined the set of all S-magic constants for certain classes of graphs for different label sets S.
For a natural number h, Salehi defines a graph G to be h-magic if there is a labeling α from the edges of G to the nonzero integers in Zh such that for each vertex v in G the sum of all α values of edges incident to v is a constant (called the magic sum index) that is independent of the choice of v. If the constant is 0, G is called a zero-sum h-magic graph. The null set of graph G is the set of all natural numbers h for which G admits a zero-sum h-magic labeling. In Salehi determines the null sets for Kn, Km,n, Cn, books, and cycles with a Pk chord. Lin and Wang determine the null sets of generalized wheels and generalized fans, and construct infinitely many examples of Zh-magic graphs with magic sum zero and present some open problems.
In 2020, Kamatchi, Paramasivam, Prajeesh, Sabeel, and Arumugam introduced the notion group vertex magic graphs as follows. For a simple undirected graph G and an additive Abelian group A with identity 0, a mapping f from the vertices of G to the nonzero elements of A is said to be an A-vertex magic labeling of G if there exists a µ in A such that w(v) = P f(u) taken over all u ∈N(v) is µ for all vertices v of G. If G admits such a labeling it is called an A-vertex magic graph. If G is an A-vertex magic for every non-trivial Abelian group A, G is called a group vertex magic graph. They obtained some necessary conditions for a graph to be group vertex magic and gave a characterization of trees with diameter at most 4 that are Z2 × Z2-vertex magic. In Karthik, Sabeel, the electronic journal of combinatorics (2023), #DS6 146 and Paramasivama provided some necessary conditions for the group vertex magicness of graphs with at least one pendant edge and for the group vertex magicness of product graphs. They also characterized group vertex magicness of trees of diameter up to 5 for all infinite Abelian groups with finitely many elements of finite order. Sabeel, Paramasivam, Prajeesh, Kamatchi, and Arumugam characterized the Z2 × Z2-vertex magicness of any tree with diameter 5. They further characterized A-vertex magic trees of diameter at most 5 for any finite Abelian group A and proved that A-vertex magic graphs do not possess any forbidden structures. They gave a method for constructing larger A-vertex magic graphs from the existing ones. Balamoorthy, Bharanedhar, and Kamatchi obtained various results about the A-vertex magicness for graphs formed using joins, tensor products, and lexicographic products of graphs. Kavitha and Stella Arputha Mary say a vertex magic labeling of V4 (Z2 × Z2) is a hefty V4-magic if its magic number is 1. They proved regular graphs, Kn (n ≥3). vertex transitive graphs, and Cm × Cn are hefty V4-magic graphs and connected acyclic graphs are not V4-magic graphs. Stella Arputha Mary, Navaneetha Krishnan, and Nagarajan proved that triangular snakes, books, the one-point union of the apexes of t fans, and the splitting graph of paths, are Z4p-magic graphs. In Liao and and Liu provided characterizations of unicyclic graphs with diameter at most 4 that are A-vertex magic and a characterization of bicyclic graphs of diameter 3 that are group vertex magic. In Khuluq, Krisnawati, and Hidayat new determined Zk-vertex-magic labelings of paths, complete graphs, cycles, and stars.
In 1976 Sedláček defined a connected graph with at least two edges to be pseudo-magic if there exists a real-valued function on the edges with the property that distinct edges have distinct values and the sum of the values assigned to all the edges incident to any vertex is the same for all vertices. Sedláček proved that when n ≥4 and n is even, the Möbius ladder Mn is not pseudo-magic and when m ≥3 and m is odd, Cm × P2 is not pseudo-magic.
A vertex magic total labeling of a graph with p vertices and q edges is a bijection from the union of the vertex set and edge set to the consecutive integers 1, 2, . . . , p + q with the property that for every vertex u, the sum of the label of u and the labels of the edges incident with u is a constant k. A vertex magic total labeling is said to be a-vertex multiple magic if the set of the labels of the vertices is {a, 2a, . . . , na} and is b-edge multiple magic b-edge multiple magic if the set of labels of the edges is {b, 2b, . . . , mb}. Nagaraj, Ponnappan, and Prabakaran provide properties of a-vertex multiple magic graphs and b-edge multiple magic graphs. In Zhang and Wang verify the existence of E-super vertex magic total labeling for odd regular graphs containing a particular 3-factor.
Listiana, Darmaji, and Slamin investigated the existence of vertex magic total labelings of directed sun graphs Sn = Cn ⊙K1 and mSn.
A vertex magic total labeling is said to be a V -super vertex magic labeling if f(V (G)) = {1, 2, 3, . . . , |V |}. A graph G is called V -super vertex magic if it admits a V -super vertex magic labeling. In Vimal Kumar and Vijayalakshmi establish the V -super vertex magic labelings of some classes of parity graphs (that is, for every two induced paths between the same two vertices both paths have odd length, or both have even length).
Kong, Lee, and Sun used the term “magic labeling” for a labeling of the edges the electronic journal of combinatorics (2023), #DS6 147 with nonnegative integers such that for each vertex v the sum of the labels of all edges incident with v is the same for all v. In particular, the edge labels need not be distinct.
They let M(G) denote the set of all such labelings of G. For any L in M(G), they let s(L) = max{L(e) | e ∈E} and define the magic strength of G as m(G) = min{s(L) | L ∈ M(G)}. To distinguish these notions from others with the same names and notation, which we will introduced in the next section for labelings from the set of vertices and edges, we call the Kong, Lee, and Sun version the edge magic strength and use em(G) for min{s(L): L in M(G)} instead of m(G). Kong, Lee, and Sun use DS(k) to denote the graph obtained by taking two copies of K1,k and connecting the k pairs of corresponding leafs. They show: for k > 1, em(DS(k)) = k −1; em(Pk + K1) = 1 for k = 1 or 2, em(Pk + K1) = k if k is even and greater than 2, and 0 if k is odd and greater than 1; for k ≥3, em(W(k)) = k/2 if k is even and em(W(k)) = (k −1)/2 if k is odd; em(P2 × P2) = 1, em(P2 × Pn) = 2 if n > 3, em(Pm × Pn) = 3 if m or n is even and greater than 2; em(C(n) 3 ) = 1 if n = 1 (Dutch windmill, – see §2.4), and em(C(n) 3 ) = 2n −1 if n > 1. They also prove that if G and H are magic graphs then G × H is magic and em(G × H) = max{em(G), em(H)} and that every connected graph is an induced subgraph of a magic graph (see also and ). They conjecture that almost all connected graphs are not magic.
In Ichishima, López, Muntaner-Batle, and Takahashi introduce the parameter l(n) as the minimum size of a graph G of order n for which all graphs of order n and size at least l(n) have µs(G) = +∞, and provide lower and upper bounds for l(G). Imran, Baig, and Feňovčiková, established that for n = 0 (mod 4), µs(Cn×K2) ≤3n/2−1.
Ichishima, López, Muntaner-Batle, and Takahashi, improve this bound by showing that µs(n) + 1 when n ≥4 is even. Enomoto, Lladó, Nakamigawa, and Ringel posed the conjecture that every nontrivial tree is super edge-magic. They propose a new approach to attack this conjecture. They believe that their approach may also help to resolve the conjecture by Graham and Sloane that every nontrivial tree is harmonious . Huang, Hanif, Siddiqui, and Nadeem, showed the super edge-magicness of certain types of generalized combs and disjoint unions of generalized combs and stars.
Recall a lexicographic product of two graphs G1 and G2 is a graph that arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 with Kn,n where n is the order of G2. Sun and Lee show that the Cartesian, conjunctive, normal, lexicographic, and disjunctive products of two magic graphs are magic and the sum of two magic graphs is magic. They also determine the edge magic strengths of the products and sums in terms of the edge magic strengths of the components graphs. In Lee, Saba, and Sun show that the edge magic strength of P k n is 0 when k and n are both odd.
In Akka and Warad define the super magic strength of a graph G, sms(G) as the minimum of all magic constants c(f) where the minimum is taken over all super magic labeling f of G if there exist at least one such super magic labeling. They determine the super magic strength of paths, cycles, wheels, stars, bistars, P 2 n, < K1,n : 2 > (the graph obtained by joining the centers of two copies of K1,n by a path of length 2), and (2n + 1)P2.
For a simple graph G(V, E) a bijection f from V ∪E to {1, 2, . . . , |V |+|E|} is said to be the electronic journal of combinatorics (2023), #DS6 148 edge-magic total labeling of G, if there exists an integer k such that f(u)+f(uv)+f(v) = k for every edge uv ∈E. If, in addition, f(V ) = {1, 2, . . . , |V |}, f is said to be an super edge-magic total labeling. Deeothi investigated the super edge-magic total strength of the family of unicyclic graphs having an odd cycle a varying number of pendant vertices adjacent to each vertex.
A Halin graph ia a planar 3-connected graphs that consist of a tree and a cycle connecting the end vertices of the tree. Let G be a (p, q)-graph in which the edges are labeled k, k + 1, . . . , k + q −1, where k ≥0. In Lee, Su, and Wang define a graph with p vertices to be k-edge-magic for every vertex v the sum of the labels of the incident edges at v are constant modulo p. They investigate some classes of Halin graphs that are k-edge-magic. Lee, Su, and Wang investigated some classes of cubic graphs that are k-edge-magic and provided a counterexample to a conjecture that any cubic graph of order p ≡2 (mod 4) is k-edge-magic for all k. Shiu and Lau gave some necessary conditions for families of wheels with certain spokes missing to admit k-edge-magic labelings.
Lau, Alikhani, Lee, and Kocay (see also ) show that maximal outerplanar graphs of orders p = 4, 5, 7 are k-edge magic if and only if k ≡2 (mod p) and determined all maximal outerplanar graphs that are k-edge magic for k = 3 and 4. They also characterize all (p, p −h)-graphs that are k-edge magic for h ≥0 and conjecture that a maximal outerplanar graph of prime order p is k-edge magic if and only if k ≡2 (mod p).
S. M. Lee and colleagues and call a graph G k-magic if there is a labeling from the edges of G to the set {1, 2, . . . , k −1} such that for each vertex v of G the sum of all edges incident with v is a constant independent of v. The set of all k for which G is k-magic is denoted by IM(G) and called the integer-magic spectrum of G. In Lee and Wong investigate the integer-magic spectrum of powers of paths. They prove: IM(P 2 4 ) is {4, 6, 8, 10, . . .}; for n > 5, IM(P 2 n) is the set of all positive integers except 2; for all odd d > 1, IM(P d 2d) is the set of all positive integers except 1; IM(P 3 4 ) is the set of all positive integers; for all odd n ≥5, IM(P 3 n) is the set of all positive integers except 1 and 2; and for all even n ≥6, IM(P 3 n) is the set of all positive integers except 2. For k > 3 they conjecture: IM(P k n) is the set of all positive integers when n = k + 1; the set of all positive integers except 1 and 2 when n and k are odd and n ≥k; the set of all positive integers except 1 and 2 when n and k are even and k ≥n/2; the set of all positive integers except 2 when n is even and k is odd and n ≥k; and the set of all positive integers except 2 when n and k are even and k ≤n/2. In Lee, Su, and Wang showed that besides the natural numbers there are two types of the integer-magic spectra of honeycomb graphs. Fu, Jhuang and Lin determine the integer-magic spectra of graphs obtained from attaching a path of length at least 2 to the end vertices of each edge of a cycle. Navas, Ajitha, and Varkey the determined the vertex integer magic new spectrum of caterpillars, super caterpillars, and extended super caterpillars.
In Lee, Lee, Sun, and Wen investigated the integer-magic spectrum of various graphs such as stars, double stars (trees obtained by joining the centers of two disjoint stars K1,m and K1,n with an edge), wheels, and fans. In Salehi and Bennett report that a number of the results of Lee et al. are incorrect and provide a detailed accounting the electronic journal of combinatorics (2023), #DS6 149 of these errors as well as determine the integer-magic spectra of caterpillars. Shiu and Low determined the integer-magic spectra and null sets of the Cartesian product of two trees.
Lee, Lee, Sun, and Wen use the notation Cm@Cn to denote the graph obtained by starting with Cm and attaching paths Pn to Cm by identifying the endpoints of the paths with each successive pairs of vertices of Cm. They prove that IM(Cm@Cn) is the set of all positive integers if m or n is even and IM(Cm@Cn) is the set of all even positive integers if m and n are odd.
Lee, Valdés, and Ho investigate the integer magic spectrum for special kinds of trees. For a given tree T they define the double tree DT of T as the graph obtained by creating a second copy T ∗of T and joining each end vertex of T to its corresponding vertex in T ∗. They prove that for any tree T, IM(DT) contains every positive integer with the possible exception of 2 and IM(DT) contains all positive integers if and only if the degree of every vertex that is not an end vertex is even. For a given tree T they define ADT, the abbreviated double tree of T, as the the graph obtained from DT by identifying the end vertices of T and T ∗. They prove that for every tree T, IM(ADT) contains every positive integer with the possible exceptions of 1 and 2 and IM(ADT) contains all positive integers if and only if T is a path.
Lee, Salehi, and Sun have investigated the integer-magic spectra of trees with diameter at most four. Among their findings are: if n ≥3 and the prime power factor-ization of n −1 = pr1 1 pr2 2 · · · prk k , then IM(K1,n) = p1N ∪p2N ∪· · · ∪pkN (here piN means all positive integer multiples of pi); for m, n ≥3, the double star IM(DS(m, m)) (that is, stars Km,1 and Kn,1 that have an edge in common) is the set of all natural num-bers excluding all divisors of m −2 greater than 1; if the prime power factorization of m −n = pr1 1 pr2 2 · · · prk k and the prime power factorization of n −2 = ps1 1 ps2 2 · · · psk k , (the ex-ponents are permitted to be 0) then IM(DS(m, n)) = A1∪A2∪· · ·∪Ak where Ai = p1+si i N if ri > si ≥0 and Ai = ∅if si ≥ri ≥0; for m, n ≥3, IM(DS(m, n)) = ∅if and only if m −n divides n −2; if m, n ≥3 and |m −n| = 1, then DS(m, n) is not magic. Lee and Salehi give formulas for the integer-magic spectra of trees of diameter four but they are too complicated to include here.
For a graph G(V, E) and a function f from the V to the positive integers, Salehi and Lee define the functional extension of G by f, as the graph H with V (H) = ∪{ui| u ∈V (G) and i = 1, 2, . . . , f(u)} and E(H) = ∪{uiuj| uv ∈E(G), i = 1, 2, . . . , f(u); j = 1, 2, . . . , f(v)}. They determine the integer-magic spectra for P2, P3, and P4.
Hungund and Akka introduce new concepts of reverse super edge-magic labeling new and reverse super edge-magic strength of a graph G as follows. A reverse edge magic (REM) labeling of a graph G(V, E) with p vertices and q edges is a bijection f : V (G) ∪ E(G) →{1, 2, . . . , p + q} such that k = f(uv) −(f(u) + f(v)) is a constant k for any edge uv ∈E(G). A REM labeling f is called reverse super edge magic (RSEM) labeling if f(V (G)) = {1, 2, 3, . . . , p} and f(E(G)) = {p+1, p+2, p+3, . . . , p+q}. They obtained the reverse super edge-magic labelings and reverse super edge-magic strength for y-trees, odd cycles, generalized Petersen graphs, and (2m+1)C3. Reddy and Sharief Basha find the electronic journal of combinatorics (2023), #DS6 150 some new classes of RSEM labeling and investigate the connection between the RSEM labeling and different classes of labelings.
More specialized results about the integer-magic spectra of amalgamations of stars and cycles are given by Lee and Salehi in .
Table 5 summarizes the state of knowledge about magic-type labelings. In the table, SM means semi-magic, M means magic, and SPM means supermagic. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová.
the electronic journal of combinatorics (2023), #DS6 151 Table 5: Summary of Magic Labelings Graph Types Notes Kn M if n = 2, n ≥5 SPM for n ≥5 iff n > 5 n ̸≡0 (mod 4) Km,n SM if n ≥3 Kn,n M if n ≥3 fans fn M iff n is odd, n ≥3 not SM if n ≥2 wheels Wn M if n ≥4 SM if n = 5 or 6 wheels with one M if n = 4, n ≥6 spoke deleted null graph with n vertices Möbius ladders Mn SPM if n ≥3, n is odd Cn × P2 not SPM for n ≥4, n even Cm[Kn] SPM if m ≥3, n ≥2 Kn, n, . . . , n | {z } p SPM n ≥3, p > 5 and p ̸≡0 (mod 4) composition of r-regular SPM if n ≥3 SPM graph and Kn Kk[Kn] SPM if k = 3 or 5, n = 2 or n odd mKn,n SPM for n ≥2 iff n is even or both n and m are odd Qn SPM iff n = 1 or n > 2 even Continued on next page the electronic journal of combinatorics (2023), #DS6 152 Table 5 – Continued from previous page Graph Types Notes Cm × Cn SPM m = n or m and n are even Cm × Cn SPM?
for all m and n connected (p, q)-graph M iff 5p/4 < q ≤p(p −1)/2 other than P2 Gi M |G| ≥5, i ≥3 G2 M G ̸= P5 and G does not have a 1-factor whose every edge is incident with an end-vertex of G K1,m,n M for all m, n P 2 n M for all n except 2, 3, 5 , G × H M iff G and H are magic 5.2 Edge-magic Total and Super Edge-magic Total Labelings In 1970 Kotzig and Rosa defined a magic valuation of a graph G(V, E) as a bijection f from V ∪E to {1, 2, . . . , |V ∪E|} such that for all edges xy, f(x)+f(y)+f(xy) is constant (called the magic constant). This notion was rediscovered by Ringel and Lladó in 1996 who called this labeling edge-magic. To distinguish between this usage from that of other kinds of labelings that use the word magic we will use the term edge-magic total labeling as introduced by Wallis in 2001. (We note that for 2-regular graphs a vertex-magic total labeling is an edge-magic total labeling and vice versa.) Kotzig and Rosa proved: Km,n has an edge-magic total labeling for all m and n; Cn has an edge-magic total labeling for all n ≥3 (see also , , , and ); and the disjoint union of n copies of P2 has an edge-magic total labeling if and only if n is odd. They further state that Kn has an edge-magic total labeling if and only if n = 1, 2, 3, 5, or 6 (see , , and ) and ask whether all trees have edge-magic total labelings.
Wallis, Baskoro, Miller, and Slamin enumerate every edge-magic total labeling of complete graphs. They also prove that the following graphs are edge-magic total: paths, crowns, complete bipartite graphs, and cycles with a single edge attached to one vertex.
Enomoto, Llado, Nakamigana, and Ringel prove that all complete bipartite graphs are edge-magic total. They also show that wheels Wn are not edge-magic total when n ≡3 the electronic journal of combinatorics (2023), #DS6 153 (mod 4) and conjectured that all other wheels are edge-magic total. This conjecture was proved when n ≡0, 1 (mod 4) by Phillips, Rees, and Wallis and when n ≡6 (mod 8) by Slamin, Bača, Lin, Miller, and Simanjuntak . Fukuchi verified all cases of the conjecture independently of the work of others. Slamin et al. further show that all fans are edge-magic total. In 2002 Lee and Kong conjectured that odd star forests are super edge magic. In 2019 Cerioli, Fernandes, Lee, Lintzmayer, Mota, and da Silva proved this conjecture for odd symmetric star forests and proved that odd uniform forests of catterpillars are edge-magic. Afzal, Ather, Baig, and Maheshwari analyzed pyramidion ladders and Cn-books for their super edge-magicness and gave some methods for finding new super edge-magic graphs from existing ones. In Darmaji and Saputro use G(+)Pm(+)H (m ≥2) to denote the graph obtained by taking one copy of the graphs G, H, and Pm, then connecting one endpoint of Pm to all vertices of G and the other endpoint of Pm to all vertices of H. For any super edge-magic total graphs G, they provide some graphs H such that G(+)Pm(+)H is also super edge-magic total. They further show how to construct a super edge-magic total graph from a super edge-magic total graph by considering a super edge-magic labeling of the original graph.
In Ichishima and Muntaner-Batle study the super edge-magicness of graphs of order n with degree sequences: 4, 2, 2, . . . , 2. They also investigate the super edge-magic properties of certain families of graphs. One such family is graph C(m, n)—the one-point union of the cycles Cm and Cn. One of their results is C(m, n) is not super edge-magic when m + n is odd. They state that an exhaustive computer search shows that C(3, 5) is not super edge-magic. However, Oshima [personal communication] pointed out new that this is not the case, and showed that C (3, 4k −3) is super edge-magic for integers k ≥2.
In Kanwal, Riasat, Imtiaz, Iftikhar, Javed, and Ashraf define a fork as the graph obtained by starting with three paths of length t with vertices x1,j, x2,j, x3,j, 1 ≤j ≤t, a single new edge x2,0 adjacent to x2,1, an edge joining x1,1 and x2,1 and an edge joining x2,1 and x3,1. They gave super edge-magic total labelings and deficiencies of forks, the disjoint union of a fork with a star, a bistar, and a path, and of trees obtained by starting with two copies of P2t+1 and adding an edge joining the middle vertex of each path. The super edge-magic total labeling strengths of forks and the trees are also determined. Girija and Karthikeyan proved that 3 copies of the jelly fish graphs are super edge magic graphs.
Inspired by Kotzig-Rosa’s notion, Enomoto, Lladó, Nakamigawa, and Ringel called a graph G(V, E) with an edge-magic total labeling that has the additional property that the vertex labels are 1 to |V | a super edge-magic total labeling (SEMT). Kanwal and Kanwal determined super edge-magic total labelings and deficiencies for forests formed by two sided generalized combs, stars, combs, and banana trees. A two-sided generalized comb Cb2 a,b, where b is odd, is obtained from a path Pa+1 by attaching two paths P(b+1)/2 to each of the vertices of degree two and one vertex of degree one of Pa+1.
In Kanwal, Azam, and Iftikhar investigate the SEMT strength of generalized comb and the SEMT labeling and deficiency of forests composed of two components, where one of the components for each forest is a generalized comb and other component is star, the electronic journal of combinatorics (2023), #DS6 154 bistar, comb, or path. In Kanwal, Imtiaz, Iftikhar, Ashraf, Arshad, Irfan, and Sumbal z studied the super edge-magic deficiency of paths, caterpillars, and the disjoint union of a 2-sided generalized comb with a bistar. They also provide the super edge-magic total strength for a 2-sided generalized comb. Javed, Riasat, and Kanwal study super edge-magic total labeling and deficiencies of forests consisting of combs, generalized combs, and stars. Their results provide the evidence to support a conjecture proposed by Figueroa-Centeno, Ichishima, and Muntaner-Bartle .
Cerioli, Fernandes, Lee, Lintzmayer proved certain forests of stars admit a super edge-magic labeling and that certain forests of caterpillars admit an edge-magic labeling.
Baskoro, Sudarsana, and Cholily provided some constructions of new super edge-magic graphs from some old ones by attaching 1, 2, or 3 pendent vertices and edges.
In Kim introduces a new construction of new super edge-magic graphs by attaching any number pendent vertices and edges under some conditions. In Darmaji and Saputro define a graph G(+)Pm(+)H where m ≥2 as a graph obtained by taking one copy of the graphs G and H and Pm, then connect an end point of Pm to all vertices of G and the other end Pm to all vertices of H. For any super edge-magic total graphs G, they provide some graphs H such that G(+)Pm(+)H is also super edge-magic total. They further show how to construct a super edge-magic total graph from a super edge-magic total graph by considering a super edge-magic labeling of the origin graph. One such instance is (P2n ∪mK1) + 2K1.
Ringel and Llado prove that a graph with p vertices and q edges is not edge-magic total if q is even and p + q ≡2 (mod 4) and each vertex has odd degree. Ringel and Llado conjecture that trees are edge-magic total. In Baskar Babujee and Rao show that the path with n vertices has an edge-magic total labeling with magic constant (5n + 2)/2 when n is even and (5n + 1)/2 when n is odd. For stars with n vertices they provide an edge-magic total labeling with magic constant 3n. In Eshghi and Azimi discuss a zero-one integer programming model for finding edge-magic total labelings of large graphs.
Santhosh proved that for n odd and at least 3, the crown Cn ⊙P2 has an edge-magic total labeling with magic constant (27n + 3)/2 and for n odd and at least 3, Cn ⊙P3 has an edge-magic total labeling with magic constant (39n + 3)/2. Ngurah and Adiwijaya investigated whether various classes of chain graphs formed from ladders, triangular ladders, diagonal ladders, C4, and K4 have an edge-magic or super edge-magic labelings. Baig and Afzal investigated the super edge-magicness of special classes of graphs having maximum magic constant k = 3p.
In Freyberg introduced a generalization of edge-magic total labeling that allows multiple labels on the vertices or edges of a graph. He used this new labeling as a tool to construct face-magic labelings of some infinite families of graphs. He then considered the question “Given a graph G, for which a, b, c ∈{0, 1} does G admit a face-magic labeling of type (a, b, c)? He completely answered this question for two families of chained cycles, ladders and subdivided ladders, fans and subdivided fans, and wheels and subdivided wheels. See for some corrections for these results. Freyberg provided (1, 1, 1)-face-magic labelings for square tilings, hexagon tilings on a torus, and a special class of the electronic journal of combinatorics (2023), #DS6 155 triangle tilings on a cylinder.
Ahmad, Baig, and Imran define a zig-zag triangle as the graph obtained from the path x1, x2, . . . , xn by adding n new vertices y1, y2, . . . , yn and new edges y1x1, ynxn−1; xiyi for 1 ≤i ≤n; yixi−1yixi+1 for 2 ≤i ≤n −1. They define a graph Cbn as one obtained from the path x1, x2, . . . , xn adding n −1 new vertices y1, y2, . . . , yn−1 and new edges yixi+1 for 1 ≤i ≤n −1. The graph Cb∗ n is obtained from the Cbn by joining a new edge x1y1. They prove that zig-zag triangles, graphs that are the disjoint union of a star and a banana tree, certain disjoint unions of stars, and for n ≥4, Cb∗ n ∪Cbn−1 are super edge-magic total. Baig, Afzal, Imran, and Javaid investigate the existence of super edge-magic labeling of volvox and pancyclic graphs.
The super edge-magic deficiency of a graph G, denoted by µs(G), is either the minimum nonnegative integer n such that G ∪nK1 is super edge-magic or +∞if there exists no such n. Krisnawati, Ngurah, Hidayat, and Alghofari investigated the super edge-magic deficiency of forests whose components are subdivided stars or paths. Imran, Afzal, and Baig investigate the super edge-magic deficiency of volvox and dumbbell type graphs in . Kanwal, Iftikhar, and Azam found super edge magic total labelings and deficiencies of forests consisting of two components, where one of the components for each forest is a generalized comb and the other component is a star, bistar, comb, or path.
They also investigated the super edge magic total strength of generalized combs.
Let G be a graph with p vertices with V (G) = {v1, v2, . . . , vp} and let Sm be the star with m leaves. If in G, every vertex vi is identified to the center vertex of Smi, for some mi ≥0, 1 ≤i ≤n, where S0 = K1, then the graph obtained is denoted by G(m1,m2,...,mp).
Let M(G) = {(m1, m2, . . . , mp) | G(m1,m2,...,mp) is a super edge-magic graph}. The star super edge-magic deficiency Sµ∗(G) is defined as Sµ∗(G) = ( min(m1,m2,...,mp)(m1 + m2 + · · · + mp) if M(G) ̸= ∅, +∞, if M(G) = ∅.
In Kathiresan and Sabarimalai Madha determine the star super edge-magic defi-ciency of certain classes of graphs. In Krisnawati, Ngurah, Hidayat, and Alghofar showed that some subdivisions of double stars have zero (consecutively) super edge-magic deficiency.
Beardon extended the notion of edge-magic total to countable infinite graphs G(V, E) (that is, V ∪E is countable). His main result is that a countably infinite tree that processes an infinite simple path has a bijective edge-magic total labeling using the integers as labels. He asks whether all countably infinite trees have an edge-magic total labeling with the integers as labels and whether the graph with the integers as vertices and an edge joining every two distinct vertices has a bijective edge-magic total labeling using the integers.
Cavenagh, Combe, and Nelson investigate edge-magic total labelings of countably infinite graphs with labels from a countable Abelian group A. Their main result is that if G is a countable graph that has an infinite set of mutually disjoint edges and A is isomorphic to a countable subgroup of the real numbers under addition then for any k in A there is an edge-magic labeling of G with elements from A that has magic constant k.
the electronic journal of combinatorics (2023), #DS6 156 Balakrishnan and Kumar proved that the join of Kn and two disjoint copies of K2 is edge-magic total if and only if n = 3. Yegnanarayanan has proved the following graphs have edge-magic total labelings: nP3 where n is odd; Pn + K1; Pn × C3 (n ≥2); the crown Cn ⊙K1; and Pm × C3 with n pendent vertices attached to each vertex of the outermost C3. He conjectures that for all n, Cn ⊙Kn, the n-cycle with n pendent vertices attached at each vertex of the cycle, and nP3 have edge-magic total labelings. In fact, Figueroa-Centeno, Ichishima, and Muntaner-Batle, have proved the stronger statement that for all n ≥3, the corona Cn ⊙Km admits an edge-magic labeling where the set of vertex labels is {1, 2, . . . , |V |}. (See also .) Yegnanarayanan also introduces several variations of edge-magic labelings and provides some results about them. Kotzig provides some necessary conditions for graphs with an even number of edges in which every vertex has odd degree to have an edge-magic total labeling. Craft and Tesar proved that an r-regular graph with r odd and p ≡4 (mod 8) vertices can not be edge-magic total. Wallis proved that if G is an edge-magic total r-regular graph with p vertices and q edges where r = 2ts + 1 (t > 0) and q is even, then 2t+2 divides p.
Figueroa-Centeno, Ichishima, and Muntaner-Batle have proved the following graphs are edge-magic total: P4 ∪nK2 for n odd; P3 ∪nK2; P5 ∪nK2; nPi for n odd and i = 3, 4, 5; 2Pn; P1 ∪P2 ∪· · · ∪Pn; mK1,n; Cm ⊙nK1; K1 ⊙nK2 for n even; W2n; K2×Kn, nK3 for n odd (the case nK3 for n even and larger than 2 is done in ); binary trees, generalized Petersen graphs (see also ), ladders (see also ), books, fans, and odd cycles with pendent edges attached to one vertex.
In Figueroa-Centeno, Ichishima, Muntaner-Batle, and Oshima, investigate super edge-magic total labelings of graphs with two components.
Among their results are: C3 ∪Cn is super edge-magic total if and only if n ≥6 and n is even; C4 ∪Cn is super edge-magic total if and only if n ≥5 and n is odd; C5 ∪Cn is super edge-magic total if and only if n ≥4 and n is even; if m is even with m ≥4 and n is odd with n ≥m/2 + 2, then Cm ∪Cn is super edge-magic total; for m = 6, 8, or 10, Cm ∪Cn is super edge-magic total if and only if n ≥3 and n is odd; 2Cn is strongly felicitous if and only if n ≥4 and n is even (the converse was proved by Lee, Schmeichel, and Shee in ); C3 ∪Pn is super edge-magic total for n ≥6; C4 ∪Pn is super edge-magic total if and only if n ̸= 3; C5 ∪Pn is super edge-magic total for n ≥4; if m is even with m ≥4 and n ≥m/2+2 then Cm∪Pn is super edge-magic total; Pm ∪Pn is super edge-magic total if and only (m, n) ̸= (2, 2) or(3,3); and Pm ∪Pn is edge-magic total if and only (m, n) ̸= (2, 2). In Rizvi, Ali, Iqbal, and Gulraze give super edge-magic total labelings of forests whose components are caterpillars and stars, forests whose components are stars and banana trees, and a new families of trees.
Enomoto, Llado, Nakamigawa, and Ringel conjecture that if G is a graph of order n + m that contains Kn, then G is not edge-magic total for n ≫m. Wijaya and Baskoro proved that Pm × Cn is edge-magic total for odd n at least 3. Ngurah and Baskoro state that P2 × Cn is not edge-magic total. Hegde and Shetty have shown that every Tp-tree (see §4.4 for the definition) is edge-magic total. Ngurah, Simanjuntak, and Baskoro show that certain subdivisions of the star K1,3 have the electronic journal of combinatorics (2023), #DS6 157 edge-magic total labelings. Ali, Hussain, Shaker, and Javaid provide super edge-magic total labelings of subdivisions of stars K1,p for p ≥5. In Ngurah, Baskoro, Tomescu gave methods for construction new (super) edge-magic total graphs from old ones by adding some new pendent edges. They also proved that K1,m ∪P m n is super edge-magic total. Wallis proves that a cycle with one pendent edge is edge-magic total. In Wallis poses a large number of research problems about edge-magic total graphs.
For n ≥3, López, Muntaner-Batle, and Rius-Font (see for (corrigen-dum) let Sn denote the set of all super edge-magic total 1-regular labeled digraphs of order n where each vertex takes the name of the label that has been assigned to it. For π ∈Sn. they define a generalization of generalized Petersen graphs that they denote by GGP(n; π), which consists of an outer n-cycle x0, x1, . . . , xn−1, x0, a set of n-spokes xiyi, 0 ≤i ≤n −1, and n inner edges defined by yiyπ(i), i = 0, . . . , n −1. Notice that, for the permutation π defined by π(i) = i + k (mod n) we have GGP(n; π) = P(n; k). They define a second generalization of generalized Petersen graphs, GGP(n; π2, . . . , πm), as the graphs with vertex sets ∪m j=1{xj i : i = 0, . . . , n −1}, an outer n-cycle x1 0, x1 1, . . . , x1 n−1, x1 0, and inner edges xj−1 i xj i and xj ixj πj(i), for j = 2, . . . , m, and i = 0, . . . , n −1. Notice that, GGP(n; π2, . . . , πm) = Pm × Cn, when πj(i) = i + 1 (mod n) for every j = 2, . . . , m.
Among their results are the Petersen graphs are super edge-magic total; for each m with 1 < l ≤m and 1 ≤k ≤2, the graph GGP(5; π2, . . . , πm), where πi = σ1 for i ̸= l and πl = σk, is super edge-magic total; for each 1 ≤k ≤2, the graph P(5n; k + 5r) where r is the smallest integer such that k + 5r = 1 (mod n) is super edge-magic total.
A w-graph, W(n), has vertices {(c1, c2, b, w, d) ∪(x1, x2, . . . , xn) ∪(y1, y2, . . . , yn)} and edges {(c1x1, c1x2, . . . , c1xn) ∪(c2y1, c2y2, . . . , c2yn) ∪(c1b, c1w) ∪(c2w, c2d)}. A w-tree, WT(n, k), is a tree obtained by taking k copies of a w-graph W(n) and a new vertex a and joining a with in each copy d where n ≥2 and k ≥3. An extended w-tree Ewt(n, k, r) is a tree obtained by taking k copies of an extended w-graph Ew(n, r) and a new vertex a and joining a with the vertex d in each of the k copies for n ≥2, k ≥3 and r ≥2. Super edge-magic total labelings for w-trees, extended w-trees, and disjoint unions of extended w-trees are given in , , and . Javaid, Hussain, Ali, and Shaker provided super edge-magic total labelings of subdivisions of K1,4 and w-trees. Shaker, Rana, Zobair, and Hussain gave a super edge-magic total labeling for a subdivided star with a center of degree at least 4.
In 1988 Godbod and Slater made the following conjecture. If n is odd, n ̸= 5, Cn has an edge magic labeling with valence k, when (5n + 3)/2 ≤k ≤(7n + 3)/2. If n is even, Cn has an edge-magic labeling with valence k when 5n/2 + 2 ≤k ≤7n/2 + 1.
Except for small values of n, very few valences for edge-magic labelings of Cn are known.
In López, Muntaner-Batle, and Rius-Font use the ⊗h-product in order to prove the following two results. Let n = pα1 1 pα2 2 · · · pαk k be the unique prime factorization of an odd number n. Then Cn admits at least 1 + Pk i=1 αi edge-magic labelings with at least 1 + Pk i=1 αi mutually different valences. Let n = 2αpα1 1 pα2 2 · · · pαk k be the unique prime factorization of an even number n, with p1 > p2 > · · · > pk. Then Cn admits at least Pk i=1 αi edge-magic labelings with at least Pk i=1 αi mutually different valences. If α ≥2 the electronic journal of combinatorics (2023), #DS6 158 this lower bound can be improved to 1 + Pk i=1 αi. In López, Muntaner-Batle, and Prabu introduce a new ⊗h labeling construction that has a wider range of applications and applies it to the magic valences of cycles and crowns.
In Swita, Rafflesia, Henni Ms, Adji, and Simanihuruk use B[(Ca, m), (Cb, n), Pt] to denote the graph that consists of m cycles Ca and n cycles Cb with a common path Pt. They proved that B[(C7, 1), (C3, n), P2] admits an edge-magic total labeling, B[(Ca, 1), (C3, n), P2] admits a super edge-magic total labeling for all a ≡3 mod 4 (a > 3), and B[(C7, 2), (C3, n), P2] admits a super edge-magic total labeling.
In 1996 Erdős asked for M (n), the maximum number of edges that an edge-magic total graph of order n can have (see ). In 1999 Craft and Tesar gave the bound ⌊n2/4⌋≤M (n) ≤⌊n(n −1)/2⌋. For large n this was improved by Pikhurko in 2006 to 2n2/7 + O(n) ≤M (n) ≤(0.489 + · · · + o(1)n2).
Enomoto, Lladó, Nakamigawa, and Muntaner-Batle proved that a super edge-magic total graph G(V, E) with |V | ≥4 and with girth at least 4 has at most 2|V | −5 edges. They prove this bound is tight for graphs with girth 4 and 5 in and .
In his Ph.D. thesis, Barrientos introduced the following notion. Let L1, L2, . . . , Lh be ordered paths in the grid Pr × Pt that are maximal straight segments such that the end vertex of Li is the beginning vertex of Li+1 for i = 1, 2, . . . , h −1. Suppose for some i with 1 < i < h we have V (Li) = {u0, v0} where u0 is the end vertex of Li−1 and the beginning vertex of Li and v0 is the end vertex of L1 and the beginning vertex of Li+1.
Let u ∈V (Li−1) −{u0} and v ∈V (Li+1) −{v0}. The replacement of the edge u0v0 by a new edge uv is called an elementary transformation of the path Pn. A tree is called a path-like tree if it can be obtained from Pn by a sequence of elementary transformations on an embedding of Pn in a 2-dimensional grid. In Bača, Lin, and Muntaner-Batle proved that if T1, T2, . . . , Tm are path-like trees each of order n ≥4 where m is odd and at least 3, then T1 ∪T2, ∪· · · ∪Tm has a super edge-magic labeling. In Bača, Lin, Muntaner-Batle and Rius-Font proved that the number of such trees grows at least exponentially with m. As an open problem Bača, Lin, Muntaner-Batle and Rius-Font ask if graphs of the form T1∪T2∪· · ·∪Tm where T1, T2, . . . , Tm are path-like trees each of order n ≥2 and m is even have a super edge-magic labeling. In Barrientos proved that all path-like trees admit an α-valuation. Using Barrientos’s result, it is very easy to obtain that all path-like trees are a special kind of super edge-magic by using a super edge-magic labeling of the path Pn, and hence they are also super edge-magic. Furthermore, Figueroa-Centeno, Ichishima, and Muntaner-Batle proved that if a tree is super edge-magic, then it is also harmonious. Therefore all path-like trees are also harmonious. In López, Muntaner-Batle, and Rius-Font also use a variation of the Kronecker product of matrices in order to obtain lower bounds for the number of non isomorphic super edge-magic labeling of some types of path-like trees. As a corollary they obtain lower bounds for the number of harmonious labelings of the same type of trees. López, Muntaner-Batle, and Rius-Font proved that if m ≥4 is an even integer and n ≥3 is an odd divisor of m, then Cm ∪Cn is super edge-magic. Lee and Kong conjecture that if n is an odd, then St(a1, a2, . . . , an) is super edge-magic, and they proved that the following graphs are super edge-magic: St(m, n) (n ≡0 mod(m + 1)), St(1, k, n)(k = 1, 2 or n), St(2, k, n) (k = the electronic journal of combinatorics (2023), #DS6 159 2, 3), St(1, 1, k, n) (k = 2, 3), St(k, 2, 2, n) (k = 1, 2). Zhenbin and Chongjin proved that St(1, m, n), St(3, m, m+1), St(n, n+1, n+2) are super edge-magic, and under some conditions St(a1, a2, . . . , a2n+1), St(a1, a2, . . . , a4n+1), St(a1, a2, . . . , a4n+3) are also super edge-magic. In Figueroa-Centeno, Ichishima, and Muntaner-Batle conjectured that the cycle books B(4, m) that consists of m cycles C4 with a common path P2 is super edge-magic total if and only if m is even or m ≡5 (mod 8). Simanihuruk, Kusmayadi, Swita, Romala, and Damanik proved this conjecture for m ≥36 and m even.
For a simple graph H we say that G(V, E) admits an H-covering if every edge in E(G) belongs to a subgraph of G that is isomorphic to H. In López, Muntaner-Batle, Rius-Font study a relationship existing among (super) magic coverings and the Kronecker product of matrices. (For a simple graph H, G(V, E) admits an H-covering if every edge in E(G) belongs to a subgraph of G that is isomorphic to H.) Their results can be applied to construct S-magic partitions. For m copies of a graph G and a fixed subgraph H of each copy the graph I(G, H, m) is formed by taking of all the Gi’s and identifying their subgraph H. Liang determined which I(G, H, m) and which mG have G supermagic coverings. Farida, Indah, and Sudiby study magic coverings of domino graphs (graphs for which every vertex is contained in at most two maximal cliques) and edge magic labelings of domino graphs. They note that coverings can be applied to secret sharing schemes and edge magic labelings can be applied to ruler models.
Bača, Lin and Muntaner-Batle in using a generalization of the Kronecker product of matrices prove that the number of non-isomorphic super edge-magic labelings of the disjoint union of m copies of the path Pn, m ≡2 (mod 4), m ≥2, n ≥4, is at least (m/2)(2n−2).
In López, Muntaner-Batle and Rius-Font proved that every super edge-magic graph with p vertices and q edges where q ≥p −1 has an even harmonious labeling (See Section 4.6.) In they stated some open problems concerning relationships among super edge-magic labelings and graceful and harmonious labelings. A Langford sequence of order m and defect d is a sequence (t1, t2, . . . , t2m) of 2m numbers such that (i) for every k ∈[d, d + m1] there exist exactly two subscripts i, j ∈[1, 2m] with ti = tj = k and (ii) the subscripts i and j satisfy the condition |ij| = k. López and Muntaner-Batle provided new lower bounds on the number of distinct Langford sequences with certain properties in terms of the number of 1-regular super edge-magic labeled digraphs of a particular order.
Lee and Lee prove the following graphs are super edge-magic: P2n + Km, (P2 ∪ nK1) + K2, graphs obtained by appending a path to the apex of a fan with at least 4 vertices (umbrella), and jelly fish graphs J(m, n) obtained from a 4-cycle v1, v2, v3, v4 by joining v1 and v3 with an edge and appending m pendent edges to v2 and n pendent edges to v4. In Afzel introduces two new familes of graphs called carrom and jukebox graphs and proves they admit super edge-magic labelings. Carroms are generalizations of Cn × P2.
Marimuthu and Balakrishnan define a graph G(p, q) to be edge magic graceful if there exists a bijection f from V (G)∪E(G) to {1, 2, . . . , p+q} such |f(u)+f(v)−f(uv)| is a constant for all edges uv of G. An edge magic graceful graph is said to be super the electronic journal of combinatorics (2023), #DS6 160 edge magic graceful if V (G) = {1, 2, . . . , p}. They present some properties of super edge magic graceful graphs, prove some classes of graphs are super edge magic graceful, and prove that every super edge magic graceful graph with either f(uv) > f(u) + f(v) for all edges uv or f(uv) < f(u) + f(v) for all edges uv is sequential, harmonious, super edge magic and not graceful. Marimuthu, Kavitha, and Balakrishnan proved that the generalized Petersen graphs P(n, 1) and P(n, (n −1)/2) are super edge magic graceful when n is odd.
Let G = (V, E) be a (p, q)-linear forest. In Bača, Lin, Muntaner-Batle, and Rius-Font call a labeling f a strong super edge-magic labeling of G and G a strong super edge-magic graph if f : V ∪E →{1, 2, . . . , p+q} with the extra property that if uv ∈E, u′, v′ ∈ V (G) and dG(u, u′) = dG(v, v′) < +∞, then we have that f(u) + f(v) = f(u′) + f(v′).
In Ahmad, López, Muntaner-Batle, and Rius-Font define the concept of strong super edge-magic labeling of a graph with respect to a linear forest as follows. Let G = (V, E) be a (p, q)-graph and let F be any linear forest contained in G. A strong super edge-magic labeling of G with respect to F is a super edge-magic labeling f of G with the extra property with if uv ∈E(F), u′, v′ ∈V (F) and dF(u, u′) = dF(v, v′) < +∞then we have that f(u) + f(v) = f(u′) + f(v′). If a graph G admits a strong super edge-magic labeling with respect to some linear forest F, they say that G is a strong super edge-magic graph with respect to F. They prove that if m is odd and G is an acyclic graph which is strong super edge-magic with respect to a linear forest F, then mG is strong super edge-magic with respect to F1 ∪F2 ∪· · · ∪Fm, where Fi ≃F for i = 1, 2, . . . , m and every regular caterpillar is strong super edge-magic with respect to its spine.
Noting that for a super edge-magic labeling f of a graph G with p vertices and q edges, the magic constant k is given by the formula: k = (P u∈V deg(u)f(u)+Pp+q i=p+1 i)/q, López, Muntaner-Batle and Rius-Font define the set SG = n (P u∈V deg(u)g(u) + Pp+q i=p+1 i)/q : the function g : V →{i}p i=1 is bijective o .
If ⌈min SG⌉≤⌊max SG⌋then the super edge-magic interval of G is the set IG = [⌈min SG⌉, ⌊max SG⌋] ∩N.
The super edge-magic set of G is σG = {k ∈IG : there exists a super edge-magic labeling of G with valence k}. López et al. call a graph G perfect super edge-magic if IG = σG. They show that the family of paths Pn is a family of perfect super edge-magic graphs with |IPn| = 1 if n is even and |IPn| = 2 if n is odd and raise the question of whether there is an infinite family F1, F2, . . . of graphs such that each member of the family is perfect super edge-magic and limi→+∞|IFi| = +∞.
They show that graphs G ∼ = Cpk ⊙Kn where p > 2 is a prime is such a family. Taka-hasi, Muntaner-Batle, and Ichishima investigated the perfect (super) edge-magic deficiency of K1,n In López et al. define the irregular crown C(n; j1, j2, . . . , jn) = (V, E), where n > 2 and ji ≥0 for all i ∈{1, 2, . . . , n} as follows: V = {vi}n i=1 ∪V1 ∪V2 ∪· · · ∪Vn, where Vk = {v1 k, v2 k, . . . , vjk k }, if jk ̸= 0 and Vk = ∅if jk = 0, for each k ∈{1, 2, . . . , n} and E = {vivi+1}n−1 i=1 ∪{v1vn} ∪(∪n k=1,jk̸=0{vkvl k}jk l=1). In particular, they denote Cn m ∼ = C(m; j1, j2, . . . , jm), where j2i−1 = n, for each i with 1 ≤i ≤(m + 1)/2, and j2i = 0, for each i, 1 ≤i ≤(m −1)/2. They prove that the graphs Cn 3 and Cn 5 are perfect edge-magic for all n > 1.
the electronic journal of combinatorics (2023), #DS6 161 López et al. define Fk-family and Ek-family of graphs as follows. The infinite family of graphs (F1, F2, . . . ) is an Fk-family if each element Fn admits exactly k different valences for super edge-magic labelings, and limn→+∞|I(Fn)| = +∞. The infinite family of graphs (F1, F2, . . . ) is an Ek-family if each element Fn admits exactly k different va-lences for edge-magic labelings, and limn→+∞|J(Fn)| = +∞. An easy observation is that (K1,2, K1,3, . . . ) is an F2-family and an E3-family. They pose the two problems: for which positive integers k is it possible to find Fk-families and Ek-families? Their main results in are that an Fk-family exits for each k = 1, 2, 3; and an Ek-family exits for each k = 3, 4 and 7.
McSorley and Trono define a relaxed version of edge-magic total labelings of a graph as follows. An edge-magic injection µ of a graph G is an injection µ from the set of vertices and edges of G to the natural numbers such that for every edge uv the sum µ(u) + µ(v) + µ(uv) is some constant kµ. They investigate κ(G), the smallest kµ among all edge-magic injections of a graph G. They determine κ(G) in the cases that G is K2, K3, K5, K6 (recall that these are the only complete graphs that have edge-magic total labelings), a path, a cycle, or certain types of trees. They also show that every graph has an edge-magic injection and give bounds for κ(Kn).
Avadayappan, Vasuki, and Jeyanthi define the edge-magic total strength of a graph G as the minimum of all constants over all edge-magic total labelings of G. We denote this by emt(G). They use the notation < K1,n : 2 > for the tree obtained from the bistar Bn,n (the graph obtained by joining the center vertices of two copies of K1,n with an edge) by subdividing the edge joining the two stars. They prove: emt(P2n) = 5n+1; emt(P2n+1) = 5n+3; emt(< K1,n : 2 >) = 4n+9; emt(Bn,n) = 5n+6; emt((2n+ 1)P2) = 9n+6; emt(C2n+1) = 5n+4; emt(C2n) = 5n+2; emt(K1,n) = 2n+4; emt(P 2 n) = 3n; and emt(Kn,m) ≤(m + 2)(n + 1) where n ≤m. Using an analogous definition for super edge-magic total strength, Swaninathan and Jeyanthi , , provide results about the super edge-magic strength of trees, fire crackers, unicyclic graphs, and generalized theta graphs. Basha determined the reverse super edge-magic strength of banana trees.
Ngurah, Simanjuntak, and Baskoro show that certain subdivisions of the star K1,3 have super edge-magic total labelings. In Enomoto, Lladó, Nakamigawa and Ringel conjectured that all trees have a super edge-magic total labeling.
Ichishima, Muntaner-Batle, and Rius-Font have shown that any tree of order p is contained in a tree of order at most 2p −3 that has a super edge-magic total labeling.
In Bača, Lin, Muntaner-Batle, and Rius-Font use a generalization of the Kro-necker product of matrices introduced by Figueroa-Centeno, Ichishima, Muntaner-Batle, and Rius-Font to obtain an exponential lower bound for the number of non-isomorphic strong super edge-magic labelings of the graph mPn, for m odd and any n, starting from the strong super edge-magic labeling of Pn. They prove that the num-ber of non-isomorphic strong super edge-magic labelings of the graph mPn, n ≥4, is at least 5 22⌊m 2 ⌋+ 1 where m ≥3 is an odd positive integer. This result allows them to gen-erate an exponential number of non-isomorphic super edge-magic labelings of the forest F ∼ = Sm j=1 Tj, where each Tj is a path-like tree of order n and m is an odd integer.
the electronic journal of combinatorics (2023), #DS6 162 López, Muntaner-Batle, and Rius-Font introduced a generalization of super edge-magic graphs called super edge-magic models and prove some results about them.
Yegnanarayanan and Vaidhyanathan use the term nice (1, 1) edge-magic labeling for a super edge-magic total labeling. They prove: a super edge-magic total labeling f of a (p, q)-graph G satisfies 2 P v∈V (G) f(v)deg(v) ≡0 mod q; if G is (p, q) r-regular graph (r > 1) with a super edge-magic total labeling then q is odd and the magic constant is (4p + q + 3)/2; every super edge-magic total labeling has at least two vertices of degree less than 4; fans Pn + K1 are edge-magic total for all n and super edge-magic total if and only if n is at most 6; books Bn are edge-magic total for all n; a super edge-magic total (p, q)-graph with q ≥p is sequential; a super edge-magic total tree is sequential; and a super edge-magic total tree is cordial. These last three results had been proved earlier by Figueroa-Centenoa, Ichishima, and Muntaner-Batle .
In Yegnanarayanan conjectured that the disjoint union of 2t copies of P3 has a (1, 1) edge-magic labeling and posed the problem of determining the values of m and n such that mPn has a (1, 1) edge-magic labeling. Manickam and Marudai prove the conjecture and partially settle the open problem.
Hegde and Shetty (see also ) define the maximum magic strength of a graph G as the maximum magic constant over all edge-magic total labelings of G. We use eMt(G) to denote the maximum magic strength of G. Hegde and Shetty call a graph G with p vertices strong magic if eMt(G) = emt(G); ideal magic if 1 ≤eMt(G)−emt(G) ≤ p; and weak magic if eMt(G) −emt(G) > p. They prove that for an edge-magic total graph G with p vertices and q edges, eMt(G) = 3(p + q + 1) −emt(G). Using this result they obtain: Pn is ideal magic for n > 2; K1,1 is strong magic; K1,2 and K1,3 are ideal magic; and K1,n is weak magic for n > 3; Bn,n is ideal magic; (2n + 1)P2 is strong magic; cycles are ideal magic; and the generalized web W(t, 3) (see §2.2 for the definition) with the central vertex deleted is weak magic.
Santhosh has shown that for n odd and at least 3, eMt(Cn ⊙P2) = (27n+3)/2 and for n odd and at least 3, (39n + 3)/2 ≤eMt(Cn ⊙P2) ≤(40n + 3)/2. Moreover, he proved that for n odd and at least 3 both Cn ⊙P2 and Cn ⊙P3 are weak magic. In Chopra and Lee provide an number of families of super edge-magic graphs that are weak magic.
In Murugan introduces the notions of almost-magic labeling, relaxed-magic la-beling, almost-magic strength, and relaxed-magic strength of a graph. He determines the magic strength of Huffman trees and twigs of odd order and the almost-magic strength of nP2 (n is even) and twigs of even order. Also, he obtains a bound on the magic strength of the path-union Pn(m) and on the relaxed-magic strength of kSn and kPn.
Enomoto, Llado, Nakamigawa, and Ringel call an edge-magic total labeling super edge-magic if the set of vertex labels is {1, 2, . . . , |V |} (Wallis calls these labelings strongly edge-magic). They prove the following: Cn is super edge-magic if and only if n is odd; caterpillars are super edge-magic; Km,n is super edge-magic if and only if m = 1 or n = 1; and Kn is super edge-magic if and only if n = 1, 2, or 3. They also prove that if a graph with p vertices and q edges is super edge-magic then, q ≤2p −3. In MacDougall and Wallis study super edge-magic (p, q)-graphs where q = 2p −3. Enomoto the electronic journal of combinatorics (2023), #DS6 163 et al. conjecture that every tree is super edge-magic. Lee and Shan have verified this conjecture for trees with up to 17 vertices with a computer. Fukuchi, and Oshima, have shown that if T is a tree of order n ≥2 such that T has diameter greater than or equal to n −5, then T has a super edge-magic labeling.
Various classes of banana trees that have super edge-magic total labelings have been found by Swaminathan and Jeyanthi and Hussain, Baskoro, and Slamin . In Ahmad, Ali, and Baskoro investigate the existence of super edge-magic labelings of subdivisions of banana trees and disjoint unions of banana trees. They pose three open problems.
Kotzig and Rosa’s ( and ) proof that nK2 is edge-magic total when n is odd actually shows that it is super edge-magic. Kotzig and Rosa also prove that every caterpillar is super-edge magic. Figueroa-Centeno, Ichishima, and Muntaner-Batle prove the following: if G is a bipartite or tripartite (super) edge-magic graph, then nG is (super) edge-magic when n is odd ; if m is a multiple of n+1, then K1,m ∪K1,n is super edge-magic ; K1,2 ∪K1,n is super edge-magic if and only if n is a multiple of 3; K1,m ∪K1,n is edge-magic if and only if mn is even ; K1,3 ∪K1,n is super edge-magic if and only if n is a multiple of 4 ; Pm ∪K1,n is super edge-magic when m ≥4 ; 2Pn is super edge-magic if and only if n is not 2 or 3; K1,m ∪2nK2 is super edge-magic for all m and n ; C3 ∪Cn is super edge-magic if and only if n ≥6 and n is even (see also ); C4 ∪Cn is super edge-magic if and only if n ≥5 and n is odd (see also ); C5 ∪Cn is super edge-magic if and only if n ≥4 and n is even ; if m is even and at least 6 and n is odd and satisfies n ≥m/2 + 2, then Cm ∪Cn is super edge-magic ; C4 ∪Pn is super edge-magic if and only if n ̸= 3 ; C5 ∪Pn is super edge-magic if n ≥4 ; if m is even and at least 6 and n ≥m/2 + 2, then Cm ∪Pn is super edge-magic ; and Pm ∪Pn is super edge-magic if and only if (m, n) ̸= (2, 2) or (3,3) . They conjecture that K1,m ∪K1,n is super edge-magic only when m is a multiple of n+1 and they prove that if G is a super edge-magic graph with p vertices and q edges with p ≥4 and q ≥2p −4, then G contains triangles. In Figueroa-Centeno et al. conjecture that Cm ∪Cn is super edge-magic if and only if m + n ≥9 and m + n is odd.
Singgih gave super edge magic total labelings for unions of books mB(n) for odd m and m(P2 ×Pn) for m and n odd. In the editions of this survey published between 2017 and 2023 it was mistakenly stated that included proofs that r(Pm × Pn) for new odd r and (m, n) ̸= (2, 2) or (3,3), r(P3 × mPn) for odd r; mPn for m ≡2 (mod 4) and n ̸= 2, 3, and mP4n for m ≡2 (mod 4), n > 1 admitted super edge magic total labelings.
In Fukuchi and Oshima describe a construction of super edge-magic labelings of some families of trees with diameter 4. Salman, Ngurah, and Izzati use Sm n (n ≥3) to denote the graph obtained by inserting m vertices in every edge of the star Sn. They prove that Sm n is super edge-magic when m = 1 or 2.
In López, Muntaner-Batle, and Ruis-Font introduce a new construction for super edge-magic labelings of 2-regular graphs which allows loops and is related to the knight jump in the game of chess. They also study the super edge-magic properties of cycles with cords.
the electronic journal of combinatorics (2023), #DS6 164 Muntaner-Batle calls a bipartite graph with partite sets V1 and V2 special su-per edge-magic if is has a super edge-magic total labeling f with the property that f(V1) = {1, 2, . . . , |V1|}. He proves that a tree has a special super edge-magic labeling if and only if it has an α-labeling (see §3.1 for the definition). Figueroa-Centeno, Ichishima, Muntaner-Batle, and Rius-Font use matrices to generate edge-magic total labeling and define the concept of super edge-magic total labelings for digraphs. They prove that if G is a graph with a super edge-magic total labeling then for every natural number d there exists a natural number k such that G has a (k, d)-arithmetic labeling (see §4.2 for the definition). In Lee and Lee prove that a graph is super edge-magic if and only if it is (k, 1)-strongly indexable (see §4.3 for the definition of (k, d)-strongly indexable graphs). They also provide a way to construct (k, d)-strongly indexable graphs from two given (k, d)-strongly indexable graphs. This allows them to obtain several existing results about super edge-magic graphs as special cases of their constructions. Acharya and Ger-mina proved that the class of strongly indexable graphs is a proper subclass of super edge-magic graphs.
In Ichishima, López, Muntaner-Batle and Rius-Font show how one can use the product ⊗h of super edge-magic 1-regular labeled digraphs and digraphs with harmonious, or sequential labelings to create new undirected graphs that have harmonious, sequential labelings or partitional labelings (see §4.1 for the definition). They define the product ⊗h as follows. Let − → D = (V, E) be a digraph with adjacency matrix A(− → D) = (aij) and let Γ = {Fi}m i=1 be a family of m digraphs all with the same set of vertices V ′. Assume that h : E − →Γ is any function that assigns elements of Γ to the arcs of D. Then the digraph − → D ⊗h Γ is defined by V (D⊗h Γ) = V ×V ′ and ((a1, b1), (a2, b2)) ∈E(D⊗h Γ) ⇐ ⇒ [(a1, a2) ∈E(D)∧(b1, b2) ∈E(h(a1, a2))]. An alternative way of defining the same product is through adjacency matrices, since one can obtain the adjacency matrix of − → D ⊗h Γ as follows: if aij = 0 then aij is multiplied by the p′ × p′ 0-square matrix, where p′ = |V ′|.
If aij = 1 then aij is multiplied by A(h(i, j)) where A(h(i, j)) is the adjacency matrix of the digraph h(i, j). They prove the following. Let − → D = (V, E) be a harmonious (p, q)-digraph with p ≤q and let h be any function from E to the set of all super edge-magic 1-regular labeled digraphs of order n, which we denote by Sn. Then the undirected graph und(− → D ⊗hSn) is harmonious. Let − → D = (V, E) be a sequential digraph and let h : E − →Sn be any function. Then und(− → D ⊗h Sn) is sequential. Let D be a partitional graph and let h : E − →Sn be any function, where − → D = (V, E) is the digraph obtained by orienting all edges from one stable set to the other one. Then und(− → D ⊗h Sn) is partitional.
Marr, Ochel, and Perez say a digraph D with v vertices and e directed edges has an in-magic total labeling if there exists a bijective function λ from V (D) ∪E(D) to {1, 2, . . . , v+e} such that for every vertex x we have λ(x)+P λ(y, x) = k for some integer k, where the sum is taken over all directed edges (y, x). They provide such labelings for trees and cycles and discuss some relationships between this labeling and other digraph labelings.
Let D(V, A) be a digraph of order p and size q. For an integer k ≥1 and for v ∈V (D), the electronic journal of combinatorics (2023), #DS6 165 let wk(v) = P e∈Ek(v) f(e), where Ek(v) is the set of all arcs that are at distance at most k from v. The digraph D is said to be Ek-regular with regularity r if |Ek(e)| = r for some integer r ≥1 and for all e ∈A(D). A Vk-super vertex out-magic total labeling (Vk-SVIMTL) is a bijection f : V (D) ∪A(D) →{1, 2, . . . , p + q} with the property that f(V (D)) = {1, 2, . . . , p} and there exists a positive integer M such that f(v) + wk(v) = M for each vertex in V (D). A digraph that admits an Vk-SVOMTL is called an Vk-super vertex out-magic total (Ek-SVOM). Mutharasu, Kumar, and Mary Berrard new characterized the digraphs that are Vk-SVOM and the unidirectional cycles and union of unidirectional cycles that are V2-SVOM.
Let D = (V, A) be a directed graph with p vertices and q arcs. For an integer k ≥1 and v ∈V (D), let wk(v) = P e∈Ek(v) f(e), where Ek(v) is the set of all in-neighborhood arcs that are at distance at most k from v.
The digraph D is said to be Ek-regular with regularity r, if |Ek(e)| = r for some integer r ≥1 and for all e ∈A(D), where Ek(e) = Ek(u, v) = {u ∈V (D) : 1 ≤d(u, v) ≤k}. An Ek-super vertex in-magic to-tal labeling (Ek-SVIMTL) is a bijection f : V (D) ∪A(D) →{1, 2, . . . , p + q} with the property that f(A(D)) = {1, 2, . . . , q} and for each v ∈V (D), f(v) + wk(v) = M for some positive integer M. A digraph that admits an Ek-SVIMTL is called Ek-super vertex in-magic total (Ek-SVIMT). Mutharasu, Bernard, and Kumar obtained a neces- new sary and sufficient condition for the existence of Ek-SVIMTL in digraphs and the magic constant for Ek-regular digraphs. They investigated the Ek-SVIMTL of unidirectional paths and star graphs and proved necessary and sufficient conditions are for the existence of a E2-SVIMTL in unidirectional cycles and union of unidirectional cycles. In new Mutharasu and Kumar gave a necessary and sufficient condition for the existence of Ek-SVIMTL in digraphs and the magic constant for Ek-regular digraphs. They also provided a necessary and sufficient conditions for the existence of a E2-SVIMTL in unidirectional cycles and union of unidirectional cycles and investigated the existence of Ek-SVIMTL of unidirectional paths and stars.
In López, Muntaner-Batle and Rius-Font introduce the concept of {Hi}i∈I-super edge-magic decomposable as follows: Let G = (V, E) be any graph and let {Hi}i∈I be a set of graphs such that G = ⊕i∈IHi (that is, G decomposes into the graphs in the set {Hi}i∈I). Then we say that G is {Hi}i∈I-super edge-magic decomposable if there is a bijection β : V →[1, |V |] such that for each i ∈I the subgraph Hi meets the following two requirements: (i) β(V (Hi)) = [1, |V (Hi)|] and (ii) {β(a) + β(b) : ab ∈E(Hi)} is a set of consecutive integers. Such function β is called an {Hi}i∈I-super edge-magic labeling of G.
When Hi = H for every i ∈I we just use the notation H-super edge-magic decomposable labeling. Among their results are the following. Let G = (V, E) be a (p, q)-graph which is {H1, H2}-super edge-magic decomposable for a pair of graphs H1 and H2. Then G is super edge-bimagic; Let n be an even integer. Then the cycle Cn is (n/2)K2-super edge-magic decomposable if and only if n ≡2 (mod 4). Let n be odd. Then for any super edge-magic tree T there exists a bipartite connected graph G = G(T, n) such that G is (nT)-super edge-magic decomposable. Let G be a {Hi}i∈I-super edge magic decomposable graph, where Hi is an acyclic digraph for each i ∈I. Assume that − → G is any orientation of G the electronic journal of combinatorics (2023), #DS6 166 and h : E(− → G) →Sp is any function. Then und(− → G ⊗h Sp) is {pHi}i∈I-super edge magic decomposable.
As a corollary of the last result they have that if G is a 2-regular, (1-factor)-super edge-magic decomposable graph and − → G is any orientation of G and h : E(− → G) →Sp is any function, then und(− → G ⊗hSp) is a 2-regular, (1-factor)-super edge-magic decomposable graph. Moreover, if we denote the 1-factor of G by F then pF is the 1-factor of und(− → G ⊗h Sp).
They pose the following two open questions: Fix p ∈N. Find the maximum r ∈N such that there is a r-regular graph of order p which is (p/2)K2-super edge-magic decomposable: and characterize the set of 2-regular graphs of order n, n ≡2 (mod 4), such that each component has even order and admits an (n/2)K2-super edge-magic decomposition. In connection to open question 1 they prove: For all r ∈N, there is n ∈N such that there exists a k-regular bipartite graph B(n), with k > r and |V (B(n))| = 2 · 3n, such that B(n) is (3nK2)-super edge-magic decomposable.
Hendy, Sugeng, Salman, and Ayunda provided a sufficient condition for Cn[Km] to have a Pt[Km]-magic decompositions, where n > 3, m > 1, and t = 3, 4, n −2.
An H-magic labeling in an H-decomposable graph G is a bijection f V (G) ∪E(G) → {1, 2, . . . , p + q} such that, for every copy H in the decomposition, P v∈V (H) f(v) + P e∈E(H) f(e) is constant.
The function f is said to be an H-V -super magic labeling if f(V (G)) = {1, 2, . . . , p}. In Murugan and Chandra Kumar find the magic con-stant for H-factorable graphs that are H-V -super magic. Also, they give a necessary and sufficient condition for an H-factorable graph to be H-V -super magic and characterize the even regular graphs with a 2-factor-V -super magic labeling. In Murugan and new Chandra Kumar determined the magic constant for H-factorable graphs that are H-V -super magic and gave a necessary and sufficient condition for an even regular graph to be 2-factor-V -super magic.
A bipartite graph G with partite sets X1 and X2 is called consecutively super edge-magic if there exists a bijective function f : V (G)∪E (G) →{1, 2, . . . , |V (G)| + |E (G)|} such that f (X1) = {1, 2, . . . , |X1|}, f (X2) = {|X1| + 1, |X1| + 2, . . . , |V (G)|} and f (u)+ f (v) + f (uv) is a constant for each uv ∈E (G). In Ichishima, Muntaner-Batle, and Oshima investigated for which bipartite graphs is it possible to add a finite number of isolated vertices so that the resulting graph is consecutively super edge-magic. If it is possible for a bipartite graph G, then they say that the minimum such number µc(G) of isolated vertices is the consecutively super edge-magic deficiency of G; otherwise, it is +∞. Thus, the consecutively super edge-magic deficiency of a graph G is a measure of how close G is to being consecutively super edge-magic. They also include a detailed discussion of other concepts that are closely related to the consecutively super edge-magic deficiency.
In Ichishima, Muntaner-Batle, and Oshima prove that α(G) = µc(G)+|V (G)|+ 1. Thus a tree has a consecutively super edge-magic if and only if it has an α-valuation.
They explore the relation between super edge-magic labelings and graceful labelings of trees.
the electronic journal of combinatorics (2023), #DS6 167 In Ichishima, Oshima, and Takamashi introduce the notion of strength sum of a non-empty graph as follows. The strength sum strsf(G) of a numbering f : V (G) → {1, 2, . . . , |V (G)|} is defined by strs(G) = min{strsf(G) | f is a numbering of G}, where strsf(G) = P uv∈E(G)(f(u) + f(v)). A numbering f of a graph G for which strsf(G) = strs(G) is called a strength sum labeling of G. They also discuss relations among invariants on super edge-magic graphs and their its strength sums. Ichishima, Muntaner-Batle, and Oshima proved that for every k ∈[1, n −1], there exists a graph G of order n satisfying δ(G) = k and str(G) = n + k, where δ(G) denotes the minimum degree of G.
Avadayappan, Jeyanthi, and Vasuki define the super magic strength of a graph G as sm(G) = min{s(L)} where L runs over all super edge-magic labelings of G. They use the notation < K1,n : 2 > for the tree obtained from the bistar Bn,n (the graph obtained by joining the center vertices of two copies of K1,n with an edge) by subdividing the edge joining the two stars. They prove: sm(P2n) = 5n + 1; sm(P2n+1) = 5n + 3; sm(< K1,n : 2 >) = 4n + 9; sm(Bn,n) = 5n + 6; sm((2n + 1)P2) = 9n + 6; sm(C2n+1) = 5n + 4; emt(C2n) = 5n + 2; sm(K1,n) = 2n + 4; and sm(P 2 n) = 3n. Note that in each case the super magic strength of the graph is the same as its magic strength.
Santhosh and Singh proved that Cn ⊙P2 and Cn ⊙P3 are super edge-magic for all odd n ≥3 and prove for odd n ≥3, sm(Cn ⊙P2) = (15n + 3)/2 and (20n + 3) ≤ sm(Cn ⊙P3) ≤(21n + 3)/2.
Gray proves that C3 ∪Cn is super edge-magic if and only if n ≥6 and C4 ∪Cn is super edge-magic if and only if n ≥5. His computer search shows that C5 ∪2C3 does not have a super edge-magic labeling.
In Wallis posed the problem of investigating the edge-magic properties of Cn with the path of length t attached to one vertex. Kim and Park call such a graph an (n, t)-kite. They prove that an (n, 1)-kite is super edge-magic if and only if n is odd and an (n, 3)-kite is super edge-magic if and only if n is odd and at least 5. Park, Choi, and Bae show that (n, 2)-kite is super edge-magic if and only if n is even. Wallis also posed the problem of determining when K2 ∪Cn is super edge-magic.
In and Park et al. prove that K2 ∪Cn is super edge-magic if and only if n is even.
Kim and Park show that the graph obtained by attaching a pendent edge to a vertex of degree one of a star is super-edge magic and that a super edge-magic graph with edge magic constant k and q edges satisfies q ≤2k/3 −3.
Lee and Kong use St(a1, a2, . . . , an) to denote the disjoint union of the n stars St(a1), St(a2), . . . , St(an).
They prove the following graphs are super edge-magic: St(m, n) where n ≡0 mod(m+1); St(1, 1, n); St(1, 2, n); St(1, n, n); St(2, 2, n); St(2, 3, n); St(1, 1, 2, n) (n ≥2); St(1, 1, 3, n); St(1, 2, 2, n); and St(2, 2, 2, n). They conjecture that St(a1, a2, . . . , an) is super edge-magic when n > 1 is odd. Gao and Fan proved that St(1, m, n); St(3, m, m + 1); and St(n, n + 1, n + 2) are super edge-magic, and under certain conditions St(a1, a2, . . . , a2n+1), St(a1, a2, . . . , a4n+1), and St(a1, a2, . . . , a4n+3) are also super edge magic.
In MacDougall and Wallis investigate the existence of super edge-magic labelings of cycles with a chord. They use Ct v to denote the graph obtained from Cv by joining two vertices that are distance t apart in Cv. They prove: Ct 4m+1 (m ≥3) has a super the electronic journal of combinatorics (2023), #DS6 168 edge-magic labeling for every t except 4m −4 and 4m −8; Ct 4m (m ≥3) has a super edge-magic labeling when t ≡2 mod 4; and that Ct 4m+2 (m > 1) has a super edge-magic labeling for all odd t other than 5, and for t = 2 and 6. They pose the problem of what values of t does Ct 2n have a super edge-magic labeling.
Enomoto, Masuda, and Nakamigawa have proved that every graph can be em-bedded in a connected super edge-magic graph as an induced subgraph. Slamin, Bača, Lin, Miller, Simanjuntak proved that the friendship graph consisting of n triangles is super edge-magic if and only if n is 3, 4, 5, or 7. Fukuchi proved the generalized Petersen graph P(n, 2) (see §2.7 and at least 5. Baskoro and Ngurah showed that nP3 is super edge-magic for n ≥4 and n even.
Hegde and Shetty showed that a graph is super edge-magic if and only if it is strongly k-indexable (see §4.1 for the definition). Figueroa-Centeno, Ichishima, and Muntaner-Batle proved that a graph is super edge-magic if and only if it is strongly 1-harmonious and that every super edge-magic graph is cordial. They also proved that P 2 n and K2 × C2n+1 are super edge-magic. In Figueroa-Centeno et al. show that the following graphs are super edge-magic: P3 ∪kP2 for all k; kPn when k is odd; k(P2 ∪Pn) when k is odd and n = 3 or n = 4; and fans Fn if and only if n ≤6. They conjecture that kP2 is not super edge-magic when k is even. This conjecture has been proved by Z.
Chen who showed that kP2 is super edge-magic if and only if k is odd. Figueroa-Centeno et al. proved that the book Bn is not super edge-magic when n ≡1, 3, 7 (mod 8) and when n = 4. They proved that Bn is super edge-magic for n = 2 and 5 and conjectured that for every n ≥5, Bn is super edge-magic if and only if n is even or n ≡5 (mod 8). Yuansheng, Yue, Xirong, and Xinhong proved this conjecture for the case that n is even. They prove that every tree with an α-labeling is super edge-magic.
Yokomura (see ) has shown that P2m+1 × P2 and C2m+1 × Pm are super edge-magic (see also ). In , Figueroa-Centeno et al. proved that if G is a (super) edge-magic 2-regular graph, then G⊙Kn is (super) edge-magic and that Cm⊙Kn is super edge-magic.
Fukuchi shows how to recursively create super edge-magic trees from certain kinds of existing super edge-magic trees. Ngurah, Baskoro, and Simanjuntak provide a method for constructing new (super) edge-magic graphs from existing ones. One of their results is that if G has an edge-magic total labeling and G has order p and size p or p −1, then G ⊙nK1 has an edge-magic total labeling.
Ichishima, Muntaner-Batle, Oshima enlarged the classes of super edge-magic 2-regular graphs by presenting some constructions that generate large classes of super edge-magic 2-regular graphs from previously known super edge-magic 2-regular graphs or pseudo super edge-magic graphs. By virtue of known relationships among other classes of labelings the 2-regular graphs obtained from their constructions are also harmonious, sequential, felicitous and equitable. Their results add credence to the conjecture of Holden et al. that all 2-regular graphs of odd order with the exceptions of C3∪C4, 3C3∪C4, and 2C3 ∪C5 possess a strong vertex-magic total labeling, which is equivalent to super edge-magic labelings for 2-regular graphs. For a 2-regular graph G with 2m + 1 vertices that has a strong vertex-magic total labeling McQuillan and McQuillan proved that G ∪2mC3, G ∪(2m + 2)C3, G ∪mC8 and G ∪(m + 1)C8 also have a strong vertex-magic the electronic journal of combinatorics (2023), #DS6 169 total labeling.
Lee and Lee investigate the existence of total edge-magic labelings and super edge-magic labelings of unicylic graphs. They obtain a variety of positive and negative results and conjecture that all unicyclic are edge-magic total.
Shiu and Lee investigated edge labelings of multigraphs. Given a multigraph G with q edges they call a bijection from the set of edges of G to {1, 2, . . . , q} with the property that for each vertex v the sum of all edge labels incident to v is a constant independent of v a supermagic labeling of G. They use K2[n] to denote the multigraph consisting of n edges joining 2 vertices and mK2[n] to denote the disjoint union of m copies of K2[n]. They prove that for m and n at least 2, mK2[n] is supermagic if and only if n is even or if both m and n are odd.
In 1970 Kotzig and Rosa defined the edge-magic deficiency, µ(G), of a graph G as the minimum n such that G ∪nK1 is edge-magic total. If no such n exists they define µ(G) = ∞. In 1999 Figueroa-Centeno, Ichishima, and Muntaner-Batle extended this notion to super edge-magic deficiency, µs(G), is the analogous way. They prove the following: µs(nK2) = µ(nK2) = n −1 (mod 2); µs(Cn) = 0 if n is odd; µs(Cn) = 1 if n ≡0 (mod 4); µs(Cn) = ∞if n ≡2 (mod 4); µs(Kn) = ∞if and only if n ≥ 5; µs(Km,n) ≤(m −1)(n −1); µs(K2,n) = n −1; and µs(F) is finite for all forests F.
They also prove that if a graph G has q edges with q/2 odd, and every vertex is even, then µs(G) = ∞and conjecture that µs(Km,n) ≤(m −1)(n −1). This conjecture was proved for m = 3, 4, and 5 by Hegde, Shetty, and Shankaran using the notion of strongly k-indexable labelings. Baig, Baskoro, and Semaničová-Feňovčíková investigated the super edge-magic deficiency of a forest consisting of stars. Ngurah investigates the (super) edge-magic deficiency of chain graphs in and Ngurah and Adiwijaya does the same in .
For an (n, t)-kite graph (a path of length t attached to a vertex of an n-cycle) G Ahmad, Siddiqui, Nadeem, and Imran proved the following: for odd n ≥5 and even t ≥4, µs(G) = 1; for odd n ≥5, t ≥5, t ̸= 11, and t ≡3, 7 (mod 8), µs(G) ≤1; for n ≥10, n ≡2 (mod 4) and t = 4, µs(G) ≤1; and for t = 5, µs(G) = 1.
In Baig, Ahmad, Baskoro, and Simanjuntak provide an upper bound for the super edge-magic deficiency of a forest formed by paths, stars, combs, banana trees, and subdivisions of K1,3. Baig, Baskoro, and Semaničová-Feňovčíková investigate the super edge-magic deficiency of forests consisting of stars.
Among their results are: a forest consisting of k ≥3 stars has super edge-magic deficiency at most k −2; for every positive integer n a forest consisting of 4 stars with exactly 1, n, n, and n + 2 leaves has a super edge-magic total labeling; for every positive integer n a forest consisting of 4 stars with exactly 1, n + 5, 2n + 6, and n + 1 leaves has a super edge-magic total labeling; and for every positive integers n and k a forest consisting of k identical stars has super edge-magic deficiency at most 1 when k is even and deficiency 0 when k is odd. In Ahmad, Javaid, Nadeem, and Hasni investigate the super edge-magic deficiency of some families of graphs related to ladder graphs. Kanwal, Javed, and Riasat give super edge-magic total labelings and the deficiency for forests consisting of extended w-trees, combs, stars and paths. In Ahmad, Nadeem, and Gupta provided bounds for the the electronic journal of combinatorics (2023), #DS6 170 super edge-magic deficiency of some Toeplitz graphs.
The generalized Jahangir graph Jn,m for m ≥3 is a graph on nm+1 vertices, consisting of a cycle Cnm with one additional vertex that is adjacent to m vertices of Cnm at distance n to each other on Cnm. In Baig, Imran, Javaid, and Semaničová-Feňovčiková study the super edge-magic deficiencies of the web graph Wbn,m, the generalized Jahangir graph J2,n, crown products Ln ⊙K1, K4 ⊙nK1, and gave the exact value of super edge-magic deficiency for one class of lobsters.
In Figueroa-Centeno, Ichishima, and Muntaner-Batle proved that µs(Pm ∪ K1,n) = 1 if m = 2 and n is odd, or m = 3 and n is not congruent to 0 mod 3, whereas in all other cases µs(Pm ∪K1,n) = 0. They also proved that µs(2K1,n) = 1 when n is odd and µs(2K1,n) ≤1 when n is even. They conjecture that µs(2K1,n) = 1 in all cases. Other results in are: µs(Pm ∪Pn) = 1 when (m, n) = (2, 2) or (3, 3) and µs(Pm ∪Pn) = 0 in all other cases; µs(K1,m ∪K1,n) = 0 when mn is even and µs(K1,m ∪K1,n) = 1 when mn is odd; µ(Pm ∪K1,n) = 1 when m = 2 and n is odd and µ(Pm ∪K1,n) = 0 in all other cases; µ(Pm ∪Pn) = 1 when (m, n) = (2, 2) and µ(Pm ∪Pn) = 0 in all other cases; µs(2Cn) = 1 when n is even and ∞when n is odd; µs(3Cn) = 0 when n is odd; µs(3Cn) = 1 when n ≡0 (mod 4); µs(3Cn) = ∞when n ≡2 (mod 4); and µs(4Cn) = 1 when n ≡0 (mod 4).
They conjecture the following: µs(mCn) = 0 when mn is odd; µs(mCn) = 1 when mn ≡0 (mod 4); µs(mCn) = ∞when mn ≡2 (mod 4); µs(2K1,n) = 1; and if F is a forest with two components, then µ(F) ≤1 and µs(F) ≤1. Santhosh and Singh proved: for n odd at least 3, µs(K2 ⊙Cn) ≤(n −3)/2; for n > 1, 1 ≤µs(Pn[P2]) = ⌈(n −1)/2⌉; and for n ≥1, 1 ≤µs(Pn × K4) ≤n.
Ichishima and Oshima prove the following: if a graph G(V, E) has an α-labeling and no isolated vertices, then µs(G) ≤|E|−|V |+1; if a graph G(V, E) has an α-labeling, is not sequential, and has no isolated vertices, then µs(G) = |E| −|V | + 1; and, if m is even, then µs(mK1,n) ≤1. As corollaries of the last result they have: µs(2K1,n) = 1; when m ≡2 (mod 4) and n is odd, µs(mK1,n) = 1; µs(mK1,3) = 0 when m ≡4 (mod 8) or m is odd; µs(mK1,3) = 1 when m ≡2 (mod 4); µs(mK2,2) = 1; for n ≥4, (n −4)2n−2 + 3 ≤ µs(Qn) ≤(n −2)2n−1 −4; and for s ≥2 and t ≥2, µs(mKs,t) ≤m(st −s −t) + 1. They conjecture that for s ≥2 and t ≥2, µs(mKs,t) = m(st −s −t) + 1 and pose as a problem determining the exact value of µs(Qn).
Ichishima and Oshima determined the super edge-magic deficiency of graphs of the form Cm ∪Cn for m and n even and for arbitrary n when m = 3, 4, 5, and 7.
They state a conjecture for the super edge-magic deficiency of Cm ∪Cn in the general case. Afzal and Aslam investigate the super edge-magic deficiency of various disjoint unions of K2,n with stars, paths and disjoint union of paths. The join product of two graphs is their graph union with additional edges that connect all vertices of the first graph to each vertex of the second graph. In Ngurah and Simanjuntak investigate the super edge-magic deficiencies of a wheel minus an edge and join products of a path, a star, and a cycle with isolated vertices. They also show that the join product of a super edge-magic graph with isolated vertices has finite super edge-magic deficiency.
A block of a graph is a maximal subgraph with no cut-vertex. The block-cut-vertex graph of a graph G is a graph H whose vertices are the blocks and cut-vertices in G; two the electronic journal of combinatorics (2023), #DS6 171 vertices are adjacent in H if and only if one vertex is a block in G and the other is a cut-vertex in G belonging to the block. A chain graph is a graph with blocks B1, B2, B3, . . . , Bk such that for every i, Bi and Bi+1 have a common vertex in such a way that the block-cut-vertex graph is a path. The chain graph with k blocks where each block is identical and isomorphic to the complete graph Kn is called the kKn-path.
Ngurah, Baskoro, and Simanjuntak investigate the exact values of µs(kKn-path) when n = 2 or 4 for all values of k and when n = 3 for k ≡0, 1, 2 (mod 4), and give an upper bound for k ≡3 (mod 4). They determine the exact super edge-magic deficiencies for fans, double fans, wheels of small order and provide upper and lower bounds for the general case as well as bounds for some complete partite graphs. They also include some open problems. Lee and Wang show that various chain graphs with blocks that are complete graphs are super edge-magic.
In investigate the super edge-magic deficiency of some kites and Cn ∪K2.
Figueroa-Centeno and Ichishima introduce the notion of the sequential number σ(G) of a graph G without isolated vertices to be either the smallest positive integer n for which it is possible to label the vertices of G with distinct elements from the set {0, 1, . . . , n} in such a way that each uv ∈E(G) is labeled f(u) + f(v) and the resulting edge labels are |E(G)| consecutive integers or +∞if there exists no such integer n.
They prove that σ(G) = µs(G) + |V (G)| −1 for any graph G without isolated vertices, and σ(Km,n) = mn, which settles the conjecture of Figueroa-Centeno, Ichishima, and Muntaner-Batle that µs(Km,n) = (m −1)(n −1).
In Ichishima and Muntaner-Batle define the strong sequential number σs(G) of G as the smallest positive integer n for which there exists an injective function from the vertices of G to [0, n] such that when each edge uv is labeled f(u)+f(v), the resulting set of edge labels is [c, c+q−1] for some positive integer c and there exists an integer λ so that min{f(u), f(v)} ≤λ < max{f(u), f(v)} for all edges uv. Note that for G to have finite σs(G), it must be bipartite. They prove for a graph G of order p, σ (G) = µs (G) + p −1.
From this it follows that the problems of determining the sequential number and super edge-magic deficiency are equivalent and that for any graph G, σ (G) is finite if and only if µs (G) is finite. They also introduced the following parameter as a measure of how close a graph G is to having an α-labeling. The alpha-number α (G) of a graph G with q edges is the smallest positive integer n for which there exists an injective function f : V (G) →[0, n] such that when each edge uv is labeled |f (u) −f (v)| the resulting set of edge labels is [c, c + q −1] for some positive integer c, and there exists an integer λ so that min {f(u), f(v)} ≤λ < max{f(u), f(v)} for each uv ∈E(G). If no such n exists the alpha-number of G is defined to be +∞. Since a graph that admits an α-labeling is necessarily bipartite, graphs with finite α (G) are bipartite.
Ichishima and Muntaner-Batle prove: if every vertex of graph G has even degree and |E (G)| ≡2 (mod 4), then σ (G) = σs (G) = +∞; for every graph G of order p, σs (G) = µc (G)+p−1; and if G is a super edge-magic graph with at least one edge, then the graph G + nK1 is sequential for every positive integer n. As corollaries they have: for every graph σs (G) = α (G); a graph G has an α-labeling if and only if σs (G) = |E(G)|; and if a graph G of order p and size q ≥1 has a super edge-magic labeling f with s = the electronic journal of combinatorics (2023), #DS6 172 min{f(u) + f(v) : uv ∈E(G)}, then σ (G + nK1) ≤s + q + (n −1) p −2; if G is a graph of order p and size q ≥1 and G has a super edge-magic labeling f with s = min{f(u) + f(v) : uv ∈E(G)}, then µs (G + nK1) ≤s + q + (n −2) (p −1) −3; and if G is a super edge-magic graph with at least one edge, then the graph G+nK1 is harmonious and felicitous for any positive integer n.
For a graph G order p and size q Ichishima, Muntaner-Batle, and Oshima prove the following: if q = p −1 and βs (G) = p −1, then β (G ⊙nK1) = βs (G ⊙nK1) = (n + 1) p −1 for every positive integer n; if q > p −1 and βs (G) = q, then there exists a supergraph H of G such that β (H ⊙nK1) = βs (H ⊙nK1) = (n + 1) (q + 1) −1 for every positive integer n; if G has a subgraph H such that βs (H) = q < p −1, then β (H ⊙nK1) = βs (H ⊙nK1) = (n + 1) (q + 1) −1 for every positive integer n; and if G has a subgraph H such that βs (H) = q < p −k (H′), where H′ is a subgraph of H without isolated vertices, then β (H ⊙nK1) = βs (H ⊙nK1) = (n + 1) (q + 1) −1 for every positive integer n.
As the concept of super magic strength is effectively defined only for super edge-magic graphs, Ichishima, Muntaner-Batle, and Oshima generalize it for any nonempty graph as follows. A numbering f of a graph G of order p is a labeling that assigns distinct elements of the set [1, p] to the vertices of G, where each edge uv of G is labeled f (u)+f (v).
The strength, strf (G), of a numbering f : V (G) →[1, p] of G is defined by strf (G) = max {f (u) + f (v) |uv ∈E (G)} , that is, strf (G) is the maximum edge label of G, and the strength, str(G), of a graph G itself is str (G) = min {strf (G) |f is a numbering of G} . A numbering f of a graph G for which strf (G) = str (G) is called a strength labeling of G.
If G is an empty graph, then str (G) is undefined. For a graph G of order p they prove the following: if G has order at least 3 and contains a path of order k (k ∈[2, p −1]) as an induced subgraph, then str (G) ≤2p −(k −1); if ∆(G) + 2 ≤str (G) ≤2p −1; and if p + m + min {p, δ (G) + m} ≤str (G + mK1) ≤str (G) + 2m for every positive integer m.
They determine the exact strength for many basic families of graphs such as paths, cycles complete graphs, ladders, books, and hypercubes. They conclude with six problems and a conjecture.
In Takahashi, Ichishima, and Muntaner-Batle provide sharp lower bounds for new the strength of a graph in terms of its domination number as well as its (edge) covering and (edge) independence numbers. They also provide a necessary and sufficient condition for the strength of a graph to attain an earlier bound in terms of its subgraph structure.
In addition, they establish a sharp lower bound for the domination number of a graph under certain conditions.
In Ichishima, Muntaner-Batle, and Oshima determined the strength of cater-pillars and complete n-ary k-level trees. The strength str(G) is also given for graphs G obtained by taking the corona of certain graphs and arbitrary number of isolated vertices.
They further proved if G is a graph of order p with δ (G) ≥1 and str (G) = p + δ (G), then str (G ⊙nK1) = (n + 1) p + 1. for every positive integer n. In Ichishima, Muntaner-Batle, Oshima show that for every k ∈[1, n −1], there exists a graph G of order n satisfying δ (G) = k and str (G) = n + k, where δ (G) denotes the minimum de-gree of G. In Ichishima, Muntaner-Batle, and Oshima investigate minimum degree the electronic journal of combinatorics (2023), #DS6 173 conditions for the strength of graphs. They determine certain degree sequences of graphs that naturally arise and proof that these degree sequences determine a unique graph re-alization. In addition, they establish a parallel bandwidth result to the one on strength of graphs and enlarge the class of k-stable properties known so far.
The following result Ichishima, Muntaner-Batle, and Oshima in shows the con-nection between the alpha-number of a graph and its consecutively super edge-magic deficiency. For every graph G of order p, α (G) = µc(G) + p −1. This result shows that the problems of determining the alpha-number and consecutively super edge-magic deficiency are equivalent.
In Ngurah and Simanjuntak proved that if G is a cycle-free graph with minimum degree one and µs(G + K1) = 0 then G is either a tree or a forest. They also prove: the join product of some classes of trees and forests with an isolated vertex has zero super edge-magic deficiency; for all but one tree of order at most 6, their join product with an isolated vertex has zero super edge-magic deficiency. For trees T of order at least 7 they proved that if µs(T + K1) = 0, then either 2K1,3 or K3 ∪K1,3 is a subgraph of T + K1.
For the super edge-magic deficiency of the join product of a tree T of order at least 2 with m ≥2 isolated vertices, they showed that µs(T + mK1) = 0 if and only if T = P2. For a tree T ̸= P2, they proved µs(T + mK1) ≥ j (m−1)(|V (T)−2|+1) 2 k .They also present results for the super edge-magic deficiency of some chain graphs.
Z. Chen has proved: the join of K1 with any subgraph of a star is super edge-magic; the join of two nontrivial graphs is super edge-magic if and only if at least one of them has exactly two vertices and their union has exactly one edge; and if a k-regular graph is super edge-magic, then k ≤3. Chen also obtained the following: there is a connected super edge-magic graph with p vertices and q edges if and only if p −1 ≤q ≤ 2p −3; there is a connected 3-regular super edge-magic graph with p vertices if and only if p ≡2 (mod 4); and if G is a k-regular edge-magic total graph with p vertices and q edges then (p + q)(1 + p + q) ≡0 (mod 2d) where d = gcd(k −1, q). As a corollary of the last result, Chen observes that nK2 + nK2 is not edge-magic total.
Another labeling that has been called “edge-magic” was introduced by Lee, Seah, and Tan in 1992 . They defined a graph G = (V, E) to be edge-magic if there exists a bijection f : E →{1, 2, . . . , |E|} such that the induced mapping f + : V →N defined by f +(u) = P (u,v)∈E f(u, v) (mod |V |) is a constant map. Lee (see ) conjectured that a cubic graph with p vertices is edge-magic if and only if p ≡2 (mod 4).
Lee, Pigg, and Cox verified this conjecture for prisms and several other classes of cubic graphs. They also show that Cn × K2 is edge-magic if and only if n is odd. Shiu and Lee showed that the conjecture is not true for multigraphs and disconnected graphs.
In Lee’s conjecture was modified by restricting it to simple connected cubic graphs.
A computer search by Lee, Wang, and Wen showed that the new conjecture was false for a graph of order 10. Using different methods, Shiu and Lee, Su, and Wang gave proofs that it is was false.
Lee, Seah, and Tan establish that a necessary condition for a multigraph with p vertices and q edges to be edge-magic is that p divides q(q + 1) and they exhibit several new classes of cubic edge-magic graphs. They also proved: Kn,n (n ≥3) is edge-magic the electronic journal of combinatorics (2023), #DS6 174 and Kn is edge-magic for n ≡1, 2 (mod 4) and for n ≡3 (mod 4) (n ≥7). Lee, Seah, and Tan further proved that following graphs are not edge-magic: all trees except P2; all unicyclic graphs; and Kn where n ≡0 (mod 4). Schaffer and Lee have proved that Cm×Cn is always edge-magic. Lee, Tong, and Seah have conjectured that the total graph of a (p, p)-graph is edge-magic if and only if p is odd. They prove this conjecture for cycles. Lee, Kitagaki, Young, and Kocay proved that a maximal outerplanar graph with p vertices is edge-magic if and only if p = 6. Shiu used matrices with special properties to prove that the composition of Pn with Kn and the composition of Pn with Kkn where kn is odd and n is at least 3 have edge-magic labelings. Boonklurb, Narissayaporn, and Singhun show that under some conditions the m-node k-uniform hyperpaths and m-node k-uniform hypercycles are super edge-magic.
For a (p, q)-graph a bijection f from V (G) ∪E(G) to {1, 2, . . . , p + q} such that for each edge xy ∈E(G) the value of f(x) + f(xy) + f(y) is either k1, k2 or k3 is said to be an edge trimagic total labeling . Regees and Jayasekaran prove that Cm × Pn, the generalized web graph, and the generalized web graph without a center are super edge trimagic total graphs. In proved that the star type graphs P3⊙Kn, Bm,n, ⟨Bm,n : 2⟩ and ⟨K1,n 3⟩admits edge trimagic total labelings and super edge trimagic total labelings.
A triangular belt is obtained from Pn ×P2 with vertices u1, u2, . . . un and v1, v2, . . . , vn1 by adding an edge {(ukvk+1) | k = 1, 2, . . . , n −1}. The braid graph is obtained from a pair of paths Pn and P ′ n with vertices u1, u2, . . . un and v1, v2, . . . un obtained by joining the ith vertex of Pn and the (i + 1)th vertex with P ′ n and joining the ith vertex of P ′ n and the (i + 2)th vertex with Pn for 1 ≤i ≤n −2. A semi Jahangir graph is connected graph with a vertex set {u, uk | 1 ≤k ≤n} ∪{sk | 1 ≤k ≤n −1} and edge set {uksk | 1 ≤k ≤n −1} ∪{skuk+1 | 1 ≤k ≤n −1} ∪{ukui | 1 ≤k ≤n} A graph obtained from a graph G replacing each edge ei by an H-graph in such a way that the ends of ei are merged with a pendant vertex in P2 and pendant vertex in P ′ 2 is called H super subdivision of G is denoted by HSS(G), where the H-graph is a tree on 6 vertices in which exactly two vertices have degree 3. Vaghela and Parmar prove that the H-graph of a path, alternate triangular belt graph, braid graph, semi Jahangir graph, F-trees, (obtained from a path v1, v2, . . . , vn by appending an edge to vn−1 and vn), H-super subdivision of a path are edge magic total graphs, and an H-graph of a path, alternate triangular belt graph, braid graphs, semi Jahangir graph, H-super subdivision of a path, F-trees, the H ⊙K1 graph of a path are edge trimagic total graphs.
In Regees, Anisha, and Nicholas introduced the notion of edge bimagic har- new monious graphs as follows. A graph G(V, E) with p vertices and q edges is called an edge bimagic harmonious graph if there exists a bijective mapping f from V ∪E onto {1, 2, . . . , p + q} such that for each edge xy the value of the (f(x) + f(y)) mod q)((f(x) + f(y))(modq) + f(xy)) is a constant c1 or c2. If there are three such constants c1, c2 and c3, G said to be an edge trimagic harmonious graph. They proved that ladders and double ladders are edge bimagic harmonious graphs and that circular ladders, triangular ladders are edge magic and bimagic harmonious graphs. If an edge bimagic harmonious labeling has the extra property that the vertex labels are 1 the number of vertices it called a su-per edge bimagic harmonious labeling . In Deen, Aboamer, and Sherbiny showed new the electronic journal of combinatorics (2023), #DS6 175 that wheels and the splitting graph of odd cycles are super edge bimagic harmonious graphs. They also showed that sunflower graphs and double sunflower graphs are super edge trimagic harmonious graphs.
Amuthavalli and Sugapriya defined a reverse edge-trimagic labeling on a graph G(V, E) with p vertices and q edges as a one-to-one map that takes the vertices and edges onto the integers 1, 2, . . . , p + q with the property that for every edge e, when the sum of all vertex labels incident to e is subtracted from edge label f(e), the result is one of three constants. A reverse edge-trimagic labeling is said to be a reverse super edge-trimagic labeling if f(V ) = {1, 2, . . . , p} and f(E) = {p + 1, p + 2, . . . , p + q}. They investigated the reverse super edge-trimagic labeling of barycentric subdivision of bistars, degree splitting graphs of K1,n + K1,n and K1,n ∪K1,n, and the splitting graphs of stars.
Chopra, Dios, and Lee investigated the edge-magicness of joins of graphs. Among their results are: K2,m is edge-magic if and only if m = 4 or 10; the only possible edge-magic graphs of the form K3,m are those with m = 3, 5, 6, 15, 33, and 69; for any fixed m there are only finitely many n such that Km,n is edge-magic; for any fixed m there are only finitely many trees T such that T + Km is edge-magic; and wheels are not edge-magic.
Lee, Ho, Tan, and Su define the edge-magic index of a graph G to be the smallest positive integer k such that the graph kG is edge-magic. They completely determined the edge-magic indices of graphs which are stars. In Shiu, Lam, and Lee give the edge-magic index set of the second power of a path.
For any graph G and any positive integer k the graph G[k], called the k-fold G, is the hypergraph obtained from G by replacing each edge of G with k parallel edges. Lee, Seah, and Tan proved that for any graph G with p vertices, G[2p] is edge-magic and, if p is odd, G[p] is edge-magic. Shiu, Lam, and Lee show that if G is an (n + 1, n)-multigraph, then G is edge-magic if and only if n is odd and G is isomorphic to the disjoint union of K2 and (n −1)/2 copies of K2. They also prove that if G is a (2m + 1, 2m)-multigraph and k ≥2, then G[k] is edge-magic if and only if 2m + 1 divides k(k −1). For a (2m, 2m −1)-multigraph G and k at least 2, they show that G[k] is edge-magic if 4m divides (2m −1)k((2m −1)k + 1) or if 4m divides (2m + k −1)k. In Shiu, Lam, and Lee characterize the (p, p)-multigraphs that are edge-magic as mK2 or the disjoint union of mK2 and two particular multigraphs or the disjoint union of K2, mK2, and four particular multigraphs. They also show for every (2m + 1, 2m + 1)-multigraph G, G[k] is edge-magic for all k at least 2. Lee, Seah, and Tan prove that the multigraph Cn[k] is edge-magic for k ≥2.
Tables 6 and 7 summarize what is known about edge-magic total labelings and super edge-magic total labelings. We use SEMT to indicate the graphs have super edge-magic total labelings and EMT to indicate the graphs have edge-magic total labelings.
A question mark following SEMT or EMT indicates that the graph is conjectured to have the corresponding property. The tables were prepared by Petr Kovář and Tereza Kovářová.
the electronic journal of combinatorics (2023), #DS6 176 Table 6: Summary of Edge-magic Total Labelings Graph Types Notes Pn EMT trees EMT?
, Cn EMT for n ≥3 , , , Kn EMT iff n = 1, 2, 3, 4, 5, or 6 , , enumeration of all EMT of Kn Km,n EMT , crowns Cn ⊙K1 EMT , Cn with a single edge EMT attached to one vertex wheels Wn EMT iff n ̸≡3 (mod 4) , fans EMT , , (p, q)-graph not EMT if q even, p + q ≡2 (mod 4) nP2 EMT iff n odd Pn + K1 EMT r-regular graph not EMT r odd and p ≡4 (mod 8) P3 ∪nK2 and P5 ∪nK2 EMT , P4 ∪nK2 EMT n odd , nPi EMT n odd, i = 3, 4, 5 , , nP3 EMT?
2Pn EMT , Continued on next page the electronic journal of combinatorics (2023), #DS6 177 Table 6 – Continued from previous page Graph Types Notes P1 ∪P2 ∪· · · ∪Pn EMT , mK1,n EMT , unicylic graphs EMT?
K1 ⊙nK2 EMT n even , K2 × Kn EMT , nK3 EMT iff n ̸= 2 odd , , binary trees EMT , P(m, n) (generalized EMT , , Petersen graph see §2.7) ladders EMT , books EMT , odd cycle with pendent EMT , edges attached to one vertex Pm × Cn EMT n odd n ≥3 Pm × P2 EMT m odd m ≥3 K1,m ∪K1,n EMT iff mn is even G ⊙Kn EMT if G is EMT 2-regular the electronic journal of combinatorics (2023), #DS6 178 Table 7: Summary of Super Edge-magic Labelings Graph Types Notes Cn SEMT iff n is odd caterpillars SEMT , , Km,n SEMT iff m = 1 or n = 1 Kn SEMT iff n = 1, 2 or 3 trees SEMT?
nK2 SEMT iff n odd nG SEMT if G is a bipartite or tripartite SEM graph and n odd mB(n) SEMT if m is odd m(P2 × Pn) SEMT if m, n are odd K1,m ∪K1,n SEMT if m is a multiple of n + 1 K1,m ∪K1,n SEMT?
iff m is a multiple of n + 1 K1,2 ∪K1,n SEMT iff n is a multiple of 3 K1,3 ∪K1,n SEMT iff n is a multiple of 4 Pm ∪K1,n SEMT if m ≥4 is even 2Pn SEMT iff n is not 2 or 3 2P4n SEMT for all n mPn SEMT if m ≡2 (mod 4), n ̸= 2, 3 mP4n SEMT ifm ≡2 (mod 4), n > 1 K1,m ∪2nK1,2 SEMT for all m and n Continued on next page the electronic journal of combinatorics (2023), #DS6 179 Table 7 – Continued from previous page Graph Types Notes C3 ∪Cn SEMT iff n ≥6 even , C4 ∪Cn SEMT iff n ≥5 odd , C5 ∪Cn SEMT iff n ≥4 even Cm ∪Cn SEMT if m ≥6 even, n odd n ≥m/2 + 2 Cm ∪Cn SEMT?
iff m + n ≥9 and m + n odd C4 ∪Pn SEMT iff n ̸= 3 C5 ∪Pn SEMT if n ̸= 4 Cm ∪Pn SEMT if m ≥6 even, n ≥m/2 + 2 Pm ∪Pn SEMT iff (m, n) ̸= (2, 2) or (3, 3) corona Cn ⊙Km SEMT n ≥3 St(m, n) SEMT n ≡0 (mod m + 1) St(1, k, n) SEMT k = 1, 2 or n St(2, k, n) SEMT k = 2, 3 St(1, 1, k, n) SEMT k = 2, 3 St(k, 2, 2, n) SEMT k = 1, 2 St(a1, . . . , an) SEMT?
for n > 1 odd Ct 4m SEMT Ct 4m+1 SEMT friendship graph of SEMT iff n = 3, 4, 5, or 7 n triangles Continued on next page the electronic journal of combinatorics (2023), #DS6 180 Table 7 – Continued from previous page Graph Types Notes generalized Petersen SEMT if n ≥3 odd graph P(n, 2) (see §2.7) nP3 SEMT if n ≥4 even P 2 n SEMT K2 × C2n+1 SEMT P3 ∪kP2 SEMT for all k kPn SEMT if k is odd k(P2 ∪Pn) SEMT if k is odd and n = 3, 4 fans Fn SEMT iff n ≤6 books Bn SEMT if n even books Bn SEMT?
if n ≡5 (mod 8) trees with α-labelings SEMT P2m+1 × P2 SEMT , C2m+1 × Pm SEMT G ⊙Kn SEMT if G is SEM 2-regular graph Cm ⊙Kn SEMT join of K1 with any SEMT subgraph of a star if G is k-regular SEMT then k ≤3 graph G is connected (p, q)-graph SEMT G exists iff p −1 ≤q ≤2p −3 G is connected 3-regular SEMT iff p ≡2 (mod 4) Continued on next page the electronic journal of combinatorics (2023), #DS6 181 Table 7 – Continued from previous page Graph Types Notes graph on p vertices nK2 + nK2 not SEMT 5.3 Vertex-magic Total Labelings MacDougall, Miller, Slamin, and Wallis introduced the notion of a vertex-magic total labeling in 1999. For a graph G(V, E) an injective mapping f from V ∪E to the set {1, 2, . . . , |V | + |E|} is a vertex-magic total labeling if there is a constant k, called the magic constant, such that for every vertex v, f(v) + P f(vu) = k where the sum is over all vertices u adjacent to v (some authors use the term “vertex-magic” for this concept).
They prove that the following graphs have vertex-magic total labelings: Cn; Pn (n > 2); Km,m (m > 1); Km,m −e (m > 2); and Kn for n odd. They also prove that when n > m + 1, Km,n does not have a vertex-magic total labeling. They conjectured that Km,m+1 has a vertex-magic total labeling for all m and that Kn has vertex-magic total labeling for all n ≥3. The latter conjecture was proved by Lin and Miller for the case that n is divisible by 4 while the remaining cases were done by MacDougall, Miller, Slamin, and Wallis . McQuillan provided many vertex-magic total labelings for cycles Cnk for k ≥3 and odd n ≥3 using given vertex-magic labelings for Ck. Gray, MacDougall, and Wallis then gave a simpler proof that all complete graphs are vertex-magic total. Krishnappa, Kothapalli, and Venkaiah gave another proof that all complete graphs are vertex-magic total. Senthil Amutha and Murugesan characterized connected vertex magic total labeling graphs through their ideals in topological spaces. Among other results, Wang and Zhang settle a 2006 conjecture raised by Slamin et al., which claims the existence of the vertex magic total labeling of disjoint union of multiple copies of Cn ⊙K1. Vimal Kumar and Vijayalakshmi investigated vertex magic total labelings of the middle and total graphs of cycles.
In MacDougall, Miller, Slamin, and Wallis conjectured that for n ≥5, Kn has a vertex-magic total labeling with magic constant h if and only if h is an integer satisfying n3 + 3n ≤4h ≤n3 + 2n2 + n. In McQuillan and Smith proved that this conjecture is true when n is odd. Armstrong and McQuillan proved that if n ≡2 (mod 4) (n ≥6) then Kn has a vertex-magic total labeling with magic constant h for each integer h satisfying n3 + 6n ≤4h ≤n3 + 2n2 −2n. If, in addition, n ≡2 (mod 8), then Kn has a vertex-magic total labeling with magic constant h for each integer h satisfying n3 + 4n ≤4h ≤n3 + 2n2. They further showed that for each odd integer n ≥5, 2Kn has a vertex-magic total labeling with magic constant h for each integer h such that n3 + 5n ≤2h ≤n3 + 2n2 −3n. If, in addition, n ≡1(mod 4), then 2Kn has a vertex-magic total labeling with magic constant h for each integer h such that n3 + 3n ≤2h ≤n3 + 2n2 −n.
the electronic journal of combinatorics (2023), #DS6 182 In McQuillan and McQuillan investigate the existence of vertex-magic labelings of nC3. They prove: for every even integer n ≥4, nC3 is vertex-magic (and therefore also edge-magic); for each even integer n ≥6, nC3 has vertex-magic total labelings with at least 2n −2 different magic constants; if n ≡2 mod 4, two extra vertex-magic total labelings with the highest possible and lowest possible magic constants exist; if n = 2·3k, k > 1, nC3 has a vertex-magic total labeling with magic constant k if and only if (1/2)(15n + 4) ≤k ≤(1/2)(21n + 2); if n is odd, there are vertex-magic total labelings for nC3 with n + 1 different magic constants. In McQuillan provides a technique for constructing vertex-magic total labelings of 2-regular graphs. In particular, if m is an odd positive integer, G = Cn1 ∪Cn2 ∪· · · ∪Cnk has a vertex-magic total labeling, and J is any subset of I = {1, 2, . . . , k} then (∪i∈J mCni) ∪(∪i∈I−J mCni) has a vertex-magic total labeling.
In Cichacz, Fronček and Singgih introduced a new method to expand some known vertex magic total labelings of 2-regular graphs. The also proved that for odd values of m, if (2r + 1) ̸≡0 (mod 3) and n ̸≡0 (mod (2r + 1)), then 2mCrn ∪mCn has a vertex magic total labeling.
Lin and Miller have shown that Km,m is vertex-magic total for all m > 1 and that Kn is vertex-magic total for all n ≡0 (mod 4). Phillips, Rees, and Wallis generalized the Lin and Miller result by proving that Km,n is vertex-magic total if and only if m and n differ by at most 1. Cattell has shown that a necessary condition for a graph of the form H +Kn to be vertex-magic total is that the number of vertices of H is at least n −1. As a corollary he gets that a necessary condition for Km1,m2,...,mr,n where n is the largest size of any partite set to be vertex-magic total is that m1+m2+· · ·+mr ≥n.
He poses as an open question whether graphs that meet the conditions of the theorem are vertex-magic total. Cattell also proves that K1,n,n has a vertex-magic total labeling when n is odd and K2,n,n has a vertex-magic total labeling when n ≡3 (mod 4). In Rahim and Slamin proved the disjoint union of coronas Ct1 ⊙K1∪Ct2 ⊙K1∪· · ·∪Ctn ⊙K1 has a vertex-magic total labeling with magic constant 6 Pn k=1 tk + 1.
Miller, Bača, and MacDougall have proved that the generalized Petersen graphs P(n, k) (see §2.7) for the definition) are vertex-magic total when n is even and k ≤n/2−1.
They conjecture that all P(n, k) are vertex-magic total when k ≤(n −1)/2 and all prisms Cn × P2 are vertex-magic total. Bača, Miller, and Slamin proved the first of these conjectures (see also for partial results) while Slamin and Miller prove the second. Slamin, Prihandoko, Setiawan, Rosita and Shaleh constructed vertex-magic total labelings for the disjoint union of two copies of P(n, k) and Silaban, Parestu, Herawati, Sugeng, and Slamin extended this to any number of copies of P(n, k).
More generally, they proved that for nj ≥3 and 1 ≤kj ≤⌊(nj −1)/2⌋, the union P(n1, k1) ∪P(n2, k2) ∪· · · ∪P(nt, kt) has a vertex-magic total labeling with vertex magic constant 10(n1 + n2 + · · · + nt) + 2. In the same article Silaban et al. define the union of t special circulant graphs ∪t j=1Cn(1, mj) as the graph with vertex set {vj i | 0 ≤i ≤n−1, 1 ≤ j ≤t} and edge set {vj i vj i+1| 0 ≤i ≤n−1, 1 ≤j ≤t}∪{vj i vj i+mj| 0 ≤i ≤n−1, 1 ≤j ≤t}.
They prove that for odd n at least 5 and mj ∈{2, 3, . . . , (n −1)/2}, the disjoint union ∪t j=1Cn(1, mj) has a vertex-magic total labeling with constant 8tn + (n −10/2 + 3.
the electronic journal of combinatorics (2023), #DS6 183 MacDougall et al. ( , and ) have shown: Wn has a vertex-magic total labeling if and only if n ≤11; fans Fn have a vertex-magic total labelings if and only if n ≤10; friendship graphs have vertex-magic total labelings if and only if the number of triangles is at most 3; Km,n (m > 1) has a vertex-magic total labeling if and only if m and n differ by at most 1. Wallis proved: if G and H have the same order and G∪H is vertex-magic total then so is G+H; if the disjoint union of stars is vertex-magic total, then the average size of the stars is less than 3; if a tree has n internal vertices and more than 2n leaves then it does not have a vertex-magic total labeling. Wallis has shown that if G is a regular graph of even degree that has a vertex-magic total labeling then the graph consisting of an odd number of copies of G is vertex-magic total. He also proved that if G is a regular graph of odd degree (not K1) that has a vertex-magic total labeling then the graph consisting of any number of copies of G is vertex-magic total.
Gray, MacDougall, McSorley, and Wallis investigated vertex-magic total label-ings of forests. They provide sufficient conditions for the nonexistence of a vertex-magic total labeling of forests based on the maximum degree and the number of internal vertices, and leaves or the number of components. They also use Skolem sequences to prove a star forest with each component a K1,2 has a vertex-magic total labeling.
Recall a helm Hn is obtained from a wheel Wn by attaching a pendent edge at each vertex of the n-cycle of the wheel. A generalized helm H(n, t) is a graph obtained from a wheel Wn by attaching a path on t vertices at each vertex of the n-cycle. A generalized web W(n, t) is a graph obtained from a generalized helm H(n, t) by joining the corresponding vertices of each path to form an n-cycle.
Thus W(n, t) has (t + 1)n + 1 vertices and 2(t + 1)n edges. A generalized Jahangir graph Jk,s is a graph on ks + 1 vertices consisting of a cycle Cks and one additional vertex that is adjacent to k vertices of Cks at distance s to each other on Cks. Rahim, Tomescu, and Slamin prove: Hn has no vertex-magic total labeling for any n ≥3; W(n, t) has a vertex-magic total labeling for n = 3 or n = 4 and t = 1, but it is not vertex-magic total for n ≥17t + 12 and t ≥0; and Jn,t+1 is vertex-magic total for n = 3 and t = 1, but it does not have this property for n ≥7t + 11 and t ≥1. Recall a flower is the graph obtained from a helm by joining each pendent vertex to the central vertex of the helm. Ahmad and Tomescu proved that flower graph is vertex-magic if and only if the underlying cycle is C3.
Fronček, Kovář, and Kovářová proved that Cn × C2m+1 and K5 × C2n+1 are vertex-magic total. Kovář furthermore proved some general results about products of certain regular vertex-magic total graphs. In particular, if G is a (2r+1)-regular vertex-magic total graph that can be factored into an (r+1)-regular graph and an r-regular graph, then G × K5 and G × Cn for n even are vertex-magic total. He also proved that if G an r-regular vertex-magic total graph and H is a 2s-regular supermagic graph that can be factored into two s-regular factors, then their Cartesian product G × H is vertex-magic total if either r is odd, or r is even and |H| is odd.
Ivančo and Polláková consider supermagic graphs having a saturated vertex (i.e., a vertex that is adjacent to every other vertex).
They characterize supermagic graphs G + K1, where G is a regular graph, using a connection to vertex-magic total graphs. They prove that if G is a d-regular graph of order n then the join G + K1 is the electronic journal of combinatorics (2023), #DS6 184 supermagic if and only if G has a VMT labeling with constant h such that (n −d −1) is a divisor of the non-negative integer (n + 1)h −n((d + 2)/2)(n(d + 2)/2) + 1). They also prove K1,n,n is supermagic if and only if n ≥2; K1,2,2,...,2 is supermagic except for K1,2; and the graph obtained from Kn,n (n ≥5) by removing all edges in a Hamilton cycle is supermagic. They also consider circulant graphs and prove that the complement of the circulant graph C2n(1, n), n ≥4, is supermagic.
In a novel algorithm is proposed based on the calculation of vertex magic total labelling value for every node in the network. Upon receiving the message from the sender node, the receiver node will quickly detect the faulty node by comparing the vertex magic total labeling pivot value. Experimental results show that the proposed approach leads to high true fault rate detection accuracy compared to the false fault rate detection.
MacDougall, Miller, and Sugeng define a super vertex-magic total labeling of a graph G(V, E) as a vertex-magic total labeling f of G with the additional property that f(V ) = {1, 2, . . . , |V |} and f(E) = {|V | + 1, |V | + 2, . . . , |V | + |E|} (some authors use the term “super vertex-magic” for this concept). They show that a (p, q)-graph that has a super vertex-magic total labeling with magic constant k satisfies the following conditions: k = (p + q)(p + q + 1)/v −(v + 1)/2; k ≥(41p + 21)/18; if G is connected, k ≥(7p −5)/2; p divides q(q +1) if p is odd, and p divides 2q(q +1) if p is even; if G has even order either p ≡0 (mod 8) and q ≡0 or 3 (mod 4) or p ≡4 (mod 8) and q ≡1 or 2 (mod 4); if G is r-regular and p and r have opposite parity then p ≡0 (mod 8) implies q ≡0 (mod 4) and p ≡4 (mod 8) implies q ≡2 (mod 4). They also show: Cn has a super vertex-magic total labeling if and only if n is odd; and no wheel, ladder, fan, friendship graph, complete bipartite graph or graph with a vertex of degree 1 has a super vertex-magic total labeling.
They conjecture that no tree has a super vertex-magic total labeling and that K4n has a super vertex-magic total labeling when n > 1. The latter conjecture was proved by Gómez in . In Gómez proved that if G is a d-regular graph that has a vertex-magic total labeling and k is a positive integer such that (k −1)(d + 1) is even, then kG has a super vertex-magic total labeling. As a corollary, we have that if n and k are odd or if n ≡0 (mod 4) and n > 4, then kKn has a super vertex-magic total labeling. Gómez also shows how graphs with super vertex-magic total labeling can be constructed from a given graph G with super vertex-magic total labeling by adding edges to G in various ways.
Gray and MacDougall establish the existence of vertex-magic total labelings for several infinite classes of regular graphs. Their method enables them to begin with any even-regular graph and from it construct a cubic graph possessing a vertex-magic total labeling.
A feature of the construction is that it produces strong vertex-magic total labelings many even order regular graphs. The construction also extends to certain families of non-regular graphs. MacDougall has conjectured (see ) that every r-regular (r > 1) graph with the exception of 2K3 has a vertex-magic total labeling. As a corollary of a general result Kovář has shown that every 2r-regular graph with an odd number of vertices and a Hamiltonian cycle has a vertex-magic total labeling.
Gómez and Kovář proved that a super vertex-magic total labeling of kKn exists for n odd and any k, for 4 < n ≡0 (mod 4) and any k, and for n = 4 and k even. They also showed kK4t+2 does not admit a super vertex-magic total labeling for k odd and the electronic journal of combinatorics (2023), #DS6 185 provide a large number of super vertex-magic total labelings of kK4t+2 for any k based on a super vertex-magic total labeling of kK4t+1.
Beardon has shown that a necessary condition for a graph with c components, p vertices, q edges and a vertex of degree d to be vertex-magic total is (d+2)2 ≤(7q2+(6c+ 5)q +c2 +3c)/p. When the graph is connected this reduces to (d+2)2 ≤(7q2 +11q +4)/p.
As a corollary, the following are not vertex-magic total: wheels Wn when n ≥12; fans Fn when n ≥11; and friendship graphs C(n) 3 when n ≥4.
Beardon has investigated how vertices of small degree effect vertex-magic total labelings. Let G(p, q) be a graph with a vertex-magic total labeling with magic constant k and let d0 be the minimum degree of any vertex. He proves k ≤(1 + d0)(p + q −d0/2) and q < (1 + d0)q. He also shows that if G(p, q) is a vertex-magic graph with a vertex of degree one and t is the number of vertices of degree at least two, then t > q/3 ≥(p−1)/3.
Beardon has shown that the graph obtained by attaching a pendent edge to Kn is vertex-magic total if and only if n = 2, 3, or 4.
Meissner and Zwierzyński used finding vertex-magic total labelings of graphs as a way to compare the efficiency of parallel execution of a program versus sequential processing.
Swaminathan and Jeyanthi prove the following graphs are super vertex-magic total: Pn if and only if n is odd and n ≥3; Cn if and only if n is odd; the star graph if and only if it is P2; and mCn if and only if m and n are odd. In they prove the following: no super vertex-magic total graph has two or more isolated vertices or an isolated edge; a tree with n internal edges and tn leaves is not super vertex-magic total if t > (n + 1)/n; if ∆is the largest degree of any vertex in a tree T with p vertices and ∆> (−3+ √1 + 16p)/2, then T is not super vertex-magic total; the graph obtained from a comb by appending a pendent edge to each vertex of degree 2 is super vertex-magic total; the graph obtained by attaching a path with t edges to a vertex of an n-cycle is super vertex-magic total if and only if n + t is odd. Ali, Bača, and Bashir proved that mP3 and mP4 have no super vertex-magic total labeling For n > 1 and distinct odd integers x, y and z in [1,n −1] Javaid, Ismail, and Salman define the chordal ring of order n CRn(x, y, z), as the graph with vertex set Zn, the additive group of integers modulo n, and edges (i, i + x), (i, i + y), (i, i + z) for all even i. They prove that CRn(1, 3, n −1) has a super vertex-magic total labeling when n ≡0 mod 4 and n ≥8 and conjecture that for an odd integer ∆, 3 ≤∆≤n −3, n ≡0 mod 4, CRn(1, ∆, n −1) has a super vertex-magic total labeling with magic constant 23n/4 + 2.
The Knödel graphs W∆,n with n even and degree ∆, where 1 ≤∆≤⌊log2n⌋have vertices pairs (i, j) with i = 1, 2 and 0 ≤j ≤n/2 −1 where for every 0 ≤j ≤n/2 −1 and there is an edge between vertex (1, j) and every vertex (2, (j + 2k −1) mod n/2), for k = 0, 1, . . . , ∆−1. Xi, Yang, Mominul, and Wong have shown that W3,n is super vertex-magic total when n ≡0 mod 4.
Marimuthu and Balakrishnan called a vertex magic total labeling of G(V, E) E-super vertex magic if f(E(G)) = {1, 2, 3, . . . , |E(G)|}.
The cocktail party graph, Hm,n (m, n ≥2), is the graph with a vertex set V = {v1, v2, . . . , vmn} partitioned into n independent sets V = {I1, I2, . . . , In} each of size m such that vivj ∈E for all the electronic journal of combinatorics (2023), #DS6 186 i, j ∈{1, 2, . . . , mn} where i ∈Ip, j ∈Iq, p ̸= q. (The graph Hn,n is the complement of the ladder graph and the dual graph of the n-cube.) Marimuthu and Balakrishnan gave some basic properties of such labelings and proved that Hm,n is E-super vertex magic.
Wang and Zhang show the following: Hamiltonian even regular graphs of odd order are E-super magic; even-regular graphs of odd order that contains a 2-factor consisting of an odd number of odd cycles with the same size are E-super vertex magic; graphs that can be decomposed into the sum of two spanning graphs where one is E-super magic and one is regular of even degree are E-supermagic; even-regular graphs of odd order that contain a 2-factor consisting of an odd number of odd cycles with the same size are E-super vertex magic; and circulant graphs with odd order are E-super vertex magic. Swaminathan and Jeyanthi proved that mCn is E-super magic if and only if both m and n are odd.
Samarathunge, Athapattu, and Perera proved that an E-super vertex labeling of edges adjacent to a leaf vertex must be greater than or equal to k −(p + q) and sum of the parent edge labelings must be greater than or equal to k −(p + q), which k is the magic constant. This implies that an E-super vertex labeling does not been proved that E-super vertex magic labeling does not exist for the perfect binary trees. A broom Bn,d is ontained by attaching n −d pendent edges to one of the pendent vertices of Pd.
Marimuthu, Suganya, Kalaivani, and Balakrishnan proved the following: Bn,n−2 is E-super vertex magic if and only if n is odd; a graph obtained by subdividing all the edges of K1,n is super E-super vertex magic if and only if n ≤4, and K1,n is E-super vertex magic if and only if n = 2.
In Marimuthu and Kumar investigate E-super vertex magic labelings of dis-connected graphs. They prove: if a graph with p vertices and q edges and even order has an E-super vertex magic labeling, then either (i) p ≡0 (mod 8) and q ≡0 or 3 (mod 4), or (ii) p ≡4 (mod 8) and q ≡1 or 2 (mod 4); if an r-regular graph G of order p has an E-super vertex magic labeling, then p and r have opposite parity and (i) if p ≡0 (mod 8), then q ≡0 (mod 4) (ii) if p ≡4 (mod 8), then q ≡2 (mod 4); mCn is E-super vertex magic if and only if Pn ∪(m −1)Cn is E-super vertex magic; Pm ∪K1,m is not E-super vertex magic; Cm ∪Pn is not E-super vertex magic if both m and n have the same parity; the disjoint union of two non-isomorphic suns is not E-super vertex magic; the disjoint union of any number of isomorphic suns is not E-super vertex magic; and mP3 is not E-super vertex magic for any integer m > 1. They conjecture that Km ∪Pm is E-super vertex magic if m = 8t + 2.
In Mutharasu and Kumar generalized the notion of super vertex-magic total labelings as follows. Let G(V, E) be a graph and k be an integer with 1 ≤k ≤diam(G).
For e ∈E(G), let Ek(e) be the set of all vertices that are at a distance at most k from e and let Ek(v) be the set of all edges that are at a distance at most k from v (u and v are at distance 1 from the edge uv). A graph G is said to be Ek-regular with regularity r if, for all edges e, |Ek(e)| = r for some positive integer r. Note that all nontrivial graphs are E1-regular.
Let G be a simple graph with p vertices and q edges.
A V -super vertex magic labeling is a bijection f : V (G) ∪E(G) →{1, 2, . . . , p + q} such that f(V (G)) = {1, 2, . . . , p} and for each vertex v ∈V (G), f(v) + P u∈N(v) f(uv) = M for some positive integer M. A Vk-super vertex magic labeling (Vk-SVML) is a bijection the electronic journal of combinatorics (2023), #DS6 187 f : V (G) ∪E(G) →{1, 2, . . . , p + q} with the property that f(V (G)) = {1, 2, . . . , p} and for each v ∈V (G), f(v) + P e∈Ek(v) f(e) = M for some positive integer M. A graph that admits a Vk-SVML is called Vk-super vertex magic. Mutharasu and Kumar gave a necessary and sufficient condition for the existence of Vk-SVML in graphs, determined the magic constant for Ek-regular graphs, and obtained results about V2-SVML labelings for cycles, complement of cycles, prisms, and a family of circulant graphs.
Balbuena, Barker, Das, Lin, Miller, Ryan, and Slamin call a vertex-magic to-tal labeling of G(V, E) a strongly vertex-magic total labeling if the vertex labels are {1, 2, . . . , |V |}. They prove: the minimum degree of a strongly vertex-magic total graph is at least 2; for a strongly vertex-magic total graph G with n vertices and e edges, if 2e ≥ √ 10n2 −6n + 1 then the minimum degree of G is at least 3; and for a strongly vertex-magic total graph G with n vertices and e edges if 2e < √ 10n2 −6n + 1 then the minimum degree of G is at most 6. They also provide strongly vertex-magic total label-ings for certain families of circulant graphs. In McQuillan provides a technique for constructing vertex-magic total labelings of 2-regular graphs. In particular, if m is an odd positive integer, G = Cn1 ∪Cn2 ∪· · · ∪Cnk has a strongly vertex-magic total labeling, and J is any subset of I = {1, 2, . . . , k} then (∪i∈J mCni) ∪(∪i∈I−J mCni) has a strongly vertex-magic total labeling.
Gray proved that if G is a graph with a spanning subgraph H that possesses a strongly vertex-magic total labeling and G −E(H) is even regular, then G also possesses a strongly vertex-magic total labeling. As a corollary one has that regular Hamiltonian graphs of odd order have a strongly vertex-magic total labelings.
In a series of papers Gray and MacDougall expand on McQuillan’s technique to obtain a variety of results. In Gray and MacDougall show that for any r ≥4, every r-regular graph of odd order at most 17 has a strong vertex-magic total labeling. They also show that several large classes of r-regular graphs of even order, including some Hamiltonian graphs, have vertex-magic total labelings. They conjecture that every 2-regular graph of odd order possesses a strong vertex-magic total labeling if and only if it is not of the form (2t −1)C3 ∪C4 or 2tC3 ∪C5. They include five open problems.
In Gray and MacDougall introduce a procedure called a mutation that trans-forms one vertex-magic totaling labeling into another one by swapping sets of edges among vertices that may result in different labeling of the same graph or a labeling of a different graph. Among their results are: a description of all possible mutations of a labeling of the path and the cycle; for all n ≥2 and all i from 1 to n −1 the graphs obtained by identifying an end points of paths of lengths i, i + 1, and 2n −2i −1 have a vertex-magic total labeling; for odd n, the graph obtained by attaching a path of length n−m to an m cycle, (such graphs are called (m; n −m)-kites ) have strong vertex-magic total labelings for m = 3, . . . , n−2; C2n+1 ∪C4n+4 and 3C2n+1 have a strong vertex-magic total labeling; and for n ≥2, C4n ∪C6n−1 has a strong vertex-magic total labeling. They conclude with three open problems.
Kimberley and MacDougall studied mutations that involve labelings of regular graphs into labelings of other regular graphs. They present results of extensive compu-tations which confirm how prolific this procedure is. These computations add weight to the electronic journal of combinatorics (2023), #DS6 188 MacDougall’s conjecture that all nontrivial regular graphs are vertex-magic.
Gray and MacDougall show how to construct vertex-magic total labelings for several families of non-regular graphs, including the disjoint union of two other graphs already possessing vertex-magic total labelings. They prove that if G is a d-regular graph of order v and H a t-regular graph of order u with each having a strong vertex magic total labeling and vd2 + 2d + 2v + 2u = 2tvd + 2t + ut2 then G ∪H possesses a strong vertex-magic total labeling. They also provide bounds on the minimum degree of a graph with a vertex-magic total labeling.
In Gray and MacDougall establish the existence of vertex-magic total labelings for several infinite classes of regular graphs. Their method enables them to begin with any even-regular graph and construct a cubic graph possessing a vertex-magic total labeling that produces strong vertex-magic total labelings for many even order regular graphs.
The construction also extends to certain families of non-regular graphs.
In Nagaraj, Ponnappan, and Prabakaran define a vertex-magic total la-beling of G to be an even vertex magic total labeling if the set of vertex labels is {2, 4, 6, . . . , 2|V (G)|}. They prove the following: Cn is even vertex magic total if and only if n is odd; rCs is even vertex magic total if and only if r and s are odd; Cn ⊙K1 is even vertex magic total; wheels are not even vertex magic total; fans (excluding C3) are not even vertex magic total; kites are not even vertex magic total; and K4n is not even vertex magic total. In they prove that C3 ∪C2t (t > 2) and C4 ∪C2t+1 (t ≥2) have even vertex magic total labelings. In Nagaraj, Ponnappan, and Prabakan prove that the union of any finite numbers of graphs of the form Cn ⊙K1 (the sizes may vary) has an even vertex magic total labeling. Saduakdee and Khemmani studied new conditions for which there exists an even vertex magic labeling of the t-fold wheel Wn,t in terms of n and t. Furthermore, those having an even vertex magic total labeling are determined.
Rahim and Slamin give the bounds for the number of vertices for Jahangir graphs, helms, webs, flower graphs and sunflower graphs when the graphs considered are not vertex-magic total. Thirusangu, Nagar, and Rajeswari show that certain Cayley digraphs of cyclic groups have vertex-magic total labelings.
Balbuena, Barker, Lin, Miller, and Sugeng call vertex-magic total labeling an a-vertex consecutive magic labeling if the vertex labels are {a, a + 1, . . . , a + |V |}. For an a-vertex consecutive magic labeling of a graph G with p vertices and q edges they prove: if G has one isolated vertex, then a = q and (p −1)2 + p2 = (2q + 1)2; if q = p −1, then p is odd and a = p −1; if p = q, then p is odd and if G has minimum degree 1, then a = (p+1)/2 or a = p; if G is 2-regular, then p is odd and a = 0 or p; and if G is r-regular, then p and r have opposite parities. They also define an b-edge consecutive magic labeling analogously and state some results for these labelings. Marimuthu and Kumar gave some results related to a-vertex consecutive magic graphs. In Setiawan, Sugeng, Silaban, and Riama provided sufficient conditions for a graph to admit a super edge magic total labeling and to have a b-edge consecutive edge magic total labeling. They also gave the super edge magic total labelings for banana trees, firecrackers, and identified several classes of connected graphs that have both labelings.
the electronic journal of combinatorics (2023), #DS6 189 Wood generalizes vertex-magic total and edge-magic total labelings by requiring only that the labels be positive integers rather than consecutive positive integers. He gives upper bounds for the minimum values of the magic constant and the largest label for complete graphs, forests, and arbitrary graphs.
Exoo, Ling, McSorley, Phillips, and Wallis call a function λ a totally magic labeling of a graph G if λ is both an edge-magic total and a vertex-magic total labeling of G. A graph with such a labeling is called totally magic. Among their results are: P3 is the only connected totally magic graph that has a vertex of degree 1; the only totally magic graphs with a component K1 are K1 and K1 ∪P3; the only totally magic complete graphs are K1 and K3; the only totally magic complete bipartite graph is K1,2; nK3 is totally magic if and only if n is odd; P3 ∪nK3 is totally magic if and only if n is even.
In Wallis asks: Is the graph K1,m ∪nK3 ever totally magic? That question was answered by Calhoun, Ferland, Lister, and Polhill who proved that if K1,m ∪nK3 is totally magic then m = 2 and K1,2 ∪nK3 is totally magic if and only if n is even.
McSorley and Wallis examine the possible totally magic labelings of a union of an odd number of triangles and determine the spectrum of possible values for the sum of the label on a vertex and the labels on its incident edges and the sum of an edge label and the labels of the endpoints of the edge for all known totally magic graphs.
Gray and MacDougall define an order n sparse semi-magic square to be an n×n array containing the entries 1, 2, . . . , m once (for some m < n2), has its remaining entries equal to 0, and whose rows and columns have a constant sum of k. They prove some basic properties of such squares and provide constructions for several infinite families of squares, including squares of all orders n ≥3. Moreover, they show how such arrays can be used to construct vertex-magic total labelings for certain families of graphs.
In Tables 8, 9, and 10, VMT means vertex-magic total labeling, SVMT means super vertex magic total, and TM means totally magic labeling. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property.
The tables were prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2007.
Table 8: Summary of Vertex-magic Total Labelings Graph Types Notes Cn VMT Pn VMT n > 2 Km,m −e VMT m > 2 Km,n VMT iff |m −n| ≤1 , , Kn VMT for n odd Continued on next page the electronic journal of combinatorics (2023), #DS6 190 Table 8 – Continued from previous page Graph Types Notes for n ≡2 (mod 4),n > 2 nK3 VMT iff n ̸= 2 , , mKn VMT m ≥1, n ≥4 Petersen P(n, k) VMT prisms Cn × P2 VMT Wn VMT iff n ≤11 , Fn VMT iff n ≤10 , friendship graphs VMT iff # of triangles ≤3 , G + H VMT |V (G)| = |V (H)| and G ∪H is VMT unions of stars VMT tree with n internal vertices not VMT and more than 2n leaves nG VMT n odd, G regular of even degree, VMT G is regular of odd degree, VMT, but not K1 Cn × C2m+1 VMT K5 × C2n+1 VMT G × C2n VMT G 2r + 1-regular VMT G × K5 VMT G 2r + 1-regular VMT G × H VMT G r-regular VMT, r odd or r even and |H| odd, H 2s-regular supermagic the electronic journal of combinatorics (2023), #DS6 191 Table 9: Summary of Super Vertex-magic Total Labelings Graph Types Notes Pn SVMT iff n > 1 is odd Cn SVMT iff n is odd and K1,n SVMT iff n = 1 mCn SVMT iff m and n are odd Wn not SVMT ladders not SVMT friendship graphs not SVMT Km,n not SVMT dragons (see §2.2) SVMT iff order is even , Knödel graphs W3,n SVMT n ≡0 (mod 4) graphs with min. deg. 1 not SVMT K4n SVMT n > 1 Table 10: Summary of Totally Magic Labelings Graph Types Notes P3 TM the only connected TM graph with vertex of deg 1 Kn TM iff n = 1, 3 Km,n TM iff Km,n = K1,2 nK3 TM iff n is odd Continued on next page the electronic journal of combinatorics (2023), #DS6 192 Table 10 – Continued from previous page Graph Types Notes P3 ∪nK3 TM iff n is even K1,m ∪nK3 TM iff m = 2 and n is even 5.4 H-Magic Labelings In 2005 Gutiérrez and Lladó introduced the notion of an H-magic labeling of a graph, which generalizes the concept of a magic valuation. Let H and G = (V, E) be finite simple graphs with the property that every edge of G belongs to at least one subgraph isomorphic to H. A bijection f : V ∪E →{1, . . . , |V | + |E|} is an H-magic labeling of G if there exists a positive integer m(f), called the magic sum, such that for any subgraph H′(V ′, E′) of G isomorphic to H, the sum P v∈V ′ f(v)+P e∈E′ f(e) is equal to the magic sum, m(f). A graph is H-magic if it admits an H-magic labeling. If, in addition, the H-magic labeling f has the property that {f(v)}v∈V = {1, . . . , |V |}, then the graph is H-supermagic. A K2-magic labeling is also known as an edge-magic total labeling. Gutiérrez and Lladó investigate the cases where G = Kn or G = Km,n and H is a star or a path. Among their results are: a d-regular graph is not K1,h for any 1 < h < d; Kn,n is K1,n-magic for all n; Kn,n is not K1,n-supermagic for n > 1; for any integers 1 < r < s, Kr,s is K1,h-supermagic if and only if h = s; Pn is Ph-supermagic for all 2 ≤h ≤n; Kn is not Ph-magic for any 2 < h ≤n; Cn is Ph-magic for any 2 ≤h < n such that gcd(n, h(h −1)) = 1. They also show that by uniformly gluing copies of H along edges of another graph G, one can construct connected H-magic graphs from a given 2-connected graph H and an H-free supermagic graph G.
Lladó and Moragas studied cycle-magic graphs. They proved: wheels Wn are C3-magic for odd n at least 5; for r ≥3 and k ≥2 the windmill graphs C(k) r (the one-point union of k copies of Cr) are Cr-supermagic; and if G is C4-free supermagic graph of odd size, then G × K2 is C4-supermagic. As corollaries of the latter result, they have that for n odd, prisms Cn × K2 and books K1,n × K2 are C4-magic. They define a subdivided wheel Wn(r, k) as the graph obtained from a wheel Wn by replacing each radial edge vvi, 1 ≤i ≤n by a vvi-path of size r ≥1, and every external edge vivi+1 by a vivi+1-path of size k ≥1. They prove that Wn(r, k) is C2r+k-magic for any odd n ̸= 2r/k +1 and that Wn(r, 1) is C2r+1-supermagic. They also prove that the graph obtained by joining the end points of any number of internally disjoint paths of length p ≥2 is C2p-supermagic.
Asif, Ali, Numan, and Semaničová-Feňovčíková proved that if G is Cr-(super)magic, then so is nG and that Pm × Pn (m, n ≥4) is C4-supermagic. In Pradipta and Salman define a calendula graph, denoted by Clm,n, as the graph constructed from Cm and m copies of Cn, Cn1, Cn2, . . . , Cnm, and grafting the i-th edge of Cm to an edge of Cni for each i. They provide some cycle-supermagic labelings of calendula graphs.
the electronic journal of combinatorics (2023), #DS6 193 Chithra, Marimuthu, and Kumar provided some basic results on the magic constant of graphs, on cycle-supermagic labelings of generalized splitting graphs, and proved that mCn is cycle -supermagic for m ≥2 and n ≥3. In Azeem provided C8-super magic new labelings of zig-zag chains, linear chains, and the disjoint union of non-isomorphic copies of both chains. Yang, Rashid, Siddiqui, and Hanif provided explicit formulas for various Cn-super magic labelings of pumpkin graphs (graphs with exactly two vertices with multiple edges joining them) and two classes of planar graphs containing 8-sided and 4-sided faces or 6-sided and 4-sided faces.
A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f from V (G) ∪E(G) onto {0, 1, . . . , |V (G)| + |E(G)|1} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. By using the results on the sumset partition problem, Inayah, Lladó, and Moragas show that K2m+1 admits T-magic decompositions by any graceful tree with m edges. They address analogous problems for complete bipartite graphs and for antimagic and (a, d)-antimagic decompositions.
An edge of H-magic graph G is said to be a good edge if it belongs to only one subgraph isomorphic to H. For s ≥1, B is the collection of good edges obtained by choosing exactly s good edges from each subgraph isomorphic to H in G. A uniform subdivided graph G of the graph G is obtained by subdividing all edges of B with k ≥1 vertices. A nonuniform subdivided graph is obtained by subdividing the edges of E(G) \ B. Rizvi, Khalid, Ali, Miller, and Ryan prove that if a graph G is a Cn-supermagic graph then its uniform subdivided graph G is Cn+sk-(super)magic for positive integers n, s, and k. Using known results on the cycle-supermagicness they immediately obtain that uniform subdivided graphs of fans, antiprisms, triangular ladders, ladders and grids are cycle-(super)magic.
They also prove that some special nonuniform subdivisions of fans and triangular ladders are cycle-supermagic.
Jeyanthi and Muthuraja established that Pm,n is C2m-supermagic for all m, n ≥2 and the splitting graph of Cn is C4-supermagic for n ̸= 4. Nirmalasari Wi-jaya, Ryan, and Kalinowski show that for odd n and arbitrary k, the firecracker Fk,n is F2,n-supermagic, the banana tree Bk,n is B1,n-supermagic, and flower graphs are C3-supermagic. Kojima proved that for two positive integers m and t with m > t ≥2, if Cm is Pt-supermagic, then C3m is also Pt-supermagic and for t = 2, 3, 4, or 9 and Cn is Pt-supermagic if and only if n is odd with n > t. Nirmalasari Wijaya, Ryan, and Kalinowski proved that every d-dimensional grid graph (d > 2) is Qd-supermagic where Qd is the d-cube. Pu, Numan, Butt, Asif, Rafique, and Shao showed that toroidal fullerenes, Klein-bottle fullerenes, and the disjoint union of toroidal and Klein-bottle fullerenes are C6-supermagic and the subdivision of toroidal fullerenes, Klein-bottle fullerenes, and any graph homeomorphic to a toroidal fullerene or Klein-bottle fullerene are cyclic-supermagic. Ulfatimah, Roswitha, and Kusmayadi proved that a star with one or more appended edges at each end-point admits a double star S2,2-supermagic labeling and Lm ⊙Pn admits supermagic labeling of the one-point union of C3 and C4 for m, n ≥2. Martini, Roswitha, and Lestari provided a C4-supermagic labeling of Pn × Pm and a C3 supermagic labeling of K1 + K2.
the electronic journal of combinatorics (2023), #DS6 194 Öner and Erol showed that the triangular book-snake graph S(3, n, s) admits a C3-supermagic labeling. Using these labelings and supermagicness of subdivisions given by Taimur, Numan, Mumtaz, and Semanic̆ová-Fen̆ovc̆íková in and the fact that S(m, n, s) can be thought of as a subdivision of S(3, n, s), Öner and Erol showed the Cm-supermagicness of polygonal book-snake graph S(m, n, s).
The edge corona path graph Gm ⋄Pn is the graph obtained from one copy of the gear graph Gm and 3m copies of Pn, P i n, by joining two end vertices of ei ∈E(Gm) to every vertex vjinV (Pn) in the i-th copy of Gm with i = 1, 2, . . . , 3m and j = 1, 2, . . . , n.
Noviati, Martini, and Indriati provided a C3 ⋄Pn-supermagic labeling for fn ⋄Pn and a P3 ⋄Pn-supermagic labeling for Sn ⋄Pn for odd n ≥3.
Rizvi, Ali, and Hussian proved: the disjoint union of two or more copies of G is C3-supermagic when G is a fan, triangular ladder, wheel, or a generalized antiprism; the disjoint union of two or more copies of G is C3-supermagic when G is a ladder or a book; sFn+1 ∪kFn is C3-supermagic; and sLn+1 ∪kLn is C4-supermagic. Khalid, Rizvi, and Ali investigated whether the disjoint union of isomorphic copies of a connected cycle-supermagic graph is cycle-supermagic or not. They also study cycle-supermagic labelings for the disjoint union of isomorphic copies of fans, ladders, triangular ladders, wheels, books, and generalized antiprisms as well as disjoint unions of non-isomorphic copies of ladders and fans.
Ali, Rizvi, Semaničová-Feňovčíková proved that the disjoint union of an arbitrary number of isomorphic copies of prisms Cn × Pm, m ≥2 and n ≥3, n ̸= 4, is C4-supermagic. They propose an open problem to find a C4-supermagic labeling of the graph t(C4 × Pm) for m ≥2 and t ≥1.
Liang proved the following: if there exist an even integer k and mi ≡0 (mod k) for every i in [1,n], then there exist Kk,k- and C2k-supermagic decompositions of Km1,...,mn; if k and tn ≥k are even integers, then for any positive integers ti ≡0 (mod k), i in [1, n−1], there exists a C2k-supermagic decomposition of Kt1,...,tn−1,tn; if there exists an even integer k and Km,n is C2k-decomposable, then there exists a C2k-supermagic decomposition of Km,n; and if G is a graph with p vertices and p edges, H is a graph with q vertices and q edges, and there is an H-supermagic decomposition of G, then there exists an H-supermagic decomposition of nG. In Wichianpaisarn and Mato gave necessary and sufficient conditions for the existence of K1,n−1-supermagic decomposition of Kn,n minus a one-factor.
In Maryati, Baskoro, and Salman provided Pn-(super) magic labelings of subdi-visions of stars, shrubs and banana trees. Ngurah, Salman, and Sudarsana construct Cn-(super) magic labelings for some fans and ladders. For any connected graph H, Mary-ati, Salman, Baskoro, and Irawati proved that the disjoint union of k isomorphic copies of a connected graph H is a H-supermagic graph if and only if |V (H)| + |E(H)| is even or k is odd. In Maryati, Baskoro, Salman, and Irawati give some necessary conditions for any Pn-magic graph and provide some Pn-supermagic labelings of a cycle with some pendent edges and its subdivisions.
The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1, G2, . . . , Gm, and then joining every vertex u in Gi, i ∈{1, 2, . . . , m −1}, to the neighbors of the corresponding vertex v in Gi+1. Agustin, Susanto, Dafik, Prihandini, the electronic journal of combinatorics (2023), #DS6 195 Alfarisi, and Sudarsana studied the H-supermagic labelings of Dm(G) where G are paths and cycles.
Kojima proved the following. Let G be a C4-free super edge-magic (p, q)-graph with the minimum degree at least one and m ≥2. If q odd and m = 2 or |p−q| ≥2, then Pm × G is C4-supermagic; if p is odd and m = 2 or |p −q| = 1 and m ≤5, then Pm × G is C4-supermagic; if n ≥3 is odd and m is even, then P2 × (Cn ⊙Km) is C4-supermagic; if n ≥3 is odd and m is odd, then P2 × (Cn ⊙Km) is not C4-supermagic; if G is a caterpillar, then Pm × G is C4-supermagic for m ≥2; and Pm × Cn is C4-supermagic for m ≥2 and n ≥3. The latter result solved an open problem in by Ngurah, Salman, and Susilowati. Kojma also proved that if a C4-free bipartite (p, p −1)-graph G with the minimum degree at least one and partite sets U and V has a super edge-magic labeling f of G such that f(U) = {1, 2, . . . , |U|}, then Pm × (2G) is C4-supermagic.
Maryati, Salman, Baskoro, Ryan, and Miller define a shackle as a graph obtained from nontrivial connected graphs G1, G2, . . . , Gk (k ≥2) such that Gs and Gt have no common vertex for every s and t in [1, k] with |s −t| ≥2, and for every i in [1, k − 1], Gi and Gi+1 share exactly one common vertex that are all distinct.
They prove that shackles and amalgamations constructed from copies of a connected graph H is H-supermagic. (Recall for finite collection of graph G1, G2, . . . , Gk with a fixed vertex vi from each Gi, an amalgamation, AmalGi, vi), is the graph obtained by identifying the vi.) Ashari and Salman gave sufficient conditions for (H1, H2)-supermagic labelings for shackles involving cycles, flowers, and prisms.
Ngurah, Salman, and Susilowati proved the following: chain graphs with identi-cal blocks each isomorphic to Cn are Cn-supermagic; fans are C3-supermagic; ladders and books are C4-supermagic; K1,n +K1 are C3-supermagic; grids Pm ×Pn are C4-supermagic for m ≥3 and n = 3, 4, and 5. They pose the case that Pm × Pn are C4-supermagic for n > 5 as an open problem. They also have some results on Pt-(super) magic labelings of cycles. Kathiresan, Marimuthu, and Chithra investigated the existence of Cm-supermagic labelings generalized fans, generalization of a graph obtained by joining of a star K1,n with one isolated vertex, grids, and generalized books. The results given in this article are the generalizations of some results given by Ngurah, Salman, and Susilowati in .
Roswitha, Baskoro, Maryati, Kurdhi, and Susanti proved: the generalized Ja-hangir graph Jk,s is Cs+2-supermagic; K2,n is C4-supermagic; and Wn for n even and n ≥4 is C3-supermagic. As an open problem they asked if Km,n, 2 < m ≤n, admits a C2m-supermagic labeling. Roswitha and Baskoro proved that double stars, caterpillars, firecrackers, and banana trees admit star-supermagic labelings.
Maryati, Salman, and Baskoro characterized all graphs G such that the disjoint union of copies of G is G-supermagic. They also showed: the disjoint union of any paths is mPn-supermagic for certain values of m and n; some subgraph amalgamations of graphs G are G-supermagic; and for any subgraph H of G Amal(G, H, k) is G-supermagic.
Salman and Maryati proved that Amal(G, Pn, k) is G-supermagic.
Selvagopal and Jeyanthi proved: for any positive integer n, a the k-polygonal snake of length n is Ck-supermagic ; for m ≥2, n = 3, or n > 4, Cn × Pm is C4-supermagic ; P2 × Pn and P3 × Pn are C4-supermagic for all n ≥2 ; the the electronic journal of combinatorics (2023), #DS6 196 one-point union of any number of copies of a 2-connected H is H-magic ; graphs obtained by taking copies H1, H2, . . . , Hn of a 2-connected graph H and two distinct edges ei, e′ i from each Hi and identifying e′ i of Hi with ei+1 of Hi+1 where |V (H)| ≥4, |E(H)| ≥4 and n is odd or both n and |V (H)|+|E(H)| are even are H-supermagic . For simple graphs H and G the H-supermagic strength of G is the minimum constant value of all H-magic total labelings of G for which the vertex labels are {1, 2, . . . , |V |}. Jeyanthi and Selvagopal found the Cn-supermagic strength of n-polygonal snakes of any length and the H-supermagic strength of a chain of an arbitrary 2-connected simple graph.
Let H1, H2, . . . , Hn be copies of a graph H. Let ui and vi be two distinct vertices of Hi for i = 1, 2, . . . , n. The chain graph Hn of H of length n is the graph obtained by identifying the vertices ui and vi+1 for i = 1, 2, . . . , n −1. In Jayanthi and Selvagopal show that a chain graph of any 2-connected simple graph H is H-supermagic and if H is a 2-connected (p, q) simple graph, then Hn is H-supermagic if p + q is even or p + q + n is even.
The antiprism on 2n vertices has vertex set {x1,1, . . . , x1,n, x2,1, . . . , x2,n} and edge set {xj,i, xj,i+1} ∪{x1,i, x2,i} ∪{x1,i, x2,i−1} (subscripts are taken modulo n). Jeyanthi, Sel-vagopal, and Sundaram proved the following graphs are C3-supermagic: antiprisms, fans, and graphs obtained from the ladders P2 × Pn with the two paths v1,1, . . . , v1,n and v2,1, . . . , v2,n by adding the edges v1,jv2,j+1.
Jeyanthi and Selvagopal show that for any 2-connected simple graph H the edge amalgamation of a finite number of copies of H is H-supermagic. They also show that the graph obtained by picking one endpoint vi from each of k copies of K1,k then creating a new graph by joining each vi to a fixed new vertex v is K1,k-supermagic.
An H-magic labeling in an H-decomposable of a graph G is a bijection f : V (G)∪E(G) onto {1, 2, . . . , p + q} such that for every copy of H in the decomposition, the sum of f(v) + f(e) over all v in V (H) and e in E(H) is constant. The labeling f is said to be H −V -super magic if f(V (G)) = {1, 2, . . . , p}. Marimuthu and Kumar prove that Kn,n (n ≥2) is H-V -super magic decomposable when H is K1,n. Marimuthu and Kumar provide a necessary and sufficient condition for the existence of V -super vertex-magic labeling and give E-super and V -super vertex-magic total labeling of certain families of generalized Petersen graphs. They also prove that no wheel is E-super vertex-magic, C3 is the only friendship graph that is V -super vertex-magic, and C3 is the only friendship graph that is E-super vertex-magic.
An H-magic labeling f is said to be an H-E-super magic labeling if f(E(G)) = {1, 2, . . . , q}. A graph that admits an H-E-super magic labeling is called an H-E-super magic decomposable graph.
Subbiah and Pandimadevi study some elementary properties of H-E-super magic labelings with H an m-factor and provide a necessary and sufficient condition for an even regular graph to be H-E-super magic decomposable where H is a 2-factor.
the electronic journal of combinatorics (2023), #DS6 197 5.5 Magic Labelings of Type (a, b, c) A magic-type method for labeling the vertices, edges, and faces of a planar graph was introduced by Lih in 1983. Lih defines a magic labeling of type (1,1,0) of a planar graph G(V, E) as an injective function from {1, 2, . . . , |V |+|E|} to V ∪E with the property that for each interior face the sum of the labels of the vertices and the edges surrounding that face is some fixed value. Similarly, Lih defines a magic labeling of type (1, 1, 1) of a planar graph G(V, E) with face set F as an injective function from {1, 2, . . . , |V |+|E|+|F|} to V ∪E ∪F with the property that for each interior face the sum of the labels of the face and the vertices and the edges surrounding that face is some fixed value. Lih calls a labeling involving the faces of a plane graph consecutive if for every integer s the weights of all s-sided faces constitute a set of consecutive integers. Lih gave consecutive magic labelings of type (1, 1, 0) for wheels, friendship graphs, prisms, and some members of the Platonic family. In Bača shows that the cylinders Cn × Pm have magic labelings of type (1, 1, 0) when m ≥2, n ≥3, n ̸= 4. In Bača proves that the generalized Petersen graph P(n, k) (see §2.7 for the definition) has a consecutive magic labeling if and only if n is even and at least 4 and k ≤n/2−1. In Ahmed and Babujee provide face bimagic labelings of type (1,1,0) for certain wheels, cylinders, and disjoint unions of m copies of prism graphs. Roopa and Shobana proved the existence of face new bimagic labelings of types (1,0,1), (1,1,0) and (0,1,1) for the graphs obtained by double duplication of all vertices by edges of ladder graphs. They also proved that if a graph G is (1, 0, 1)-face bimagic, except for three sided faces, then double duplication of all vertices by edges of G is face bimagic.
Bača gave magic labelings of type (1, 1, 1) for fans , ladders , planar bipyra-mids (that is, 2-point suspensions of paths) , grids , hexagonal lattices , Möbius ladders , and Pn × P3 . Kathiresan and Ganesan show that the graph Pa,b consisting of b ≥2 internally disjoint paths of length a ≥2 with common end points has a magic labeling of type (1, 1, 1) when b is odd, and when a = 2 and b ≡0 (mod 4). They also show that Pa,b has a consecutive labeling of type (1, 1, 1) when b is even and a ̸= 2. Ali, Hussain, Ahmad, and Miller study magic labeling of type (1, 1, 1) for wheels and subdivided wheels. They prove: wheels admits a magic labeling of type and (1, 1, 1) and (0, 1, 1), for odd n wheels Wn n admit a magic labeling of type (0, 1, 0), and subdivided wheels admit a magic labeling of type (1, 1, 0). As an open problem they ask for a magic labeling of type (1, 1, 0) for Wn and n even. Ahmad proves that subdivided ladders admit magic labelings of type (1,1,1) and admit consecutive magic labelings of type (1,1,0).
Bača , , , , , and Bača and Holländer gave magic labelings of type (1, 1, 1) and type (1, 1, 0) for certain classes of convex polytopes. Kathire-san and Gokulakrishnan provided magic labelings of type (1, 1, 1) for the families of planar graphs with 3-sided faces, 5-sided faces, 6-sided faces, and one external infinite face. Bača also provides consecutive and magic labelings of type (0, 1, 1) (that is, an injective function from {1, 2, . . . , |E| + |F|} to E ∪F with the property that for each interior face the sum of the labels of the face and the edges surrounding that face is some the electronic journal of combinatorics (2023), #DS6 198 fixed value) and a consecutive labeling of type (1, 1, 1) for a kind of planar graph with hexagonal faces. Tabraiz and Hussain provide a super magic labeling of type (1, 0, 0) for ladders and a super magic labeling of type (1, 0, 0) for subdivided ladders.
A magic labeling of type (1,0,0) of a planar graph G with vertex set V is an injective function from {1, 2, . . . , |V |} to V with the property that for each interior face the sum of the labels of the vertices surrounding that face is some fixed value. Kathiresan, Muthuvel, and Nagasubbu define a lotus inside a circle as the graph obtained from the cycle with consecutive vertices a1, a2, . . . , an and the star with central vertex b0 and end vertices b1, b2, . . . , bn by joining each bi to ai and ai+1 (an+1 = a1). They prove that these graphs (n ≥5) and subdivisions of ladders have consecutive labelings of type (1, 0, 0). Devaraj proves that graphs obtained by subdividing each edge of a ladder exactly the same number of times has a magic labeling of type (1, 0, 0).
In Table 11 we use following abbreviations M(a, b, c) magic labeling of type (a, b, c) CM(a, b, c) consecutive magic labeling of type (a, b, c).
A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová.
the electronic journal of combinatorics (2023), #DS6 199 Table 11: Summary of Magic Labelings of Type (a, b, c) Graph Labeling Notes Wn CM(1,1,0) friendship graphs CM(1,1,0) prisms CM(1,1,0) cylinders Cn × Pm M(1,1,0) m ≥2, n ≥3, n ̸= 4 fans Fn M(1,1,1) ladders M(1,1,1) planar bipyramids (see §5.3) M(1,1,1) grids M(1,1,1) hexagonal lattices M(1,1,1) Möbius ladders M(1,1,1) Pn × P3 M(1,1,1) certain classes of M(1,1,1) , , , convex polytopes M(1,1,0) , certain classes of planar graphs M(0,1,1) with hexagonal faces CM(0,1,1) CM(1,1,1) lotus inside a circle (see §5.3) CM(1,0,0) n ≥5 subdivisions of ladders M(1,0,0) CM(1,0,0) the electronic journal of combinatorics (2023), #DS6 200 5.6 Sigma Labelings/1-vertex magic labelings/Distance Magic In 1987 Vilfred (see also ) defined a sigma-labeling of a graph G with n vertices as a bijection f from the vertices of G to {1, 2, . . . , n} such that there is a constant k with the property that, at any vertex v the sum P f(u) taken over all neighbors u of v is k. The concept of sigma labeling was independently studied in 2003 by Miller, Rodger, and Simanjuntak in under the name 1-vertex magic. In a 2009 article Sugeng, Fronček, Miller, Ryan, and Walker used the term distance magic labeling.
For convenience, we will use the term distance magic. In Vilfred and Jinnah give a number of necessary conditions for a graph to have a distance magic labeling. One of them is that if u and v are vertices of a graph with a distance labeling, then the order of the symmetric difference of N(u) and N(v) (neighborhoods of u and v) is not 1 or 2.
This condition rules out a large class of graphs as having distance magic labelings. Rao, Singh, and Parameswaran have shown Cm × Cn has a distance magic labeling if and only if m = n ≡2 (mod 4) and Km × Kn, m ≥2, n ≥3 does not have a distance magic labeling. In Benna gives necessary and sufficient condition for Km,n to be a distance magic graph and proves that if G1 and G2 are connected graphs with minimum degree 1 and at least three vertices, then G1 ×G2 does not have a distance magic labeling.
Rao, Sighn, and Parameswaran prove that every graph is an induced subgraph of a regular graph that has a distance magic labeling. As open problems, Rao asks for a characterize 4-regular graphs that have distance magic labelings and which graphs of the form Cm × Cn, m = n ≡2 (mod 4) have distance magic labelings. Kovář, Fronček, and Kovářová classified all orders n for which a 4-regular distance magic graph exists and also showed that there exists a distance magic graph with k = 2t for every integer t ≥6. Acharaya, Rao, Signh, and Parameswaran proved Pm × Cn does not have a distance magic labeling when m is at least 3 and provide necessary and sufficient conditions for Km,n to have a distance magic labeling. However, in 2024 Rozmana new and S̆parl classify all distance magic Cartesian products of two cycles, thereby correcting an error in paper cited above by Rao, Singh, and Parameswaren .
Rozmana and S̆parl further show that each distance magic labeling of a Cartesian product of cycles is determined by a pair or quadruple of suitable sequences, thereby obtaining a complete characterization of all distance magic labelings of these graphs. They also determine a lower bound on the number of all distance magic labelings of Cm × C2m for m ≥3 odd.
Kovár and Silber proved that an (n −3)-regular distance magic graph with n vertices exists if and only if n ≡3 (mod 6) and that its structure is determined uniquely.
Moreover, they reduce constructions of Fronček to a single construction and provide an-other sufficient condition for the existence a distance magic graph with an odd number of vertices. Fronček, Kovář, and Kovářová provide a construction for distance magic graphs arising from arbitrary regular graphs based on an application of magic rectan-gles. They also solve a problem posed by Shafiq, Ali, and Simanjuntak . In new Kovářa, Silber, Kabelíıkova–-Hrušsková, and Kravčenko classify of all feasible odd orders of r-regular distance magic graphs for r = 6, 8, 10, and 12. Godinho and Singh in-vestigate the distance magic labelings for neighborhood expansions of graphs and present the electronic journal of combinatorics (2023), #DS6 201 a method for embedding regular graphs into distance magic graphs.
Among the results of Miller, Rodger, and Simanjuntak in : the only trees that have a distance magic labeling are P1 and P3; Cn has a distance magic labeling if and only if n = 4; Kn has a distance magic labeling if and only if n = 1; the wheel Wn = Cn + P1 has a distance magic labeling if and only if n = 4; the complete graph Kn,n,...,n with p partite sets has a distance magic labeling if and only if n is even or both n and p are odd; an r-regular graph where n is odd does not have a distance magic labeling; and G × K2n has a distance magic labeling for any regular graph G. They also give necessary and sufficient conditions for complete tripartite graphs to have a distance magic labeling.
For a given a graph G = (V, E) and a positive integer t, the generalized Mycielskian graph Mt(G) is the graph with vertex set {V ×{0, 1, . . . , t−1}}∪{u}, with edges (x, 0)(y, 0) and (x, i)(y, i + 1) where there is an edge xy ∈E, and an edge (x, t −1)u for all x ∈V .
In Pawar and Singh gave some results on distance magic labeling of generalized Mycielskian graphs.
Generalizing definitions given by Vilfred in and , Vilfred Kamalappan defines a labeling f : V (G) →{1, 2, . . . , n} on a graph G of order n ≥3 as a k-distance magic labeling (k-DML ) if P w∈∂Nk(u) f(w) is a constant and independent of u ∈V (G) where ∂Nk(u) = {v ∈V (G) : d(u, v) = k}, k ∈N. A graph G is called a k-distance magic (k-DM) if it has a k-DML. A long brush, denoted by LPn,m, is a graph with vertex set {u1, u2, . . . , un, v1, v2, . . . , vm}, a path Pn = u1 u2 · · · un and edge set E(Pn) ∪{u1vi : i = 1 to m} ∪E(< v1, v2, . . . , vm >), m + n ≥3 and m, n ∈N. Using partition techniques, he obtain families of k-DM graphs and prove that (i) For k, n ≥3, m ≥2 and k, m, n ∈N, LPn,m is k-DM if and only if m(m −1) ≤2n and k = n; (ii) For a given m ≥2 and for every k ∈N0, LP m(m−1) 2 +k,m is a ( m(m−1) 2 + k)-DM graph, and (iii) For m ≥3, x ≥2, LP1,m = K1(u1) + (Km1 ∪Km2 ∪... ∪Kmx), 1 ≤m1 ≤m2 ≤... ≤mx, m1 + m2 + ... + mx = m, m1 + m2 ≥3 and m1, m2, ..., mx, x ∈N, LP1,m is 2-DM if and only if u1 is assigned with a suitable j and Jm+1 \ {j} is partitioned into x constant sum partites of orders m1, m2, ..., mx, 1 ≤j ≤m + 1.
In Fronček defined the notion of a Γ-distance magic graph as one that has a bijective labeling of vertices with elements of an Abelian group Γ resulting in constant sums of neighbor labels. A graph that is Γ-distance magic for an Abelian group Γ is called group distance magic. Cichacz and Fronček showed that for an r-regular distance magic graph G on n vertices, where r is odd there does not exist an Abelian group Γ of order n having exactly one involution (i.e., an element that is its own inverse) that is Γ-distance magic. Fronček proved that Cm × Cn is a Zmn-distance magic graph if and only if mn is even. He also showed that C2n ×C2n has a Z22n-distance magic labeling.
In Cichacz showed some Γ-distance magic labelings for Cm ×Cn where Γ ̸≈Zmn and Γ ̸≈Z22n. Anholcer, Cichacz, Peterin, and Tepeh proved that if an r1-regular graph G1 is Γ1-distance magic and an r2-regular graph G2 is Γ2-distance magic, then the direct product of graphs G1 and G2 is Γ1 × Γ2-distance magic. Moreover they showed that if G is an r-regular graph of order n and m = 4 or m = 8 and r is even, then Cm × G is group distance magic. They proved that Cm × Cn is Zmn-distance magic if and only if m ∈{4, 8} or n ∈{4, 8} or m, n ≡0 (mod 4). They also showed that if m, n ̸≡0 (mod 4) the electronic journal of combinatorics (2023), #DS6 202 then Cm ×Cn is not Γ-distance magic for any Abelian group Γ of order mn. Cichacz gave necessary and sufficient conditions for complete k-partite graphs of odd order p to be Zp-distance magic. Moreover she showed that if p ≡2 (mod 4) and k is even, then there does not exist a group Γ of order p that has a Γ-distance labeling for a k-partite complete graph of order p. She also proved that Km,n is a group distance magic graph if and only if n + m ̸≡2 (mod 4). In Cichacz proved that if G is an Eulerian graph, then the lexicographic product of G and C4 is group distance magic. In the same paper she also showed that if m + n is odd, then the lexicographic product of Km,n and C4 is group distance magic. In Cichacz gave necessary and sufficient conditions for direct product of Km,n and C4 for m + n odd and for Km,n × C8 to be group distance magic.
In Cichacz proved that for n even and r > 1 the Cartesian product the complete r-partitie graph Kn,n,...,n and C4 is group distance magic. Godinho and Singh obtain group distance magic labelings of Cr n for certain classes of Abelian groups and provide necessary conditions for existence of such labelings.
An orientable Γ-distance magic labeling of a graph, was introduced by Cichacz, Frey-berg and Fronček as a generalization of group distance magic labeling for oriented graphs. They showed that an even regular circulant graph of order n is orientable Zn-distance magic, the direct product Cn × Cm is orientable Znm-distance magic.
They also considered some products of circulant graphs. Moreover they proved that if G has order n ≡2 (mod 4) and all vertices of odd degree, then there does not exist an ori-entable Γ-distance magic labeling of G for any Abelian group Γ of order n.
Dyrlaga and Szopa in gave necessary and sufficient conditions for lexicographic product Km ◦Kn ∼ = Km, m, . . . , m | {z } n to be oriantable ζmn-distance magic. As a consequence, they provide an infinite family of odd regular graphs possessing orientable ζn-distance magic labeling. In and Freyberg and Keranen found orientable Zn-distance magic labelings of the Cartesian product of cycles.
In they studied Zn-distance magic labelings for the strong product of cycles.
In Mukherjee and Pawa use L = {1, 2, . . . , np} to denote the multiset obtained by reducing 1, 2, . . . , n modulo p and replacing 0’s, if any, by p. They call a graph G p-distance magic if there is a bijective map f from the vertex set V to the multiset L such that the weight of each vertex w(x) is equal to the same element µ (mod p), where w(x) = P f(y) is taken over all y ∈N(x). They proved that a graph G is distance magic if and only if it is p-distance magic for all p ≥1. They also proved that for p = n, p-distance magic labeling becomes a Zn-distance magic labeling. In some cases, they provide ways to construct different magic constants in for Zn-distance magic labelings. In some cases, they prove the uniqueness of the magic constant for Zn-distance magic labeling.
Anholcer, Cichacz, Peterin, and Tepeh proved that the direct product of two cycles Cm and Cn is distance magic if and only if m = 4 or n = 4, or m, n ≡0 (mod 4) (the direct product of graphs G and H has the vertex set V (G) × V (H) and (g, h) is adjacent to (g′, h′) if g is adjacent to g′ in G and h is adjacent to h′ in H). Zeng, Deng, and Luo characterized group distance magic labelings of the Cartesian product of new two cycles. In Cichacz gave necessary and sufficient conditions for circulant graph the electronic journal of combinatorics (2023), #DS6 203 Cn(1, 2, . . . , p) to be distance magic for p odd. In Cichacz and Fronček characterized all distance magic circulant graphs Cn(1, p) for p odd.
Cichacz, Fronček, Krop, and Raridan proved that r-partite graph Kn,n,...,n × C4 is distance magic if and only if r > 1 and n > 2 is even. Anholcer and Cichacz gave necessary and sufficient conditions for lexicographic product of an r-regular graph G and Km,n to be distance magic. Cichacz and Görlich gave necessary and sufficient conditions for the direct product of an r-regular graph G and Km,n to be distance magic. Kamatchi, Ramalakshmi, and Nilavarasi proved that P 2 n and the graphs obtained from Pn (n ̸= 5) by joining each internal vertex v with an end vertex at even distance from v are distance antimagic.
In Shrimali and Parmar discuss the existence of distance magic labelings for the product, direct product, strong product, and carona product of graphs involving Ct 3 and C4. In necessary and sufficient conditions for complete tripartite graphs to be group distance magic was given by Cichacz. In Arumugam, Kamatchi, and Kovář give several results on distance magic graphs and open problems.
A graph Γ(V, E) of order n is said to be a closed distance magic labeling of Γ is a bijection f : V →{1, 2, . . . , n} for which there exists a positive integer r such that P x∈N[u] f(x) = r for all vertices u ∈V , where N[u] is the closed neighborhood of u. A graph is said to be closed distance magic if it admits a closed distance magic labeling.
Fernández, Maleki, Miklavič, and Razafimahatratra classified all connected closed distance magic circulants with valency at most 5, that is, Cayley graphs Cay(Zn; S), where |S| ≤5 and S generates Zn.
A finite r-regular graph G has a p-partition (resp. closed p-partition) (p ≥2) if there exists a partition of the set V (G) into V1, V2, . . . , Vp such that for every x ∈V (G), all V (x) ∩Vi (respectively, V [x] ∩V1) have the same size. In Cichacz and Nikodem proved the following for finite r-regular graphs G. If G is distance magic (resp. closed distance magic) graph with a p-partition and p(t −1) even then tG is also distance (resp.
closed distance) magic. If G has order t and H is p-regular such that tH is distance (resp.
closed distance) magic, then the lexicographic product of G and H is distance (resp.
closed distance) magic. If G has order t and H is such that tH is distance magic, then the lexicographic product of G and H and the direct product of G and H are distance (resp.
closed distance) magic. If H is a p-regular distance magic graph with a 2-partition, then the lexicographic product of G and H and the direct product of G and H are distance magic. They further proved that if G = C3 or G is the strong product of Cn and Cm for n = 3 and m odd, or m, n ≡3 (mod 6), then tG is closed distance magic if and only if t is odd. (The strong direct product of G and H has vertex set V (G) × V (H) and (g, h) is adjacent to (g ′, h ′) if g = g and h is adjacent to h ′ in H, or h = h and g is adjacent to g in G.) In Seoud, Maqsoud, and Aldiban determined whether or not the following fam-ilies of graphs have a distance magic vertex labeling: Kn−{e}; Kn−{2e}; P k n; C2 n; Km× Cn; Cm + Pn; Cm + Cn; Pm + Pn; K1,r,s; K1,r,m,n; K2,r,m,n; Km,n + Pk; Km,n + Ck; Cm + Kn; Pm + Kn; Pm × Pn; Km,n × Pk; Km × Pn; the splitting graph of Km,n; Kn + G; Km + Kn; Km + Cn; Km + Pn; Km,n + Kr; Cm × Pn; Cm × K1,n; Cm × Kn,n; Cm × Kn,n+1; Km × Kn,r; and Km × Kn. Typically, distance magic labelings exist the electronic journal of combinatorics (2023), #DS6 204 only a few low parameter cases.
Miklavic and Sparl provided a sufficient condition for a Hamming graph to be distance magic and as a they corollary provide an infinite number of pairs (D, q) for which the corresponding Hamming graph with words of length D and over an alphabet of size q is distance magic. They classify distance magic folded hypercubes (a graph obtained from a hypercube by identifying pairs of vertices at maximal distance) by showing that the dimension-D folded hypercube is distance magic if and only if is divisible by 4.
Anholcer, Cichacz, Froncek, Simanjuntak, and Qiu proved that for n odd, there does not exist a Γ-distance magic labeling of Qn for any Abelian group Γ of order |V (Qn)|, whereas for n even there exists a Γ-distance magic labeling of Qn for every Abelian group Γ of order |V (Qn)|. They also study similar distance antimagic and Γ-distance antimagic labelings where one finds a bijection such that the sums of labels are pairwise distinct for all the vertices. They show that there exists a Γ-distance antimagic labeling of Qn for any Abelian group Γ of order 2n where n is odd for these labelings and give some relationships between Γ-closed distance magic and antimagic labelings and Γ-distance antimagic labelings.
Cichacz showed there exists an infinite family of odd regular graphs possessing Γ-distance magic labeling for groups Γ with more than one involution. In Cichacz using a notion of a Γ-magic rectangle set MRSΓ(a, b; c) showed group distance labeling for Cartesian and direct product of complete r-partite graphs. These results supported a conjecture in that says that if G is a distance magic graph, then G is group distance magic.
A directed Γ-distance magic labeling of an oriented graph − → G = (V, A) of order n is a bijective mapping f from the vertex set of G to an Abelian group Γ of order n with the property that there exists a constant c ∈Γ such that, for every vertex v ∈V (− → G), w(v) = P u∈Nin G (v) f(u) −P u∈Nout G (v) f(u) = c, where N in G (v) is the open in-neighborhood and N out G (v) is the open out-neighborhood of vertex v, that is N in G (v) = {u : uv ∈A} and N out G (v) = {u : vu ∈A}. If for a graph G there exists an orientation − → G such that there is a directed Γ-distance magic labeling f for − → G the graph G is called orientable Γ-distance magic.
Freyberg and Keranen proved that Cm × Cn is orientable Zmn-distance magic for all m, n ≥3.
In Anholcer, Cichacz, Peterin, and Tepeh introduce the notion of balanced dis-tance magic graphs.
They say that a distance magic graph G with an even number of vertices is balanced if there exists a bijection f from V (G) to {1, 2, . . . , |V (G)|} such that for every vertex w the following holds: If u ∈N(w) with f(u) = i, then there exists v ∈N(w), u ̸= v with f(v) = |V (G)| −i + 1.
They prove that a graph G is balanced distance magic if and only if G is regular and V (G) can be partitioned in pairs (ui, vi), i ∈{1, 2, . . . , |V (G)|/2, such that N(ui) = N(vi) for all i. Using this characteri-zation, the following theorems are proved: if G is a regular graph and H is a graph not isomorphic to Kn where n is odd, then G ⊙H is a balanced distance magic graph if and only if H is a balanced distance magic graph; G×H is balanced distance magic if and only if one of G and H is balanced distance magic and the other one is regular; and Cm × Cn the electronic journal of combinatorics (2023), #DS6 205 is distance magic if and only if n = 4 or m = 4 or m, n ≡0 (mod 4) and Cm × Cn is balanced distance magic if and only if n = 4 or m = 4. In they prove that every balanced distance magic graph is also group distance magic; the Cartesian product of a balanced distance magic graph and a regular graph is group distance magic; the direct product of C4 or C8 and a regular graph is group distance magic; and they show that C8 × G is also group distance magic for any even-regular graph G. They also prove that C4s × C4t is A × B-distance magic for any Abelian groups A and B of order 4s and 4t, respectively. Moreover, they conjecture that C4m × C4n is a group distance magic graph.
They prove that Cm × Cn is Zmn-distance magic if and only if m ∈{4, 8} or n ∈{4, 8} or both n and m are divisible by 4, and that Cm × Cn with orders not divisible by 4 is not Γ-distance magic for any Abelian group Γ of order mn.
Let G = (V, E) be a graph on n vertices. A bijection f from the vertices of graph G to {1, 2, . . . , |V (G)|} is called a nearly distance magic labeling of G if there exists a positive integer k such that P f(x) over all x ∈N(v) = k or k + 1 for all v . The constant k is called a magic constant of the graph and any graph which admits such a labeling is called a nearly distance magic graph. Godinho, Singh, and Arumugam give several basic results on nearly distance magic graphs and compute the magic constant k in terms of the fractional total domination number of the graph.
In Marr and Simanjunak defined D-magic labelings for oriented graphs where D is a distance set as follows.
The vertices of a graph G are distinct integers in {1, 2, . . . , |V (G)|} such that the sum of all the vertex labels that are a distance in D from a given vertex is the same across all vertices. They gave some results related to the magic constant, constructed a few infinite families of D-magic graphs, and examined trees, cycles, and multipartite graphs.
In Kang, Chen, Li, and Hou investigated D-magic labelings of the halved n-cube (n ≥2) (that is, all binary strings of length n with even number of 1’s as vertices and edges between any two strings of Hamming distance 2). They show that the folded n-cube has a {1}-magic labeling (resp., a {0, 1}-magic labeling) if and only if n = 0 (mod 4) (resp., n = 3 (mod 4)).
A survey of results on distance magic (sigma, 1-vertex) labelings through 2009 is given in .
5.7 Other Types of Magic Labelings For a graph G = (V, E) naturally embedded in the torus, let F(G) denote the set of faces of G. Then, G is called a Cn-face-magic toroidal graph if there exists a bijection f : V (G) →{1, 2, . . . , |V (G)|} such that for every F ∈F(G) with F ∼ = Cn, the sum of all the vertex labels along Cn is a constant S.
Let xv = f(v) for all v ∈V (G).
Then {xv | v ∈V (G)} is called a Cn-face-magic toroidal labeling on G. Curran, Low, and Locke show that, for all m, n ≥2, Cm×Cn admits a C4-face-magic toroidal labeling if and only if either m = 2 or n = 2, or both m and n are even. They say that a C4-face-magic toroidal labeling {xi,j | (i, j) ∈V (C2m × C2n)} on C2m × C2n is antipodal balanced if xi,j + xi+m,j+n = S/2, for all (i, j) ∈V (C2m × C2n). They show that there exists an the electronic journal of combinatorics (2023), #DS6 206 antipodal balanced C4-face-magic toroidal labeling on C2m×C2n if and only if the parity of m and n are the same. Furthermore, when both m and n are even, an antipodal balanced C4-face-magic toroidal labeling on C2m × C2n is both row-sum balanced and column-sum balanced. In addition, when m = n is even, an antipodal balanced C4-face-magic toroidal labeling on C2n × C2n is diagonal-sum balanced. In Curran and Locke proved that there are 144 distinct C4-face-magic labelings on the 4 × 4 projective grid graph P4,4 (up to symmetries on the projective plane).
Curran showed that an m × n projective grid graph admits a C4-face-magic labeling if and only if both m and n have the same parity. When m and n are even, the C4-face-magic value of a C4-face-magic labeling on an m × n projective grid graph must be 2mn + 2. Also, when m and n are odd, he proved that the C4-face-magic value of a C4-face-magic labeling on an m × n projective grid graph is either 2mn + 1, 2mn + 2, or 2mn + 3. In Curran determined a category of the C4-face-magic labelings on Pm,n having C4-face-magic value 2mn + 1 or 2mn + 3. He conjectured that these are the only C4-face-magic labelings on Pm,n having C4-face-magic value 2mn + 1 or 2mn + 3.
In 2004 Baskar Babujee and introduced the notion of vertex-bimagic labeling in which there exist two constants k1 and k2 such that the sums involved in a specified type of magic labeling is k1 or k2. Thus a vertex-bimagic total labeling with bimagic constants k1 and k2 is the same as a vertex-magic total labeling except for each vertex v the sum of the label of v and all edges adjacent to v may be k1 or k2. Murugesan and Senthil Amutha proved that the bistar Bn,n is vertex-bimagic total labeling for n > 2. An edge bimagic total labeling edge bimagic total of a graph G(V, E) with p vertices and q edges is a bijection f from the set of vertices and edges to such that for every edge uv ∈E, f(u) + f(uv) + f(v) is one of two oconstants k1 or k2, independent of the choice of the edge. A bimagic labeling is of interest for graphs that do not have a magic labeling of a particular type. Bimagic labelings for which the number of sums equal to k1 and the number of sums equal to k2 differ by at most 1 are called equitable.
When all sums except one are the same the labeling is called almost magic. Although the wheel Wn does not have an edge-magic total labeling when when n ≡3 (mod 4), Marr, Phillips and Wallis showed that these wheels have both equitable bimagic and almost magic labelings. They also show that whereas nK2 has an edge-magic total labeling if and only if n is odd, nK2 has an edge-bimagic total labeling when n is even and although even cycles do not have super edge-magic total labelings all cycles have super edge-bimagic total labelings. They conjecture that there is a constant N such that Kn has a edge-bimagic total labeling if and only if n is at most N. They show that such an N must be at least 8. They also prove that if G has an edge-magic total labeling then 2G has an edge-bimagic total equitable labeling. Amara Jothi, David, and Baskar Babujee provide edge-bimagic labelings for switching of paths, cycles, stars, crowns and helms. They also examine whether operations on edge magic graphs results in edge bimagic graphs or not.
Baskar Babujee and Babitha call a graph with p vertices 1-vertex bimagic if there is a bijective labeling f from the vertices to {1, 2, . . . , p} such that for each vertex u the sum of all f(v) where v is adjacent to u is either a constant k1 or a constant k2 the electronic journal of combinatorics (2023), #DS6 207 and k1 ̸= k2. A graph with p vertices is called odd 1-vertex bimagic if there is a bijective labeling f from the vertices to {1, 3, . . . , 2p −1} such that for each vertex u the sum of all f(v) where v is adjacent to u is either a constant k1 or a constant k2 and k1 ̸= k2. A graph with p vertices is called even 1-vertex bimagic if there is a bijective labeling f from the vertices to {0, 2, . . . , 2(p −1)} such that for each vertex u the sum of all f(v) where v is adjacent to u is either a constant k1 or a constant k2 and k1 ̸= k2.
Baskar Babujee and Babitha prove that a necessary condition for the existence of a 1-vertex bimagic vertex labeling f of a graph G is P x∈V (G) d(x)f(x) = k1p1 + k2p2 where d(x) is the degree of vertex x and p1 and p2 are the number of vertices with common count k1 and k2, respectively. Among their results are: if G has a 1-vertex bimagic vertex labeling and G ̸= C4, then G+K1 admits a 1- vertex bimagic vertex labeling; Cn a 1-vertex bimagic if and only if n = 4; Km,n is 1-vertex bimagic; graphs obtained from Pn (n ≥3) by adding edges joining every pair of vertices an odd distance apart are 1-vertex bimagic; n-partite graphs of the form Kp,p,...,p are 1-vertex bimagic for all p > 1 when n is even and 1-vertex bimagic for all even p when n is odd; a regular or biregular graph admits a 1-vertex bimagic labeling if and only if it the admits an odd 1-vertex bimagic labeling and if and only it admits an even 1-vertex bimagic labeling.
In Semenyuta, Nedilko, and Nedilko introduce the notion of the equivalence of vertex labelings on a given graph. They prove the equivalence of three bimagic labelings for regular graphs and obtain a particular solution for the problem of the existence a 1-vertex bimagic vertex labeling for graphs of isomorphic Kn,n,m. They prove that the sequence of biregular graphs Kn(ij) = ((Kn−1 −M) + K1) −(unui) −(unuj) admits a 1-vertex bimagic vertex labeling, where ui, uj is any pair of nonadjacent vertices in the graph Kn−1 −M, un is the vertex of K1, and M is the perfect matching of the complete graph Kn−1.
They show that if an r-regular graph G of order n is a distance magic graph, then the graph G + G has a 1-vertex bimagic vertex labeling with magic constants (n+1)(n+r)/2+n2 and (n+1)(n+r)/2+nr. They also define two new types of graphs that do not admit 1-vertex bimagic vertex labeling.
Baskar Babujee and Jagadesh , , , and proved the following graphs have super edge bimagic labelings: cycles of length 3 with a nontrivial path attached; P3 ⊙K1,n n even; Pn + K2 (n odd); P2 + mK1 (m ≥2); 2Pn (n ≥2); the disjoint union of two stars; 3K1,n (n ≥2); Pn ∪Pn+1 (n ≥2); C3 ∪K1,n; Pn; K1,n; K1,n,n; the graphs obtained by joining the centers of any two stars with an edge or a path of length 2; the graphs obtained by joining the centers of two copies of K1,n (n ≥3) with a path of length 2 then joining the center one of copies of K1,n to the center of a third copy of K1,n with a path of length 2; combs Pn ⊙K1; cycles; wheels; fans; gears; Kn if and only if n ≤5.
Given positive integers k and λ, Yao, Chen, Yao, and Cheng say that a total labeling f of a connected graph G(V, E) from V ∪E to {1, 2, . . . , |V | + |E|} such that f(x) ̸= f(y) for distinct x, y ∈V ∪E and f(u)+f(v) = k+λf(uv) for each edge uv in E is a (k, λ)-magically total labeling of G. They provide necessary and sufficient conditions for graphs with (k, λ) )-magically total labelings to also have graceful, odd-graceful, felicitous, and (a, d)-edge antimagic total labelings (see §6.2).
In López, Muntaner-Batle, and Rius-Font give a necessary condition for a com-the electronic journal of combinatorics (2023), #DS6 208 plete graph to be edge bimagic in the case that the two constants have the same parity.
In Baskar Babujee, Babitha, and Vishnupriya make the following definitions. For any natural number a, a graph G(p, q) is said to be a-additive super edge bimagic if there exists a bijective function f from V (G)∪E(G) to {a+1, a+2, . . . , a+p+q} such that for every edge uv, f(u)+f(v)+f(uv) = k1 or k2. For any natural number a, a graph G(p, q) is said to be a-multiplicative super edge bimagic if there exists a bijective f from V (G)∪E(G) to {a, 2a, . . . , (p+q)a} such that for every edge uv, f(u)+f(v)+f(uv) = k1 or k2. A graph G(p, q) is said to be super edge-odd bimagic if there exists a bijection f from V (G)∪E(G) to {1, 3, 5, . . . , 2(p + q) −1} such that for every edge uv f(u) + f(v) + f(uv) = k1 or k2.
If f is a super edge bimagic labeling, then a function g from E(G) to {0, 1} with the property that for every edge uv, g(uv) = 0 if f(u) + f(v) + f(uv) = k1 and g(uv) = 1 if f(u) + f(v) + f(uv) = k2 is called a super edge bimagic cordial labeling if the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. They prove: super edge bimagic graphs are a-additive super edge bimagic; super edge bimagic graphs are a-multiplicative super edge bimagic; if G is super edge-magic, then G + K1 is super edge bimagic labeling; the union of two super edge magic graphs is super edge bimagic; and Pn, C2n and K1,n are super edge bimagic cordial.
For any nontrivial Abelian group A under addition a graph G is said to be A-magic if there exists a labeling f of the edges of G with the nonzero elements of A such that the vertex labeling f + defined by f +(v) = Σf(vu) over all edges vu is a constant. In and Stanley noted that Z-magic graphs can be viewed in the more general context of linear homogeneous diophantine equations. Shiu, Lam, and Sun have shown the following: the union of two edge-disjoint A-magic graphs with the same vertex set is A-magic; the Cartesian product of two A-magic graphs is A-magic; the lexicographic product of two A-magic connected graphs is A-magic; for an Abelian group A of even order a graph is A-magic if and only if the degrees of all of its vertices have the same parity; if G and H are connected and A-magic, G composed with H is A-magic; Km,n is A-magic when m, n ≥2 and A has order at least 4; Kn with an edge deleted is A-magic when n ≥4 and A has order at least 4; all generalized theta graphs (§4.4 for the definition) are A-magic when A has order at least 4; Cn + Km is A-magic when n ≥3, m ≥2 and A has order at least 2; wheels are A-magic when A has order at least 4; flower graphs Cm@Cn are A-magic when m, n ≥2 and A has order at least 4 (Cm@Cn is obtained from Cn by joining the end points of a path of length m−1 to each pair of consecutive vertices of Cn).
When the constant sum of an A-magic graph is zero the graph is called zero-sum A-magic. The null set N(G) of a graph G is the set of all positive integers h such that G is zero-sum Zh-magic. Akbari, Ghareghani, Khosrovshahi, and Zare and Akbari, Kano, and Zare proved that the null set N(G) of an r-regular graph G, r ≥3, does not contain the numbers 2, 3 and 4. Akbari, Rahmati, and Zare proved the following: if G is an even regular graph then G is zero-sum Zh-magic for all h; if G is an odd r-regular graph, r ≥3 and r ̸= 5 then N(G) contains all positive integers except 2 and 4; if an odd regular graph is also 2-edge connected then N(G) contains all positive integers except 2; and a 2-edge connected bipartite graph is zero-sum Zh-magic for h ≥6. They the electronic journal of combinatorics (2023), #DS6 209 also determine the null set of 2-edge connected bipartite graphs, describe the structure of some odd regular graphs, r ≥3, that are not zero-sum 4-magic, and describe the structure of some 2-edge connected bipartite graphs that are not zero-sum Zh-magic for h = 2, 3, and 4. They conjecture that every 5-regular graph admits a zero-sum 3-magic labeling.
In Lee, Saba, Salehi, and Sun investigate graphs that are A-magic where A = V4 ≈Z2 ⊕Z2 is the Klein four-group. Many of theorems are special cases of the results of Shiu, Lam, and Sun given in the previous paragraph. They also prove the following are V4-magic: a tree if and only if every vertex has odd degree; the star K1,n if and only if n is odd; Km,n for all m, n ≥2; Kn −e (edge deleted Kn) when n > 3; even cycles with k pendent edges if and only if k is even; odd cycles with k pendent edges if and only if k is odd; wheels; Cn + K2; generalized theta graphs; graphs that are copies of Cn that share a common edge; and G + K2 whenever G is V4-magic. In Libeeshkumar and Anil Kumar discussed induced V4-magic labelings of some cycle related graphs.
In Choi, Georges, and Mauro explore Zk 2-magic graphs in terms of even edge-coverings, graph parity, factorability, and nowhere-zero 4-flows.
They prove that the minimum k such that bridgeless G is zero-sum Zk 2-magic is equal to the minimum number of even subgraphs that cover the edges of G, known to be at most 3. They also show that bridgeless G is zero-sum Zk 2-magic for all k ≥2 if and only if G has a nowhere-zero 4-flow, and that G is zero-sum Zk 2-magic for all k ≥2 if G is Hamiltonian, bridgeless pla-nar, or isomorphic to a bridgeless complete multipartite graph, and establish equivalent conditions for graphs of even order with bridges to be Zk 2-magic for all k ≥4. In Georges, Mauro, and Wang utilized well-known results on edge-colorings in order to con-struct infinite families that are V4-magic but not Z4-magic. In Libeeshkumar and Kumar discussed some induced V4-magic labelings of some subdivision graphs.
Low and Roberts use the combinatorial nullstellensatz to show the existence of Zp-magic labelings (prime p ≥3) for various graphs, without having to construct the Zp-magic labelings. They give examples illustrating the usefulness and limitations in applying the combinatorial nullstellensatz to the integer-magic labeling problem. They show vari-ous classes of graphs have Z3-magic labelings. Baskar Babujee and Shobana prove that the following graphs have Z3-magic labelings: C2n; Kn (n ≥4); Km,2m (m ≥3); ladders Pn × P2 (n ≥4); bistars B3n−1,3n−1; and cyclic, dihedral, and symmetric Cayley digraphs for certain generating sets. Siddiqui proved that generalized prisms, gen-eralized antiprisms, fans and friendship graphs are Z3k-magic for k ≥1. In Chou and Lee investigated Z3-magic graphs.
Chou and Lee showed that every graph is an induced subgraph of an A-magic graph for any nontrivial Abelian group A. Thus it is impossible to find a Kuratowski type characterization of A-magic graphs. Low and Lee have shown that if a graph is A1-magic then it is A2-magic for any subgroup A2 of A1 and for any nontrivial Abelian group A every Eulerian graph of even size is A-magic. For a connected graph G, Low and Lee define T(G) to be the graph obtained from G by adding a disjoint uv path of length 2 for every pair of adjacent vertices u and v. They prove that for every finite nontrivial Abelian group A the graphs T(P2k) and T(K1,2n+1) are A-magic. Shiu and Low show that Kk1,k2,...,kn(ki ≥2) is A-magic, for all A where |A| ≥3. In Shiu and Low the electronic journal of combinatorics (2023), #DS6 210 analyze the A-magic property for complete n-partite graphs and composition graphs with deleted edges. Lee, Salehi and Sun have shown that for m, n ≥3 the double star DS(m, n) is Z-magic if and only if m = n.
S. M. Lee calls a graph G fully magic if it is A-magic for all nontrivial Abelian groups A. Low and Lee showed that if G is an Eulerian graph of even size, then G is fully magic. In Lee gives several constructions that produce infinite families of fully magic graphs and proves that every graph is an induced subgraph of a fully magic graph.
In Kwong and Lee call the set of all k for which a graph is Zk-magic the integer-magic spectrum of the graph. They investigate the integer-magic spectra of the coronas of some specific graphs including paths, cycles, complete graphs, and stars. Low and Sue have obtained some results on the integer-magic spectra of tessellation graphs. Shiu and Low provide the integer-magic spectra of sun graphs. Chopra and Lee determined the integer-magic spectra of all graphs consisting of any number of pairwise disjoint paths with common end vertices (that is, generalized theta graphs).
Low and Lee show that Eulerian graphs of even size are A-magic for every finite nontrivial Abelian group A whereas Wen and Lee provide two families of Eularian graphs that are not A-magic for every finite nontrivial Abelian group A and eight infinite families of Eulerian graphs of odd sizes that are A-magic for every finite nontrivial Abelian group A. Low and Lee also prove that if A is an Abelian group and G and H are A-magic, then so are G×H and the lexicographic product of G and H. Low and Shiu prove: K1,n ×K1,n has a Zn+1-magic labeling with magic constant 0; if G×H is Z2-magic, then so are G and H; if G is Zm-magic and H is Zn-magic, then the integer-magic spectra of G × H contains all common multiples of m and n; if n is even and ki ≥3 then the integer-magic spectra of Pk1 × Pk2 × · · · × Pkn = {3, 4, 5, . . .}. In Shiu and Low determine all positive integers k for which fans and wheels have a Zk-magic labeling with magic constant 0. Shiu and Low determined for which k ≥2 a connected bicyclic graph without a pendent has a Zk-magic labeling.
Jeyanthi and Jeya Daisy prove that P 2 n (n > 4), C2 n, the total graph of Cn, and the splitting graph of C2n are Zk-magic graphs. They also prove: the splitting graph of Cn is Zk-magic when n is even and n is odd and k is even, the middle graph of Cn is Zk-magic when n and k are odd, the m∆2n-snake graph is Zk-magic when k > m, the graph obtained by joining the vertices ui and ui+1 of Cn by a path of length mi for 1 ≤i ≤n−1, and u1 and un by a path of length mn is Zk-magic if either all m1, m2, . . . , mn are even or all are odd. In Jeyanthi and Jeya Daisy prove total graphs of the paths, flower graphs, and Cm × Pn are Zk-magic. They also prove closed helms are Zk-magic when k > 4 is even, lotuses inside a circle are Z4k-magic, and graphs consisting of two cycles with a common edge are Zk-magic when at least one cycle is even. In Jeyanthi prove the following graphs are Zk-magic: two odd cycles connected by a path; the graph obtained by identifying a vertex of Cn with a pendent vertex of a star, m-splitting graphs of paths, and m-middle graphs of paths. They prove that if G is Zm-magic with magic constant a then G ⊙Km is Zm-magic.
Jeyanthi and Jeya Daisy prove that the subdivision graphs of the following the electronic journal of combinatorics (2023), #DS6 211 families of graphs are Zk-magic: ladders, triangular ladders, the shadow graph of paths, the total graph of paths, flowers, generalized prisms Cm × Pn for m even, m∆n-snakes, lotuses inside a circle, the square graph of paths, gears of even cycles, closed helms of even cycles, and antiprisms Am n for m even.
Let G be a graph and let G1, G2, . . . , Gn be n ≥2 copies of G. The graph obtained by replacing each endpoint vertex of K1,n by the graphs G1, G2, . . . , Gn is called the open star of G. Jeyanthi and Jeya Daisy proved that the open star graphs of shells, flowers, double wheels, cylinders, wheels, generalized Peterson graphs, lotuses inside a circle, and closed helms are Zk-magic graphs. They also prove that the super subdivision of any graph is Zk-magic.
Jeyanthi and Jeya Daisy proved that the path union of n ≥2 copies of the following families of graphs are Zk-magic: odd cycles; generalized Peterson graphs P(r, m) when r is odd and 1 ≤m ≤r 2; shell graphs Sr when r > 3; wheels Wr when r > 3; closed helms CHr when (i) r > 3 is odd and (ii) r is even and k is even; double wheels DWr when r > 3 is odd; flowers Flr when r > 2; Cr × P2 when r is odd; total graphs of paths T(Pr) for all n, r > 4; lotuses inside a circle LCr when r > 3; and Cr ⊙K1 for odd r.
Jeyanthi and Jeya Daisy proved that the following graphs are k-magic: shell graphs Sn when n is odd or n is even and k is even; generalized Jahangir graphs Jn,s when n and s have the same parity or n is even, s is odd, and k is even; (Pn + P1) × P2 when n is odd; double wheels 2Cn + K1; mongolian tents M(m, n) when m is even; flower snark graphs; slanting ladders (that is, graphs obtained from two paths u1, u2, . . . , un and v1, v2, . . . , vn by joining each ui with vi+1, 1 ≤i ≤n −1) when n is even; double step grid graphs; double arrow graphs obtained from Pm × Pn by joining a new vertex with the m vertices of the first copy of Pm and another new vertex with the m vertices of the last copy of Pm when m is even; semi Jahangir graphs (the connected graph with vertex set {p, xi, yk : 1 ≤i ≤n + 1, 1 ≤k ≤n} and the edge set {pxi : 1 ≤i ≤n + 1} ∪{xiyi : 1 ≤ i ≤n} ∪{yixi+1 : 1 ≤i ≤n}); graphs obtained by connecting double wheels DWn1 and DWn2 by a path when n1 and n2 are odd; graphs obtained by joining two copies of shell graphs by a path; and the splitting graph of a Zk magic graph with magic constant 0.
Let G be a graph with n vertices {u1, u2, . . . , un} and consider n copies of G, G1, G2, . . . , Gn, with vertex sets V (Gi) = {uj i : 1 ≤i ≤n, 1 ≤j ≤n}. The cycle of a graph G, denoted by C(n.G), is obtained by identifying the vertex uj 1 of Gj with ui of G for 1 ≤i ≤n, 1 ≤j ≤n. Jeyanthi and Jeya Daisy prove that the following graphs are Zk-magic: C(n.Cr) except r is even, n is odd, and k is odd; generalized Peter-son graphs C(n.P(r, m)) except r is even, n is odd, and k is odd; cycles of shell graphs; cycles of wheel graphs; cycles of closed helms; cycles of double wheels C(n.DWr) except r is even, n is odd, and k is odd; cycles of triangular ladder graphs; cycles of flower graphs; and cycles of lotus inside a circle graphs. Jeyanthi and Jeya Daisy also prove that if G is Zk-magic then C(n.G) is Zk-magic if n or k are even.
Shiu and Low have introduced the notion of ring-magic as follows. Given a commutative ring R with unity, a graph G is called R-ring-magic if there exists a labeling f of the edges of G with the nonzero elements of R such that the vertex labeling f + defined by f +(v) = Σf(vu) over all edges vu and vertex labeling f × defined by f ×(v) = Πf(vu) the electronic journal of combinatorics (2023), #DS6 212 over all edges vu are constant. They give some results about R-ring-magic graphs.
In Cahit says that a graph G(p, q) is total magic cordial (TMC) provided there is a mapping f from V (G) ∪E(G) to {0, 1} such that (f(a) + f(b) + f(ab)) mod 2 is a constant modulo 2 for all edges ab ∈E(G) and |f(0) −f(1)| ≤1 where f(0) denotes the sum of the number of vertices labeled with 0 and the number of edges labeled with 0 and f(1) denotes the sum of the number of vertices labeled with 1 and the number of edges labeled with 1. He says a graph G is total sequential cordial (TSC) if there is a mapping f from V (G) ∪E(G) to {0, 1} such that for each edge e = ab with f(e) = |f(a) −f(b)| it is true that |f(0) −f(1)| ≤1 where f(0) denotes the sum of the number of vertices labeled with 0 and the number of edges labeled with 0 and f(1) denotes the sum of the number of vertices labeled with 1 and the number of edges labeled with 1. He proves that the following graphs have a TMC labeling: Km,n (m, n > 1), trees, cordial graphs, and Kn if and only if n = 2, 3, 5, or 6. He also proves that the following graphs have a TSC labeling: trees; cycles; complete bipartite graphs; friendship graphs; cordial graphs; cubic graphs other than K4; wheels Wn (n > 3); K4k+1 if and only if k ≥1 and √ k is an integer; K4k+2 if and only if √ 4k + 1 is an integer; K4k if and only if √ 4k + 1 is an integer; and K4k+3 if and only if √ k + 1 is an integer. In Jeyanthi, Angel Benseera, and Cahit prove mP2 is TMC if m ̸≡2 ( mod 4), mPn is TMC for all m ≥1 and n ≥3, and obtain partial results about TMC labelings of mKn. Neela and Selvaraj proved that the complete tripartite graphs are TMC and complete multipartite graphs are TMC when the partite sets have even sizes Parameswari and Jayalakshmi e proved that octopus graphs (the join of a fan Fn and K1,n (n > 1)), vanessa graphs 2Fn + K1,n, lilly graphs 2K1,n + 2Pn, and lotus graphs admit cordial totally magic labelings. Parameswari proved that the square graphs and shadow graphs of bistars admit cordial totally magic labelings. In and Parameswari and Rajeswari proved that Paley digraphs, fans, gears, and shadow graphs of paths and stars are total magic cordial. Neela and Selvaraj showed that complete tripartite graphs and complete multipartite graphs are total magic cordial when the partite sets are of even sizes.
Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of a totally magic cordial graph thereby ruling out any possibility of obtaining any forbidden subgraph characterization for totally magic cordial graphs.
They also proved that every connected graph can be embedded as an induced subgraph of a totally magic cordial connected graph and every planar graph can be embedded as an induced subgraph of a totally magic cordial planner graph. Similar results were also obtained for total sequential cordial labeling.
Jeyanthi and Angel Benseera investigated the existence of TMC labelings of the one-point unions of copies of cycles, complete graphs and wheels. In Jeyanthi and Angel Benseera prove that if Gi(pi, qi), i = 1, 2, 3, . . . , n are totally magic cordial graphs with C = 0 such that pi +qi, i = 1, 2, 3, ..., n are even, and |pi −2mi| ≤1, where mi is the number of vertices labeled with 0 in Gi, i = 1, 2, . . . , n, then G1 + G2 + · · · + Gn is TMC; if G is an odd graph with p + q ≡2 ( mod 4), then G is not TMC; fans Fn are TMC for n ≥2; wheels Wn (n ≥3) are TMC if and only if n ̸≡3 (mod 4); mW4t+3 is TMC if the electronic journal of combinatorics (2023), #DS6 213 and only if m is even; mWn is TMC if n ̸≡3 (mod 4); Cn + K2m+1 is TMC if and only if n ̸≡3 (mod 4); C2n+1 ⊙Km is TMC if and only if m is odd; the disjoint union of K1,m and K1,n is TMC if and only if m or n is even.
For a bijection f : V (G) ∪E(G) →Zk such that for each edge uv ∈E(G), f(u) + f(v) + f(uv) is constant (mod k) nf(i) denotes the number vertices and edges labeled by i under f. If |nf(i) −nf(j)| ≤1 for all 0 ≤i < j ≤k −1, f is called a k-totally magic cordial labeling of G. A graph is said to be k-totally magic cordial if it admits a k-totally magic cordial labeling. In Jeyanthi, Angel Benseera, and Lau provide some ways to construct new families of k-totally magic cordial (k-TMC) graphs from a known k-totally magic cordial graph. Let G (respectively, H) be a (p, q)-graph (respectively, an (n, m)-graph) that admits a k-TMC labeling f (respectively, g) with constant C such that nf(i) and vf(i) = p k (or ng(i) and vg(i) = n k) are constants for all 0 ≤i ≤k −1, they show that G + H also admits a k-TMC labeling with constant C. They prove the following.
If G is an edge magic total graph, then G is k-TMC for k ≥2; if G is an odd graph with p + q ≡k (mod 2k) and k ≡2 (mod 4), then G is not k-TMC; if n ≡7 (mod 8), Kn ⊙K1 is not 2n-TMC; if n ≡2 (mod 4), Cn ⊙C3 is not n-TMC; if n ≡1 (mod 2), Cn ⊙K5 is not 2n-TMC; if n ≡2 (mod 4), Cn × P2 is not n-TMC; Kn (n ≥3) is n-TMC; Kn ⊙K1 (n ≥3) is n-TMC; Sn is n-TMC for all n ≥1; Km,n (m ≥n ≥2) is both m-TMC and n-TMC; Wn is n-TMC for all odd n ≥3 and is 3-TMC for n ≡0 (mod 6); mKn (n ≥2) is n-TMC if n ≥3 is odd; Kn + Kn is n-TMC if n ≥3 is odd; Sn + Sn (n ≥1) is (n + 1)-TMC; and if m ≥3 and n is odd, Cn × Pm (n ≥3) is n-TMC. In Jeyanthi, Angel Benseera, and Lau call a graph G hypo-k-TMC if G −{v} is k-TMC for each vertex v in V (G) and establish that some families of graphs admit and do not admit hypo-k-TMC labeling.
A graph G(V, E) where V = {vi, 1 ≤i ≤n} and E = {vivi+1, 1 ≤i ≤n} is 0-edge magic if there exists a bijection f : V (G) →{1, −1} such that the induced edge la-beling defined by f ∗(uv) = f(u) + f(v) is 0 for all uv ∈E. Paths, cycles, complete n-ary pseudo trees, Pm × Cn where n ≡0 (mod 2), Qn, the graph Cm attached to mK1, m ≡0 (mod 2), friendship graphs C(m) n , and the graph Pm × Pm × Pm are 0-edge magic graphs , , . Jayapriya proved the splitting graphs spl(Pn), spl(Cn), spl(K1,n), spl(Bm,n), and splitting graph of any tree admits 0-edge magic labelings. Lau-rejas and Pedrano determine the 0-edge magic labeling of Pm × Pn, Cm × Cn, and the generalized Petersen graph. They also prove that odd cycles are not 0-edge magic.
A binary magic total labeling of a graph G is a function f : V (G) ∪E(G) →{0, 1} such that f(a) + f(b) + f(ab) ≡C (mod 2) for all ab ∈E(G).
Jeyanthi and Angel Benseera define the totally magic cordial deficiency of G as the minimum number of vertices taken over all binary magic total labeling of G that is necessary to add so that that the resulting graph is totally magic cordial. The totally magic cordial deficiency of G is denoted by µT(G). They provide µT(Kn) for some cases.
Let G be a graph rooted at a vertex u and fi be a binary magic total labeling of G and fi(u) = 0 for i = 1, 2, . . . , k and nfi(0) = αi, nfi(1) = βi for i = 1, 2, . . . , k. Jeyanthi and Angel Benseera determine the totally magic cordial deficiency of the one-point union G(n) of n copies of G. They show that for n ≡3 (mod 4) the totally magic cordial the electronic journal of combinatorics (2023), #DS6 214 deficiency of Wn, W (4t+1) n , W (n) 4t+1 and Cn + K2m+1 is 1; for m odd, µT(mW4t+3) = 1; and for n ≡1 (mod 4), µT(K(n) 4 ) = 1.
In 2001, Simanjuntak, Rodgers, and Miller defined a 1-vertex magic (also known as distance magic labeling vertex labeling of G(V, E) as a bijection from V to {1, 2, . . . , |V |} with the property that there is a constant k such that at any vertex v the sum P f(u) taken over all neighbors of v is k. Among their results are: H × K2k has a 1-vertex-magic vertex labeling for any regular graph H; the symmetric complete multipartite graph with p parts, each of which contains n vertices, has a 1-vertex-magic vertex labeling if and only if whenever n is odd, p is also odd; Pn has a 1-vertex-magic vertex labeling if and only if n = 1 or 3; Cn has a 1-vertex-magic vertex labeling if and only if n = 4; Kn has a 1-vertex-magic vertex labeling if and only if n = 1; Wn has a 1-vertex-magic vertex labeling if and only if n = 4; a tree has a 1-vertex-magic vertex labeling if and only if it is P1 or P3; and r-regular graphs with r odd do not have a 1-vertex-magic vertex labeling.
Miller, Rogers, and Simanjuntak the complete p-partite (p > 1) graph Kn,n,...,n (n > 1) has a 1-vertex-magic vertex labeling if and only if either n is even or np is odd. Shafiq, Ali, Simanjuntak proved mKn,n,...,n has a 1-vertex-magic ver-tex labeling if n is even or mnp is odd and m ≥1, n > 1, p > 1; and mKn,n,...,n does not have a 1-vertex-magic vertex labeling if np is odd, p ≡3 (mod 4), and m is even.
Recall if V (G) = {v1, v2, . . . , vp} is the vertex set of a graph G and H1, H2, . . . , Hp are isomorphic copies of a graph H, then G[H] is the graph obtained from G by replacing each vertex vi of G by Hi and joining every vertex in Hi to every neighbor of vi. Shafiq, Ali, Simanjuntak proved if G is an r-regular graph (r ≥1) then G[Cn] has a 1-vertex-magic vertex labeling if and only if n = 4. They also prove that for m ≥1 and n > 1, mCp[Kn] has 1-vertex-magic vertex labeling if and only if either n is even or mnp is odd or n is odd and p ≡3 (mod 4).
For a graph G Jeyanthi and Angel Benseera define a function f from V (G) ∪ E(G) to {0, 1} to be a totally vertex-magic cordial labeling (TVMC) with a constant C if f(a)+P b∈N(a) f(ab) ≡C (mod 2) for all vertices a ∈V (G) and |nf(0)−nf(1)| ≤1, where N(a) is the set of vertices adjacent to the vertex a and nf(i) is the sum of the number of vertices and edges with label i. They prove the following graphs have totally vertex-magic cordial labelings: vertex-magic total graphs; trees; Kn; Km,n whenever |m−n| ≤1; Pn + P2; friendship graphs with C = 0; and flower graphs Fln for n ≥3 with C= 0. They also proved that if G is TVMC with C = 1, then the graph obtained by identifying any vertex of G with any vertex of a tree is TVMC with C = 1; if G is a (p, q) graph with |p −q| ≤1, then G is TVMC with C = 1; and if G(p, q) is a TVMC graph with constant C = 0 where p is odd, then G + K2m is TVMC with C = 1 if m is odd and with C = 0 if m is even.
Jeyanthi, Angel Benseera, and Immaculate Mary showed that the following graphs have totally magic cordial labelings: (p, q) graphs with |p −q| ≤1; flower graphs Fln for n ≥3; ladders; and graphs obtained by identifying a vertex of Cm with each vertex of Cn. They also proved that if G1(p1, q1) and G2(p2, q2) are two disjoint totally magic cordial graphs with p1 = q1 or p2 = q2 then G1 ∪G2 is totally magic cordial. In Theorem the electronic journal of combinatorics (2023), #DS6 215 10 in Cahit stated that Kn is totally magic cordial if and only if n ∈{2, 3, 5, 6}.
Jeyanthi and Angel Benseera proved that Kn is totally magic cordial if and only if √ 4k + 1 has an integer value when n = 4k; √ k + 1 or √ k have an integer value when n = 4k + 1; √ 4k + 5 or √ 4k + 1 have an integer value when n = 4k + 2; or √ k + 1 has an integer value when n = 4k + 3.
A graph G is said to have a totally magic cordial TMC labeling with constant C if there exists a mapping f : V (G) ∪E(G) →{0, 1} such that f(a) + f(b) + f(ab) ≡C (mod 2) for all ab ∈E(G) and |nf(0) −nf(1)| ≤1, where nf(i) (i = 0, 1) is the sum of the number of vertices and edges with label i. In Jeyanthi and Angel Benseera prove that if Gi(pi, qi), i = 1, 2, 3, . . . , n are totally magic cordial graphs with C = 0 such that pi + qi, i = 1, 2, 3, . . . , n are even, and |pi −2mi| ≤1, where mi is the number of vertices labeled with 0 in Gi, i = 1, 2, . . . , n, then G1 + G2 + · · · + Gn is TMC. They also prove the following. If G be an odd graph with p + q ≡2 (mod 4), then G is not TMC; fan graph Fn is TMC for n ≥2; the wheel graph Wn (n ≥3) is TMC if and only if n ̸≡3(mod 4); mW4t+3 is TMC if and only if m is even; mWn is TMC if n ̸≡3 (mod 4) and m ≥1; Cn + K2m+1 is TMC if and only if n ̸≡3 (mod 4); C2n+1 ⊙Km is TMC if and only if m is odd; and the disjoint union of K1,m and K1,n is TMC if and only if m or n is even.
Balbuena, Barker, Lin, Miller, and Sugeng call a vertex-magic total labeling of a graph G(V, E) an a-vertex consecutive magic labeling if the vertex labels are {a + 1, a + 2, . . . , a + |V |} where 0 ≤a ≤|E|. They prove: if a tree of order n has an a-vertex consecutive magic labeling then n is odd and a = n −1; if G has an a-vertex consecutive magic labeling with n vertices and e = n edges, then n is odd and if G has minimum degree 1, then a = (n + 1)/2 or a = n; if G has an a-vertex consecutive magic labeling with n vertices and e edges such that 2a ≤e and 2e ≥ √ 6n −1, then the minimum degree of G is at least 2; if a 2-regular graph of order n has an a-vertex consecutive magic labeling, then n is odd and a = 0 or n; and if a r-regular graph of order n has an a-vertex consecutive magic labeling, then n and r have opposite parities.
Balbuena et al. also call a vertex-magic total labeling of a graph G(V, E) a b-edge consecutive magic labeling if the edge labels are {b+1, b+2, . . . , b+|E|} where 0 ≤b ≤|V |.
They prove: if G has n vertices and e edges and has a b-edge consecutive magic labeling and one isolated vertex, then b = 0 and (n −1)2 + n2 = (2e + 1)2; if a tree with odd order has a b-edge consecutive magic labeling then b = 0; if a tree with even order has a b-edge consecutive magic labeling then it is P4; a graph with n vertices and e edges such that e ≥7n/4 and b ≥n/4 and a b-edge consecutive magic labeling has minimum degree 2; if a 2-regular graph of order n has a b-edge consecutive magic labeling, then n is odd and b = 0 or b = n; and if a r-regular graph of order n has an b-edge consecutive magic labeling, then n and r have opposite parities.
Sugeng and Miller prove: If (V, E) has an a-vertex consecutive edge magic labeling, where a ̸= 0 and a ̸= |E|, then G is disconnected; if (V, E) has an a-vertex consecutive edge magic labeling, where a ̸= 0 and a ̸= |E|, then G cannot be the union of three trees with more than one vertex each; for each nonnegative a and each positive n, there is an a-vertex consecutive edge magic labeling with n vertices; the union of r stars and a set of r −1 isolated vertices has an s-vertex consecutive edge magic labeling, where the electronic journal of combinatorics (2023), #DS6 216 s is the minimum order of the stars; for every b every caterpillar has a b-edge consecutive edge magic labeling; if a connected graph G with n vertices has a b-edge consecutive edge magic labeling where 1 ≤b ≤n −1, then G is a tree; the union of r stars and a set of r −1 isolated vertices has an r-edge consecutive edge magic labeling.
Baskar Babujee, Vishnupriya, and Jagadesh introduced a labeling called a-vertex consecutive edge bimagic total as a graph G(V, E) for which there are two positive integers k1 and k2 and a bijection f from V ∪E to {1, 2, . . . , |V | + |E|} such that f(u) + f(v) + f(uv) = k1 or k2 for all edges uv and f(V ) = {a + 1, a + 2, . . . , a + |V |}, 0 ≤a ≤|V |.
They proved the following graphs have such labelings: Pn, K1,n, combs, bistars Bm,n, trees obtained by adding a pendent edge to a vertex adjacent to the end point of a path, trees obtained by joining the centers of two stars with a path of length 2, trees obtained from P5 by identifying the center of a copy K1,n with the two end vertices and the middle vertex. In Baskar Babujee and Jagadesh proved that cycles, fans, wheels, and gear graphs have a-vertex consecutive edge bimagic total labelings. Baskar Babujee, Jagadesh, Vishnupriya study the properties of a-vertex consecutive edge bimagic total labeling for P3 ⊙K1,2n, Pn +K2 (n is odd and n ≥3), (P2 ∪mK1)+K2, (P2 +mK1) (m ≥2), Cn, fans Pn + K1, double fans Pn + 2K1, and graphs obtained by appending a path of length at least 2 to a vertex of C3. Baskar Babujee and Jagadesh prove the following graphs have a-vertex consecutive edge bimagic total labelings: 2Pn (n ≥2), Pn ∪Pn+1 (n ≥2), K2,n, Cn ⊙K1, and that C3 ∪K1,n an a-vertex consecutive edge bimagic labeling for a = n + 3.
Vishnupriya, Manimekalai, and Baskar Babujee define a labeling f of a graph G(p, q) to be a edge bimagic total labeling if there exists a bijection f from V (G)∪E(G) → {1, 2, . . . , p+q} such that for each edge e = (u, v) ∈E(G) we have f(u)+f(e)+f(v) = k1 or k2, where k1 and k2 are two constants. They provide edge bimagic total labelings for Bm,n, K1,n,n, and trees obtained from a path by appending an edge to one of the vertices adjacent to an endpoint of the path. An edge bimagic total labeling is G(V, E) is called an a-vertex consecutive edge bimagic total labeling if the vertex labels are {a+1, a+2, . . . , a+ |V |} where 0 ≤a ≤|E|. Baskar Babujee and Jagadesh prove the following graphs a-vertex consecutive edge-bimagic total labelings: the trees obtained from K1,n by adding a new pendent edge to each of the existing n pendent vertices; the trees obtained by adding a pendent path of length 2 to each of the n pendent vertices of K1,n; the graphs obtained by joining the centers of two copies of identical stars by a path of length 2; and the trees obtained from a path by adding new pendent edges to one pendent vertex of the path. Baskar Babujee, Vishnupriya, and Jagadesh proved the following graphs have such labelings: Pn, K1,n, combs, bistars Bm,n, trees obtained by adding a pendent edge to a vertex adjacent to the end point of a path, trees obtained by joining the centers of two stars with a path of length 2, trees obtained from P5 by identifying the center of a copy K1,n with the two end vertices and the middle vertex. In Baskar Babujee and Jagadesh proved that cycles, fans, wheels, and gear graphs have a-vertex consecutive edge bimagic total labelings.
Baskar Babujee, Jagadesh, Vishnupriya study the properties of a-vertex consecutive edge bimagic total labeling for P3 ⊙K1,2n, Pn + K2 (n is odd and n ≥3), (P2 ∪mK1) + K2, (P2 + mK1) (m ≥2), Cn, fans Pn + K1, double the electronic journal of combinatorics (2023), #DS6 217 fans Pn + 2K1, and graphs obtained by appending a path of length at least 2 to a vertex of C3.
Vishnupriya, Manimekalai, and Baskar Babujee prove that bistars, trees ob-tained by adding a pendent edge to a vertex adjacent to the end point of a path, and trees obtained subdividing each edge of a star have edge bimagic total labelings. Prathap and Baskar Babujee obtain all possible edge magic total labelings and edge bimagic total labelings for the star K1,n. Jayasekaran1 and Flower proved that the shadow graph and the splitting graph of paths stars and cycles are edge trimagic total and super edge trimagic total.
Let D be a digraph of order p and size q. For an integer k ≥1 and v ∈V (D), let wk(v) = P f(e) over the set of all in-arcs that are at distance at most k from v.
A Vk-super vertex in-magic labeling (Vk-SVIML) of D is an one-to-one onto function f from V (D) ∪A(D) to {1, 2, . . . , p + q} such that f(V (D)) = {1, 2, . . . , p} and for every v ∈V (D), f(v) + wk(v) = M for some positive integer M. A digraph that admits a Vk-SVIML is called Vk-super vertex in-magic (Vk-SVIM). In Mutharasu, Mary Bernard, Duraisamy kumar study some properties of Vk-SVIML in digraphs. They characterized the digraphs that are Vk-SVIM and find the magic constant for Ek-regular digraphs. They furthermore characterized the unidirectional cycles and union of unidirectional cycles which are V2-SVIML.
More about magic labelings of directed graphs can be found in and .
the electronic journal of combinatorics (2023), #DS6 218 6 Antimagic-type Labelings 6.1 Antimagic Labelings Hartsfield and Ringel introduced antimagic graphs in 1990. A graph with q edges is called antimagic if its edges can be labeled with 1, 2, . . . , q without repetition such that the sums of the labels of the edges incident to each vertex are distinct.3 Among the graphs they prove are antimagic are: Pn (n ≥3), cycles, wheels, and Kn (n ≥3). T. Wang has shown that the toroidal grids Cn1 ×Cn2 ×· · ·×Cnk are antimagic and, more generally, graphs of the form G × Cn are antimagic if G is an r-regular antimagic graph with r > 1.
Cheng proved that all Cartesian products or two or more regular graphs of positive degree are antimagic and that if G is j-regular and H has maximum degree at most k, minimum degree at least one (G and H need not be connected), then G × H is antimagic provided that j is odd and j2 −j ≥2k, or j is even and j2 > 2k. Wang and Hsiao prove the following graphs are antimagic: G × Pn (n > 1) where G is regular; G × K1,n where G is regular; compositions G[H] (see §2.3 for the definition) where H is d-regular with d > 1; and the Cartesian product of any double star (two stars with an edge joining their centers) and a regular graph. In Cheng proved that Pn1 ×Pn2 ×· · ·×Pnt (t ≥2) and Cm × Pn are antimagic.
In Solairaju and Arockiasamy prove that various families of subgraphs of grids Pm ×Pn are antimagic. Liang and Zhu proved that if G is k-regular (k ≥2), then for any graph H with |E(H)| ≥|V (H)|−1 ≥1, the Cartesian product H × G is antimagic. They also showed that if |E(H)| ≥|V (H)| −1 and each connected component of H has a vertex of odd degree, or H has at least 2|V (H)|−2 edges, then the prism of H is antimagic. Shang showed that all spiders are antimagic.
Lee, Lin, and Tsai proved that C2 n is antimagic and the vertex sums form a set of successive integers when n is odd. Shang, Lin, and Liaw show that a star forest containing no S1 and at most one S2 as components is antimagic. They also prove that if a star forest mS2 is antimagic then m = 1 and mS2 ∪Sn (n ≥3) is antimagic if and only if m ≤min{2n + 1, 2n −5 + √ 8n2 −24n + 17/2}. Wang, Miao, and Li show that certain graphs with even factors are antimagic. Li gives antimagic labelings for Ck n for k = 2, 3, and 4. In Wang and Zhang showed that certain classes of regular graphs of odd degree with particular type of perfect matchings are antimagic. As a by-product, they get that generalized Petersen graphs and a subclass of Cayley graphs of Zn are antimagic. Deng and Li proved that caterpillars with maximum degree 3 are antimagic.
Let τ(G) to denote the maximum integer such that G∪tP3 is antimagic for all t ≤τ(G).
The existence of such an integer was proved by Chang, Chen, Li, and Pan in .
Shang, Lin, Liaw and Li gave tight bounds of τ(G) for star forests and 3A comprehensive expository treatment of antimagic labelings is given by Bača, Miller, Ryan, and Semaničová-Feňovčíkováin .
the electronic journal of combinatorics (2023), #DS6 219 balanced double stars. Their general bound is also tight for many other families of graphs, including cycles Cn (3 ≤9) and 2C3. Moreover, they discuss properties of τ(G) and pose open questions, including whether their bound is always tight. Chavez, Le, Der-Fen Liu, Shurman generalized the results of Shang et alby providing an upper bound of τ(G) for all graphs without isolated vertices and P2 as components.
For a graph G and a vertex v of G, the vertex switching graph Gv is the graph obtained from G by removing all edges incident to v and adding edges joining v to every vertex not adjacent to v in G. Vaidya and Vyas proved that the graphs obtained by the switching of a pendent vertex of a path, a vertex of a cycle, a rim vertex of a wheel, the center vertex of a helm, or a vertex of degree 2 of a fan are antimagic graphs.
Phanalasy, Miller, Rylands, and Lieby in 2011 showed that there is a relation-ship between completely separating systems and labeling of regular graphs. Based on this relationship they proved that some regular graphs are antimagic. Phanalasy, Miller, Iliopoulos, Pissis, and Vaezpour proved the Cartesian product of regular graphs obtained from is antimagic. Ryan, Phanalasy, Miller, and Rylands introduced the generalized web and flower graphs in and proved that these families of graphs are antimagic. Rylands, Phanalasy, Ryan, and Miller extended the concept of generalized web graphs to the single apex multi-generalized web graphs and they proved these graphs to be antimagic in . Ryan, Phanalasy, Rylands and Miller extended the concept of generalized flower to the single apex multi-(complete) generalized flower graphs and constructed antimagic labeling for this family of graphs in . For more about an-timagicness of generalized web and flower graphs see . Phanalasy, Ryan, Miller and Arumugam introduced the concept of generalized pyramid graphs and they con-structed antimagic labeling for these graphs. Bača, Miller, Phanalasy, and Feňovčíková proved that some join graphs and incomplete join graphs are antimagic in . More-over, in they proved that the complete bipartite graph Km,m and complete 3-partite graph Km,m,m are antimagic and if G is a k-regular (connected or disconnected) graph with p vertices and k ≥2, then the join of G and (p −k)K1 is antimagic. In new Jane and Myom modeled a dynamic encryption with implicit key exchange for end-to-end encryption using totally antimagic labeling of complete bipartite graphs over the network that makes it impossible to successfully intercept the message.
Arumugam, Miller, Phanalasy, and Ryan provided antimagic labelings for a family of generalized pyramid graphs. Daykin, Iliopoulas. Miller, and Phanalasy show several families of graphs recursively defined from a sequence of graphs that are generalizations of corona graphs are antimagic. Lozano, Mora, Seara, and Tey proved that caterpillars are antimagic.
Let G be a k-regular graph with p vertices and q edges. The generalized sausage graph, denoted by S(G; m), is the graph obtained from G × Pm (G × P1 = G), by joining each end vertex of the Pm to a new vertex (which we call apexes) with an edge. In particular, when m = 1, each vertex of the graph G joins to two vertices with two edges.
The mixed generalized sausage graph, denoted by MS(G; m), is the graph obtained from the generalized sausage graph S(G; m), m ≥3, by joining each vertex of each copy of the ⌈m/2⌉copies of G on the left hand side to the left hand side apex, except the nearest copy the electronic journal of combinatorics (2023), #DS6 220 to the apex, and similarly for the right hand side apex. The complete mixed generalized sausage graph, denoted by CMS(G; m) is the graph obtained from the generalized sausage graph by joining each vertex of each copy of G, except the two nearest copies of G to the apexes, to each apex with an edge, and each corresponding pair of vertices of the two nearest copies of G to the apexes with an edge. The complete mixed generalized sausage graph CMS−(G; m) is the graph obtained from CMS(G; m) by deleting the edge connecting each corresponding pair of vertices of the two nearest copies of G to the apexes.
In Phanalasy proved a families of generalized sausage graphs, mixed generalized sausage graphs, and complete mixed generalized sausage graphs are antimagic.
A split graph is a graph that has a vertex set that can be partitioned into a clique and an independent set. Tyshkevich (see ) defines a canonically decomposable graph as follows. For a split graph S with a given partition of its vertex set into an independent set A and a clique B (denoted by S(A, B)), and an arbitrary graph H the composition S(A, B) ◦H is the graph obtained by taking the disjoint union of S(A, B) and H and adding to it all edges having an endpoint in each of B and V (H). If G contains nonempty induced subgraphs H and S and vertex subsets A and B such that G = S(A, B)◦H, then G is canonically decomposable; otherwise G is canonically indecomposable. Barrus proved that every connected graph on at least 3 vertices that is split or canonically decomposable is antimagic.
Hartsfield and Ringel conjecture that every tree except P2 is antimagic and, moreover, every connected graph except P2 is antimagic. In 2004 Alon, Kaplan, Lev, Roditty, and Yuster use probabilistic methods and analytic number theory to show that this conjecture is true for all graphs with n vertices and minimum degree Ω(log n).
They also prove that if G is a graph with n ≥4 vertices and ∆(G) ≥n −2, then G is antimagic and all complete partite graphs except K2 are antimagic. Slíva proved the conjecture for graphs with a regular dominating subgraph. In 2016 Eccles improved the result of Alon et al. by proving that there exists an absolute constant d0 such that if G is a graph with average degree at least d0 and G contains no isolated edge and at most one isolated vertex, then G is antimagic.
Chawathe and Krishna proved that every complete m-ary tree is antimagic.
Yilma extended results on antimagic graphs that contain vertices of large degree by proving that a connected graph with ∆(G) ≥|V (G)| −3, |V (G)| ≥9 is antimagic and that if G is a graph with ∆(G) =deg(u) = |V (G)| −k, where k ≤|V (G)|/3 and there exists a vertex v in G such that the union of neighborhoods of the vertices u and v forms the whole vertex set V (G), then G is antimagic. Sethuraman and Shermily proved that binomial trees and Fibonacci trees are antimagic.
Vasuki, Shobana, and Ahmed proved the existence of face antimagic labelings for double duplication of barycentric subdivisions of cycles and some other graphs. Vasuki ·and Shobana proved the existence of face antimagic labeling of types (0, 1, 0) and (0, 1, 1) for crown related graphs.
For connnected graphs G, H1, H2, . . . , Hn with G having the edges e1, e2, . . . , en Nivedha and Yamini define the generalized edge corona of G and H1, H2, . . . , Hn as new the graph obtained by adding all possible edges between the end vertices of e1, e2, . . . en the electronic journal of combinatorics (2023), #DS6 221 and the vertices of H1, H2, . . . , Hn. They prove that such graphs are antimagic under certain conditions where G has only one vertex of maximum degree three and each Hi has at least 2 vertices.
In Ahmed and Babujee defined a strong face plane graph as one that is obtained from a plane graph by adding a new vertex to every face, in such way that the faces of the resulting graph are three-sided.
If the faces of original plane graphs are three sided faces, then the number of faces increase. They investigated the existence of (super) d-antimagic labeling of type (1, 1, 1) for some strong face plane graphs. Shawkat and Ahmed investgated the existence of vertex antimagic edge labelings for strong face ladders, strong face wheels, strong face fans, strong face prisms Cm × P2, and strong face friendship graphs. Kuppan and Shobana proved the existence of face antimagic labelings for the double duplication of all vertices by the edges of gear graphs Gn for n ≥3, P3 × P2n (n ≥2), Cn × P2 (n ≥5), and the double duplication of all vertices by the edges of a strong face of the triangular snake graph Tn (n ≥3). They noted that an (a, d)-face antimagic labeling for double duplication of special graphs can be used to encrypt and decrypt the messages in real time. Jesintha, Vinodhini, Lakshmi proved that triangular books and double fans graph admit antimagic labelings.
Let G = (V, E) be a graph of order n. Let f : V →{1, 2, . . . , n} be a bijection. The weight w(v) of a vertex v with respect to the labeling f is defined by w(v) = P u∈N(v) f(u), where N(v) is the open neighborhood of v. The labeling f is called a distance antimagic labeling if w(v1) ̸= w(v2) for any two distinct vertices v1, v2 in V . Cutinho, Sudha, and Arumugam proved that Kn × Kn is distance antimagic if and only if n ̸= 2 and K3 × Cn is distance antimagic when n ≥3 is odd. They included the case K3 × Cn when n is even as a problem.
Fronček defines a handicap incomplete tournament of n teams with r rounds, HIT(n, r), as a tournament in which every team plays r other teams and the total strength of the opponents that team i plays is ⃗ Sn,r(i) = t −i for every i and some fixed constant t. (This means that the strongest team plays strongest opponents, and the lowest ranked team plays weakest opponents.) In terms of distance magic graphs this restriction cor-responds to finding a distance antimagic graph with the additional property that the sequence w(1), w(2), . . . , w(n) (where team i is again the i-th ranked team) is an increas-ing arithmetic progression with difference one. These graphs are called handicap distance antimagic graphs. A handicap distance d-antimagic labeling of a graph G(V, E) with n vertices is a bijection ⃗ f : V →{1, 2, . . . , n} with the property that ⃗ f(xi) = i and the sequence of the weights w(x1), w(x2), . . . , w(xn) forms an increasing arithmetic progres-sion with difference d. A graph G is a handicap distance d-antimagic graph if it admits a handicap distance d-antimagic labeling, and handicap distance antimagic graph when d = 1. In Fronček establishes a relationship between handicap incomplete tourna-ments and distance antimagic graphs and construct some new infinite classes of distance antimagic graphs and infinite classes of handicap incomplete round robin tournaments.
Fronček and Shepanik construct r-regular handicap distance antimagic graphs of order n ≡0 (mod 8) for all feasible values of r.
Fronček proved that regular handicap distance antimagic graphs exist for every feasible odd order by proving that the electronic journal of combinatorics (2023), #DS6 222 there exists a regular handicap graph of an odd order n if and only if n = 9 or n ≥13.
In Fronček constructed a class of regular 2-handicap distance antimagic graphs for every order n ≡0 (mod 16). In he proved that a k-regular 2-handicap distance antimagic graph of order n ≡0 (mod 16) exists if and only if n ≥16 and 4 ≤k ≤n −6.
Fronček and Shepanik constructed k-regular handicap distance antimagic graphs of order n = 4 (mod 8) for all feasible values of k. In Shrimali and Parmar discuss the existence distance antimagic labelings for the product, direct product, strong product, and carona product of graphs involving C t 3 and C4.
Cranston proved that for k ≥2, every k-regular bipartite graph is antimagic.
For non-bipartite regular graphs, Liang and Zhu proved that every cubic graph is antimagic. That result was generalized by Cranston, Liang, and Zhu , who proved that odd degree regular graphs are antimagic. Hartsfield and Ringel proved that every 2-regular graph is antimagic. Bérczi, Bernäth, and Vizer use a slight modifica-tion of an argument of Cranston et al. to prove that k-regular graphs are antimagic for k ≥2. The same was done by Chang, Liang, Pan, and Zhu proved that every even degree regular graph is antimagic.
Tai, Chia, and Ong proved that the graphs obtained by starting with two verticies and joining them with at least r ≥3 edges then subdividing the edges of this graph arbitrarily so that at most one edge (multi-bridge graphs) is not subdivided are antimagic. For a connected, undirected, simple graph G(V, E) a bijection f from V to {1, 2, . . . , |V |} is called a rainbow antimagic vertex labeling if, for any two edges uv and u′v′ in the same path, w(uv) ̸= w(u′v′), where w(uv) = f(u) + f(v). The rainbow antimagic connection number of G is the smallest number of colors taken over all rainbow colorings induced by a rainbow antimagic labeling of G. In Septory, Utoyo, Dafik, Sulistiyono, and Agustin determined the exact value of the rainbow antimagic connection numbers of Jahangir graphs, firecrackers, K2,m, and double stars. A rainbow antimagic labeling f of a graph G is called a strong rainbow antimagic labeling of G if for every two vertices u and v there exists path from u to v in which no two edges of the path have the same weight.
The strong rainbow antimagic connection number of G is the smallest number of colors taken over all strong rainbow colorings induced by strong rainbow antimagic labelings of G. In Lestari1, Dafik, Susanto, and Wahab determined the connection number of strong rainbow antimagic coloring for Jahangir graphs, Jahangir semi graphs, friendship graphs, fans, stars, and paths.
A graph G(V, E) is k-shifted antimagic if there exists a bijection f from E to {k + 1, . . . , k +|E|} such that the vertex sums of all verticies are distinct; G(V, E) is absolutely antimagic if it is k-shifted antimagic for every integer k. In Chang, Li, Daphne Liu, and Pan proved that certain spider forests (a graph where each component is a spider) are k-shifted antimagic for all k ≥0. In addition, they showed that for a spider forest G with m edges, there exists a positive integer k0 ≤m such that G is k-shifted antimagic for all k ≥k0 and k ≤(m+k0+11). Li and Wang proved that every tree of diameter four or five, except for two casess, is k-shifted antimagic for every integer k. Chang, Chen, Li, and Pan proved: forest without a component isomorphic to K2 are k-shifted-antimagic, graphs consisting of vertices of odd degrees and containing no component isomorphic to the electronic journal of combinatorics (2023), #DS6 223 K2 are k-shifted-antimagic for sufficiently large k, and Pn (n ≥6) are k-shifted for all k.
They also gave necessary and sufficient conditions for stars, double stars, kP3, and 2K1,3 to be k-shifted-antimagic. Dhananjaya and Li proved that P2, P3, P4-free linear forests and K1,2-free forests are k-shifted-antimagic with a few exceptions. This extended the results by Shang et al. in and . Moreover, they prove that the odd tree forests are k-shifted-antimagic for all k.
Chang, Chen, Li, and Pan established connections among various concepts pro-posed in the literature of antimagic labelings and extend previous results in three ways: some classes of graphs, including trees and graphs whose vertices are of odd degrees, that have not been verified to be antimagic are shown to be k-shifted-antimagic for sufficiently large k; some graphs are proved k-shifted-antimagic for all k, whereas some are proved not for some particular k; and disconnected graphs are also considered.
Beck and Jackanich showed that every connected bipartite graph except P2 with |E| edges admits an edge labeling with labels from {1, 2, . . . , |E|}, with repetition allowed, such that the sums of the labels of the edges incident to each vertex are distinct. They call such a graph weak antimagic.
Wang, Liu, and Li proved: mP3 (m ≥2) is not antimagic; Pn ∪Pn (n ≥4) is antimagic; Sn ∪Pn is antimagic; Sn ∪Pn+1 is antimagic; Cn ∪Sm is antimagic for m ≥2√n + 2; mSn is antimagic; if G and H are graphs of the same order and G ∪H is antimagic, then so is G + H; and if G and H are r-regular graphs of even order, then G+H is antimagic. In Wang, Liu, and Li proved that if G is an n-vertex graph with minimum degree at least r and H is an m-vertex graph with maximum degree at most 2r −1 (m ≥n), then G+H is antimagic. Bača, Kimáková, Semaničová-Feňovčikovǎ, and Umar prove the disjoint union of multiple copies of a (a, 1)-(super)-tree-antimagic graph is also a (b, 1)-(super)-tree-antimagic for certain a and b.
For any given degree sequence pertaining to a tree, Miller, Phanalasy, Ryan, and Rylands gave a construction for two vertex antimagic edge trees with the given degree sequence and provided a construction to obtain an antimagic unicyclic graph with a given degree sequence pertaining to a unicyclic graph.
Kaplan, Lev, and Roditty prove that every non-trivial rooted tree for which every vertex that is a not a leaf has at least two children is antimagic (see ) for a correction of a minor error in the the proof). For a graph G with m vertices and an Abelian group A they define G to be A-antimagic if there is a one-to-one mapping from the edges of G to the nonzero elements of A such that the sums of the labels of the edges incident to v, taken over all vertices v of G, are distinct. For any n ≥2 they show that a non-trivial rooted tree with n vertices for which every vertex that is a not a leaf has at least two children is Zn-antimagic if and only if n is odd. They also show that these same trees are A-antimagic for elementary Abelian groups G with prime exponent congruent to 1 (mod 3).
In Chan, Low, and Shiu use [G, A] to denote the class of distinct A-antimagic labelings of G. They prove: for a non-trivial Abelian group A that underlies some com-mutative ring R with unity, if d is a unit in R and f ∈[G, A], then d f ∈[G, A]; if A is an Abelian group that contains a subgroup isomorphic to B and a graph G is B-antimagic, the electronic journal of combinatorics (2023), #DS6 224 then G is A-antimagic; P4m+r and C4m+r are Zk-antimagic for k ≥4m + r and r = 0, 1, 3; P4m+2 is Zk-antimagic for k ≥4m + 3; regular Hamiltonian graphs of order 4m + r are Zk-antimagic for k ≥4m + r and r = 0, 1, 3, and Zk-antimagic for k ≥4m + 3 and r = 2; for odd n, Sn is Zk-antimagic for k ≥n > 4; for even n, Sn is Zk-antimagic for k ≥n + 2 ≥6 but not Zn-antimagic or Zn+1-antimagic; trees of order n with exactly one vertex of even degree are Zk-antimagic for k ≥n; trees of order n with exactly two vertices of even degree are Zk-antimagic for k ≥n + 1; and double stars of order are Zk-antimagic for k ≥n + 1 when n ≡2 (mod 4) and Zk-antimagic for k ≥n when n ̸≡2 (mod 4).
Chang and Liu call a graph G R-antimagic if for each subset A of R with |A| = |E(G)|, there is an edge labeling such that the sums of the labels assigned to edges incident to distinct vertices are different. They proved that wheels, cycles, and complete graphs of order at least 3 are R-antimagic, and that Cartesian products with at least two terms are R-antimagic when each term is a complete graph of order at least 2 or a cycle.
The integer-antimagic spectrum of a graph G is the set {k | G is Zk-antimagic (k ≥2}.
Shiu, Sun, and Low determine the integer-antimagic spectra of tadpoles and lol-lipops. Shiu and Low determine the integer-antimagic spectra of complete bipartite graphs and complete bipartite graphs with a deleted edge. Shiu determined the integer-antimagic spectra of disjoint unions of cycles. In Odabasi, Roberts, amd Low determine the integer-antimagic spectra for all Hamiltonian graphs.
Liang, Wong, and Zhu study trees with many degree 2 vertices with a restriction on the subgraph induced by degree 2 vertices and its complement. Denoting the set of degree 2 vertices of a tree T by V2(T) Liang, Wong, and Zhu proved that if V2(T) and V \ V2(T) are both independent sets, or V2(T) induces a path and every other vertex has an odd degree, then T is antimagic. In Lozano, Mora, Seara, and Tey extended this result by showing that trees whose vertices of even degree induce a path are antimagic.
In Vaidya and Vyas proved that the middle graphs, total graphs, and shadow graphs of paths and cycles are antimagic. In and Krishnaa provided some results for antimagic labelings for graphs derived from wheels and antimagic labelings of helm related graphs.
Bertault, Miller, Pé-Rosés, Feria-Puron, and Vaezpour approached labeling prob-lems as combinatorial optimization problems. They developed a general algorithm to de-termine whether a graph has a magic labeling, antimagic labeling, or an (a, d)-antimagic labeling (see Section 6.3). They verified that all trees with fewer than 10 vertices are super edge magic and all graphs of the form P r 2 × P s 3 with less than 50 vertices are an-timagic. Kuppan, Shobana, and Cangul used an (a, d)-face antimagic labeling of a strong face of the duplication of all vertices by edges of a tree Tn (n ≥2) to encrypt and decrypt thirteen secret numbers that can be extended to the double duplication of graphs to encode and decode the numbers that can be used in applications.
In Bača, MacDougall, Miller, Slamin, and Wallis survey results on antimagic, edge-magic total, and vertex-magic total labelings.
A total labeling of a graph G is a bijection f from V (G) ∪E(G) to {1, 2, . . . , |V (G)| + |E(G)|}. When f(V (G)) = {1, 2, . . . , |V (G)|}, we say the total labeling is super. For a the electronic journal of combinatorics (2023), #DS6 225 labeling f the associated edge-weight of an edge uv is defined by wtf(uv) = f(uv)+f(u)+ f(v). The associated vertex-weight of a vertex v is defined by wtf(v) = P u∈N(v) f(uv) + f(v), where N(v) is the set of the neighbors of v. A labeling f is called edge-antimagic total (vertex-antimagic total) if all edge-weights (vertex-weights) are pairwise distinct.
A graph that admits an edge-antimagic total (vertex-antimagic total) labeling is called an edge-antimagic total (vertex-antimagic total) graph. A labeling that is simultaneously edge-antimagic total and vertex-antimagic total is called a totally antimagic total labeling.
A graph that admits a totally antimagic total labeling is called a totally antimagic total graph. A labeling g is said to be ordered (sharp ordered) if wtg(u) ≤wtg(v) (wtg(u) < wtg(v)) holds for every pair of vertices u, v ∈V (G) such that g(u) < g(v). A graph that admits a (sharp) ordered labeling is called a (sharp) ordered graph.
Miller, Phanalasy, and Ryan proved that all graphs have vertex-antimagic to-tal labelings. Bača, Miller, Phanalasy, Ryan, Semaničová-Feňovčíková, and Abildgaard Sillasen prove that mK1, mK2, Pn (n ≥2), and Cn are sharp ordered super totally antimagic total. They prove if G is an ordered super edge-antimagic total graph then G + K1 is a totally antimagic total graph. As a corollary they get that stars, friendship graphs nK2+K1, fans, and wheels are totally antimagic total. They also prove that if G is a regular ordered super edge-antimagic total graph then G ⊙nK1 is totally antimagic to-tal. As a corollary of this result, they have double-stars K2 ⊙nK1 and crowns Cm ⊙nK1 are totally antimagic total. They show that a union of regular totally antimagic total graphs is a totally antimagic total graph.
Ahmed and Baskar Baskar proved that complete bipartite graphs admit a totally antimagic total labeling.
The same result was proved by Akwu and Ajayi who also showed that the join of a complete bipartite graph and K1 is a totally antimagic total graph. Ahmed, Baskar Babujee, Bača, Semaničová-Feňovčíková proved that complete graphs admit totally antimagic total labeling. They also considered the problem of finding total labelings for prisms and for two special classes of graphs related to paths that are simultaneously edge-magic and vertex-antimagic.
Miller, Phanalasy, Ryan, and Rylands provide a method whereby, given any degree sequence pertaining to a tree, one can construct an antimagic tree based on this sequence. By swapping the roles of edges and vertices with respect to a labeling, they provide a method to construct an edge antimagic vertex labeling for any tree. In new Selvarasu and Murugan showed that twings and combs admit vertex and edge antimagic labelings.
Ahmad, Semaničová-Feňovčíková, Siddiqui, and Kamran construct α-labelings from graceful labelings of smaller trees and transform this labeling to edge-antimagic vertex labeling of trees. Shang shows that linear forests without either of the paths P2 or P3 as components are antimagic. Shang proved that P2, P3, and P4-free linear forests are antimagic. In Shiu and Low analyzed various antimagic properties for Cartesian products, hexagonal nets and theta graphs.
A Fibonacci mean antimagic labeling of a graph G is an injective function g : V (G) →{f2, f3, . . . , fn+1}, where fn is nth Fibonacci number, and the induced a function g∗: E(G) →N defined by g∗(uv) = ⌈(g(u) + g(v))/2⌉is injective. A graph is called the electronic journal of combinatorics (2023), #DS6 226 Fibonacci mean antimagic if it admits a Fibonacci mean antimagic labeling.
Thirug-nanasambandam and Chitra proved the following graphs have Fibonacci mean antimagic labelings: caterpillars, bistars, triangular snakes, quadrilateral snakes, ladders, and the corona product of last three graphs and K1. Barasara and Prajapati stud- new ied antimagic labelings of double triangular snakes, alternate triangular snakes, double alternate triangular snakes, quadrilateral snakes, double quadrilateral snakes, alternate quadrilateral snakes, double alternate quadrilateral snakes.
In Hefetz, Mütze, and Schwartz investigate antimagic labelings of directed graphs. An antimagic labeling of a directed graph D with n vertices and m arcs is a bijection from the set of arcs of D to the integers {1, . . . , m} such that all n oriented vertex sums are pairwise distinct, where an oriented vertex sum is the sum of labels of all edges entering that vertex minus the sum of labels of all edges leaving it. Hefetz et al. raise the questions “Is every orientation of any simple connected undirected graph antimagic? and “Given any undirected graph G, does there exist an orientation of G which is antimagic?” They call such an orientation an antimagic orientation of G. Re-garding the first question, they state that, except for K1,2 and K3, they know of no other counterexamples. They prove that there exists an absolute constant C such that for every undirected graph on n vertices with minimum degree at least C log n every orientation is antimagic. They also show that every orientation of Sn, n ̸= 2, is antimagic; every orientation of Wn is antimagic; and every orientation of Kn, n ̸= 3, is antimagic. For the second question they prove: for odd r, every undirected r-regular graph has an an-timagic orientation; for even r every connected undirected r-regular graph that admits a matching that covers all but at most one vertex has an antimagic orientation; and if G is a graph with 2n vertices that admits a perfect matching and has an independent set of size n such that every vertex in the independent set has degree at least 3, then G has an antimagic orientation. They conjecture that every connected undirected graph admits an antimagic orientation and ask if it true that every connected directed graph with at least 4 vertices is antimagic. Shan supported this conjecture by proving that every bipartite graph with no vertex of degree 0 or 2 admits an antimagic orientation and every graph with minimum degree at least 33 admits an antimagic orientation. In Yang, Carlson, Owens, Perry, Singgih, Song, Zhang, and Zhang proved that a graph G admits an antimagic orientation if ∆(G) ≥|G| −3 or ∆(G) = |G| −t ≥4 for each t = 4, 5.
For a directed graph D = (V, A) with p vertices and q arcs. Kumar, Nirmala, and Mutharasu say that D has a V-super vertex in-antimagic total graceful labeling new (V -SVIAMTG labeling) if there is a bijection f : V (D) ∪A(D) to {1, 2, . . . , p + q} with the property that f(V (D)) = {1, 2, . . . , p} and for each v ∈V (D), |w(v) −f(v)| consists of distinct integers, where w(v) is P(f(u, v) over all u incident to v. They call D a V-super vertex in-antimagic total graceful digraph (V -SVIAMTG digraph) if D admits a V -SVIAMTG labeling. They prove existence of V -SVIAMTG for directed paths, cycles, coronas, fans, some cylinders, and some grids.
Motivated by the Hartsfield and Ringel on antimagic labelings of graphs, in 2010 Hefetz, Mütze, and Schwartz initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. The con-the electronic journal of combinatorics (2023), #DS6 227 jecture has been verified to be true for regular graphs (see [ , , ]), and biregular bipartite graphs with minimum degree at least two by Shan and Yu . Yang, Carlson, Owens, Perry, Singgih, Song, Zhang, Zhang proved that every connected graph G on n ≥9 vertices with maximum degree at least n −5 admits an antimagic orientation. Li, Song, Wang, Yang, and Zhang proved that every 2-regular graph has an antimagic orientation and for all integers d ≥2, every connected 2d-regular graph has an antimagic orientation. Gao, Yuping and Shan proved that every lobster admits an antimagic orientation. Ferraro, Newkirk, and Shan prove that subdivided caterpillars in which all pendant edges are replaced by paths of the same length admit an antimagic orientation.
Sonntag investigated antimagic labelings of hypergraphs. He shows that certain classes of cacti, cycle, and wheel hypergraphs have antimagic labelings. Javaid and Bhatti extended some of Sonntag’s results to disjoint unions of hypergraphs. In Nalliah investigated the existence of antimagic labelings of some families of digraphs using hooked Skolem sequences. Marimuthu, Raja Durga, and Durga Devi investigated the existence of super vertex in-antimagic total labelings of generalized de Bruijn digraphs.
Hefetz calls a graph with q edges k-antimagic if its edges can be labeled with 1, 2, . . . , q + k such that the sums of the labels of the edges incident to each vertex are distinct. In particular, antimagic is the same as 0-antimagic. More generally, given a weight function ω from the vertices to the natural numbers Hefetz calls a graph with q edges (ω, k)-antimagic if its edges can be labeled with 1, 2, . . . , q + k such that the sums of the labels of the edges incident to each vertex and the weight assigned to each vertex by ω are distinct. In particular, antimagic is the same as (ω, 0)-antimagic where ω is the zero function. Using Alon’s combinatorial nullstellensatz as his main tool, Hefetz has proved the following: a graph with 3m vertices and a K3 factor is antimagic; a graph with q edges and at most one isolated vertex and no isolated edges is (ω, 2q−4)-antimagic; a graph with p > 2 vertices that admits a 1-factor is (p −2)-antimagic; a graph with p vertices and maximum degree n−k, where k ≥3 is any function of p is (3k−7)-antimagic and, in the case that p ≥6k2, is (k−1)-antimagic. Hefetz, Saluz, and Tran improved the first of Hefetz’s results by showing that a graph with pm vertices, where p is an odd prime and m is positive, and a Cp factor is antimagic.
A vertex in-out-antimagic total labeling of a directed graph (digraph) D(V, A) with n vertices and m) arcs is a bijection from V ∪A to {1, 2, . . . , m + n} such that all n vertex in-weights are pairwise distinct, and simultaneously, all n vertex out-weights are pairwise distinct. If the smallest labels are used on vertices, the labeling is called a super vertex in-out-antimagic total labeling. In Bača, Kovář, Kovář, and Semaničová-Feňovčíková new provided a general way for determining if dense digraphs and certain sparse digraphs admit a super vertex in-out-antimagic total labelings. They also include constructions of vertex in-out-antimagic total labelings for three large infinite classes of digraphs and conjecture that all digraphs allow such a labeling.
A graph G = (V, E) is strongly antimagic if there is a bijective mapping f : E → 1, 2, . . . , |E| such that for any two vertices u ̸= v, not only P e∈E(u) f(e) ̸= P e∈E(v) f(e) and also P e∈E(u) f(e) < P e∈E(v) f(e) whenever deg(u) < deg(v), where E(u) is the set the electronic journal of combinatorics (2023), #DS6 228 of edges incident to u. Chang, Chin, Li, and Pan proved double spiders (the trees contains exactly two vertices of degree at least 3) are strongly antimagic. They raise the following two questions. Does there exist a strongly antimagic labelings for every antimagic graph? Is there a k-antimagic graph but not (k + 1)-antimagic?
A strong edge antimagic total labeling of a simple graph G(V, E) is a labeling in which the vertex labels are consecutive integers from 1 to |V | such that the total of the labels of the vertices incident to an edge form an ascending arithmetic sequence. Prasetyo proved that mK1,n has a strong edge antimagic total labeling.
For a graph G(V, E) of order p and size q having no isolated vertices, a bijection f : V →{1, 2, 3, . . . , p} is called a local edge antimagic labeling if for any two adjacent edges uv and vw of G, we have w(uv) ̸= w(vw), where the edge weight w(uv) = f(u)+f(v) and w(vw) = f(v) + f(w). A graph G is called a local edge antimagic if G has a local edge antimagic labeling. The local edge antimagic chromatic number number of G is the minimum number of colors taken over all colorings induced by local edge antimagic labelings of G. Agustin, Hasan, Dafik, Alfarisi and Prihandini found the local edge antimagic chromatic numbers of paths, cycles, friendship graphs, ladders, stars, wheels, complete graphs, prisms, Cm ⊙mK1, and G ⊙mK1, where G is any graph of size at least 3. Moreover, they found a relation between local edge antimagic chromatic number and local antimagic vertex chromatic number. Rajkumar and Nalliah determined the local edge antimagic chromatic numbers for a friendship graphs, wheels, fans, helms, flower graphs, and closed helms. In Hadiputra and Maryati investigated characterizations new of graphs with small local edge chromatic numbers, the relationship between local edge antimagic chromatic number and edge independence number, and bounds for the local edge chromatic number for any graph.
Ahmad, Bača, Lascsáková and Semaničová-Feňovčíková call a labeling of a plane graph d-antimagic if for every positive integer s, the set of s-sided face weights is Ws = {as, as + d, as + 2d, . . . , as + (fs −1)d} for some positive integers as as and d, where fs is the number of the s-sided faces. (They allow different sets Ws for different s). A d-antimagic labeling is called super if the smallest possible labels appear on the vertices.
In they investigated the existence of super d-antimagic labelings of type (1, 1, 0) for disjoint union of plane graphs for several values of difference d. Bača, Numan, and Semaničová-Feňovčíková investigated the existence of super d-antimagic labelings of generalized prisms. Hussainn and Tabraiz investigated super d-antimagic labeling of type (1,1,1) on the snakes kC5; subdivided kC5; and isomorphic copies of kC5 for strings (1, 1, . . . , 1) and (2, 2, . . . , 2).
In Li, Lau, and Shiu defined an edge labeling of a connected graph G(V, E) new of order p and size q to be a modulo local antimagic labeling if it is a bijection π : E → {1, . . . , q} such that for any pair of adjacent vertices u and v, π+(u) ̸= π+(v), where the induced vertex label π+(u) = P π(e) (mod p), with e ranging over all the edges incident to u. The modulo local antimagic chromatic number of G, denoted πla(G), is the minimum number of distinct induced vertex labels over all modulo local antimagic labelings of G.
They investgated the relationship among chromatic number, local antimagic chromatic number, and modulo local antimagic chromatic number of graphs. They also determined the electronic journal of combinatorics (2023), #DS6 229 the modulo local antimagic chromatic numbers of Km,m, Pn, tadpoles, coconut trees and combs.
Bača, Baskoro, Jendroľ, and Miller investigated various k-antimagic labelings for graphs in the shape of hexagonal honeycombs. They use Hm n to denote the honeycomb graph with m rows, n columns, and mn 6-sided faces.
They prove: for n odd Hm n , has a 0-antimagic vertex labeling and a 2-antimagic edge labeling, and if n is odd and mn > 1, Hm n has a 1-antimagic face labeling.
In Shiu and Low show how to construct k-antimagic graphs from existing graphs G with particular labeling properties by joining G to cycles and dumbbell related graphs with an edge.
Huang, Wong, and Zhu say a graph G is weighted-k-antimagic if for any vertex weight function w from the vertices of G to the natural numbers there is an injection f from the edges of G to {1, 2, . . . , |E| + k} such that for any two distinct vertices u and v, P(f(e) + w(v)) ̸= P(f(e) + w(u)) over all edges incidence to v. They proved that if G has odd prime power order pz and has total domination number 2 with the degree of one vertex in the total dominating set not a multiple of p, then G is weighted-1-antimagic, and if G has odd prime power order pz, p ̸= 3 and has maximum degree at least |V (G)| −3, then G is weighted-1-antimagic. Wong and Zhu proved: graphs that have a vertex that is adjacent to all other vertices are weighted-2-antimagic; graphs with a prime number of vertices that have a Hamiltonian path are weighted-1-antimagic; and connected graphs G ̸= K2 on n vertices are weighted-⌊3n/2⌋-antimagic. Matamala and Zamora proved that Km,n, 3 ≤m ≤n, n ≥3, is weighted-0-antimagic and described a polynomial time algorithm that computes a (w, 0)-antimagic labeling of Km,n.
They also prove the following. Let H be an arbitrary complete partite graph with n ≥5 vertices not isomorphic to K1,n. Then, any graph containing H as a spanning subgraph is weighted-0-antimagic and given a weight function w, a (w, 0)-antimagic labeling can be computed in polynomial time. They prove that each connected graph G on n ≥3 vertices having K1,n as a spanning subgraph is weighted-1-antimagic unless G is isomorphic to K1,n and n is even.
A graph G is weighted k-list-antimagic if for any vertex weighting ω : V (G) →R and any list assignment L : E(G) →2R with |L(e)| ≥|E(G)| + k, there exists an edge labeling f such that f(e) ∈L(e) for all e ∈E(G), labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. Berikkyzy, Brandt, Jahanbekam, Larsen, and Rorabaugh proved that every graph on n vertices having no K1 or K2 component is weighted-⌊4n 3 ⌋-list-antimagic.
A distance k-antimagic labeling of a graph G(V, E) is a bijection ¯f from V to {1, 2, . . . , |V |} with the property that there exists an ordering of the vertices of G such that the sequence of the weights w(x1), w(x2), . . . , w(xn) forms an arithmetic progression with difference k. When k = 1, then ¯f is simply called a distance antimagic labeling.
A distance k-antimagic graph is a distance k-antimagic graph that admits a distance k-antimagic labeling, and is called distance antimagic when k = 1. Cichacz, Froncek, Sugeng and Zhou in gave a necessary condition for a graph with an even number of vertices to be distance antimagic with respect to an Abelian group with a unique in-the electronic journal of combinatorics (2023), #DS6 230 volution. They also gave sufficient conditions for a Cayley graph on an Abelian group to be distance antimagic or magic with respect to the same group, and discussed the consequences of these results to Cayley graphs on elementary Abelian groups. In Handa, Godinho, and Singh investigate the existence of distance antimagic labelings of ladders.
For a positive integer k, define fk : V (G) − →{1 + k, 2 + k, . . . , n + k} by fk(x) = f(x) + k. If wfk(u) ̸= wfk(v) for every pair of vertices u, v ∈V , for any k ≥0 then f is said to be an arbitrarily distance antimagic labeling and the graph which admits such a labeling is said to be an arbitrarily distance antimagic graph. Handa, Godinho, and Singh provide arbitrarily distance antimagic labelings for rPn, the generalized Petersen graph P(n, k), n ≥5, the Harary graph H4,n for n ̸= 6 and prove that join of these graphs is distance antimagic.
For an arbitrary set of distances D ⊆{0, 1, . . . , diam(G)}, a D-weight of a vertex x in a graph G under a vertex labeling f : V →{1, 2, . . . , v} is defined as wD(x) = P y∈ND(x) f(y), where ND(x) = {y ∈V | d(x, y) ∈D}. A graph G is said to be D-distance magic if all vertices has the same D-vertex-weight, it is said to be D-distance antimagic if all vertices have distinct D-vertex-weights. In Simanjuntak and Wijaya gave some necessary conditions for the existence of D-distance antimagic graphs and conjectured that those conditions are sufficient. They also gave {1}-distance antimagic labelings for cycles, suns, prisms, complete graphs, wheels, fans, and friendship graphs.
In Arumugam and Kamatchi introduced the notion of (a, d)-distance antimagic graphs as follows. Let G be a graph with vertex set V and f : V →{1, 2, . . . , |V |} be a bijection. If for all v in G the set of sums P f(u) taken over all neighbors u of v is the arithmetic progression {a, a + d, a + 2d, . . . , a + (|V | −1)d}, f is called an (a, d)-distance antimagic labeling and G is called a (a, d)-distance antimagic graph. Arumugam and Kamatchi proved: Cn is (a, d)-distance antimagic if and only if n is odd and d = 1; there is no (1, d)-distance antimagic labeling for Pn when n ≥3; a graph G is (1, d)-distance antimagic graph if and only if every component of G is K2; Cn × K2 is (n + 2, 1)-distance antimagic; and the graph obtained from C2n = (v1, v2, . . . , v2n) by adding the edges v1vn+1 and viv2n+2−i for i = 2, 3, . . . , n is (2n + 2, 1)-distance antimagic.
In and Froncek proved that disjoint copies of the Cartesian product of two complete graphs and its complement are (a, 2)-distance antimagic and (a, 1)-distance antimagic.
He also proved that disjoint copies of the hypercube Q3 is (a, 1)-distance antimagic. Semeniuta proved that the crown Pn ⊙P1 does not admit an (a, 1)-distance antimagic labeling for n ≥2 and a ≥2 and determines the values of a for which Pn can be an (a, 1)-distance antimagic graph. The circulant graph is also investigated.
Semenyuta proved that Pn ⊙P1 is not an (a, d)-distance antimagic graph for all a and d and that Qn is a (2n + n −1, n −2)-distance antimagic graph. He found two types of graphs that do not allow 1-vertex bimagic vertex labeling and established a relation between the distance magic labeling of a regular graph G with 1-vertex bimagic vertex labeling G ∪G. Meganingtyas, Dafik, and Slamin investigated the existence of super (a, d)-vertex antimagic labeling of directed cycles.
Patel and Vasva use Circ(n, k) n ≥3 to denote the graph Circ(n, {k}) defined the electronic journal of combinatorics (2023), #DS6 231 as follows: for a subset S ⊂{1, 2, . . . , n}, the circulant graph Circ(n, S) is the graph with vertex set {v1, v2, . . . , vn} and there is an edge between vertices vi and vj if and only if |i −j| ∈S S{1, n −1}). In they proved the existence or non-existence of (a, d)-distance antimagic labelings for the following graphs: Circ(2n, {1, n}) is (2n+2, 1)-distance antimagic for all even n; mK2n is (n(2mn −2m + 1), 1)-distance antimagic for all m and n; 3K2n+1 is (6n2 + n −1, 1)-distance antimagic for all n; 2K2n+1 is not (a, d)-distance antimagic for all n; helms Hn is not (a, d)-distance antimagic for any n; books Bn = Sn × P2 of order 2n + 2 are not (a, d)-distance antimagic for any n; and Kn ⊙K1 is not (a, d)-distance antimagic for n > 1.
Kamatchi, Vijayakumar, Ramalakshmi, Nilavarasi, and Arumugam prove that the hypercube is (a, d)-distance antimagic and the bistar K2(n, n) is distance antimagic.
They also show that if G is a regular distance antimagic graph, then 2G is also distance antimagic and several families of disconnected graphs are distance antimagic graphs.
A connected graph G = (V, E) with m edges is called universal antimagic if for each set B of m positive integers there is an bijective function f : E →B such that the function ˜ f : V →N defined at each vertex v as the sum of all labels of edges incident to v is injective. Matamala and Zamora proved that paths, cycles, split graphs, and graphs that contains the complete bipartite graph K2,n as a spanning subgraph are universal antimagic.
A universal antimagic graph is weight universal antimagic if, in addition, for any weight function w on the vertices, all w(u)+f u are distinct. Generalizing their previous result , in the authors show constructively that if a graph has a complete bipartite graph Km,n as a spanning subgraph with m, n ≥3, then it is weighted universal antimagic (hence universal antimagic). They also show that for all other values of m and n, the graph is universal antimagic.
In 2019 Bača, Miller, Ryan, and Semaničová-Feňovčíková published a mono-graph that focuses on variations of magic and antimagic type labelings and includes new results, techniques, constructions, and open problems and conjectures.
In Table 12 we use the abbreviation A to mean antimagic. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property.
The table was prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2014.
Table 12: Summary of Antimagic Labelings Graph Labeling Notes Pn A for n ≥3 Cn A Wn A Continued on next page the electronic journal of combinatorics (2023), #DS6 232 Table 12 – Continued from previous page Graph Labeling Notes Kn A for n ≥3 every tree except K2 A?
caterpillars A regular graphs A , , every connected graph A?
except K2 n ≥4 vertices A ∆(G) ≥n −2 all complete partite A graphs except K2 Cm × Pn A Pm1 × Pm2 × · · · × Pmk A Cm1 × Cm2 × · · · × Cmk A C2 n A mP3 m ≥2 not A 6.2 (a, d)-Antimagic Labelings The concept of an (a, d)-antimagic labelings was introduced by Bodendiek and Walther in 1993.
A connected graph G = (V, E) is said to be (a, d)-antimagic if there exist positive integers a, d and a bijection f : E →{1, 2, . . . , |E|} such that the induced mapping gf : V →N, defined by gf(v) = P{f(uv)| uv ∈E(G)}, is injective and gf(V ) = {a, a+d, . . . , a+(|V |−1)d}. (In Lin, Miller, Simanjuntak, and Slamim called these (a, d)-vertex-antimagic edge labelings). Bodendick and Walther ( and ) prove the Herschel graph is not (a, d)-antimagic and obtain both positive and negative results about (a, d)-antimagic labelings for various cases of graphs called parachutes Pg,p. (Pg,p is the graph obtained from the wheel Wg+p by deleting p consecutive spokes.) In Bača and Holländer prove that necessary conditions for Cn ×P2 to be (a, d)-antimagic are the electronic journal of combinatorics (2023), #DS6 233 d = 1, a = (7n+4)/2 or d = 3, a = (3n+6)/2 when n is even, and d = 2, a = (5n+5)/2 or d = 4, a = (n + 7)/2 when n is odd.
Bodendiek and Walther conjectured that Cn × P2 (n ≥3) is ((7n + 4)/2, 1)-antimagic when n is even and is ((5n + 5)/2, 2)-antimagic when n is odd. These conjectures were verified by Bača and Holländer who further proved that Cn × P2 (n ≥3) is ((3n + 6)/2, 3)-antimagic when n is even.
Bača and Holländer conjecture that Cn × P2 is ((n + 7)/2, 4)-antimagic when n is odd and at least 7. Bodendiek and Walther also conjectured that Cn ×P2 (n ≥7) is ((n+7)/2, 4)-antimagic. Miller and Bača prove that the generalized Petersen graph P(n, 2) is ((3n+6)/2, 3)-antimagic for n ≡0 (mod 4), n ≥8 and conjectured that P(n, k) is ((3n + 6)/2, 3)-antimagic for even n and 2 ≤k ≤n/2 −1 (see §2.7 for the definition of P(n, k)). This conjecture was proved for k = 3 by Xu, Yang, Xi, and Li . Jirimutu and Wang proved that P(n, 2) is ((5n + 5)/2, 2)-antimagic for n ≡3 (mod 4) and n ≥7.
Xu, Xu, Lü, Baosheng, and Nan proved that P(n, 2) is ((3n + 6)/2, 2)-antimagic for n ≡2 (mod 4) and n ≥10. Xu, Yang, Xi, and Li proved that P(n, 3) is ((3n + 6)/2, 3)-antimagic for even n ≥10 and for n ≡0 (mod 4), n ≥8. In Lingqi, Linna, Yuan show that P(n, 3) is (5n + 5)/2, 2)-antimagic for odd n ≥7. Feng, Hong, Yang, and Jirimutu show that P(n, 5) is (3n + 6)/2, 3)-antimagic for even n ≥12.
Bao, Zhao, Yang, Feng, and Jirimutu proved that P(n, 7) is ( 3n+6 2 , 3)-antimagic for even n ≥16. Ivančo investigated (a, 1)-antimagic labelings and their connection with supermagic generalized double graphs. Bodendiek and Walther proved that the following graphs are not (a, d)-antimagic: even cycles; paths of even order; stars; C(k) 3 ; C(k) 4 ; trees of odd order at least 5 that have a vertex that is adjacent to three or more end vertices; n-ary trees with at least two layers when d = 1; the Petersen graph; K4 and K3,3. They also prove: P2k+1 is (k, 1)-antimagic; C2k+1 is (k +2, 1)-antimagic; if a tree of odd order 2k + 1 (k > 1) is (a, d)-antimagic, then d = 1 and a = k; if K4k (k ≥2) is (a, d)-antimagic, then d is odd and d ≤2k(4k −3) + 1; if K4k+2 is (a, d)-antimagic, then d is even and d ≤(2k + 1)(4k −1) + 1; and if K2k+1 (k ≥2) is (a, d)-antimagic, then d ≤(2k + 1)(k −1). Lin, Miller, Simanjuntak, and Slamin show that no wheel Wn (n > 3) has an (a, d)-antimagic labeling. In Susanto provided super (a, d)-Cn-antimagic total labelings for various cases of mCn. In , Kathiresan and Laurence posed the problem of characterizing the super (a, 1)−P3-antimagic total labeling of the stars Sn, where n = 6, 7, 8, and 9. This problem was completely solved in by Rajkumar, Nalliah, and Venkataraman. In Yatin1, Awanis, and Wardhana showed new that for odd m, Pn ⊙Pm is super (9m2n + 4mn + mn + 3, 1) −P2 ⊙Pm-antimagic, and for m even, Pn ⊙Pm is super (9m2n + 4mn + m2n + 5, 3) −P2 ⊙Pm-antimagic.
The Hill cipher is a cryptographic algorithm that uses modulo arithmetic and a matrix as a key to perform encryption and decryption.
In Prihandini and Adawiyah1 discussed how to use a super (3n + 5, 2)- edge antimagic total labeling to construct the Hill cipher algorithm. They stated that variations of the edge weight function and the corresponding edge label on the graph will make the constructed lock more complicated to hack.
In Ivančo, and Semaničová show that a 2-regular graph is super edge-magic if and only if it is (a, 1)-antimagic. As a corollary we have that each of the following graphs the electronic journal of combinatorics (2023), #DS6 234 are (a, 1)-antimagic: kCn for n odd and at least 3; k(C3 ∪Cn) for n even and at least 6; k(C4 ∪Cn) for n odd and at least 5; k(C5 ∪Cn) for n even and at least 4; k(Cm ∪Cn) for m even and at least 6, n odd, and n ≥m/2 + 2. Extending a idea of Kovář they prove if G is (a1, 1)-antimagic and H is obtained from G by adding an arbitrary 2k-factor then H is (a2, 1)-antimagic for some a2. As corollaries they observe that the following graphs are (a, 1)-antimagic: circulant graphs of odd order; 2r-regular Hamiltonian graphs of odd order; and 2r-regular graphs of odd order n < 4r. They further show that if G is an (a, 1)-antimagic r-regular graph of order n and n −r −1 is a divisor of the non-negative integer a + n(1 + r −(n + 1)/2), then G ⊕K1 is supermagic. As a corollary of this result they have if G is (n −3)-regular for n odd and n ≥7 or (n −7)-regular for n odd and n ≥15, then G ⊕K1 is supermagic.
Bertault, Miller, Feria-Purón, and Vaezpour approached labeling problems as combinatorial optimization problems. They developed a general algorithm to determine whether a graph has a magic labeling, antimagic labeling, or an (a, d)-antimagic labeling.
They verified that all trees with fewer than 10 vertices are super edge magic and all graphs of the form P r 2 × P s 3 with less than 50 vertices are antimagic. Javaid, Hussain, Ali, and Dar and Javaid, Bhatti, and Hussain constructed super (a, d)-edge-antimagic total labelings for w-trees and extended w-trees (see 5.2 for the definitions) as well as super (a, d)-edge-antimagic total labelings for disjoint union of isomorphic and non-isomorphic copies of extended w-trees. In Javaid and Bhatt defined a generalized w-tree and proved that they admit a super (a, d)-edge-antimagic total labeling. In Wang, Li, and Wang proved that some classes of graphs derived from regular or regular bipartite graphs are antimagic. A subdivided star T(n1, n2, . . . , nr) is a tree obtained by inserting ni ≥1, 1 ≤i ≤r with r ≥3 vertices. In Raheem, Javaid, and Baig study a super (a, d)-edge-antimagic total labelings of the subdivided stars T(n, n + 1, n3, . . . , nr) when n is even and T(n, n, n + 1, n4, . . . , nr) when n is odd for all possible values of d.
In Raheem and Baig proved the super edge antimagicness of subdivided stars for all possible values of d. Bhatti, Tahir, and Javaid give super (a, d)-edge antimagic total labelings of some wheel-like graphs. In investigate the existence of super (a, d)-edge antimagic total labeling for friendship graphs and generalized friendship graphs. Girija and Karthikeyan proved that 3 copies of the jelly fish graphs are super (a, d)-edge antimagic vertex graphs and super (a, d)-edge antimagic total graphs.
In Afzal, Javaid, Alanazi, and Alshehri investigated the super (a, 0) edge-antimagicness of the union of the networks of stars, paths, and copies of paths and the rooted product of Cn with K2,n. They also provided super (a, 0) edge-antimagic labelings of the rooted product of cycles and planar pancyclic networks, and give super (a, 0) edge-antimagic labelings for a pancyclic network containing chains of C6, and three different symmetrically designed lattices.
For graphs G and F, if every edge of G belongs to a subgraph of G isomorphic to F and there exists a total labeling λ of G such that for every subgraph F ′ of G that is isomorphic to F, the set {Σλ(F ′) : F ′ ∼ = F, F ′ ⊆G} forms an arithmetic progression starting with a with common difference d, Lee, Tsai, and Lin say that G is (a, d)-F-antimagic. Furthermore, if λ(V (G)) = {1, 2, . . . , |V (G)|} then G is said to be the electronic journal of combinatorics (2023), #DS6 235 super (a, d)-F-antimagic and λ is said to be a super (a, d)-F-antimagic labeling of G. Lee, Tsai, and Lin proved that Pm×Pn (m, n ≥2) is super (a, 1)-C4-antimagic. In Selvagopal, Jeyanthi, Muthuraja, and Semaničová-Feňovčiková investigated the existence super (a, d)-star-antimagic labelings of a particular class of banana trees and construct a star-antimagic graph. In Jeyanthi, Selvi, and Ramya proved the existence of super (a, d)-Cn-antimagic labelings of fan graphs and ladders. Inayah proved the existence of an (a, b) −P4-antimagic decomposition of a generalized Petersen graph GPz(n, 3) for several values of b. Taimur, Ali, Numan, Aslam, and Anoh Yannick provided super (a, d)-C3-antimagic total labelings for the generalized antiprism for d = 0 and 1 and a super (a, d)-C8-antimagic total labeling for the toroidal octagonal map for d = 1, 2, . . . , 7.
The edge corona path graph Gm ⋄Pn is the graph obtained from one copy of the gear graph Gm and 3m copies of Pn, P i n, by joining two end vertices of ei ∈E(Gm) to every vertex vj ∈V (Pn) in the i-th copy of Gm with i = 1, 2, . . . , 3m and j = 1, 2, . . . , n.
The graph Gm · Cn is the graph obtained from Gm and 2m + 1 copies of Cn namely Ci n by joined every vertex vi ∈Gm to all vertices vi ∈Cn for i ∈{1, 2, . . . , 2m + 1}.
Nistyawati and Martini proved that for every odd m, the gear edge corona path graph Gm ⋄Pn is super C4 ⋄Pn-antimagic and for every odd m, the gear corona cycle graph Gm · Cn is super C4 · Cn-antimagic. Roswitha, Martini, and S. A. Nugroho proved: for n ≥5 the double cone DCn = Cn + K2 is (14 + 7n + (n + 1)2, 1)-C3-antimagic and (a, 1)-C3-antimagic; DCn is (a, d)-Wn-antimagic; DC2n is (a, 1)-W2n-antimagic; and DC2n+1 is (a, 2)-W2n+1-antimagic.
Yegnanarayanan introduced several variations of antimagic labelings and pro-vides some results about them.
The antiprism on 2n vertices has vertex set {x1,1, . . . , x1,n, x2,1, . . . , x2,n} and edge set {xj,i, xj,i+1} ∪{x1,i, x2,i} ∪{x1,i, x2,i−1} (subscripts are taken modulo n). For n ≥3 and n ̸≡2 (mod 4) Bača gives (6n + 3, 2)-antimagic labelings and (4n + 4, 4)-antimagic labelings for the antiprism on 2n vertices. He conjectures that for n ≡2 (mod 4), n ≥6, the antiprism on 2n vertices has a (6n+3, 2)-antimagic labeling and a (4n+4, 4)-antimagic labeling.
Nicholas, Somasundaram, and Vilfred prove the following: If Km,n where m ≤n is (a, d)-antimagic, then d divides ((m −n)(2a + d(m + n −1)))/4 + dmn/2; if m + n is prime, then Km,n, where n > m > 1, is not (a, d)-antimagic; if Kn,n+2 is (a, d)-antimagic, then d is even and n + 1 ≤d < (n + 1)2/2; if Kn,n+2 is (a, d)-antimagic and n is odd, then a is even and d divides a; if Kn,n+2 is (a, d)-antimagic and n is even, then d divides 2a; if Kn,n is (a, d)-antimagic, then n and d are even and 0 < d < n2/2; if G has order n and is unicylic and (a, d)-antimagic, then (a, d) = (2, 2) when n is even and (a, d) = (2, 2) or (a, d) = ((n + 3)/2, 1) when n is odd; a cycle with m pendent edges attached at each vertex is (a, d)-antimagic if and only if m = 1; the graph obtained by joining an endpoint of Pm with one vertex of the cycle Cn is (2, 2)-antimagic if m = n or m = n −1; if m + n is even the graph obtained by joining an endpoint of Pm with one vertex of the cycle Cn is (a, d)-antimagic if and only if m = n or m = n−1. They conjecture that for n odd and at least 3, Kn,n+2 is ((n + 1)(n2 −1)/2, n + 1)-antimagic and they have obtained several results about (a, d)-antimagic labelings of caterpillars.
the electronic journal of combinatorics (2023), #DS6 236 In Lozano, Mora, and Seara prove that any caterpillar of order n is (⌊(n − 1)/2, ⌋−2)-antimagic. Furthermore, if C is a caterpillar with a spine of order s, they prove that when C has at least ⌊(3s + 1)/2⌋leaves or ⌊(s −1)/2⌋consecutive vertices of degree at most 2 at one end of a longest path, then C is antimagic. As a consequence of a result of Wong and Zhu , they also prove that if p is a prime number, any caterpillar with a spine of order p, p −1 or p −2 is 1-antimagic.
In Vilfred and Florida proved the following: the one-sided infinite path is (1, 2)-antimagic; P2n is not (a, d)-antimagic for any a and d; P2n+1 is (a, d)-antimagic if and only if (a, d) = (n, 1); C2n+1 has an (n + 2, 1)-antimagic labeling; and that a 2-regular graph G is (a, d)-antimagic if and only if |V (G)| = 2n + 1 and (a, d) = (n + 2, 1). They also prove that for a graph with an (a, d)-antimagic labeling, q edges, minimum degree δ and maximum degree ∆, the vertex labels lie between δ(δ + 1)/2 and ∆(2q −∆+ 1)/2.
Chelvam, Rilwan, and Kalaimurugan proved that Cayley digraph of any finite group admits a super vertex (a, d)-antimagic labeling depending on d and the size of the generating set. They provide algorithms for constructing the labelings. Thirusangu and Bhrathiraja proved the existence of super vertex (a, d)- antimagic labeling and vertex magic total labeling for Cayley digraphs arising from the groups Zpn ⊕Zpn and Zm ⊕Zn.
Irfan and Semaničová-Feňovčiková provide some classes of graphs that admit a labeling that is simultaneously a super edge-magic total and a super vertex-antimagic total and give some results for fans, sun graphs, caterpillars, and prisms.
For n > 1 and distinct odd integers x, y and z in [1,n −1] Javaid, Ismail, and Salman define the chordal ring of order n, CRn(x, y, z), as the graph with vertex set Zn, the additive group of integers modulo n, and edges (i, i+x), (i, i+y), (i, i+z) for all even i. They prove that CRn(1, 3, 7) and CRn(1, 5, n−1) have (a, d)-antimagic labelings when n ≡0 mod 4 and conjecture that for an odd integer ∆, 3 ≤∆≤n −3, n ≡0 mod 4, CRn((1, ∆, n −1) has an ((7n + 8)/4, 1)-antimagic labeling.
For an arbitrary set of distances D ⊆{0, 1, . . . , diam(G)}, a D-weight of a vertex x in a graph G under a vertex labeling f : V →{1, 2, . . . , v} is defined as wD(x) = P y∈ND(x) f(y), where ND(x) = {y ∈V |d(x, y) ∈D}.
A graph G is said to be D-distance magic if all vertices have the same D-vertex-weight, it is said to be D-distance antimagic indexD-distance antimagic if all vertices have distinct D-vertex-weights, and it is called (a, d) −D-distance antimagic if the D-vertex-weights constitute an arithmetic progression with difference d and starting value a. In Simanjuntak and Wijaya gave some necessary conditions for the existence of D-distance antimagic graphs and conjectured that those conditions are sufficient. They also gave {1}-distance antimagic labelings for cycles, suns, prisms, complete graphs, wheels, fans, and friendship graphs.
Arumugam and Kamatchi characterized (a, d)-distance antimagic cycles and (a, d)-distance antimagic labelings for paths and prisms. In and Fronček proved that disjoint copies of the Cartesian product of two complete graphs and its complement are (a, 2)-distance antimagic and (a, 1)-distance antimagic. He also proved that disjoint copies of the hypercube Q3 is (a, 1)-distance antimagic. In Handa, Godinho and Singh investigate the existence of distance antimagic labeling of ladders.
the electronic journal of combinatorics (2023), #DS6 237 In Vilfred and Florida call a graph G = (V, E) odd antimagic if there exist a bijection f : E →{1, 3, 5, . . . , 2|E| −1} such that the induced mapping gf : V →N, defined by gf(v) = P{f(uv)| uv ∈E(G)}, is injective and odd (a, d)-antimagic if there exist positive integers a, d and a bijection f : E →{1, 3, 5, . . . , 2|E| −1} such that the induced mapping gf : V →N, defined by gf(v) = P{f(uv)| uv ∈E(G)}, is injective and gf(V ) = {a, a + d, a + 2d, . . . , a + (|V | −1)d}.
Although every (a, d)-antimagic graph is antimagic, C4 has an antimagic labeling but does not have an (a, d)-antimagic labeling. They prove: P2n+1 is not odd (a, d)-antimagic for any a and d; C2n+1 has an odd (2n+2, 2)-antimagic labeling; if a 2-regular graph G has an odd (a, d)-antimagic labeling, then |V (G)| = 2n + 1 and (a, d) = (2n + 2, 2); C2n is odd magic; and an odd magic graph with at least three vertices, minimum degree δ, maximum degree ∆, and q ≥2 edges has all its vertex labels between δ2 and ∆(2q −∆).
Combining the notions of 1-vertex-magic vertex labelings and antimagic labelings Swaminathan and Jeyanthi introduced a new labeling as follows.
For a graph with p vertices a 1-1 mapping from the vertices to {1, 2, . . . , p} is called an (a, d)-1-vertex-antimagic vertex labeling if the sums of the labels of the vertices adjacent to each vertex taken over all vertices form the set {a, a+d, a+2d, . . . , a+(p−1)d}. They give some basic properties of such labelings and provide some results for some classes of regular graphs.
For a graph G = (V, E), a bijection g from V (G)∪E(G) into {1, 2, . . . , |V (G)|+|E(G)|} is called a (a, d)-edge-antimagic graceful labeling of G if the edge-weights w(xy) = |g(x)+ g(y) −g(xy)|, xy ∈E(G), form an arithmetic progression starting from a and having a common difference d. An (a, d)-edge-antimagic graceful labeling is called super (a, d)-edge-antimagic graceful if g(V (G)) = {1, 2, . . . , |V (G)|}. Marimuthu and Krishnaveni proved mCn has a super (0, 1)-edge-antimagic graceful labeling for every m ≥2 and n ≥3; and mKn and MPn have a super (a, 1)-edge-antimagic graceful labeling for every m ≥2 and n ≥2.
In Ahmed, Semaničová-Feňovčíková, Bača, Baskar Babujee, and Shobana in-troduced the notion of graceful antimagic graphs as follows. A graceful labeling that is simultaneously antimagic (that is, the sums of labels of all edges incident to a given vertex are pairwise distinct for different vertices) is said to be graceful antimagic . They obtain results about paths, stars, double stars, cycles, complete graphs, complete graphs with deleted edges, and complete bipartite graphs. They include many conjectures and open problems.
In Arumugam, Premalatha, Bacǎ, and Semaničová-Feňovčíková introduced a new graph coloring parameter as follows. Let f be a local antimagic labeling of a connected graph G. The assignment of w(u) to u for each vertex u of G induces naturally a proper vertex coloring of G, called a of G. The , denoted χla(G), is the minimum number of colors taken over all local antimagic colorings of G. Arumugam, Lee, Premalatha, Wang proved that for m > 1, χla(C3⊙Om) = 3m+3, where Om is the null graph on m verticies, and for n > 1, χla(Kn⊙K1) = 2n−1. In Lau and Nalliah correct a mistake in for the lower bounds of the local antimagic chromatic number of the corona product of friendship and fan graphs with null graph and obtain a sharp lower bound that gives the exact local antimagic chromatic number of the corona product of friendship and null the electronic journal of combinatorics (2023), #DS6 238 graph. In Lau, Ng, and Shiu give counterexamples to the lower bound of χla(G∨O2) that was obtained in . A sharp lower bound of χla(G ∨On) and sufficient conditions for the given lower bound to be attained are obtained. Moreover, they improve a theorem and solve a problem stated in and determine the local antimagic chromatic number of complete bipartite graphs. In Lau, Shiu, and Ng obtain a sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. They also determine the exact value of the local antimagic chromatic number of many families of graphs with cut-vertices (possibly given by pendant edges). In Lau, Shiu, and Ng provide sharp upper and lower bounds of χla(G) for G with pendant vertices, and give sufficient conditions for equality. They show that there are infinitely many graphs with k ≥χ(G) −1 pendant vertices and χla(G) = k + 1. They conjecture that every tree Tk, other than certain caterpillars, spiders and lobsters, with k pendant vertices has χla(Tk) = k + 1. Bača, Semaničová-Feňovčíková, Lai, and Wang verified a conjecture by Arumugam, Lee, Premalatha, and Wang that for every tree T the local antimagic chromatic number satisfies l + 1 ≤χla(T) ≤l + 2, where l is the number of leaves of T, by determining the exact value for the local antimagic chromatic number of all complete full t-ary trees is l + 1 for odd t. In Priyadharshini and Nalliah new determined the graphs whose local distance antimagic chromatic number is 2. In Adawiyah, Makhfudloh, and Prihandini obtained results about the local (a, d)-antimagic chromatic number of sunflower graphs and umbrella graphs.
Lau, Shiu, and Ng obtained sharp upper and lower bounds of χla(G) for G new with pendant vertices, and gave sufficient conditions for the bounds to equal. They also showed that for k ≥1, there are infinitely many graphs with k ≥χ(G) −1 pendant vertices and χla(G) = k + 1. In Lau, Premalatha, Shiu, and Nalliah determined new the local antimagic chromatic numbers of the join product G ∨H, where G is a circulant graph and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.
Lau, Shiu, and Soo showed that a d-leg spider graph has d + 1 ≤χla ≤d + 2; new each 3-leg spider has χla = 4 if not all legs are of odd length; and that there are no 3-leg spiders with all odd leg lengths and χla = 5. These results provide partial solutions to the characterization of k-pendant trees T with χla(T) = k + 1 or k + 2. They also obtained many sufficient conditions such that both the values are attainable. They conjectured that almost all d-leg spiders of size q that satisfy d(d + 1) ≤2(2q −1) when each leg length at least 2 has χla = d + 1.
In Lau and Shiu showed the existence of non-complete regular graphs with new arbitrarily large order, regularity, and local antimagic chromatic numbers.
They also determended the local antimagic chromatic number of regular graphs of order at most 8. In for graphs G and H Lau and Shiu denote the lexicographic product of new graphs G and H by G[H]. They obtained sharp upper bound of χla(G[On]) n ≥3), the lexicographic product of graph G with the null graph. and determined the local antimagic chromatic number of infinitely many (connected and disconnected) regular graphs that partially support the conjecture of the existence of an r-regular graph G of order p such that (i) χla(G) = χ(G) = k, and (ii) χla(G) = χ(G)+1 = k for each possible r, p, k. They the electronic journal of combinatorics (2023), #DS6 239 further provided sufficient conditions for even regular bipartite and tripartite graphs G to have χla(G) = 3.
In Lau, Shiu, Premalatha, Zhang and Nalliah obtained sufficient conditions for new χla(G[H]) ≤χla(G)χla(H). They also gave examples of G and H such that χla(G[H]) = χ(G)χ(H), where χ(G) is the chromatic number of G. Premalatha, Lau, Arumugam, and Shiu gave sufficient conditions under which χla(G ∨H), where H is either a cycle new or the null graph, satisfies a sharp upper bound. They further determined the value of χla(G∨H) for many wheel-related graph G. Lau, Li, and Shiu showed the existence new of infinitely many bipartite and tripartite graphs with local antimagic chromatic number 2 or 3. They also investigatea the local antimagic chromatic number of some circulant graphs.
For a connected graph G with q edges a bijection f : E →{1, 2, . . . , q} is called a local antimagic labeling if for any two adjacent vertices u and v, w(u) ̸= w(v), where w(u) = P e∈E(u) f(e), and E(u) is the set of edges incident to u. In Arumugam, Premalatha, Bača, and Semaničová-Feňovčíková proved several basic results on this new parameter and conjectured that any connected graph other than K2 admits a local antimagic labeling.
This conjecture was proved by Haslegrave using the probabilistic method, proves that the local antimagic conjecture is true. Lau proved that every graph admits a local antimagic total labeling. For any graph G, the graph H = G ∨On, n ≥1, is defined by V (H) = V (G) ∪{vi : 1 ≤i ≤n} and E(H) = E(G) ∪{uvi : u ∈V (G)}.
In [238, Theorem 2.16], it was claimed that for any G with order m ≥4, χla(G) + 1 ≤χla(G ∨O2) ≤ ( χla(G) + 1 if m is even, χla(G) + 2 if m is odd.
Lau and Shiu determined the local antimagic total chromatic number of graphs that are the amalgamation of complete graphs.
In Lau, Schaffer, and Shiu proved every graph is local antimagic total and new provided sharp upper bound for χlat(G), the local antimagic total chromatic number of G.
They also determined χlat(G), where G is a complete bipartite graph, a path, or the Cartesian product of two cycles, and χlat(G ∨K1) and χlat(G) for a class of 2-regular graphs G. In Lau and Shiu determined the local antimagic total chromatic new number the amalgamation of complete graphs. They also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of K1 and amalgamation of complete graphs under various conditions.
The number of distinct induced vertex labels under a local antimagic labeling f is denoted by c(f), and is called the color number of f. The local antimagic chromatic number of G, denoted by χla(G), is min{c(f) : f is a local antimagic labeling of G}. In , Haslegrave proved that the local antimagic chromatic number is well-defined for every connected graph other than K2. Thus, for every connected graph G ̸= K2, χla(G) ≥χ(G), the chromatic number of G.
In Lau, Shiu, and Ng provided several sufficient conditions for χla(H) ≤χla(G), where H is obtained from G with a certain edge deleted or added. The further determined the exact value of the local antimagic chromatic number of many cycle-related join graphs.
the electronic journal of combinatorics (2023), #DS6 240 Let G(V, E) be a simple graph and f be a bijection f : V ∪E →{1, 2, . . . , |V | + |E|} where f(V ) = {1, 2, . . . , |V |}. For a vertex x ∈V , define the weight of x, w(x), as the sum of labels of all edges incident with x and the vertex label itself. Such an f is called a super vertex local antimagic total labeling (SLAT) if for every two adjacent vertices their weights are different. The super vertex local antimagic total chromatic number χslat(G) is the minimum number of colors taken over all colorings induced by super vertex local antimagic total labelings of G. Hadiputra, Sugeng, Silaban, Maryati, and Fronček classify all trees T that have χslat(T) = 2, present a class of trees that have χslat(T) = 3, and show that for any positive integer n ≥2 there is a tree T with χslat(T) = n.
In Table 13 we use the abbreviation (a, d)-A to mean that the graph has an (a, d)-antimagic labeling. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The table was prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2008.
the electronic journal of combinatorics (2023), #DS6 241 Table 13: Summary of (a, d)-Antimagic Labelings Graph Labeling Notes P2n not (a, d)-A P2n+1 iff (n, 1)-A C2n not (a, d)-A C2n+1 (n + 2, 1)-A stars not (a, d)-A C(k) 3 , C(k) 4 not (a, d)-A K3,3 not (a, d)-A K4 not (a, d)-A Petersen graph not (a, d)-A Wn not (a, d)-A n > 3 antiprism on 2n (6n + 3, 2)-A n ≥3, n ̸≡2 (mod 4) vertices (see §6.2) (4n + 4, 4)-A n ≥3, n ̸≡2 (mod 4) (2n + 5, 6)-A?
n ≥4 (6n + 3, 2)-A?
n ≥6, n ̸≡2 (mod 4) (4n + 4, 4)-A?
n ≥6, n ̸≡2 (mod 4) Hershel graph (see ) not (a, d)-A , parachutes Pg,p (see §6.2) (a, d)-A for certain classes , prisms Cn × P2 ((7n + 4)/2, 1)-A n ≥3, n even , ((5n + 5)/2, 2)-A n ≥3, n odd , ((3n + 6)/2, 3)-A n ≥3, n even ((n + 7)/2, 4)-A?
n ≥7, , generalized Petersen ((3n + 6)/2, 3)-A n ≥8, n ≡0 (mod 4) graph P(n, 2) the electronic journal of combinatorics (2023), #DS6 242 6.3 (a, d)-Antimagic Total Labelings Bača, Bertault, MacDougall, Miller, Simanjuntak, and Slamin introduced the notion of a (a, d)-vertex-antimagic total labeling in 2000.
For a graph G(V, E), an injective mapping f from V ∪E to the set {1, 2, . . . , |V | + |E|} is a (a, d)-vertex-antimagic total labeling if the set {f(v) + P f(vu)} where the sum is over all vertices u adjacent to v for all v in G is {a, a+d, a+2d, . . . , a+(|V |−1)d}. In the case where the vertex labels are 1,2, …, |V |, (a, d)-vertex-antimagic total labeling is called a super (a, d)-vertex-antimagic total labeling. Among their results are: every super-magic graph has an (a, 1)-vertex-antimagic total labeling; every (a, d)-antimagic graph G(V, E) is (a+|E|+1, d+1)-vertex-antimagic total; and, for d > 1, every (a, d)-antimagic graph G(V, E) is (a + |V | + |E|, d −1)-vertex-antimagic total. They also show that paths and cycles have (a, d)-vertex-antimagic total labelings for a wide variety of a and d. In Bača et al. use their results in to obtain numerous (a, d)-vertex-antimagic total labelings for prisms, and generalized Petersen graphs (see §2.7 for the definition). (See also and for more results on generalized Petersen graphs.) Sugeng, Miller, Lin, and Bača prove: Cn has a super (a, d)-vertex-antimagic total labeling if and only if d = 0 or 2 and n is odd, or d = 1; Pn has a super (a, d)-vertex-antimagic total labeling if and only if d = 2 and n ≥3 is odd, or d = 3 and n ≥3; no even order tree has a super (a, 1)-vertex antimagic total labeling; no cycle with at least one tail and an even number of vertices has a super (a, 1)-vertex-antimagic labeling; and the star Sn, n ≥3, has no super (a, d)-super antimagic labeling. As open problems they ask whether Kn,n has a super (a, d)-vertex-antimagic total labeling and the generalized Petersen graph has a super (a, d)-vertex-antimagic total labeling for specific values a, d, and n. In Raheem proved that various subclasses of stars admit super (a, d)-edge antimagic total labelings for d = 1, 2, and 3. Lin, Miller, Simanjuntak, and Slamin have shown that for n > 20, Wn has no (a, d)-vertex-antimagic total labeling. Tezer and Cahit proved that neither Pn nor Cn has (a, d)-vertex-antimagic total labelings for a ≥3 and d ≥6. Kovář has shown that every 2r-regular graph with n vertices has an (s, 1)-vertex antimagic total labeling for s ∈{(rn + 1)(r + 1) + tn | t = 0, 1, . . . , r}.
Dafik, Slamin, Romdhani, and Arianti studied the super (a, d)-antimagicness of generalized flower and disk brake graphs. Dafik, Slamin, Eka, and Sya’diyah proved that triangular books and diamond ladder graphs admit a super (a, d)-edge-antimagic total labeling of graph for d ∈{0, 1, 2}.
Several papers have been written about vertex-antimagic total labeling of graphs that are the disjoint union of suns. The sun graph Sn is Cn ⊙K1. Rahim and Sugeng proved that Sn1 ∪Sn2 ∪· · · ∪Snt is (a, 0)-vertex-antimagic total (or vertex magic total).
Parestu, Silaban, and Sugeng and proved Sn1 ∪Sn2 ∪· · · ∪Snt is (a, d)-vertex-antimagic total for d = 1, 2, 3, 4, and 6 and particular values of a.
In Rahim, Ali, Kashif, and Javaid provide (a, d)-vertex antimagic total labelings of disjoint unions of cycles, sun graphs, and disjoint unions of sun graphs. In Enomoto et al. proposed the conjecture that every tree is a super (a, 0)-edge-antimagic total graph.
Javaid gave (a, d)-edge-antimagic total labelings for certain subclasses of subdivided the electronic journal of combinatorics (2023), #DS6 243 stars. Javaid gave a super (a, d)-edge-antimagic total labeling for the subdivided star T(n, n, n+4, n+4, n5, n6, . . . , nr) for d = 0, 1, 2, where np = 2p−4(n+3)+1, 5 ≤p ≤r and n ≥3 is odd.
In Ngurah, Baskova, and Simanjuntak provide (a, d)-vertex-antimagic total la-belings for the generalized Petersen graphs P(n, m) for the cases: n ≥3, 1 ≤m ≤ ⌊(n −1)/2⌋, (a, d) = (8n + 3, 2); odd n ≥5, m = 2, (a, d) = ((15n + 5)/2, 1); odd n ≥5, m = 2, (a, d) = ((21n + 5)/2, 1); odd n ≥7, m = 3, (a, d) = ((15n + 5)/2, 1); odd n ≥7, m = 3, (a, d) = ((21n + 5)/2, 1); odd n ≥9, m = 4, (a, d) = ((15n + 5)/2, 1); and (a, d) = ((21n + 5)/2, 1). They conjecture that for n odd and 1 ≤m ≤⌊(m −1)/2⌋, P(n, m) has an ((21n + 5)/2, 1)-vertex-antimagic labeling. In Sugeng and Silaban show: the disjoint union of any number of odd cycles of orders n1, n2, . . . , nt, each at least 5, has a super (3(n1 + n2 + · · · + nt) + 2, 1)-vertex-antimagic total labeling; for any odd positive integer t, the disjoint union of t copies of the generalized Petersen graph P(n, 1) has a super (10t + 2)n −⌊n/2⌋+ 2, 1)-vertex-antimagic total labeling; and for any odd positive integers t and n (n ≥3), the disjoint union of t copies of the generalized Petersen graph P(n, 2) has a super (21tn + 5)/2, 1)-vertex-antimagic total labeling. Ahmad, Ali, Bača, Kovar, and Semaničová-Feňovčíková, investigated the vertex-antimagicness of reg-ular graphs and the existence of (super) (a, d)-vertex antimagic total labelings for regular graphs in general.
Ali, Bača, Lin, and Semaničová-Feňovčiková investigated super-(a, d)-vertex an-timagic total labelings of disjoint unions of regular graphs. Among their results are: if m and (m −1)(r + 1)/2 are positive integers and G is an r-regular graph that admits a super-vertex magic total labeling, then mG has a super-(a, 2)-vertex antimagic total labeling; if G has a 2-regular super-(a, 1)-vertex antimagic total labeling, then mG has a super-(m(a −2) + 2, 1), 1)-vertex antimagic total labeling; mCn has a super-(a, d)-vertex antimagic total labeling if and only if either d is 0 or 2 and m and n are odd and at least 3 or d = 1 and n ≥3; and if G is an even regular Hamilton graph, then mG has a super-(a, 1)-vertex antimagic total labeling for all positive integers m.
In Bača, A. Semaničová-Feňovčíková, Wang, and Zhang investigate the exis-tence of (a, 1)-vertex-antimagic edge labelings for disconnected 3-regular graphs. As an extension of (a, d)-vertex-antimagic edge labeling they also introduce the concept of (a, d)-vertex-antimagic edge deficiency for measuring how close a graph is away from being an (a, d)-antimagic graph. In Arumugam and Nalliah investigate the existence of a super (a, d)-edge-antimagic total labelings of disconnected graphs.
Ahmad, Ali, Bača, Kovář and Semaničová-Feňovčíková provided a technique that allows one to construct several (a, r)-vertex-antimagic edge labelings for any 2r-regular graph G of odd order provided the graph is Hamiltonian or has a 2-regular factor that has (b, 1)-vertex-antimagic edge labeling. A similar technique allows them to construct a super (a, d)-vertex-antimagic total labeling for any 2r-regular Hamiltonian graph of odd order with differences d = 1, 2, . . . , r and d = 2r + 2.
For n ≥2 Dafik, Setiawani, and Azizah define a shackle as a graph constructed from connected graphs G1, G2, . . . , Gn, all isomorphic to G, such that Gs and Gt are disjoint when |s −t| ≥2 and for every i = 1, 2, . . . , n −1, Gi and Gi+1 share exactly one the electronic journal of combinatorics (2023), #DS6 244 common vertex v. In a generalized shackle a common subgraph is shared by each Gi and Gi+1. Dafik, Setiawani, and Azizah prove that the generalized shackle of a fan of order four and five admits a super (a, d)-edge antimagic total labeling for d = 0, 1, 2.
Sugeng and Bong show how to construct super (a, d)-vertex antimagic total labelings for the circulant graphs Cn(1, 2, 3), for d = 0, 1, 2, 3, 4, 8. Thirusangu, Nagar, and Rajeswari show that certain Cayley digraphs of dihedral groups have (a, d)-vertex-magic total labelings.
For a simple graph H we say that G(V, E) admits an H-covering if every edge in E(G) belongs to a subgraph of G that is isomorphic to H. Inayah, Salman, and Siman-juntak define an (a, d)-H-antimagic total labeling of G as a bijective function ξ from V ∪E →{1, 2, . . . , |V | + |E|} such that for all subgraphs H′ isomorphic to H, the H-weights w(H′) = P v∈V (H′) ξ(v) + P e∈E(H′) ξ(e) constitute an arithmetic progression a, a + d, a + 2d, . . . , a + (t −1)d where a and d are positive integers and t is the number of subgraphs of G isomorphic to H. Such a labeling ξ is called a super (a, d)-H-antimagic total labeling, if ξ(V ) = {1, 2, . . . , |V |}. Inayah et al. study some basic properties of such labeling and give (a, d)-cycle-antimagic labelings of fans. Taimur, Numan, Mumtaz, and Semaničová-Feňovčíková proved that if a graph G is super cycle-antimagic then the subdivided graph of G also admits a super cycle-antimagic labeling and they showed that the subdivided wheel is super (a, d)-cycle-antimagic for wide range of values. Lau-rence and Kathiresan investigated super (a, d)-Pn-antimagic total labeling of stars.
In Prihandini, Dafik, Agustin, Alfarisi, Adawiyah, and Santoso investigated the ex-istence of super (a, d)-P2▷H−antimagic total labelings of the disjoint union graph Cn△H.
Baˇ ca, Jeyanthi, Selvagopal, Muthu Raja, and Semaničová-Feňovčíková proved the existence of super (a, d)-H-antimagic labelings of fan graphs and ladders for H isomorphic to a cycle.
In Semaničová-Feňovčíková, Bača, Lascsáková, Miller, and Ryan investigated the super (a, d)-Cn-antimagic total labelings of wheels and super (a, d)-Pn-antimagic to-tal labelings of cycles and paths. Ovais, Umar, Bača, and Semaničová-Feňovčíková proved that fans admits a super (a, d)-Ck-antimagic labeling for d = 1, 3, 2k −5, 2k − 1, 3k −1, k −7, k + 1, 3k −9.
They also prove that fans admits a super (a, d)-C3-antimagic labeling for d = 0, 1, 2, 3, 4, 5, 6, 8, and a super (a, d)-C4-antimagic labeling for d = 0, 1, 2, 3, 4, 5, 6, 7, 11. They propose an open problem to find a super (a, d)-Ck-antimagic labeling of fans for d ̸= 1, 3, k −7, k + 1, 2k −5, 2k −1, 3k −1, 3k −9. Bača, Miller, Ryan, and Semaničová-Feňovčíková study super (a, d)-H-antimagic labelings of a disjoint union of graphs for d = |E(H)| −|V (H)|.
For a vertex u of a graph G, Gu[Sn] is the graph obtained by identifying u with the center of Sn. Then for any vertex w of Sn G + e, e = uw is a subgraph of Gu[Sn].
Kathiresan and Laurence prove that the graph Gu[Sn] admits a super-(a, d)-(G+e)-antimagic total labeling if and only if d ∈{0, 1, 2, . . . , |V (G)| + |E(G)| + 2}. Moreover, they show that a caterpillar Sn1,n2,...,nk has a super-(a, 4n2)-Sn,n-antimagic total labeling for n1 = n2 = · · · = nk = n. In Rajkumar, Nalliah, Uma Maheswari provided a partial solution to the problem of characterizing the super (a, d) −G + e-antimagic total labeling of the graph Gu[Sn], where n ≥3 and 4 ≤d ≤p + q + 2 posed in 2015 by the electronic journal of combinatorics (2023), #DS6 245 Kathiresan and Laurence in .
Jeyanthi, Muthuraja, Semaničová-Feňovčíková, and Dharshikha proved proved that fans, triangular ladders, and middle graphs of cycles are super (a, d)-C3-antimagic for some values of a and d. They also proved that ladder are super (a, d)-C4-antimagic for 1 ≤d ≤8. Inayah, Simanjuntak, and Salman proved that there exists a super (a, d) −H-antimagic total labelings for shackles of a connected graph H. Nadzima and Martini determined (a, d)-H-antimagic total labeling for certain cases of Wn ⊙Pn with H as C3 ⊙Pn and Wn ⊙Cn with H as C3 ⊙Cn.
A graph G is said to have an (H1, H2, . . . , Hk)-covering if every edge in G belongs to at least one of the Hi’s. Susilowati, Sania, and Estuningsih investigated such antimagic labelings for the ladders Pn × P2 with Ct-coverings for t = 4, 6, and 8 for some value of d.
Simanjuntak, Bertault, and Miller define an (a, d)-edge-antimagic vertex labeling for a graph G(V, E) as an injective mapping f from V onto the set {1, 2, . . . , |V |} such that the set {f(u)+f(v)|uv ∈E} is {a, a+d, a+2d, . . . , a+(|E|−1)d}. (The equivalent notion of (a, d)-indexable labeling was defined by Hegde in 1989 in his Ph. D. thesis– see .) Similarly, Simanjuntak et al. define an (a, d)-edge-antimagic total labeling for a graph G(V, E) as an injective mapping f from V ∪E onto the set {1, 2, . . . , |V | + |E|} such that the set {f(v) + f(vu) + f(v)|uv ∈E} where v ranges over all of V is {a, a + d, a + 2d, . . . , a + (|V | −1)d}. Among their results are: C2n has no (a, d)-edge-antimagic vertex labeling; C2n+1 has a (n + 2, 1)-edge-antimagic vertex labeling and a (n + 3, 1)-edge-antimagic vertex labeling; P2n has a (n+2, 1)-edge-antimagic vertex labeling; Pn has a (3, 2)-edge-antimagic vertex labeling; Cn has (2n + 2, 1)- and (3n + 2, 1)-edge-antimagic total labelings; C2n has (4n + 2, 2)- and (4n + 3, 2)-edge-antimagic total labelings; C2n+1 has (3n + 4, 3)- and (3n + 5, 3)-edge-antimagic total labelings; P2n+1 has (3n + 4, 2)-, (3n + 4, 3)-, (2n + 4, 4)-, (5n + 4, 2)-, (3n + 5, 2)-, and (2n + 6, 4)-edge-antimagic total labelings; P2n has (6n, 1)- and (6n+2, 2)-edge-antimagic total labelings; and several parity conditions for (a, d)-edge-antimagic total labelings. They conjecture: C2n has a (2n+3, 4)-or a (2n + 4, 4)-edge-antimagic total labeling; C2n+1 has a (n + 4, 5)- or a (n + 5, 5)-edge-antimagic total labeling; paths have no (a, d)-edge-antimagic vertex labelings with d > 2; and cycles have no (a, d)-antimagic total labelings with d > 5. The first and last of these conjectures were proved by Zhenbin in and the last two were verified by Bača, Lin, Miller, and Simanjuntak who proved that a graph with v vertices and e edges that has an (a, d)-edge-antimagic vertex labeling must satisfy d(e−1) ≤2v−1−a ≤2v−4. As a consequence, they obtain: for every path there is no (a, d)-edge-antimagic vertex labeling with d > 2; for every cycle there is no (a, d)-edge-antimagic vertex labeling with d > 1; for Kn (n > 1) there is no (a, d)-edge-antimagic vertex labeling (the cases for n = 2 and n = 3 are handled individually); Kn,n (n > 3) has no (a, d)-edge-antimagic vertex labeling; for every wheel there is no (a, d)-edge-antimagic vertex labeling; for every generalized Petersen graph there is no (a, d)-edge-antimagic vertex labeling with d > 1. They also study the relationship between graphs with (a, d)-edge-antimagic labelings and magic and antimagic labelings. They conjecture that every tree has an (a, 1)-edge-antimagic total labeling.
the electronic journal of combinatorics (2023), #DS6 246 Bača and Barrientos prove that if a tree T has an α-labeling and {A, B} is the bipartition of the vertices of T, then T also admits an (a, 1)-edge-antimagic vertex labeling and it admits a (3, 2)-edge-antimagic vertex labeling if and only if ||A| −|B|| ≤1.
In Bača, Lin, Miller, and Simanjuntak prove: if Pn has an (a, d)-edge-antimagic total labeling, then d ≤6; Pn has (2n + 2, 1)-, (3n, 1)-, (n + 4, 3)-, and (2n + 2, 3)-edge-antimagic total labelings; P2n+1 has (3n + 4, 2)-,(5n + 4, 3)-, (2n + 4, 4)-, and (2n + 6, 4)-edge-antimagic total labelings; and P2n has (3n+3, 2)- and (5n+1, 2)-edge-antimagic total labelings. Ngurah proved P2n+1 has (4n + 4, 1)-, (6n + 5, 3)-,(4n + 4, 2)-,(4n + 5, 2)-edge-antimagic total labelings and C2n+1 has (4n + 4, 2)- and (4n + 5, 2)-edge-antimagic total labelings.
Silaban and Sugeng prove: Pn has (n + 4, 4)- and (6, 6)-edge-antimagic total labelings; if Cm ⊙Kn has an (a, d)-edge-antimagic total labeling, then d ≤5; Cm ⊙Kn has (a, d)-edge-antimagic total labelings for m > 3, n > 1 and d = 2 or 4; and Cm ⊙Kn has no (a, d)-edge-antimagic total labelings for m and d and n ≡1 mod 4. They conjecture that Pn (n ≥3) has (a, 5)-edge-antimagic total labelings. In Sugeng and Xie use adjacency methods to construct super edge magic graphs from (a, d)-edge-antimagic vertex graphs. Pushpam and Saibulla determined super (a, d)-edge antimagic total labelings for graphs derived from copies of generalized ladders, fans, gen-eralized prisms and web graphs. Liu, Aslam, Javaid, and Raheen compute bounds of the minimum and maximum edge-weights for super (a, d)-edge-antimagic labelings on a generalized class of subdivided caterpillars. They also investigate the existence of super (a, d)-edge-antimagic total labeling for the validation of the obtained bounds.
In Bača and Youssef used parity arguments to find a large number of conditions on p, q and d for which a graph with p vertices and q edges cannot have an (a, d)-edge-antimagic total labeling or vertex-antimagic total labeling. Bača and Youssef made the following connection between (a, d)-edge-antimagic vertex labelings and sequential labelings: if G is a connected graph other than a tree that has an (a, d)-edge-antimagic vertex labeling, then G + K1 has a sequential labeling.
In Sudarsana, Ismaimuza, Baskoro, and Assiyatun prove: for every n ≥2, Pn ∪ Pn+1 has a (6n + 1, 1)- and a (4n + 3, 3)-edge-antimagic total labeling, for every odd n ≥3, Pn ∪Pn+1 has a (6n, 1)- and a (5n + 1, 2)-edge-antimagic total labeling, for every n ≥2, nP2 ∪Pn has a (7n, 1)- and a (6n + 1, 2)-edge-antimagic total labeling. In the same authors show that Pn ∪Pn+1, nP2 ∪Pn (n ≥2), and nP2 ∪Pn+2 are super edge-magic total. They also show that under certain conditions one can construct new super edge-magic total graphs from existing ones by joining a particular vertex of the existing super edge-magic total graph to every vertex in a path or every vertex of a star and by joining one extra vertex to some vertices of the existing graph. Baskoro, Sudarsana, and Cholily also provide algorithms for constructing new super edge-magic total graphs from existing ones by adding pendent vertices to the existing graph. A corollary to one of their results is that the graph obtained by attaching a fixed number of pendent edges to each vertex of a path of even length is super edge-magic. Baskoro and Cholily show that the graphs obtained by attaching any numbers of pendent edges to a single vertex or a fix number of pendent edges to every vertex of the following graphs are super edge-magic total graphs: odd cycles, the generalized Petersen graphs P(n, 2) (n odd and the electronic journal of combinatorics (2023), #DS6 247 at least 5), and Cn × Pm (n odd, m ≥2).
Arumugam and Nalliah proved: the friendship graph C(n) 3 with n ≡0, 8 (mod 12) has no super (a, 2)-edge-antimagic total labeling; C(n) n with n ≡2 (mod 4) has no super (a, 2)-edge-antimagic total labeling; and the generalized friendship graph F2,p consisting of 2 cycles of various lengths, having a common vertex, and having order p where p ≥5, has a super (2p + 2, 1)-edge-antimagic total labeling if and only if p is odd.
An (a, d)-edge-antimagic total labeling of G(V, E) is called a super (a, d)-edge-antimagic total if the vertex labels are {1, 2, . . . , |V (G)|} and the edge labels are {|V (G)| + 1, |V (G)| + 2, . . . , |V (G)| + |E(G)|}. Bača, Baskoro, Simanjuntak, and Sug-eng prove the following: Cn has a super (a, d)-edge-antimagic total labeling if and only if either d is 0 or 2 and n is odd, or d = 1; for odd n ≥3 and m = 1 or 2, the gener-alized Petersen graph P(n, m) has a super (11n + 3)/2, 0)-edge-antimagic total labeling and a super ((5n + 5)/2, 2)-edge-antimagic total labeling; for odd n ≥3, P(n, (n −1)/2) has a super ((11n + 3)/2, 0)-edge-antimagic total labeling and a super ((5n + 5)/2, 2)-edge-antimagic total labeling. They also prove: if P(n, m), n ≥3, 1 ≤m ≤⌊(n −1)/2⌋ is super (a, d)-edge-antimagic total, then (a, d) = (4n + 2, 1) if n is even, and either (a, d) = ((11n + 3)/2, 0), or (a, d) = (4n + 2, 1), or (a, d) = ((5n + 5)/2, 2), if n is odd; and for odd n ≥3 and m = 1, 2, or (n −1)/2, P(n, m) has an (a, 0)-edge-antimagic total labeling and an (a, 2)-edge-antimagic total labeling.
(In a personal communica-tion MacDougall argues that “edge-magic” is a better term than “(a, 0)-edge-antimagic” for while the latter is technically correct, “antimagic” suggests different weights whereas “magic” emphasizes equal weights and that the edge-magic case is much more important, interesting, and fundamental rather than being just one subcase of equal value to all the others.) They conjecture that for odd n ≥9 and 3 ≤m ≤(n−3)/2, P(n, m) has a (a, 0)-edge-antimagic total labeling and an (a, 2)-edge-antimagic total labeling. Ngurah and Baskoro have shown that for odd n ≥3, P(n, 1) and P(n, 2) have ((5n + 5)/2, 2)-edge-antimagic total labelings and when n ≥3 and 1 ≤m < n/2, P(n, m) has a super (4n + 2, 1)-edge-antimagic total labeling. In Ngurah, Baskova, and Simanjuntak provide (a, d)-edge-antimagic total labelings for the generalized Petersen graphs P(n, m) for the cases m = 1 or 2, odd n ≥3, and (a, d) = ((9n + 5)/2, 2).
In Sudarsana, Baskoro, Uttunggadewa, and Ismaimuza show how to construct new larger super (a, d)-edge-antimagic-total graphs from existing smaller ones.
In Ngurah, Baskoro, and Simanjuntak prove that mCn (n ≥3) has an (a, d)-edge-antimagic total in the following cases: (a, d) = (5mn/2 + 2, 1) where m is even; (a, d) = (2mn+2, 2); (a, d) = ((3mn+5)/2, 3) for m and n odd; and (a, d) = ((mn+3), 4) for m and n odd; and mCn has a super (2mn + 2, 1)-edge-antimagic total labeling.
Bača and Barrientos have shown that mKn has a super (a, d)-edge-antimagic total labeling if and only if (i) d ∈{0, 2}, n ∈{2, 3} and m ≥3 is odd, or (ii) d = 1, n ≥2 and m ≥2, or (iii) d ∈{3, 5}, n = 2 and m ≥2, or (iv) d = 4, n = 2, and m ≥3 is odd. In Bača and Barrientos proved the following: if a graph with q edges and q + 1 vertices has an α-labeling, than it has an (a, 1)-edge-antimagic vertex labeling; a tree has a (3, 2)-edge-antimagic vertex labeling if and only if it has an α-labeling and the number of vertices in its two partite sets differ by at most 1; if a tree with at least two vertices the electronic journal of combinatorics (2023), #DS6 248 has a super (a, d)-edge-antimagic total labeling, then d is at most 3; if a graph has an (a, 1)-edge-antimagic vertex labeling, then it also has a super (a1, 0)-edge-antimagic total labeling and a super (a2, 2)-edge-antimagic total labeling.
Bača and Youssef proved the following: if G is a connected (a, d)-edge-antimagic vertex graph that is not a tree, then G+K1 is sequential; mCn has an (a, d)-edge-antimagic vertex labeling if and only if m and n are odd and d = 1; an odd degree (p, q)-graph G cannot have a (a, d)-edge-antimagic total labeling if p ≡2 (mod 4) and q ≡0 (mod 4), or p ≡0 (mod 4), q ≡2 (mod 4), and d is even; a (p, q)-graph G cannot have a super (a, d)-edge-antimagic total labeling if G has odd degree, p ≡2 (mod 4), q is even, and d is odd, or G has even degree, q ≡2 (mod 4), and d is even; Cn has a (2n + 2, 3)- and an (n+4, 3)-edge-antimagic total labeling; a (p, q)-graph is not super (a, d)-vertex-antimagic total if: p ≡2 (mod 4) and d is even; p ≡0 (mod 4), q ≡2 (mod 4), and d is odd; p ≡0 (mod 8) and q ≡2 (mod 4).
In Sudarsana, Ismaimuza, Baskoro, and Assiyatun prove: for every n ≥2, Pn ∪ Pn+1 has super (n + 4, 1)- and (2n + 6, 3)-edge antimagic total labelings; for every odd n ≥3, Pn ∪Pn+1 has super (4n + 5, 1)-,(3n + 6, 2)-, (4n + 3, 1)- and (3n + 4, 2)-edge antimagic total labelings; for every n ≥2, nP2 ∪Pn has super (6n +2, 1)- and (5n +3, 2)-edge antimagic total labelings; and for every n ≥1, nP2 ∪Pn+2 has super (6n+6, 1)- and (5n + 6, 2)-edge antimagic total labelings. They pose a number of open problems about constructing (a, d)-edge antimagic labelings and super (a, d)-edge antimagic labelings for the graphs Pn ∪Pn+1, nP2 ∪Pn, and nP2 ∪Pn+2 for specific values of d.
Dafik, Miller, Ryan, and Bača investigated the super edge-antimagicness of the disconnected graph mCn and mPn. For the first case they prove that mCn, m ≥2, has a super (a, d)-edge-antimagic total labeling if and only if either d is 0 or 2 and m and n are odd and at least 3, or d = 1, m ≥2, and n ≥3. For the case of the disjoint union of paths they determine all feasible values for m, n and d for mPn to have a super (a, d)-edge-antimagic total labeling except when m is even and at least 2, n ≥2, and d is 0 or 2. In Dafik, Miller, Ryan, and Bača obtain a number of results about super edge-antimagicness of the disjoint union of two stars and state three open problems. Nalliah and Arumugam proved that K1,6 ∪K1,5 does not have such a labeling and prove that some special cases of K1,n+1 ∪K1,n do have them.
Sudarsana, Hendra, Adiwijaya, and Setyawan show that the t-joint copies of wheel Wn have a super edge antimagic ((2n+2)t+2, 1)-total labeling for n ≥4 and t ≥2.
In Bača, Lascsáková, and Semaničová investigated the connection between graphs with α-labelings and graphs with super (a, d)-edge-antimagic total labelings. Among their results are: If G is a graph with n vertices and n −1 edges (n ≥3) and G has an α-labeling, then mG is super (a, d)-edge-antimagic total if either d is 0 or 2 and m is odd, or d = 1 and n is even; if G has an α-labeling and has n vertices and n −1 edges with vertex bipartition sets V1 and V2 where |V1| and |V2| differ by at most 1, then mG is super (a, d)–edge-antimagic total for d = 1 and d = 3. In the same paper Bača et al. prove: caterpillars with odd order at least 3 have super (a, 1)-edge-antimagic total labelings; if G is a caterpillar of odd order at least 3 and G has a super (a, 1)-edge-antimagic total labeling, then mG has a super (b, 1)-edge-antimagic total labeling for some b that is a the electronic journal of combinatorics (2023), #DS6 249 function of a and m.
In Dafik, Miller, Ryan, and Bača investigated the existence of antimagic labelings of disjoint unions of s-partite graphs. They proved: if s ≡0 or 1 (mod 4), s ≥4, m ≥ 2, n ≥1 or mn is even , m ≥2, n ≥1, s ≥4, then the complete s-partite graph mKn,n,...,n has no super (a, 0)-edge-antimagic total labeling; if m ≥2 and n ≥1, then mKn,n,n,n has no super (a, 2)-antimagic total labeling; and for m ≥2 and n ≥1, mKn,n,n,n has an (8mn + 2, 1)-edge-antimagic total labeling. They conjecture that for m ≥2, n ≥1 and s ≥5, the complete s-partite graph mKn,n,...,n has a super (a, 1)-antimagic total labeling.
In Bača, Muntaner-Batle, Semaničová-Feňovčiková, and Shafiq investigate super (a, d)-edge-antimagic total labelings of disconnected graphs. Among their results are: If G is a (super) (a, 2)-edge-antimagic total labeling and m is odd, then mG has a (super) (a′, 2)-edge-antimagic-total labeling where a′ = m(a −3) + (m + 1)/2 + 2; and if d a positive even integer and k a positive odd integer, G is a graph with all of its vertices having odd degree, and the order and size of G have opposite parity, then 2kG has no (a, d)-edge-antimagic total labeling. Bača and Brankovic have obtained a number of results about the existence of super (a, d)-edge-antimagic totaling of disjoint unions of the form mKn,n. In Bača, Dafik, Miller, and Ryan provide (a, d)-edge-antimagic vertex labelings and super (a, d)-edge-antimagic total labelings for a variety of disjoint unions of caterpillars. Bača and Youssef proved that mCn has an (a, d)-edge-antimagic vertex labeling if and only if m and n are odd and d = 1. Bača, Dafik, Miller, and Ryan constructed super (a, d)-edge-antimagic total labeling for graphs of the form m(Cn ⊙Ks) and mPn ∪kCn while Dafik, Miller, Ryan, and Bača do the same for graphs of the form mKn,n,n and K1,m ∪2sK1,n. Both papers provide a number of open problems. In Bača, Lin, and Muntaner-Batle provide super (a, d)-edge-antimagic total labeling of forests in which every component is a specific kind of tree. In Bača, Kov́ǎr, Semaničová-Feňovčiková, and Shafiq prove that every even regular graph and every odd regular graph with a 1-factor are super (a, 1)-edge-antimagic total and provide some constructions of non-regular super (a, 1)-edge-antimagic total graphs.
Bača, Lin, and Semaničová-Feňovčiková show: the disjoint union of m graphs with super (a, 1)-edge antimagic total labelings have super (m(a −2) + 2, 1)-edge antimagic total labelings; the disjoint union of m graphs with super (a, 3)-edge antimagic total labelings have super (m(a −3) + 3, 3)-edge antimagic total labelings; if G has a (a, 1)-edge antimagic total labelings then mG has an (b, 1)-edge antimagic total labeling for some b; and if G has a (a, 3)-edge antimagic total labelings then mG has an (b, 3)-edge antimagic total labeling for some b.
Bača, Miller, Ryan, and Semaničová-Feňovčíková prove that if G admits a (super) (a, d)-H-antimagic labeling, where d = |E(H)| −|V (H)|, then mG admits a (super) (b, d)-H-antimagic labelling. By considering special H-coverings of a given H-antimagic graph G they derive many corollaries.
In Semaničová-Feňovčíková, Bača, and Lascsáková provide two constructions of (super) H-antimagic graphs obtained from smaller (super) H-antimagic graphs. Dafik, Slamin, Tana, Semaničová-Feňovčíková, and Bača show a connection between a constructions of H-antimagic labelings of graph and edge-antimagic total labelings and describe how to obtain the H-antimagic the electronic journal of combinatorics (2023), #DS6 250 graph using smaller edge-antimagic graph. Bača, Semaničová-Feňovčikovǎ, Umar, and Welyyanti gave sufficient conditions for G1 × G2 to admit an H-supermagic or a su-per (a, d)-H-antimagic labeling but provide no examples of graphs that satisfy the given conditions.
For t ≥2 and n ≥4 the Harary graph, Ct p, is the graph obtained by joining every two vertices of Cp that are at distance t in Cp. In Rahim, Ali, Kashif, and Javaid provide super (a, d)-edge antimagic total labelings for disjoint unions of Harary graphs and disjoint unions of cycles. In Hussain, Ali, Rahim, and Baskoro construct various (a, d)-vertex-antimagic labelings for Harary graphs and disjoint unions of identical Harary graphs. For p odd and at least 5, Balbuena, Barker, Das, Lin, Miller, Ryan, Slamin, Sugeng, and Tkac give a super ((17p + 5)/2)-vertex-antimagic total labeling of Ct p.
MacDougall and Wallis have proved the following: Ct 4m+3, m ≥1, has a super (a, 0)-edge-antimagic total labeling for all possible values of t with a = 10m + 9 or 10m + 10; Ct 4m+1, m ≥3, has a super (a, 0)-edge-antimagic total labeling for all possible values except t = 5, 9, 4m −4, and 4m −8 with a = 10m + 4 and 10m + 5; Ct 4m+1, m ≥1, has a super (10m + 4, 0)-edge-antimagic total labeling for all t ≡1 (mod 4) except 4m −3; Ct 4m, m > 1, has a super (10m + 2, 0)-edge-antimagic total labeling for all t ≡2 (mod 4); Ct 4m+2, m > 1, has a super (10m + 7, 0)-edge-antimagic total labeling for all odd t other than 5 and for t = 2 or 6. In Hussain, Baskoro, and Ali prove the following: for any p ≥4 and for any t ≥2, Ct p admits a super (2p + 2, 1)-edge-antimagic total labeling; for n ≥4, k ≥2 and t ≥2, kCt n admits a super (2nk +2, 1)-edge-antimagic total labeling; and for p ≥5 and t ≥2, Ct p admits a super (8p + 3, 1)-vertex-antimagic total labeling, provided if p ̸= 2t.
Bača and Murugan have proved: if Ct n, n ≥4, 2 ≤t ≤n −2, is super (a, d)-edge-antimagic total, then d = 0, 1, or 2; for n = 2k + 1 ≥5, Ct n has a super (a, 0)-edge-antimagic total labeling for all possible values of t with a = 5k + 4 or 5k + 5; for n = 2k+1 ≥5, Ct n has a super (a, 2)-edge-antimagic total labeling for all possible values of t with a = 3k+3 or 3k+4; for n ≡0 (mod 4), Ct n has a super (5n/2+2, 0)-edge-antimagic total labeling and a super (3n/2+2, 0)-edge-antimagic total labeling for all t ≡2 (mod 4); for n = 10 and n ≡2 (mod 4), n ≥18, Ct n has a super (5n/2 + 2, 0)-edge-antimagic total labeling and a super (3n/2 + 2, 0)-edge-antimagic total labeling for all t ≡3 (mod 4) and for t = 2 and 6; for odd n ≥5, Ct n has a super (2n + 2, 1)-edge-antimagic total labeling for all possible values of t; for even n ≥6, Ct n has a super (2n + 2, 1)-edge-antimagic total labeling for all odd t ≥3; and for even n ≡0 (mod 4), n ≥4, Ct n has a super (2n+2, 1)-edge-antimagic total labeling for all t ≡2 (mod 4). They conjecture that there is a super (2n + 2, 1)-edge-antimagic total labeling of Ct n for n ≡0 (mod 4) and for t ≡0 (mod 4) and for n ≡2 (mod 4) and for t even.
In Bača, Lin, Miller, and Youssef prove: if the friendship C(n) 3 is super (a, d)-antimagic total, then d < 3; C(n) 3 has an (a, 1)-edge antimagic vertex labeling if and only if n = 1, 3, 4, 5, and 7; C(n) 3 has a super (a, d)-edge-antimagic total labelings for d = 0 and 2; C(n) 3 has a super (a, 1)-edge-antimagic total labeling; if a fan Fn (n ≥2) has a super (a, d)-edge-antimagic total labeling, then d < 3; Fn has a super (a, d)-edge-antimagic total labeling if 2 ≤n ≤6 and d = 0, 1 or 2; the wheel Wn has a super (a, d)-edge-antimagic the electronic journal of combinatorics (2023), #DS6 251 total labeling if and only if d = 1 and n ̸≡1 (mod 4); Kn, n ≥3, has a super (a, d)-edge-antimagic total labeling if and only if either d = 0 and n = 3, or d = 1 and n ≥3, or d = 2 and n = 3; and Kn,n has a super (a, d)-edge antimagic total labeling if and only if d = 1 and n ≥2.
Bača, Lin, and Muntaner-Batle have shown that if a tree with at least two vertices has a super (a, d)-edge-antimagic total labeling, then d is at most three and Pn, n ≥2, has a super (a, d)-edge-antimagic total labeling if and only if d = 0, 1, 2, or 3. They also characterize certain path-like graphs in a grid that have super(a, d)-edge-antimagic total labelings.
In Sugeng, Miller, and Bača prove that the ladder, Pn ×P2, is super (a, d)-edge-antimagic total if n is odd and d = 0, 1, or 2 and Pn × P2 is super (a, 1)-antimagic total if n is even. They conjecture that Pn × P2 is super (a, 0)- and (a, 2)-edge-antimagic when n is even. Sugeng, Miller, and Bača prove that Cm × P2 has a super (a, d)-edge-antimagic total labeling if and only if either d = 0, 1 or 2 and m is odd and at least 3, or d = 1 and m is even and at least 4. They conjecture that if m is even, m ≥4, n ≥3, and d = 0 or 2, then Cm × Pn has a super (a, d)-edge-antimagic total labeling. In M.-J. Lee studied super (a, 1)-edge-antimagic properties of m(P4 × Pn) for m, n ≥1 and m(Cn ⊙Kt) for n even and m, t ≥1. He also proved that for n ≥2 the graph P4 × Pn has a super (8n + 2, 1)-edge antimagic total labeling.
In and Azizu, Yulianti, Sy, Narwen proved that (Cm ×P2)⊙Kn (branched-prism) for odd m ≥3 and n ≥1 admits an edge magic and a super (a, d)-edge antimagic total labelings for odd m ≥3 and n ≥1.
Sugeng, Miller, and Bača define a variation of a ladder, Ln, as the graph ob-tained from Pn × P2 by joining each vertex ui of one path to the vertex vi+1 of the other path for i = 1, 2, . . . , n−1. They prove Ln, n ≥2, has a super (a, d)-edge-antimagic total labeling if and only if d = 0, 1, or 2.
In Dafik, Miller, and Ryan investigate the existence of super (a, d)-edge-antimagic total labelings of mKn,n,n and K1,m ∪2sK1,n. Among their results are: for d = 0 or 2, mKn,n,n has a super (a, d)-edge-antimagic total labeling if and only if n = 1 and m is odd and at least 3; K1,m ∪2sK1,n has a super (a, d)-edge-antimagic labeling for (a, d) = (4n + 5)s + 2m + 4, 0), ((2n + 5)s + m + 5, 2), ((3n + 5)s + (3m + 9)/2, 1) and (5s + 7, 4).
In Bača, Bashir, and Semaničová showed that for n ≥4 and d = 0, 1, 2, 3, 4, 5, and 6 the antiprism An has a super d-antimagic labeling of type (1, 1, 1). The generalized antiprism An m is obtained from Cm×Pn by inserting the edges {vi,j+1, vi+1,j} for 1 ≤i ≤m and 1 ≤j ≤n −1 where the subscripts are taken modulo m. Sugeng et al. prove that An m, m ≥3, n ≥2, is super (a, d)-edge-antimagic total if and only if d = 1.
A toroidal polyhex (toroidal fullerene) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. Note that the torus is a closed surface that can carry a toroidal polyhex such that all its vertices have degree 3 and all faces of the embedding are hexagons. Bača and Shabbir proved the toroidal polyhex Hn m with mn hexagons, m, n ≥2, admits a super (a, d)-edge-antimagic total labeling if and only if d = 1 and a = 4mn + 2.
the electronic journal of combinatorics (2023), #DS6 252 Bača, Miller, Phanalasy, and A. Semaničová-Feňovčíková investigated the exis-tence of (super) 1-antimagic labelings of type (1, 1, 1) for disjoint union of plane graphs.
They prove that if a plane graph G(V, E, F) has a (super) 1-antimagic labeling h of type (1, 1, 1) such that h(zext) = |V (G)| + |E(G)| + |F(G)| where zext denotes the unique external face then, for every positive integer m, the graph mG also admits a (super) 1-antimagic labeling of type (1, 1, 1); and if a plane graph G(V, E, F) has 4-sided inner faces and h is a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = |V (G)| + |E(G)| + |F(G)| where d = 1, 3, 5, 7, 9 then, for every positive integer m, the graph mG also admits a (super) d-antimagic labeling of type (1, 1, 1). They also give a similar result about plane graphs with inner faces that are 3-sided.
Sugeng, Miller, Slamin, and Bača proved: the star Sn has a super (a, d)-antimagic total labeling if and only if either d = 0, 1 or 2, or d = 3 and n = 1 or 2; if a nontrivial caterpillar has a super (a, d)-edge-antimagic total labeling, then d ≤3; all caterpillars have super (a, 0)-, (a, 1)- and (a, 2)-edge-antimagic total labelings; all cater-pillars have a super (a, 1)-edge-antimagic total labeling; if m and n differ by at least 2 the double star Sm,n (that is, the graph obtained by joining the centers of K1,m and K1,n with an edge) has no (a, 3)-edge-antimagic total labeling.
Sugeng and Miller show how to manipulate adjacency matrices of graphs with (a, d)-edge-antimagic vertex labelings and super (a, d)-edge-antimagic total labelings to obtain new (a, d)-edge-antimagic vertex labelings and super (a, d)-edge-antimagic total labelings. Among their results are: every graph can be embedded in a connected (a, d)-edge-antimagic vertex graph; every (a, d)-edge-antimagic vertex graph has a proper (a, d)-edge-antimagic vertex subgraph; if a graph has a (a, 1)-edge-antimagic vertex labeling and an odd number of edges, then it has a super (a, 1)-edge-antimagic total labeling; every super edge magic total graph has an (a, 1)-edge-antimagic vertex labeling; and every graph can be embedded in a connected super (a, d)-edge-antimagic total graph.
Rahmawati, Sugeng, Silaban, Miller, and Bača construct new larger (a, d)-edge-antimagic vertex graphs from an existing (a, d)-edge-antimagic vertex graph using adjacency matrix for difference d = 1, 2. The results are extended for super (a, d)-edge-antimagic total graphs with differences d = 0, 1, 2, 3.
Ajitha, Arumugan, and Germina show that (p, p−1) graphs with α-labelings (see §3.1) and partite sets with sizes that differ by at most 1 have super (a, d)-edge antimagic total labelings for d = 0, 1, 2 and 3. They also show how to generate large classes of trees with super (a, d)-edge-antimagic total labelings from smaller graceful trees.
Bača, Lin, Miller, and Ryan define a Möbius grid, M m n , as the graph with vertex set {xi,j| i = 1, 2, . . . , m + 1, j = 1, 2, . . . , n} and edge set {xi,jxi,j+1| i = 1, 2, . . . , m + 1, j = 1, 2, . . . , n −1} ∪{xi,jxi+1,j| i = 1, 2, . . . , m, j = 1, 2, . . . , n} ∪{xi,nxm+2−i,1| i = 1, 2, . . . , m + 1}. They prove that for n ≥2 and m ≥4, M m n has no d-antimagic vertex labeling with d ≥5 and no d-antimagic-edge labeling with d ≥9.
Ali, Bača, and Bashir, investigated super (a, d)-vertex-antimagic total labelings of the disjoint unions of paths. They prove: mP2 has a super (a, d)-vertex-antimagic total labeling if and only if m is odd and d = 1; mP3, m > 1, has no super (a, 3)-vertex-antimagic total labeling; mP3 has a super (a, 2)-vertex-antimagic total labeling for m ≡1 the electronic journal of combinatorics (2023), #DS6 253 (mod 6); and mP4 has a super (a, 2)-vertex-antimagic total labeling for m ≡3 (mod 4).
Lee, Tsai, and Lin denote the subdivision of a star Sn obtained by inserting m vertices into every edge of the star Sn by Sn m. They proved that for n ≥3, the graph kSn m is super (a, d)-edge antimagic total for certain values. In Ichishima, López, Muntaner-Batle and Rius-Font proved that if G is tripartite and has a (super) (a, d)-edge antimagic total labeling, then nG (n ≥3) has a (super) (a, d)-edge antimagic total labeling for d = 1 and for d = 0, 2 when n is odd.
Let p, t1, t2, . . . , tk be integers such that 1 ≤t1 < t2 < · · · < tk < p. A Toeplitz graph, denoted by Tp⟨t1 . . . , tk⟩, is a graph with vertex set {v1, v2, . . . , vp} and edge set {vivj : |i −j| ∈{t1, t2, . . . , tk}}.
Bača, Bashir, Nadeem, and Shabbir give an upper bound on the difference d when a Toeplitz graph Tp⟨t1, t2, . . . tk⟩is super (a, d)-edge-antimagic total. They also construct a super (a, 1)-edge-antimagic total labeling for an arbitrary Toeplitz graph without isolated vertices and prove that the Toeplitz graph Tp⟨t1⟩admits a super (a, 3)-edge-antimagic total labeling. Moreover, when p and t1 satisfy certain conditions Tp⟨t1⟩also admits a super (a, d)-edge-antimagic total labeling for d = 0 and d = 2. When k = 2 they show the existence of a super (a, 2)-edge-antimagic total labeling for the Toeplitz graph Tp⟨t1, t1 + 1⟩.
In Amudha, Jayapriya, and Gowri provide an algorithmic encryption method that employs antimagic labelings of graphs.
Pandimadevi and Subbiah show the existence and nonexistence of (a, d)-vertex antimagic total labeling for several class of digraphs and show how to construct labelings for generalized de Bruijn digraphs.
In Getzimah and Palani define vertex antimagic total labeling, edge antimagic total labeling on Zp+q, and discuss these labelings for cycles, stars, complete bipartite graphs, the subdivision graphs of ladders, and combs. They also investigate totally (a, d)-edge antimagic graphs, totally super vertex graphs, edge antimagic graphs, and determine the bounds for the vertices and the edges under total labelings.
The book by Bača and Miller has a wealth of material and open problems on super edge-antimagic labelings. In Bača, Baskoro, Miller, Ryan, Simanjuntak, and Sugeng provide detailed survey of results on edge antimagic labelings and include many conjectures and open problems. In 2015 Nalliah published a list of open problems on super (a, d)-edge antimagic total labelings of graphs. In 2017 Brankovic, Jendrol’, Lin, Phanalasy, Ryan, Semaničová-Feňovčíková, Slamin, and Sugeng provided a survey of recent results on face-antimagic labelings. It was dedicated to the memory of Mirka Miller, who introduced the concept of face-antimagic labeling of plane graphs in 2003.
In Tables 14, 15, 16 and 17 we use the abbreviations (a, d)-VAT (a, d)-vertex-antimagic total labeling (a, d)-SVAT super (a, d)-vertex-antimagic total labeling (a, d)-EAT (a, d)-edge-antimagic total labeling (a, d)-SEAT super (a, d)-edge-antimagic total labeling the electronic journal of combinatorics (2023), #DS6 254 (a, d)-EAV (a, d)-edge-antimagic vertex labeling A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The tables were prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2008.
Table 14: Summary of (a, d)-Vertex-Antimagic Total and Super (a, d)-Vertex-Antimagic Total Labelings Graph Labeling Notes Pn (a, d)-VAT wide variety of a and d Pn (a, d)-SVAT iff d = 3, d = 2, n ≥3 odd or d = 3, n ≥3 Cn (a, d)-VAT wide variety of a and d Cn (a, d)-SVAT iff d = 0, 2 and n odd or d = 1 generalized Petersen (a, d)-VAT graph P(n, k) (a, 1)-VAT n ≥3, 1 ≤k ≤n/2 prisms Cn × P2 (a, d)-VAT antiprisms (a, d)-VAT Sn1 ∪. . . ∪Snt (a, d)-VAT d = 1, 2, 3, 4, 6 , Wn not (a, d)-VAT for n > 20 K1,n not (a, d)-SVAT n ≥3 the electronic journal of combinatorics (2023), #DS6 255 Table 15: Summary of (a, d)-Edge-Antimagic Total Labelings Graph Labeling Notes trees (a, 1)-EAT?
Pn not (a, d)-EAT d > 2 P2n (6n, 1)-EAT (6n + 2, 2)-EAT P2n+1 (3n + 4, 2)-EAT (3n + 4, 3)-EAT (2n + 4, 4)-EAT (5n + 4, 2)-EAT (3n + 5, 2)-EAT (2n + 6, 4)-EAT Cn (2n + 2, 1)-EAT (3n + 2, 1)-EAT not (a, d)-EAT d > 5 C2n (4n + 2, 2)-EAT (4n + 3, 2)-EAT (2n + 3, 4)-EAT?
(2n + 4, 4)-EAT?
C2n+1 (3n + 4, 3)-EAT (3n + 5, 3)-EAT (n + 4, 5)-EAT?
(n + 5, 5)-EAT?
Kn not (a, d)-EAT d > 5 Kn,n (a, d)-EAT iff d = 1, n ≥2 caterpillars (a, d)-EAT d ≤3 Wn not (a, d)-EAT d > 4 generalized Petersen not (a, d)-EAT d > 4 graph P(n, k) ((5n + 5)/2, 2)-EAT n ≥3 odd, k = 1, 2 super (4n + 2, 1)-EAT n ≥3, 1 ≤k ≤n/2 the electronic journal of combinatorics (2023), #DS6 256 Table 16: Summary of (a, d)-Edge-Antimagic Vertex Labelings Graph Labeling Notes Pn (3, 2)-EAV not (a, d)-EAV d > 2 P2n (n + 2, 1)-EAV Cn not (a, d)-EAV d > 1 C2n not (a, d)-EAV C2n+1 (n + 2, 1)-EAV (n + 3, 1)-EAV Kn not (a, d)-EAV for n > 1 Kn,n not (a, d)-EAV for n > 3 Wn not (a, d)-EAV C(n) 3 (friendship graph) (a, 1)-EAV iff n = 1, 3, 4, 5, 7 generalized Petersen not (a, d)-EAV d > 1 graph P(n, k) the electronic journal of combinatorics (2023), #DS6 257 Table 17: Summary of (a, d)-Super-Edge-Antimagic Total Labelings Graph Labeling Notes Cn ⊙K1 (a, d)-SEAT variety of cases , Pn × P2 (ladders) (a, d)-SEAT n odd, d ≤2 n even, d = 1 (a, d)-SEAT?
d = 0, 2, n even Cn × P2 (a, d)-SEAT iff d ≤3 n odd or d = 1, n ≥4 even Cm × Pn (a, d)-SEAT?
m ≥4 even, n ≥3, d = 0, 2 caterpillars (a, 1)-SEAT C(n) 3 (friendship graphs) (a, d)-SEAT d = 0, 1, 2 Fn (n ≥2) (fans) (a, d) SEAT only if d < 3 (a, d)-SEAT 2 ≤n ≤6, d = 0, 1, 2 Wn (a, d)-SEAT iff d = 1, n ̸≡1 (mod 4) Kn (n ≥3) (a, d) SEAT iff d = 0, n = 3 d = 1, n ≥3 d = 2, n = 3 trees (a, d)-SEAT only if d ≤3 Pn (n > 1) (a, d)-SEAT iff d ≤3 mKn (a, d)-SEAT iff d ∈{0, 2}, n ∈{2, 3}, m ≥3 odd d = 1, m, n ≥2 d = 3 or 5,n = 2, m ≥2 d = 4, n = 2, m ≥3 odd Cn (a, d)-SEAT iff d = 0 or 2, n odd d = 1 P(m, n) (a, d)-SEAT many cases the electronic journal of combinatorics (2023), #DS6 258 6.4 Face Antimagic Labelings and d-antimagic Labeling of Type (1,1,1) Bača defines a connected plane graph G with edge set E and face set F to be (a, d)-face antimagic if there exist positive integers a and d and a bijection g: E →{1, 2, . . . , |E|} such that the induced mapping ψg : F →{a, a + d, . . . , a + (|F(G)| −1)d}, where for a face f, ψg(f) is the sum of all g(e) for all edges e surrounding f is also a bijection.
In Bača proves that for n even and at least 4, the prism Cn × P2 is (6n + 3, 2)-face antimagic and (4n + 4, 4)-face antimagic. He also conjectures that Cn × P2 is (2n + 5, 6)-face antimagic. In Bača, Lin, and Miller investigate (a, d)-face antimagic labelings of the convex polytopes Pm+1 ×Cn. They show that if these graphs are (a, d)-face antimagic then either d = 2 and a = 3n(m + 1) + 3, or d = 4 and a = 2n(m + 1) + 4, or d = 6 and a = n(m + 1) + 5. They also prove that if n is even, n ≥4 and m ≡1 (mod 4), m ≥3, then Pm+1 × Cn has a (3n(m + 1) + 3, 2)-face antimagic labeling and if n is at least 4 and even and m is at least 3 and odd, or if n ≡2 (mod 4), n ≥6 and m is even, m ≥4, then Pm+1 × Cn has a (3n(m + 1) + 3, 2)-face antimagic labeling and a (2n(m + 1) + 4, 4)-face antimagic labeling. They conjecture that Pm+1 × Cn has (3n(m + 1) + 3, 2)- and (2n(m + 1) + 4, 4)-face antimagic labelings when m ≡0 (mod 4), n ≥4, and for m even and m ≥4, that Pm+1 ×Cn has a (n(m+1)+5, 6)-face antimagic labeling when n is even and at least 4. Bača, Baskoro, Jendroľ, and Miller proved that graphs in the shape of hexagonal honeycombs with m rows, n columns, and mn 6-sided faces have d-antimagic labelings of type (1, 1, 1) for d = 1, 2, 3, and 4 when n odd and mn > 1.
In Bača and Miller define the class Qm n of convex polytopes with vertex set {yj,i : i = 1, 2, . . . , n; j = 1, 2, . . . , m + 1} and edge set {yj,iyj,i+1 : i = 1, 2, . . . , n; j = 1, 2, . . . , m + 1} ∪{yj,iyj+1,i : i = 1, 2, . . . , n; j = 1, 2, . . . , m} ∪{yj,i+1yj+1,i : 1 + 1, 2, . . . , n; j = 1, 2, . . . , m, j odd} ∪{yj,iyj+1,i+1 : i = 1, 2, . . . , n; j = 1, 2, . . . , m, j even} where yj,n+1 = yj,1. They prove that for m odd, m ≥3, n ≥3, Qm n is (7n(m+1)/2+2, 1)-face antimagic and when m and n are even, m ≥4, n ≥4, Qm n is (7n(m+1)/2+2, 1)-face antimagic.
They conjecture that when n is odd, n ≥3, and m is even, then Qm n is ((5n(m + 1) + 5)/2, 2)−face antimagic and ((n(m + 1) + 7)/2, 4)-face antimagic. They further conjecture that when n is even, n > 4, m > 1 or n is odd, n > 3 and m is odd, m > 1, then Qm n is (3n(m + 1)/2 + 3, 3)-face antimagic. In Bača proves that for the case m = 1 and n ≥3 the only possibilities for (a, d)-antimagic labelings for Qm n are (7n+2, 1) and (3n+3, 3). He provides the labelings for the first case and conjectures that they exist for the second case. Bača and Bača and Miller describe (a, d)-face antimagic labelings for a certain classes of convex polytopes.
In Bača et al. provide a detailed survey of results on face antimagic labelings and include many conjectures and open problems.
For a plane graph G, Bača and Miller call a bijection h from V (G)∪E(G)∪F(G) to {1, 2, . . . , |V (G)| + |E(G)| ∪|F(G)|} a d-antimagic labeling of type (1, 1, 1) if for every number s the set of s-sided face weights is Ws = {as, as +d, as +2d, . . . , as +(fs −1)d} for some integers as and d, where fs is the number of s-sided faces (Ws varies with s). They show that the prisms Cn × P2 (n ≥3) have a 1-antimagic labeling of type (1, 1, 1) and the electronic journal of combinatorics (2023), #DS6 259 that for n ≡3 (mod 4), Cn ×P2 have a d-antimagic labeling of type (1, 1, 1) for d = 2, 3, 4, and 6. They conjecture that for all n ≥3, Cn × P2 has a d-antimagic labeling of type (1, 1, 1) for d = 2, 3, 4, 5, and 6. This conjecture has been proved for the case d = 3 and n ̸= 4 by Bača, Miller, and Ryan (the case d = 3 and n = 4 is open). The cases for d = 2, 4, 5, and 6 were done by Lin, Slamin, Bača, and Miller . Bača, Lin, and Miller prove: for m, n > 8, Pm ×Pn has no d-antimagic edge labeling of type (1, 1, 1) with d ≥9; for m ≥2, n ≥2, and (m, n) ̸= (2, 2), Pm × Pn has d-antimagic labelings of type (1, 1, 1) for d = 1, 2, 3, 4, and 6. They conjecture the same is true for d = 5. Butt, Numan, Shah, and Ali prove that the generalized prisms Cn × Pm have d-antimagic face labelings of type (1,1,1) for n ≥5 and m ≥2.
Bača, Miller, and Ryan also prove that for n ≥4 the antiprism (see §6.1 for the definition) on 2n vertices has a d-antimagic labeling of type (1, 1, 1) for d = 1, 2, and 4.
They conjecture the result holds for d = 3, 5, and 6 as well. Lin, Ahmad, Miller, Sugeng, and Bača did the cases that d = 7 for n ≥3 and d = 12 for n ≥11. Sugeng, Miller, Lin, and Bača did the cases: d = 7, 8, 9, 10 for n ≥5; d = 15 for n ≥6; d = 18 for n ≥7; d = 12, 14, 17, 20, 21, 24, 27, 30, 36 for n odd and n ≥7; and d = 16, 26 for n odd and n ≥9.
Baca, Numan, and Semaničová-Feňovčíková investigated the problem of labeling the vertices, edges, and faces of a disjoint union of r copies Cn × Pm by the consecutive integers starting from 1 in such a way that the sum of the labels of a face and the labels of vertices and edges surrounding that face for all s-sided faces form an arithmetic progression with common difference d.
Ali, Bača, Bashir, and Semaničová-Feňovčíková investigated antimagic labelings for disjoint unions of prisms and cycles. They prove: for m ≥2 and n ≥3, m(Cn × P2) has no super d-antimagic labeling of type (1, 1, 1) with d ≥30; for m ≥2 and n ≥ 3, n ̸= 4, m(Cn × P2) has super d-antimagic labeling of type (1, 1, 1) for d = 0, 1, 2, 3, 4, and 5; and for m ≥2 and n ≥3, mCn has (m(n + 1) + 3, 3)- and (2mn + 2, 2)-vertex-antimagic total labeling. Bača and Bashir proved that for m ≥2 and n ≥3, n ̸= 4, m(Cn × P2) has super 7-antimagic labeling of type (1, 1, 1) and for n ≥3, n ̸= 4 and 2 ≤m ≤2n m(Cn × P2) has super 6-antimagic labeling of type (1, 1, 1).
Bača, Numan and Siddiqui investigated the existence of the super d-antimagic labeling of type (1, 1, 1) for the disjoint union of m copies of antiprism mAn.
They proved that for m ≥2, n ≥4, mAn has super d-antimagic labelings of type (1, 1, 1) for d = 1, 2, 3, 5, 6. Ahmad, Bača, Lascsáková, and Semaničová-Feňovčíková investigated super d-antimagicness of type (1, 1, 0) for mG in a more general sense. They prove: if there exists a super 0-antimagic labeling of type (1, 1, 0) of a plane graph G then, for every positive integer m, the graph mG also admits a super 0-antimagic labeling of type (1, 1, 0); if a plane graph G with 3-sided inner faces admits a super d-antimagic labeling of type (1, 1, 0) for d = 0, 6 then, for every positive integer m, the graph mG also admits a super d-antimagic labeling of type (1, 1, 0); if a plane graph G with 3-sided inner faces is a tripartite graph with a super d-antimagic labeling of type (1, 1, 0) for d = 2, 4 then, for every positive integer m, the graph mG also admits a super d-antimagic labeling of type (1, 1, 0); if a plane graph G with 4-sided inner faces admits a super d-antimagic labeling of the electronic journal of combinatorics (2023), #DS6 260 type (1, 1, 0) for d = 0, 4, 8 then the disjoint union of arbitrary number of copies of G also admits a super d-antimagic labeling of type (1, 1, 0); if a plane graph G with k-sided inner faces, k ≥3, admits a super d-antimagic labeling of type (1, 1, 0) for d = 0, 2k then, for every positive integer m, the graph mG also admits a super d-antimagic labeling of type (1, 1, 0); if a plane graph G with k-sided inner faces admits a super k-antimagic labeling of type (1, 1, 0) for k even then, for every positive integer m, the graph mG also admits a super k-antimagic labeling of type (1, 1, 0).
Bača, Jendraľ, Miller, and Ryan prove: for n even, n ≥6, the generalized Petersen graph P(n, 2) has a 1-antimagic labeling of type (1, 1, 1); for n even, n ≥6, n ̸= 10, and d = 2 or 3, P(n, 2) has a d-antimagic labeling of type (1, 1, 1); and for n ≡0 (mod 4), n ≥8 and d = 6 or 9, P(n, 2) has a d-antimagic labeling of type (1, 1, 1). They conjecture that there is an d-antimagic labeling of type (1,1,1) for P(n, 2) when n ≡2 (mod 4), n ≥6, and d = 6 or 9.
In Bača, Brankovic, and A. Semaničová-Feňovčikovǎ provide super d-antimagic labelings of type (1,1,1) for friendship graphs Fn (n ≥2) and several other families of planar graphs.
Bača, Brankovic, Lascsáková, Phanalasy, and Semaničová-Feňovčíková provided super d-antimagic labeling of type (1, 1, 0) for friendship graphs Fn, n ≥2, for d ∈ {1, 3, 5, 7, 9, 11, 13}.
Moreover, they show that for n ≡1 (mod 2) the graph Fn also admits a super d-antimagic labeling of type (1, 1, 0) for d ∈{0, 2, 4, 6, 8, 10}.
Bača, Baskoro, and Miller have proved that hexagonal planar honeycomb graphs with an even number of columns have 2-antimagic and 4-antimagic labelings of type (1, 1, 1). They conjecture that these honeycombs also have d-antimagic labelings of type (1, 1, 1) for d = 3 and 5. They pose the odd number of columns case for 1 ≤d ≤5 as an open problem. Bača, Baskoro, and Miller give d-antimagic labelings of a special class of plane graphs with 3-sided internal faces for d = 0, 2, and 4. Bača, Lin, Miller, and Ryan prove for odd n ≥3, m ≥1 and d = 0, 1, 2 or 4, the Möbius grid M m n has an d-antimagic labeling of type (1, 1, 1). Siddiqui, Numan, and Umar examined the existence of super d-antimagic labelings of type (1,1,1) for Jahangir graphs for certain differences d.
Bača, Numan, and Shabbir studied the existence of super d-antimagic labelings of type (1, 1, 1) for the toroidal polyhex Hn m. They labeled the edges of a 1-factor by consecutive integers and then in successive steps they labeled the edges of 2m-cycles (respectively 2n-cycles) in a 2-factor by consecutive integers.
This technique allowed them to construct super d-antimagic labelings of type (1, 1, 1) for Hn m with d = 1, 3, 5.
They suppose that such labelings exist also for d = 0, 2, 4.
Kathiresan and Ganesan define a class of plane graphs denoted by P b a (a ≥ 3, b ≥2) as the graph obtained by starting with vertices v1, v2, . . . , va and for each i = 1, 2 . . . , a −1 joining vi and vi+1 with b internally disjoint paths of length i + 1. They prove that P b a has d-antimagic labelings of type (1, 1, 1) for d = 0, 1, 2, 3, 4, and 6. Lin and Sugen prove that P b a has a d-antimagic labeling of type (1, 1, 1) for d = 5, 7a − 2, a +1, a −3, a −7, a +5, a −4, a +2, 2a−3, 2a−1, a −1, 3a−3, a +3, 2a+1, 2a+3, 3a+ 1, 4a−1, 4a−3, 5a−3, 3a−1, 6a−5, 6a−7, 7a−7, and 5a−5. Similarly, Bača, Baskoro, the electronic journal of combinatorics (2023), #DS6 261 and Cholily define a class of plane graphs denoted by Cb a as the graph obtained by starting with vertices v1, v2, . . . , va and for each i = 1, 2 . . . , a joining vi and vi+1 with b internally disjoint paths of length i+1 (subscripts are taken modulo a). In and they prove that for a ≥3 and b ≥2, Cb a has a d-antimagic labeling of type (1, 1, 1) for d = 0, 1, 2, 3, a + 1, a −1, a + 2, and a −2.
In Bača, Brankovic, and Semaničová-Feňovčikovǎ investigated the existence of super d-antimagic labelings of type (1,1,1) for plane graphs containing a special kind of Hamilton path. They proved: if there exists a Hamilton path in a plane graph G such that for every face except the external face, the Hamilton path contains all but one of the edges surrounding that face, then G is super d-antimagic of type (1,1,1) for d = 0, 1, 2, 3, 5; if there exists a Hamilton path in a plane graph G such that for every face except the external face, the Hamilton path contains all but one of the edges surrounding that face and if 2(|F(G)| −1) ≤|V (G)|, then G is super d-antimagic of type (1, 1, 1) for d = 0, 1, 2, 3, 4, 5, 6; if G is a plane graph with M = ⌊ |V (G)| |F(G)|−1⌋and a Hamilton path such that for every face, except the external face, the Hamilton path contains all but one of the edges surrounding that face, then for M = 1, G admits a super d-antimagic labeling of type (1,1,1) for d = 0, 1, 2, 3, 5; and for M ≥2, G admits a super d-antimagic labeling of type (1,1,1) for d = 0, 1, 2, 3, . . . , M + 4. They also proved that Pn × P2 (n ≥3) admits a super d-antimagic labeling of type (1,1,1) for d ∈{0, 1, 2, . . . , 15} and the graph obtained from Pn×Pm (n ≥2) by adding a new edge in every 4-sided face such that the added edges are “parallel” admits a super d-antimagic labeling of type (1,1,1) for d ∈{0, 1, 2, . . . , 9}.
In Imran, Siddiqui, and Numan examine the existence of super d-antimagic labelings of type (1,1,1) for uniform subdivision of wheel for certain differences d.
In the following tables we use the abbreviations (a, d)-FA (a, d)-face antimagic labeling d-AT(1,1,1) d-antimagic labeling of type (1, 1, 1).
A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property. The tables were prepared by Petr Kovář and Tereza Kovářová and updated by J. Gallian in 2008.
the electronic journal of combinatorics (2023), #DS6 262 Table 18: Summary of Face Antimagic Labelings Graph Labeling Notes Qm n (see §6.4) (7n(m + 1)/2 + 2, 1)-FA m ≥3, n ≥3, m odd (7n(m + 1)/2 + 2, 1)-FA m ≥4, n ≥4, m, n even ((5n(m + 1) + 5)/2, 2)-FA?
m ≥2, n ≥3, m even, n odd ((n(m + 1) + 7)/2, 4)-FA?
m ≥2, n ≥3, m even, n odd (3n(m + 1)/2 + 3, 3)-FA?
m > 1, n > 4, n even (3n(m + 1)/2 + 3, 3)-FA?
m > 1, n > 3, m odd, n odd Cn × P2 (6n + 3, 2)-FA n ≥4, n even (4n + 4, 4)-FA n ≥4, n even (2n + 5, 6)-FA?
Pm+1 × Cn (3n(m + 1) + 3, 2)-FA n ≥4, n even and m ≥3, m ≡1 (mod 4), (3n(m + 1) + 3, 2)-FA and n ≥4, n even and (2n(m + 1) + 4, 4)-FA m ≥3, m odd , or n ≥6, n ≡2 (mod 4) and m ≥4, m even (3n(m + 1) + 3, 2)-FA?
m ≥4, n ≥4, m ≡0 (mod 4) (2n(m + 1) + 4, 4)-FA?
m ≥4, n ≥4, m ≡0 (mod 4) (n(m + 1) + 5, 6)-FA?
n ≥4, n even Table 19: Summary of d-antimagic Labelings of Type (1,1,1) Graph Labeling Notes Pm × Pn not d-AT(1,1,1) m, n, d ≥9, Pm × Pn d-AT(1,1,1) d = 1, 2, 3, 4, 6; m, n ≥2, (m, n) ̸= (2, 2) Pm × Pn 5-AT(1,1,1) m, n ≥2, (m, n) ̸= (2, 2) Cn × P2 1-AT(1,1,1) d-AT(1,1,1) d = 2, 3, 4 and 6 for n ≡3 (mod 4) d-AT(1,1,1) d = 2, 4, 5, 6 for n ≥3 Continued on next page the electronic journal of combinatorics (2023), #DS6 263 Table 19 – Continued from previous page Graph Labeling Notes d-AT(1,1,1) d = 3 for n ≥5 Pm × Pn 5-AT(1,1,1)?
not d-AT m, n > 8, d ≥9 antiprism on 2n d-AT(1,1,1) d = 1, 2 and 4 for n ≥4 vertices d-AT(1,1,1)?
d = 3, 5 and 6 for n ≥4 M m n (Möbius grids) d-AT(1,1,1) n ≥3 odd, d = 0, 1, 2, 4 d = 7, n ≥3 d = 12, n ≥11 d = 7, 8, 9, 10, n ≥5 d = 15, n ≥6 d = 18 n ≥7 P(n, 2) d-AT(1,1,1) d = 1; d = 2, 3, n ≥6, n ̸= 10 P(4n, 2) d-AT(1,1,1) d = 6, 9, n ≥2, n ̸= 10 P(4n + 2, 2) d-AT(1,1,1)?
d = 6, 9, n ≥1, n ̸= 10 honeycomb graphs d-AT(1,1,1) d = 2, 4 with even number d-AT(1,1,1)?
d = 3, 5 of columns Cn × P2 d-AT(1,1,1) d = 1, 2, 4, 5, 6 , Cn × P2 3-AT(1,1,1) n ̸= 4 6.5 Product Antimagic Labelings Figueroa-Centeno, Ichishima, and Muntaner-Batle have introduced multiplicative analogs of magic and antimagic labelings. They define a graph G of size q to be product magic if there is a labeling from E(G) onto {1, 2, . . . , q} such that, at each vertex v, the product of the labels on the edges incident with v is the same. They call a graph G of size q product antimagic if there is a labeling f from E(G) onto {1, 2, . . . , q} such that the products of the labels on the edges incident at each vertex v are distinct. They prove: a graph of size q is product magic if and only if q ≤1 (that is, if and only if it is K2, Kn or the electronic journal of combinatorics (2023), #DS6 264 K2 ∪Kn); Pn (n ≥4) is product antimagic; every 2-regular graph is product antimagic; and, if G is product antimagic, then so are G + K1 and G ⊙Kn. They conjecture that a connected graph of size q is product antimagic if and only if q ≥3. Kaplan, Lev, and Roditty proved the following graphs are product antimagic: the disjoint union of cycles and paths where each path has least three edges; connected graphs with n vertices and m edges where m ≥4nln n; graphs G = (V, E) where each component has at least two edges and the minimum degree of G is at least 8 p ln |E| ln (ln |E|); all complete k-partite graphs except K2 and K1,2; and G ⊙H where G has no isolated vertices and H is regular. Wang and Gao show that caterpillars with at least three edges are product antimagic by an O(m log m) algorithm.
In Pikhurko characterizes all large graphs that are product antimagic graphs.
More precisely, it is shown that there is an n0 such that a graph with n ≥n0 vertices is product antimagic if and only if it does not belong to any of the following four classes: graphs that have at least one isolated edge; graphs that have at least two isolated vertices; unions of vertex-disjoint of copies of K1,2; graphs consisting of one isolated vertex; and graphs obtained by subdividing some edges of the star K1,k+l.
In Figueroa-Centeno, Ichishima, and Muntaner-Batle also define a graph G with p vertices and q edges to be product edge-magic if there is a labeling f from V (G) ∪E(G) onto {1, 2, . . . , p+q} such that f(u)·f(v)·f(uv) is a constant for all edges uv and product edge-antimagic if there is a labeling f from V (G) ∪E(G) onto {1, 2, . . . , p + q} such that for all edges uv the products f(u)·f(v)·f(uv) are distinct. They prove K2∪Kn is product edge-magic, a graph of size q without isolated vertices is product edge-magic if and only if q ≤1 and every graph other than K2 and K2 ∪Kn is product edge-antimagic.
the electronic journal of combinatorics (2023), #DS6 265 7 Miscellaneous Labelings 7.1 Sum Graphs In 1990, Harary introduced the notion of a sum graph. A graph G(V, E) is called a sum graph if there is an bijection f from V to a set of positive integers S such that xy ∈E if and only if f(x) + f(y) ∈S. Since the vertex with the highest label in a sum graph cannot be adjacent to any other vertex, every sum graph must contain isolated vertices.
In 1991 Harary, Hentzel, and Jacobs defined a real sum graph in an analogous way by allowing S to be any finite set of positive real numbers. However, they proved that every real sum graph is a sum graph. Bergstrand, Hodges, Jennings, Kuklinski, Wiener, and Harary defined a product graph analogous to a sum graph except that 1 is not permitted to belong to S. They proved that every product graph is a sum graph and vice versa.
For a connected graph G, let σ(G), the sum number of G, denote the minimum number of isolated vertices that must be added to G so that the resulting graph is a sum graph (some authors use s(G) for the sum number of G). A labeling that makes G together with σ(G) isolated points a sum graph is called an optimal sum graph labeling.
Ellingham proved the conjecture of Harary that σ(T) = 1 for every tree T ̸= K1. Smyth proved that there is no graph G with e edges and σ(G) = 1 when n2/4 < e ≤n(n −1)/2. Smyth conjectures that the disjoint union of graphs with sum number 1 has sum number 1. More generally, Kratochvil, Miller, and Nguyen conjecture that σ(G ∪H) ≤σ(G) + σ(H) −1.
Hao has shown that if d1 ≤d2 ≤· · · ≤dn is the degree sequence of a graph G, then σ(G) > max(di −i) where the maximum is taken over all i. Bergstand et al. proved that σ(Kn) = 2n −3.
Hartsfield and Smyth claimed to have proved that σ(Km,n) = ⌈3m+n−3⌉/2 when n ≥m but Yan and Liu found counterexamples to this assertion when m ̸= n.
Pyatkin , Liaw, Kuo, and Chang , Wang, and Liu , and He, Shen, Wang, Chang, Kang, and Yu have shown that for 2 ≤m ≤n, σ(Km,n) = ⌈n p + (p+1)(m−1) 2 ⌉ where p = ⌈ q 2n m−1 + 1 4 −1 2⌉is the unique integer such that (p−1)p(m−1) 2 < n ≤(p+1)p(m−1) 2 .
Miller, Ryan, Slamin, and Smyth proved that σ(Wn) = n 2 + 2 for n even and σ(Wn) = n for n ≥5 and n odd (see also ). Miller, Ryan, and Smyth prove that the complete n-partite graph on n sets of 2 nonadjacent vertices has sum number 4n −5 and obtain upper and lower bounds on the complete n-partite graph on n sets of m nonadjacent vertices. Fernau, Ryan, and Sugeng proved that the generalized friendship graphs C(t) n (see §2.2) has sum number 2 except for C4. Gould and Rödl investigated bounds on the number of isolated points in a sum graph. A group of six undergraduate students proved that σ(Kn −edge) ≤2n −4. The same group of six students also investigated the difference between the largest and smallest labels in a sum graph, which they called the spum. They proved spum of Kn is 4n −6 and the spum of Cn is at most 4n −10. Kratochvil, Miller, and Nguyen have proved that every sum graph on n vertices has a sum labeling such that every label is at most 4n. Konečný, Kučera, Novotná, Pekárek, Šimsa, and Töpfer showed that if one allows for non-the electronic journal of combinatorics (2023), #DS6 266 injective labelings or graphs with loops then there are sum graphs without a minimal sum labeling, which partially answers the question posed by Miller, Ryan and Smyth in .
At a conference in 2000 Miller posed the following two problems: Given any graph G, does there exist an optimal sum graph labeling that uses the label 1; Find a class of graphs G that have sum number of the order |V (G)|s for s > 1. (Such graphs were shown to exist for s = 2 by Gould and Rödl in ). Fernau and Gajjar provided a complete characterization of the sum number of graphs of maximum degree two. As an immediate corollary they have that all graphs that are the disjoint unions of paths and cycles are sum labeled graphs.
Elizeche and Tripathi characterized sum and integral sum labelings of complete graphs, symmetric complete bipartite graphs, star graphs, and deduced the spum, integral spum, and integral radius for these classes of graphs.
In Slamet, Sugeng, and Miller show how one can use sum graph labelings to distribute secret information to set of people so that only authorized subsets can reconstruct the secret.
Chang generalized the notion of sum graph by permitting x = y in the definition of sum graph. He calls graphs that have this kind of labeling strong sum graphs and uses i∗(G) to denote the minimum positive integer m such that G∪mK1 is a strong sum graph.
Chang proves that i∗(Kn) = σ(Kn) for n = 2, 3, and 4 and i∗(Kn) > σ(Kn) for n ≥5. He further shows that for n ≥5, 3nlog23 > i∗(Kn) ≥12⌊n/5⌋−3.
In Fernau and Gajjar initiate the study of sum graphs from the viewpoint of computational complexity. They show that every n-vertex, m-edge, d-degenerate graph (a graph is d-degenerate if all its subgraphs have a vertex of degree at most d) can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as Θ(n2d). As a consequence, we have that such a graph can be stored using Θ(m log n) bits of memory. The previously best known upper bound on the numbers needed for labeling general graphs with the minimum number of isolated vertices was Θ(4n), due to Kratochvíl, Miller and Nguyen in 2001.
Moreover, Fernau and Gajjar’s labeling can be constructed in polynomial time. Their results show the ever-expanding database that is gradually built up can be efficiently implemented and then accessed using their sum-labeling scheme.
In 1994 Harary generalized sum graphs by permitting S to be any set of integers.
He calls these graphs integral sum graphs. Unlike sum graphs, integral sum graphs need not have isolated vertices. Sharary has shown that Cn and Wn are integral sum graphs for all n ̸= 4. Chen proved that trees obtained from a star by extending each edge to a path and trees all of whose vertices of degree not 2 are at least distance 4 apart are integral sum graphs. He conjectures that all trees are integral sum graphs. In and Chen gives methods for constructing new connected integral sum graphs from given integral sum graphs by identifying vertices. Chen has shown that every graph is an induced subgraph of a connected integral sum graph. Chen calls a vertex of a graph saturated if it is adjacent to every other vertex of the graph. He proves that every integral sum graph except K3 has at most two saturated vertices and gives the exact structure of all integral sum graphs that have exactly two saturated vertices. Chen the electronic journal of combinatorics (2023), #DS6 267 also proves that a connected integral sum graph with p > 1 vertices and q edges and no saturated vertices satisfies q ≤p(3p −2)/8 −2. Wu, Mao, and Le proved that mPn are integral sum graphs. They also show that the conjecture of Harary that the sum number of Cn equals the integral sum number of Cn if and only if n ̸= 3 or 5 is false and that for n ̸= 4 or 6 the integral sum number of Cn is at most 1. Vilfred and Nicholas prove that graphs G of order n with ∆(G) = n −1 and |V∆(G)| > 2 are not integral sum graphs, except K3, and that integral sum graphs G of order n with ∆(G) = n −1 and |V∆(G)| = 2 exist and are unique up to isomorphism. Chen proved that if G(V, E) is an integral sum other than K3 that has vertex of degree |V | −1, then the edge-chromatic number of G is |V | −1.
In 2021 Singla, Tiwari, and Tripathi were the first to investigated the spum and integral spum (denoted ispum(G). They provided results on complete graphs, symmetric complete bipartite graphs, star graphs, cycles, and paths and gave sharp lower bounds for the spum and the integral spum of connected graphs. Shortly after that paper appeared, Li used some of the techniques in , as well as his own, to correct multiple errors and improve many results in . The paper by Singla, Tiwari, and Tripathi and the one by Li open up new areas for research.
Following are some of the corrected and improved results. For 3 ≤n ≤6, spum(Pn) = 2n−3, and for n ≥7, 2n−2 ≤spum(Pn) ≤2n+1 if n is odd, and 2n−2 ≤spum(Pn) ≤ 2n −1 if n is even. Li conjectures that for n ≥8, we have spum(Pn) = 2n + 1 if n is odd and spum(Pn) = 2n −1 if n is even. He verified the conjecture for n up to 15. In Singla, Tiwari, and Tripathi gave a flawed proof that spum(Cn) = 2n −1 for all n ≥13.
Li proved spum(C3) = 6 and spum(Cn) = 2n −1 for n ≥4.
Li improved the previous best upper bounds given for ispum(Cn) in as follows. For n ≥12, we have ispum(Cn) ≤8(n−9) if n is odd and 3 2(3n−14) if n is even.
In 1990 Harary showed that ζ(nK2) = 0 for all positive integers n, but Li is the first to investigate the spum and ispum of nK2. He proved that for all positive integers n σ(nK2) = 1, spum(nK2) = 4n −2, and ispum(2K2) = 4 and ispum(nK2) = 4n −3 for n ̸= 2.
Motivated by the fact that there is no obvious relationship between ispum(G) and spum(G), due to ζ(G) possibly being strictly smaller than σ(G), i.e., when the inequality ζ(G) ≤σ(G) is strict, Li introduced the following modification of spum.
The sum-diameter of a graph G, denoted sd(G), is the minimum possible value of range(L) for a set L of positive integer labels, such that the induced sum graph of L consists of the disjoint union of G with any number of isolated vertices. The difference between the definition of sum-diameter and spum is that in spum, one is required to use the minimum number of additional isolated vertices possible. Similarly, Li defines the integral sum-diameter of a graph G, denoted isd(G), by allowing the labels to be arbitrary distinct integers. He notes that sd(G) ≤spum(G), as any set of labels L from the definition of spum(G), i.e., on n + σ(G) vertices, is a valid labeling for the definition of sd(G).
Similarly, isd(G) ≤sd(G). Noting that any labeling L from the definition of sd(G) is a valid labeling for the definition of isd(G), yields isd(G) ≤sd(G) for all graphs G.
Li observes that the following stronger result, which was initially stated for spum(G) the electronic journal of combinatorics (2023), #DS6 268 in rather than sd(G), follows from the same proof. For graphs G of order n without any isolated vertices, with maximum and minimum vertex degree ∆and δ, respectively, sd(G) ≥2n −(∆−δ) −2.
Similarly, he noticed that the proof of Theorem 2.2 in does not use the assump-tion that exactly ζ(G) isolated vertices are present when bounding ispum(G), and thus the following strengthening is true. For graphs G of order n without any isolated vertices, with maximum degree ∆, we have isd(G) ≥2n −∆−3.
Li uses the notion of Sidon sets to prove that if G is a graph with n vertices, then sd(G) ≤64n2 −64n + 9. Because isd(G) ≤sd(G), we also have that isd(G) ≤64n2 − 64n +9. To show this upper bound is asymptotically tight, for a graph G with n vertices, he lets f(n) be the maximum value of sd(G) over all graphs G with n vertices. Then Li proves f(n) ≥n2 4 −O(n log n) = Ω(n2).
For Kn, Li proves sd(Kn) = spum(Kn) = 4n −6 for n ≥2 and isd(Kn) = spum(Kn) = 4n −6 for all n ≥2 and isd(Kn) = ispum(Kn) = n −1 for n ≤3 and ispum(Kn) = 4n −6 when n ≥4.
For cycles Li proves: for all n ≥4, 2n−2 ≤sd(Cn) ≤2n−1 and 2n−5 ≤isd(Cn) ≤ 2n −1.
The current best bounds for ispum(Pn) are due to Singla, Tiwari, and Tripathi : 2n −5 ≤ispum(Pn) ≤5 2(n −3) if n is odd and 2n −5 ≤ispum(Pn) ≤2n −3 if n is even. For the sum-diameter, Li obtains a much tighter result that for n ≥3, we have 2n −3 ≤sd(Pn) ≤2n −2. An exhaustive computer search yields that sd(Pn) = 2n −3 for 3 ≤n ≤6 and sd(Pn) = 2n −2 for 7 ≤n ≤13. This led Li to conjecture that sd(Pn) = 2n −3 for 3 ≤n ≤6 and sd(Pn) = 2n −2 for n ≥7. For isd(Pn) Li obtains: 2n −5 ≤isd(Pn) ≤2n −2 if n is odd and 2n −5 ≤isd(Pn) ≤2n −3 if n is even.
Li provides upper bounds on the sum-diameter for the disjoint union of two graphs and the join of two graphs as follows. For disjoint graphs G1 and G2 with no isolated vertices sd(G1 ∪G2) ≤10 max{sd(G1), sd(G2)} + sd(G1) + sd(G2) + 2. If one wishes to add isolated vertices to a graph G, Li proves that if G is a graph with no isolated vertices, and Nk is the empty graph on k vertices, then sd(G ∪Nk) ≤max{k, 4sd(G)} + k −5. If one wishes to add a vertex with arbitrary edges incident to a graph G with no isolated vertices to form a new graph G′ obtained by adding a vertex to G along with any desired edges incident to this new vertex, then sd(G′) ≤4sd(G) −1. For two graphs G1 and G2 with no isolated vertices, where without loss of generality sd(G1) ≤sd(G2), Li proves sd(G1 + G2) ≤8sd(G2) + 11sd(G1) −5.
For a subset U of vertices from a graph G(V, E) with no isolated vertices and G[U] denoting the subgraph of G consisting of U and all of the edges connecting pairs of vertices in U, Li proves that sd(G[U]) ≤2sd(G) −2. As an open question he asks is: For any induced subgraph G[U] of G, is sd(G[U]) ≤sd(G)? For graphs obtained by deleting an edge e from a graph G with no isolated vertices (denoted by G \ e) or by contracting an edge e from G (denoted by G/e), Li proves that sd(G \ e) ≤4sd(G) −1 and sd(G/e) ≤ 4sd(G) −1. For a graph G with no isolated vertices and an edge e = (u, v) between two vertices u and v of G that is not present in G, Li proves that sd(G + e) ≤4sd(G) −1.
Li concludes his paper by introducing the generalization of sum-diameter to hyper-the electronic journal of combinatorics (2023), #DS6 269 graphs, and generalizes his previous general upper and lower bounds to provide prelimi-nary bounds on both sides for arbitrary k-uniform hypergraphs and with some remarks on areas for further research and some open questions.
He, Wang, Mi, Shen, and Yu say that a graph has a tail if the graph contains a path for which each interior vertex has degree 2 and an end vertex of degree at least 3.
They prove that every tree with a tail of length at least 3 is an integral sum graph.
B. Xu has shown that the following are integral sum graphs: the union of any three stars; T ∪K1,n for all trees T; mK3 for all m; and the union of any number of integral sum trees. Xu also proved that if 2G and 3G are integral sum graphs, then so is mG for all m > 1. Xu poses the question as to whether all disconnected forests are integral sum graphs. Nicholas and Somasundaram prove that all banana trees (see Section 2.1 for the definition) and the union of any number of stars are integral sum graphs.
Liaw, Kuo, and Chang proved that all caterpillars are integral sum graphs (see also and for some special cases of caterpillars). This shows that the assertion by Harary in that K(1, 3) and S(2, 2) are not integral sum graphs is incorrect. They also prove that all cycles except C4 are integral sum graphs and they conjecture that every tree is an integral sum graph. Singh and Santhosh show that the crowns Cn ⊙K1 are integral sum graphs for n ≥4 and that the subdivision graphs of Cn ⊙K1 are integral sum graphs for n ≥3 . Wang, Li, and Wei proved that there exists a connected integral sum graph with any minimum degree and give an upper bound for the relation between the vertex number and the edge number of a connected integral sum graph with no saturated vertex.
For graphs with n vertices, Tiwari and Tripathi show that there exist sum graphs with m edges if and only if m ≤⌊(n −12)/4⌋and that there exists integral sum graphs with m edges if and only if m ≤⌈3(n −1)2/8⌉+ ⌊(n −1)/2⌋, except for m = ⌈3(n −1)2/8⌉+ ⌊(n −1)/2⌋−1 when n is of the form 4k + 1. They also characterize sets of positive integers (respectively, integers) that are in bijection with sum graphs (respectively, integral sum graphs) of maximum size for a given order.
The integral sum number, ζ(G), of G is the minimum number of isolated vertices that must be added to G so that the resulting graph is an integral sum graph.
Thus, by definition, G is a integral sum graph if and only if ζ(G) = 0. Harary conjectured that ζ(Kn) = 2n −3 for n ≥4. This conjecture was verified by Chen , by Sharary , and by B. Xu . Yan and Liu proved: ζ(Kn−E(Kr)) = n−1 when n ≥6, n ≡ 0 (mod 3) and r = 2n/3 −1 ; ζ(Km.m) = 2m −1 for m ≥2 ; ζ(Kn\ −edge) = 2n −4 for n ≥4 , ; if n ≥5 and n −3 ≥r, then ζ(Kn\E(Kr)) ≥n −1 ; if ⌈2n/3⌉−1 > r ≥2, then ζ(Kn\E(Kr)) ≥2n −r −2 ; and if 2 ≤m < n, and n = (i + 1)(im −i + 2)/2, then σ(Km,n) = ζ(Km,n) = (m −1)(i + 1) + 1 while if (i + 1)(im −i + 2)/2 < n < (i + 2)[(i + 1)m −i + 1]/2, then σ(Km,n) = ζ(Km,n) = ⌈((m−1)(i+1)(i+2)+2n)/(2i+2)⌉. Wang proved that σ(Kn+1\E(K1,r)) = ζ(Kn+1\E(K1,r)) = 2n −2 when r + 1, 2n −3 when 2 ≤r ≤n −1, and 2n −4 when r = n.
Nagamochi, Miller, and Slamin have determined upper and lower bounds on the sum number a graph. For most graphs G(V, E) they show that σ(G) = Ω(|E|). He, the electronic journal of combinatorics (2023), #DS6 270 Yu, Mi, Sheng, and Wang investigated ζ(Kn\E(Kr)) where n ≥5 and r ≥2. They proved that ζ(Kn\E(Kr)) = 0 when r = n or n −1; ζ(Kn\E(Kr)) = n −2 when r = n −2; ζ(Kn\E(Kr)) = n −1 when n −3 ≥r ≥ ⌈2n/3⌉−1; ζ(Kn\E(Kr)) = 3n−2r−4 when ⌈2n/3⌉−1 > r ≥n/2; ζ(Kn\E(Kr)) = 2n−4 when ⌈2n/3⌉−1 ≥n/2 > r ≥2. Moreover, they prove that if n ≥5, r ≥2, and r ̸= n−1, then σ(Kn\E(Kr)) = ζ(Kn\E(Kr)).
Dou and Gao prove that for n ≥3, the fan Fn = Pn + K1 is an integral sum graph, ρ(F4) = 1, ρ(Fn) = 2 for n ̸= 4, and σ(F4) = 2, σ(Fn) = 3 for n = 3 or n ≥6 and n even, and σ(Fn) = 4 for n ≥6 and n odd.
Wang and Gao and determined the sum numbers and the integral sum numbers of the complements of paths, cycles, wheels, and fans as follows: 0 = ζ(P4) < σ(P4) = 1; 1 = ζ(P5) < σ(P5) = 2; 3 = ζ(P6) < σ(P6) = 4; ζ(Pn) = σ(Pn) = 0, n = 1, 2, 3; ζ(Pn) = σ(Pn) = 2n −7, n ≥7. ζ(Cn) = σ(Cn) = 2n −7, n ≥7. ζ(Wn) = σ(Wn) = 2n −8, n ≥7. 0 = ζ(F5) < σ(F5) = 1; 2 = ζ(F6) < σ(F6) = 3; ζ(Fn) = σ(Fn) = 0, n = 3, 4; ζ(Fn) = σ(Fn) = 2n −8, n ≥7.
Wang, Yang and Li proved: ζ(Kn\E(Cn−1) = 0 for n = 4, 5, 6, 7; ζ(Kn\E(Cn−1) = 2n −7 for n ≥8; σ(K4\E(Cn−1) = 1; σ(K5\E(Cn−1) = 2; σ(K6\E(Cn−1) = 5; σ(K7\E(Cn−1) = 7; σ(Kn\E(Cn−1) = 2n −7 for n ≥8.
Wang and Li proved: a graph with n ≥6 vertices and degree greater than (n + 1)/2 is not an integral sum graph; for n ≥8, ζ(Kn \ E(2P3)) = σ(Kn \ E(2P3)) = ϵ(Kn \ E(2P3)) = ϵ(Kn \ E(2P3)) = 2n −7; for n ≥7, ζ(Kn \ E(K2)) = σ(Kn \ E(K2)) = 2n −4; and for n ≥7 and 1 ≤r ≤⌈n 2⌉, ζ(Kn \ E(rK2)) = σ(Kn \ E(rK2)) = 2n −5.
Chen has given some properties of integral sum labelings of graphs G with ∆(G) < |V (G)| −1 whereas Nicholas, Somasundaram, and Vilfred provided some general properties of connected integral sum graphs G with ∆(G) = |V (G)| −1. They have shown that connected integral sum graphs G other than K3 with the property that G has exactly two vertices of maximum degree are unique and that a connected integral sum graph G other than K3 can have at most two vertices with degree |V (G)| −1 (see also ).
Vilfred and Florida have examined one-point unions of pairs of small complete graphs. They show that the one-point union of K3 and K2 and the one-point union of K3 and K3 are integral sum graphs whereas the one-point union of K4 and K2 and the one-point union of K4 and K3 are not integral sum graphs. In Vilfred and Florida defined and investigated properties of maximal integral sum graphs.
Vilfred and Nicholas have shown that the following graphs are integral sum graphs: banana trees, the union of any number of stars, fans Pn + K1 (n ≥2), Dutch windmills K(m) 3 , and the graph obtained by starting with any finite number of integral sum graphs G1, G2, . . . , Gn and any collections of n vertices with vi ∈Gi and creating a graph by identifying v1, v2, . . . , vn. The same authors also proved that G + v where G is a union of stars is an integral sum graph.
Melnikov and Pyatkin have shown that every 2-regular graph except C4 is an integral sum graph and that for every positive integer r there exists an r-regular integral sum graph. They also show that the cube is not an integral sum graph. For any integral the electronic journal of combinatorics (2023), #DS6 271 sum graph G, Melnikov and Pyatkin define the integral radius of G as the smallest natural number r(G) that has all its vertex labels in the interval [−r(G), r(G)]. For the family of all integral sum graphs of order n they use r(n) to denote maximum integral radius among all members of the family. Two questions they raise are: Is there a constant C such that r(n) ≤Cn and for n > 2, is r(n) equal to the (n −2)th prime?
The concepts of sum number and integral sum number have been extended to hyper-graphs. Sonntag and Teichert prove that every hypertree (i.e., every connected, non-trivial, cycle-free hypergraph) has sum number 1 provided that a certain cardinality condition for the number of edges is fulfilled. In the same authors prove that for d ≥3 every d-uniform hypertree is an integral sum graph and that for n ≥d + 2 the sum number of the complete d-uniform hypergraph on n vertices is d(n −d) + 1. They also prove that the integral sum number for the complete d-uniform hypergraph on n vertices is 0 when d = n or n −1 and is between (d −1)(n −d −1) and d(n −d) + 1 for d ≤n −2. They conjecture that for d ≤n −2 the sum number and the integral sum number of the complete d-uniform hypergraph are equal.
Teichert proves that hypercycles have sum number 1 when each edge has cardinality at least 3 and that hyperwheels have sum number 1 under certain restrictions for the edge cardinalities.
(A hypercycle Cn = (Vn, En) has Vn = ∪n i=1{vi 1, vi 2, . . . , vi di−1}, En = {e1, e2, . . . , en} with ei = {vi 1, . . . , vi di = vi+1 1 } where i + 1 is taken modulo n. A hyperwheel Wn = (V′ n, E′ n) has V′ n = Vn ∪{c} ∪n i=1 {v2n+i, . . . , vdn+i−1n+i}, E′ n = En ∪{en+1, . . . , e2n} with en+i = {v1n+i = c, v2n+i, . . . , vdn+i−1n+i, vdn+i n+i = v1i}.) Teichert determined an upper bound for the sum number of the d-partite com-plete hypergraph Kd n1,...,nd. In Teichert defines the strong hypercycle Cd n to be the d-uniform hypergraph with the same vertices as Cn where any d consecutive vertices of Cn form an edge of Cd n.
He proves that for n ≥2d + 1 ≥5, σ(Cd n) = d and for d ≥2, σ(Cd d+1) = d. He also shows that σ(C3 5) = 3; σ(C3 6) = 2, and he conjectures that σ(Cd n) < d for d ≥4 and d + 2 ≤n ≤2d.
In Nicholas and Vilfred define the edge reduced sum number of a graph as the minimum number of edges whose removal from the graph results in a sum graph.
They show that for Kn, n ≥3, this number is (n(n −1)/2 + ⌊n/2⌋)/2. They ask for a characterization of graphs for which the edge reduced sum number is the same as its sum number. They conjecture that an integral sum graph of order p and size q exists if and only if q ≤3(p2 −1)/8 −⌊(p −1)/4⌋when p is odd and q ≤3(3p −2)/8 when p is even. They also define the edge reduced integral sum number in an analogous way and conjecture that for Kn this number is (n −1)(n −3)/8 + ⌊(n −1)/4⌋when n is odd and n(n −2)/8 when n is even.
For certain graphs G Vilfred and Florida investigated the relationships among σ(G), ζ(G), χ(G), and χ′(G) where χ(G) is the chromatic number of G and χ′(G) is the edge chromatic number of G. They prove: σ(C4) = ζ(C4) > χ(C4) = χ′(C4); for n ≥ 3, ζ(C2n) < σ(C2n) = χ(C2n) = χ′(C2n); ζ(C2n+1) < σ(C2n+1) < χ(C2n+1) = χ′(C2n+1); for n ≥4, χ′(Kn) ≤χ(Kn) < ζ(Kn) = σ(Kn); and for n ≥2, χ(Pn×P2) < χ′(Pn×P2) = ζ(Pn × P2) = σ(Pn × P2).
Alon and Scheinermann generalized sum graphs by replacing the condition the electronic journal of combinatorics (2023), #DS6 272 f(x) + f(y) ∈S with g(f(x), f(y)) ∈S where g is an arbitrary symmetric polynomial.
They called a graph with this property a g-graph and proved that for a given symmetric polynomial g not all graphs are g-graphs. On the other hand, for every symmetric poly-nomial g and every graph G there is some vertex labeling such that G together with at most |E(G)| isolated vertices is a g-graph.
Boland, Laskar, Turner, and Domke investigated a modular version of sum graphs. They call a graph G(V, E) a mod sum graph (MSG) if there exists a positive integer n and an injective labeling from V to {1, 2, . . . , n −1} such that xy ∈E if and only if (f(x)+f(y)) (mod n) = f(z) for some vertex z. Obviously, all sum graphs are mod sum graphs. However, not all mod sum graphs are sum graphs. Boland et al. have shown the following graphs are MSG: all trees on 3 or more vertices; all cycles on 4 or more vertices; and K2,n. They further proved that Kp (p ≥2) is not MSG (see also ) and that W4 is MSG. They conjecture that Wp is MSG for p ≥4. This conjecture was refuted by Sutton, Miller, Ryan, and Slamin who proved that for n ̸= 4, Wn is not MSG (the case where n is prime had been proved in 1994 by Ghoshal, Laskar, Pillone, and Fricke . In the same paper Sutton et al. also showed that for n ≥3, Kn,n is not MSG. Ghoshal, Laskar, Pillone, and Fricke proved that every connected graph is an induced subgraph of a connected MSG graph and any graph with n vertices and at least two vertices of degree n −1 is not MSG.
In Beste, de Wiljes, and Kreh investigated the sum and mod sum graphs using arithmetic progressions and prime numbers and characterized the induced sum graphs and mod sum graphs.
Sutton, Miller, Ryan, and Slamin define the mod sum number, ρ(G), of a con-nected graph G to be the least integer r such that G ∪Kr is MSG. Recall the cocktail party graph Hm,n, m, n ≥2, as the graph with a vertex set V = {v1, v2, . . . , vmn} par-titioned into n independent sets V = {I1, I2, . . . , In} each of size m such that vivj ∈E for all i, j ∈{1, 2, . . . , mn} where i ∈Ip, j ∈Iq, p ̸= q. The graphs Hm,n can be used to model relational database management systems (see ). Sutton and Miller prove that Hm,n is not MSG for n > m ≥3 and ρ(Kn) = n for n ≥4. In Sutton, Draganova, and Miller prove that for n odd and n ≥5, ρ(Wn) = n and when n is even, ρ(Wn) = 2. Wang, Zhang, Yu, and Shi proved that fan Fn(n ≥2) are not mod sum graphs and ρ(Fn) = 2 for even n at least 6. They also prove that ρ(Kn,n) = n for n ≥3.
Dou and Gao obtained exact values for ρ(Km,n) and ρ(Km −E(Kn)) for some cases of m and n and bounds in the remaining cases. They call a graph G(V, E) a mod integral sum graph if there exists a positive integer n and an injective labeling from V to {0, 1, 2, . . . , n −1} (note that 0 is included) such that xy ∈E if and only if (f(x) + f(y)) (mod n) = f(z) for some vertex z. They define the mod integral sum number, ψ(G), of a connected graph G to be the least integer r such that G ∪Kr is a mod integral sum graph. They prove that for m + n ≥3, ψ(Km,n) = ρ(Km,n) and obtained exact values for ψ(Km −E(Kn)) for some cases of m and n and bounds in the remaining cases.
Wallace has proved that Km,n is MSG when n is even and n ≥2m or when n is odd and n ≥3m −3 and that ρ(Km,n) = m when 3 ≤m ≤n < 2m. He also proves that the electronic journal of combinatorics (2023), #DS6 273 the complete m-partite Kn1,n2,...,nm is not MSG when there exist ni and nj such that ni < nj < 2ni. He poses the following conjectures: ρ(Km,n) = n when 3m −3 > n ≥m ≥3; if Kn1,n2,...,nm where n1 > n2 > · · · > nm, is not MSG, then (m −1)nm ≤ρ(Kn1,n2,...,nm) ≤ (m −1)n1; if G has n vertices, then ρ(G) ≤n; and determining the mod sum number of a graph is NP-complete (Sutton has observed that Wallace probably meant to say ‘NP-hard’). Miller has asked if it is possible for the mod sum number of a graph G be of the order |V (G)|2.
In a sum graph G, a vertex w is called a working vertex if there is an edge uv in G such that w = u+v. If G = H ∪Hr has a sum labeling such that H has no working vertex the labeling is called an exclusive sum labeling of H with respect G. The exclusive sum number, ϵ(H), of a graph H is the smallest integer r such that G ∪Kr has an exclusive sum labeling.
The exclusive sum number is known in the following cases (see and ): for n ≥3, ϵ(Pn) = 2; for n ≥3, ϵ(Cn) = 3; for n ≥3, ϵ(Kn) = 2n −3; for n ≥4, ϵ(Fn) = n (fan of order n + 1); for n ≥4, ϵ(Wn) = n; ϵ(C(n) 3 ) = 2n (friendship graph–see §2.2); m ≥2, n ≥2, ϵ(Km,n) = m + n −1; for n ≥2, Sn = n (star of order n + 1); ϵ(Sm,n) = max{m, n} (double star); H2,n = 4n −5 (cocktail party graph); and ϵ(caterpillar G) = ∆(G). Dou showed that Hm,n is not a mod sum graph for m ≥3 and n ≥3; ρ(Hm,3) = m for m ≥3; Hm,n ∪ρ(Hm,n)K1 is exclusive for m ≥3 and ≥4; and m(n −1) ≤ρ(Hm,n) ≤mn(n −1)/2 for m ≥3 and n ≥4. Vilfred and Florida proved that ϵ(P3 × P3) = 4 and ϵ(Pn × P2) = 3. In Hegde and Vasudeva provide an O(n2) algorithm that produces an exclusive sum labeling of a graph with n vertices given its adjacency matrix.
In 2001 Kratochvil, Miller, and Nguyen proved that σ(G ∪H) ≤σ(G) + σ(H) −1.
In 2003 Miller, Ryan, Slamin, Sugeng, and Tuga posed the problem of finding the exclusive sum number of the disjoint union of graphs. In 2010 Wang and Li proved the following. Let G1 and G2 be graphs without isolated vertices, Li be an exclusive sum labeling of Gi ∪ϵ(Gi)K1, and Ci be the isolated set of Li for i = 1 and 2. If maxC1 and minC2 are relatively prime, then ϵ(G1∪G2) ≤ϵ(G1)+ϵ(G2)−1. Wang and Li also proved the following: ϵ(Kr,s) = s+r−1; ϵ(Kr,s−E(K2)) = s−1; for s ≥r ≥2, ϵ(Kr,s−E(rK2)) = s + r −3. For n ≥5 they prove: ϵ(Kn −E(Kn)) = 0; ϵ(Kn −E(Kn−1)) = n −1; for 2 ≤r < n/2, ϵ(Kn −E(Kr)) = 2n−4; for n/2 ≤r ≤n−2, ϵ(Kn −E(Kr)) = 3n−2r−4, and ϵ(Cn ⊙K1) is 3 or 4. They show that ϵ(C3 ⊙K1) = 3 and guess that for n ≥4 the class Ek of hypergraphs with a k-exclusive sum labeling is hereditary, but nontrivial to characterize even for k = 1. In Purcell, Ryan, Ryjááček, and Skyvová provided a complete description of the minimal forbidden induced subhypergraphs of E1 that are 3-uniform with maximum vertex degree 2. They also show that every hypertree has a 1-exclusive sum labeling and every combinatorial design does not.
A survey of exclusive sum labelings of graphs is given by Ryan in .
If ϵ(G) = ∆(G), then G is said to be an ∆-optimum summable graph. An exclusive sum labeling of a graph G using ∆(G) isolates is called a ∆-optimum exclusive sum labeling of G. Tuga, Miller, Ryan, and Ryjáček show that some families of trees that are ∆-optimum summable and some that are not. They prove that if G is a tree that has at least one vertex that has two or more neighbors that are not leaves then ϵ(G) = ∆(G).
the electronic journal of combinatorics (2023), #DS6 274 Koh, Miller, Smyth, and Wang show the following: the graphs obtained by identifying one end of a q-path with a vertex of a p-cycle are 1-optimum summable, and that two of these graphs can be joined via a new edge to create a 2-optimum summable graph; generalized θ-graphs are 2-optimum summable; θ(p, q, r) which consists of a pair of vertices joined by 3 independent paths of lengths p, q and r (with a few small exceptions) are 2-optimum summable; there exists a 3-optimum summable graph of order 4l + 3 for all l ≥1; how to construct for all k ≥4 a k-optimum summable graph; and if G is a k-optimum summable graph of order n, then n ≥2k.
In Javaid, Khalid, Ahmad, and Imran introduce a weaker version of sum labeling of graphs as follows. Let H = (V, E) be a simple, finite, undirected graph with |V | = p.
H is a weak sum graph if there exists a labeling L (called a w-sum) of the vertices of V by distinct positive integers such that (u, v) ∈E if there exists a vertex w ∈V such that L(w) = L(u)+L(v). (A sum graph also requires the “only if” condition). If H is a w-sum graph with the additional constraint that the labels L all fall in the range 1, . . . , p, then H is called a super weak sumgraph (sw-sumgraph). Because sumgraphs must have isolated vertices we may write H = G + Kδ, where G is connected and Kδ denotes δ isolated vertices If δ is a minimum with respect to G, we say that the sumgraph (respectively, w-sumgraph, sw-sumgraph) H is δ-optimal and that G is δ-optimal summable (respectively, w-summable, sw-summable). Javaid et al. prove: paths are 1-optimal sw-summable; cycles are 2-optimal sw-summable; wheels are 3-optimal sw-summable; Kn is (n −1)-optimal sw-summable; and G = Kn1,n2,...,nq are t-optimal sw-summable, where t is the minimum degree of any vertex in G. They also prove that for n ≥5, the Cayley graph Cay(Zn, ±1, ±2) is 4-optimal w-summable. They conjecture that all connected graphs are δ-optimal w-summable for some δ. See also and .
Grimaldi has investigated labeling the vertices of a graph G(V, E) with n vertices with distinct elements of the ring Zn so that xy ∈E whenever (x + y)−1 exists in Zn.
In his 2001 Ph. D. thesis Sutton introduced two methods of graph labelings with applications to storage and manipulation of relational database links specifically in mind. He calls a graph G = (Vp ∪Vi, E) a sum graph of Gp = (Vp, Ep) if there is an injective labeling λ of the vertices of G with non-negative integers with the property that uv ∈Ep if and only if λ(u) + λ(v) = λ(z) for some vertex z ∈G. The sum∗number, σ∗(Gp), is the minimum cardinality of a set of new vertices Vi such that there exists a sum graph of Gp on the set of vertices Vp ∪Vi. A mod sum graph of Gp is defined in the identical fashion except the sum λ(u) + λ(v) is taken modulo n where the vertex labels of G are restricted to {0, 1, 2, . . . , n −1}. The mod sum number, ρ∗(Gp), of a graph Gp is defined in the analogous way. Sum graphs are a generalization of sum graphs and mod sum graphs are a generalization of mod sum graphs. Sutton shows that every graph is an induced subgraph of a connected sum graph. Sutton poses the following conjectures: ρ(Hm,n) ≤mn for m, n ≥2; σ∗(Gp) ≤|Vp|; and ρ∗(Gp) ≤|Vp|.
The following table summarizes what is known about sum graphs, mod sum graphs, sum graphs, and mod sum graphs is reproduced from Sutton’s Ph. D. thesis . It was updated by J. Gallian in 2006. A question mark indicates the value is unknown. The results on sum and mod sum graphs are found in .
the electronic journal of combinatorics (2023), #DS6 275 Table 20: Summary of Sum Graph Labelings Graph σ(G) ρ(G) σ∗(G) ρ∗(G) K2 = S1 1 1 0 0 stars, Sn, n ≥2 1 0 0 0 trees Tn, n ≥3 when Tn ̸= Sn 1 0 1 0 C3 2 1 1 0 C4 3 0 2 0 Cn, n > 4 2 0 2 0 W4 4 0 2 0 Wn, n ≥5, n odd n n 2 0 Wn, n ≥6, n even n 2 + 2 2 2 0 fan, F4, 2 1 1 0 fans, Fn, n ≥5, n odd ?
2 1 0 fans, Fn, n ≥6, n even 3 2 1 0 Kn, n ≥4 2n −3 n n −2 0 cocktail party graphs, H2,n 4n −5 0 ?
0 C(t) n (n, t) ̸= (4, 1) (see §2.2) 2 ?
?
?
Kn,n 4n−3 2 n(n ≥3) ?
?
Km,n, 2nm ≥n ≥3 ?
n ?
?
Km,n m ≥3n −3, n ≥3, m odd ?
0 ?
0 Km,n, m ≥2n, n ≥3, m even ?
0 ?
0 Km,n, m < n kn−k 2 + m k−1 ?
?
?
k = ⌈ p 1 + (8m + n −1)(n −1)/2 ⌉ Kn,n −E(nK2), n ≥6 2n −3 n −2 ?
?
the electronic journal of combinatorics (2023), #DS6 276 7.2 Prime and Vertex Prime Labelings4 The notion of a prime labeling originated with Entringer and was introduced in a paper by Tout, Dabboucy, and Howalla .
A graph with vertex set V is said to have a prime labeling if its vertices are labeled with distinct integers 1, 2, . . . , |V | such that for each edge xy the labels assigned to x and y are relatively prime.
Around 1980, Entringer conjectured that all trees have a prime labeling. Little progress was made on this conjecture until 2011 when Haxell, Pikhurko, Taraz proved that all large trees are prime. Also, their method allowed them to determine the smallest size of a non-prime connected order-n graph for all large n, proving a conjecture of Rao in this range.
Among the classes of trees known to have prime labelings are: paths, stars, complete binary trees, spiders (i.e., trees with one vertex of degree at least 3 and with all other vertices with degree at most 2), olive trees (i.e., a rooted tree consisting of k branches such that the ith branch is a path of length i), all trees of order up to 50, palm trees (i.e., trees obtained by appending identical stars to each vertex of a path), banana trees, and binomial trees (the binomial tree B0 of order 0 consists of a single vertex; the binomial tree Bn of order n has a root vertex whose children are the roots of the binomial trees of order 0, 1, 2, . . . , n−1 (see , , , , and ). Tout, Dabboucy, and Howalla showed t-toe caterpillars (the internal vertices on the spine are regular in degree) are prime and that all caterpillars with maximum degree at most 5 are prime.
Seoud, Sonbaty, and Mahran provide necessary and sufficient conditions for a graph to be prime. They also give a procedure to determine whether or not a graph is prime. Other graphs with prime labelings include all cycles and the disjoint union of C2k and Cn . The complete graph Kn does not have a prime labeling for n ≥4 and Wn is prime if and only if n is even (see ). Lee, Wui, and Yeh proved that friendship graphs have prime labelings. Diefenderfer et al. and proved that the graph obtained by identifying a vertex of Cn with an endpoint of the star Sm where 1 ≤m ≤9, chains of Cn where n = 4, 6, or 8, Cn × P2 where n −1 is prime and n ≥4, generalized books Sn × Pm where 3 ≤m ≤7, and other families of uncylic graphs have prime vertex labelings.
Seoud, Diab, and Elsakhawi have shown the following graphs are prime: fans; helms; flowers (see §2.2); stars; K2,n; and K3,n unless n = 3 or 7. They also shown that Pn + Km (m ≥3) is not prime. Berliner, Dean, Hook, Marr, Mbirka, and McBee give consecutive cyclic prime labelings of certain classes of ladders. Although Kn,n does not have a prime labeling when n > 2, Berliner et al. give minimal prime labelings for all n-values 1 ≤n ≤23 and give conditions on m and n for which Km,n are prime. They provide specific values of n for m up to 13. Dissanayake, Abeysekara, Dhananjaya, Perera, and Ranasinghe provide necessary and sufficient conditions for K1,m,n to have a prime labeling. In Bigham, Donovan, Pack, Turley, and Wiglesworth investigated the existence of prime labelings of certain snake graphs.
Tout, Dabboucy, and Howalla proved that Cm ⊙Kn is prime for all m and n.
4I am grateful to John Asplund and N. Bradley Fox for their helpful comments on the results in this section.
the electronic journal of combinatorics (2023), #DS6 277 Vaidya and Prajapati proved that the graphs obtained by duplication of a vertex by a vertex in Pn and K1,n are prime graphs and the graphs obtained by duplication of a vertex by an edge, duplication of an edge by a vertex, duplication of an edge by an edge in Pn, K1,n, and Cn are prime graphs. They also proved that graph obtained by duplication of every vertex by an edge in Pn, K1,n, and Cn are not prime graphs. Ghorbani and Kamali proved that ladders have prime labelings. In Kowsalya and new Keerthika proved C3 ⊙K1,n and C4 ⊙K1,n are prime graphs.
For m and n at least 3, Seoud and Youssef define S(m) n , the (m, n)-gon star, as the graph obtained from the cycle Cn by joining the two end vertices of the path Pm−2 to every pair of consecutive vertices of the cycle such that each of the end vertices of the path is connected to exactly one vertex of the cycle. Seoud and Youssef have proved the following graphs have prime labelings: books; S(m) n ; Pn + K2 if and only if n = 2 or n is odd; Cn⊙K1 with a complete binary tree of order 2k −1 (k ≥2) attached at each pendent vertex, and that Cm-snakes are prime (see §2.2) for the definition). They also prove that every spanning subgraph of a prime graph is prime and every graph is a subgraph of a prime graph. They conjecture that all unicycle graphs have prime labelings. Diefenderfer, Hastings, Heath, Prawzinsky, Preston, White, and Whittemore proved that certain families of graphs that are special cases of Seoud and Youssef’s conjecture have prime labelings. Seoud and Youssef proved the following graphs are not prime: Cm + Cn; C2 n for n ≥4; P 2 n for n = 6 and for n ≥8; and Möbius ladders Mn for n even (see §2.3 for the definition). They also give an exact formula for the maximum number of edges in a prime graph of order n and an upper bound for the chromatic number of a prime graph.
Youssef and Elsakhawi have shown: the union of stars Sm ∪Sn, are prime; the union of cycles and stars Cm ∪Sn are prime; Km ∪Pn is prime if and only if m is at most 3 or if m = 4 and n is odd; Kn ⊙K1 is prime if and only if n ≤7; Kn ⊙K2 is prime if and only if n ≤16; 6Km ∪Sn is prime if and only if the number of primes less than or equal to m + n + 1 is at least m; and that the complement of every prime graph with order at least 20 is not prime. Michael and Youssef determined all self-complementary graphs that have prime labelings.
For positive integers m, k, q with k ≥3 and 1 ≤q ≤⌊k/2⌋, the uniform cycle snake graph Cm k,q is constructed by taking a path with m edges and replacing each edge by a k-cycle by identifying two vertices at distance q in the cycle with the vertices of the original path edge.
Salmasian has shown that every tree with n vertices (n ≥50) can be labeled with n integers between 1 and 4n such that every two adjacent vertices have relatively prime labels. Pikhurko has improved this by showing that for any c > 0 there is an N such that any tree of order n > N can be labeled with n integers between 1 and (1 + c)n such that labels of adjacent vertices are relatively prime.
Baskar Babujee and Vishnupriya proved the following graphs have prime label-ings: nP2, Pn ∪Pn ∪· · · ∪Pn, bistars (that is, the graphs obtained by joining the centers of two identical stars with an edge), and the graph obtained by subdividing the edge joining edge of a bistar. Baskar Babujee obtained prime labelings for the graphs: the electronic journal of combinatorics (2023), #DS6 278 (Pm ∪nK1) + K2, (Cm ∪nK1) + K2, (Pm ∪Cn ∪Kr) + K2, Cn ∪Cn+1, (2n −2)C2n (n > 1), Cn ∪mPk and the graph obtained by subdividing each edge of a star once. In Baskar Babujee and Jagadesh prove the following graphs have prime labelings: bistars Bm, n; P3 ⊙K1,n; the union of K1,n and the graph obtained from K1,n by appending a pendent edge to every pendent edge of K1,n; and the graph obtained by identifying the center of K1,n with the two endpoints and the middle vertex of P5.
In Vaidya and Prajapati prove the following graphs have prime labelings: a t-ply graph of prime order; graphs obtained by joining center vertices of wheels Wm and Wn to a new vertex w where m and n are even positive integers such that m + n + 3 = p and p and p−2 are twin primes; the disjoint union of the wheel W2n and a path; the graph obtained by identifying any vertex of a wheel W2n with an end vertex of a path; the graph obtained from a prime graph of order n by identifying an end vertex of a path with the vertex labeled with 1 or n; the graph obtained by identifying the center vertices of any number of fans (that is, a “multiple shell”); the graph obtained by identifying the center vertices of m wheels Wn1, Wn2, . . . , Wnm where each ni ≥4 is an even integer and each ni is relatively prime to 2 + Pi−1 k=1 nk for each i ∈{2, 3, . . . , m}. Prajapati and Suther provided results about the existence of prime labelings of graphs obtained from K2,n by the duplication of vertices and edges. In Prajapati and Gajjar provided conditions under which the disjoint union of two graphs admit a prime labeling.
They showed that C2n+1 × P2 is not prime, Wn is prime if and only if 3 ≤n ≤6, and, for a prime p ≥3 , Cp−1 ×P2 is prime and a wheel graph of odd order is switching invariant. In they proved that generalized Petersen graph P(n, k) is prime then n must be even and k must be odd and found some classes of generalized Petersen graphs that admit prime labelings.
In Wilson and Jini define the torch On as the graph that has n + 4 vertices and 2n + 3 edges with V (On) = {v −i | 1 ≤i ≤n + 4} and E(On) = {vivn+1 | 2 ≤i ≤ n−2}∪{vivn+3 | 2 ≤i ≤n−2}∪{v1vi | n ≤i ≤n+4}∪{vn−1vn, vnvn+2, vnvn+4, vn+1vn+3}.
They proved that the torch graph On is a prime graph.
The Knödel graphs W∆,n with n even and degree ∆, where 1 ≤∆≤⌊log2n⌋have vertices pairs (i, j) with i = 1, 2 and 0 ≤j ≤n/2 −1 where for every 0 ≤j ≤n/2 −1 and there is an edge between vertex (1, j) and every vertex (2, (j + 2k −1) mod n/2), for k = 0, 1, . . . , ∆−1. Haque, Lin, Yang, and Zhao have shown that W3,n is prime when n ≤130.
In Ezhal investigated prime labelings for some bipartiate and cycle related new graphs and prime labelings the joint sum of paths joining of bipartiate graphs and cycles.
Abughazaleh and Abughneim investigated the existence prime labelings of graphs new obtained by connecting wheels to paths and cycles.
In Schuchter and Wilson gave evidence for their conjecture that a generalized Petersen graph P(n, k) is prime if and only if it is bipartite, which occurs for n even and k. They show that it is true for all even n and odd k such that n ≤9000 and 1 ≤k ≤n 2.
They conjectured that all cubic bipartite graphs with at least 8 vertices are prime and verified it for all such graphs G, connected or not, satisfying 8 ≤V (G) ≤22.
In the 24th edition of this survey, the following was stated.“Schroeder proved the electronic journal of combinatorics (2023), #DS6 279 that every bipartite graph is prime except K3,3. This result establishes that the generalized Petersen graph P(n, k) is prime precisely when it is bipartite, the Knődel graph W3,n is prime for all even n ≥4, and the union of any number of even cycles is prime.” Schroeder’s result about bipartite graphs was later modified to state that for 4 ≤n ≤32, cubic bipartite graphs on 2n vertices are prime. Consequently, the statements about the generalized Petersen graph, the Knődel graph, and the union of even cycles are open. In Mostafa and Ghorbani characterized Hamiltonian k-coprime graphs, which implied a conjecture of Schroeder in that all 2-regular graphs are prime if every union of even cycles is prime. They then cited Schroeder’s original result to claim the 1982 conjecture by Tout, Dabboucy, Howalla that all 2-regular graphs are prime. So, that conjecture is still open as well.
Prajapati and Shah investigated the existence of prime labelings for graphs obtained by the duplication of paths, cycles, stars, and wheels. Sundaram, Ponraj, and Somasundaram investigated the prime labeling behavior of all graphs of order at most 6 and established that only one graph of order 4, one graph of order 5, and 42 graphs of order 6 are not prime.
Kansagara and Patel define the web graph without a center as the graph obtained by joining the n pendent vertices of Cn ⊙K1 to form an n-cycle and then adding a single pendent edge to each vertex of this outer cycle. They investigated prime labelings for web graphs without a center and graphs resulting from the subdivision of some specific edges in it. They also provide results on prime labeling of the union of a web graph without center with a wheel, generalized Jahangir graphs, and drum graphs.
Given a collection of graphs G1, . . . , Gn and some fixed vertex vi from each Gi, Lee, Wui, and Yeh define Amal{(Gi, vi)}, the amalgamation of {(Gi, vi)| i = 1, . . . , n}, as the graph obtained by taking the union of the Gi and identifying v1, v2, . . . , vn. They proved Amal{(Gi, vi)} has a prime labeling when Gi are paths and when Gi are cycles.
They also showed that the amalgamation of any number of copies of Wn, n odd, with a common vertex is not prime. They conjecture that for any tree T and any vertex v from T, the amalgamation of two or more copies of T with v in common is prime. They further conjecture that the amalgamation of two or more copies of Wn that share a common point is prime when n is even (n ̸= 4). Vilfred, Somasundaram, and Nicholas have proved this conjecture for the case that n ≡2 (mod 4) where the central vertices are identified.
Dean , and independently Ghorbani and Kamali , showed that the ladder Pn ×P2 has a prime labeling. In Curran and Ollis pointed out a flaw in Dean’s proof and showed that a stronger condition is needed for it to hold. They conjecture that this stronger condition is true. They also offer an alternative construction inspired by Dean’s approach that shows that if the Even Goldbach Conjecture and a particular strengthening of Lemoine’s Conjecture are true, then the Prime Ladder Conjecture follows.
Vilfred, Somasundaram, and Nicholas proved the following graphs are prime: helms; Pm × Pn where n is prime, m ≤3; double fans Pn + K2 if and only if n is odd; and cycles with a Pk-chord. They conjecture that Pm×Pn where m < n and n is prime is prime and ladders Pn × P2 are prime. The conjecture about grids was proved by Sundaram, Ponraj, and Somasundaram . In the same article they also showed that Pn × Pn is the electronic journal of combinatorics (2023), #DS6 280 prime when n is prime. Kanetkar proved: P6 × P6 is prime; Pn+1 × Pn+1 is prime when n is a prime with n ≡3 or 9 (mod 10) and (n+1)2+1 is also prime; and Pn×Pn+2 is prime when n is an odd prime with n ̸≡2 (mod 7). Curran showed that Pp ×Pn has a prime labeling for any odd prime p and any integer n such that p < n ≤p2. Combining the results of Sundaram, Ponraj, and Somasundaram and Curran, we have that Pp ×Pn has a prime labeling for any odd prime p and any integer n such that 1 ≤n ≤p2.
Seoud, El Sonbaty, and Abd El Rehim proved that for m = pn+t−1 −(t + n) where pi is the ith prime number in the natural order Kn ∪Kt,m is prime and graphs obtained from K2,n, (n ≥2) by adding p and q edges out from the two vertices of degree n of K2,n are prime. They also proved that if G is not prime, then G ∪K1,n is prime if π(n + m + 1) ≥m where m is the order of G and π(x) is the number of primes less than or equal to x.
Recall that C(k) n is the graph obtained from the k ≥2 copies of the cycle Cn by identifying exactly one vertex of each of these k copies of Cn. Patel and Vasava proved the following: C(j) n ∪C(k) m is a prime graph if and only if either n is even or m is even; C(2) 2n ∪C(2) 2m∪C(2) k is a prime graph for all n, m and k; C2n∪C2n∪C2n∪C2n∪C2m∪Ck is a prime graph for all n, m and k; and G = SN k=1 C(2) nk ∪ SM j=1 C(2) mj is not a prime graph if M ≤N −2 They also provided conditions for which G = C(2) 2n ∪C(2) 2m+1 ∪C(2) 2k+1 is a prime graph.
For any finite collection {Gi, uivi} of graphs Gi, each with a fixed edge uivi, Carlson defines the edge amalgamation Edgeamal{(Gi, uivi)} as the graph obtained by taking the union of all the Gi and identifying their fixed edges. The case where all the graphs are cycles she calls generalized books. She proves that all generalized books are prime graphs.
Moreover, she shows that graphs obtained by taking the union of cycles and identifying in each cycle the path Pn are also prime.
In Baskar Babujee proves that the maximum number of edges in a simple graph with n vertices that has a prime labeling is Pn k=2 φ(k). He also shows that the planar graphs having n vertices and 3(n −2) edges (i.e., the maximum number of edges for a planar graph with n vertices) obtained from Kn (n ≥5) with vertices v1, v2, . . . , vn by deleting the edges joining vs and vt for all s and t satisfying 3 ≤s ≤n−2 and s+2 ≤t ≤n has a prime labeling if and only if n is odd.
By showing that for every even n ≤2.468 × 109 there exists 1 ≤s ≤n −1 such that both n + s and 2n + s are prime, Schluchter, Schroeder, Cokus, Ellingson, Harris, Rarity, and Wilson prove the generalized Peterson graph P(n, 1) (which is isomorphic to Cn × P2) is prime for all even 4 ≤n ≤2.468 × 109. For a fixed n they also describe a method for labeling P(n, k) that is a prime labeling for multiple values of k. Using this method, they prove P(n, k) is prime for all even n ≤50 and odd k < n/2.
Yao, Cheng, Zhongfu, and Yao have shown: a tree of order p with maximum degree at least p/2 is prime; a tree of order p with maximum degree at least p/2 has a vertex subdivision that is prime; if a tree T has an edge u1u2 such that the two components T1 and T2 of T −u1u2 have the properties that dT1(u1) ≥|T1|/2 and dT2(u2) ≥|T2|/2, then T is prime when |T1| + |T2| is prime; if a tree T has two edges u1u2 and u2u3 the electronic journal of combinatorics (2023), #DS6 281 such that the three components T1, T2, and T3 of T −{u1u2, u2u3} have the properties that dT1(u1) ≥|T1|/2, dT2(u2) ≥|T2|/2, and dT3(u3) ≥|T3|/2, then T is prime when |T1| + |T2| + |T3| is prime.
Vaidya and Prajapati define a vertex switching Gv of a graph G as the graph obtained by taking a vertex v of G, removing all the edges incident to v and adding edges joining v to every other vertex that is not adjacent to v in G. They say a prime graph G is switching invariant if for every vertex v of G, the graph Gv obtained by switching the vertex v in G is also a prime graph. They prove: Pn and K1,n are switching invariant; the graph obtained by switching the center of a wheel is a prime graph; and the graph obtained by switching a rim vertex of Wn is a prime graph if n + 1 is a prime. They also prove that the graph obtained by switching a rim vertex in Wn is not a prime graph if n + 1 is an even integer greater than 9.
Prajapati and Gajjar prove the following graphs are prime: graphs obtained from Pm+1 and m copies of Cn by identifying each edge of Pm+1 with an edge of a corresponding copy of Cn; graphs obtained from Cm and m copies of Cn by identifying each edge of Cm with an edge of corresponding copy of Cn; for a prime p ≥3 and p −2 copies of Cp+1, the graph obtained by identifying one vertex of each copy of Cp+1 with corresponding pendent vertex of K1,p−2; for a prime p ≥3, Cp−1 × P2; and for a prime p ≥3, the graphs obtained by joining every rim vertex of a wheel graph Wp−1 with the corresponding vertex of Cp−1. They also prove that the complement of Wn is prime if and only if 3 ≤n ≤6; for odd n ≥3 Cn × P2 is not prime; and W2n is switching invariant.
Selvaraju and Moha proved that the one-point union of any number of cycles and the one-point union of any number of wheels at the center are prime graphs. Haque, Xiaohui, Yuansheng, and Pingzhong proved that the generalized Petersen graph P(n, k) is prime for all even n ≤2500 when k = 1 and for all even n ≤100 when k = 3 .
They show P(n, 3) is not prime for odd n and conjecture that P(n, 3) are prime for all even n.
In Seoud, El-Sonbaty, and Mahran discuss the primality of some corona graphs G ⊙H and conjecture that Kn ⊙Km is prime if and only if n ≤π(nm + n) + 1, where π(x) is the number of primes less than or equal to x. For m ≤20 they give the exact values of n for which Kn ⊙Km is prime. They also show that Km,n is prime if and only if min{m, n} ≤π(m + n) −π((m + n)/2) + 1.
In Patel and Vasava provided the following results about prime graphs: Wn∪Cm is a prime graph if and only if n and m both are even; (Pn + K2) ∪Cm is a prime graph if and only if either n = 2, or n is odd and m is even; C(2) n ∪Cm is a prime graph if and only if at least one of n and m is even; S(m) n ∪S(j) k is not a prime graph if m and j are even and n and k are odd (S(m) n denotes the (m, n)-gon star obtained from Cn and n copies of Pm−2 by joining the two end vertices of Pm−2 to each pair of consecutive vertices of the cycle such that each of the end vertices of the path is adjacent to exactly one vertex of the cycle); S(2m) 2n ∪S(2m) k is a prime graph for all n, m, and k; Hn S Bn is a prime graph for all n and m; Circ(n, k) is not a prime graph when n and k both are even; and Circ(n, k) is not a prime graph when n is odd.
In Patel and Kansagara prove that the following graphs are prime: (Cn ⊙K1)∪ the electronic journal of combinatorics (2023), #DS6 282 (Cn ⊙K1) for all n; Gn ∪Gn for all n (Gn is the gear graph); Hn ∪Hn for all n (Hn is the helm graph); Bn ∪(K1,n × P2) for all n; Hn ∪Gn for all n; (Cn ⊙K1) ∪Gn for all n; (Cn ⊙K1) ∪Hn for all n; and Cn(Cn) ∪Cn(Cn) if n ̸≡1 (mod 3) (Cn(Cn) is the graph obtained by taking barycentric subdivision of Cn and joining each newly inserted vertices of incident edges by an edge).
In Patel, Kansagava, and Vasva provided prime labeling of graphs obtained by taking barycentric subdivision of Wn, flower graphs, and (Cn ⊙K1) ∪(Cn ⊙K1); and the graphs by obtained by taking subdivision of certain edges Wn, Hn, (Cn ⊙K1) and (Cn ⊙K1) ∪(Cn ⊙K1).
Hamm, Hamm, and Way proved that if a complete k-partite k-uniform hyper-graph has enough vertices and every pod of vertices is large enough, then it does not have a prime labeling. They also proved that if a pod of vertices in a complete k-partite k-uniform is small enough, the graph it does have a prime labeling.
Klee, Lehmann, and Park we extended the notion of prime labeling to the Gaussian integers. They showed that paths, stars, spiders, graphs obtained by joining the centers of two stars with a path, and some firecrackers admit Gaussian prime labelings.
In Shrimali and Singh proved the following graphs have Gaussian vertex prime labelings: books; kayaks; the one point union of k copies of Wn; the one point union of k copies of a gear graph; the one point union of k copies a graph obtained from a wheel by replacing each cycle edge by Pn; and the one point union of k copies a graph obtained from a wheel by replacing each spoke by Pn. Kavitha and Jayalalitha proved the following graphs admit Gaussian prime distance labeling: grids, paths, bipartite graphs, non-trivial Dutch windmills, non-trivial stars, double stars, and spider trees.
The Prime Ladder Conjecture states that every ladder Pn × P2 is prime. This was proved by Dean in 2017. He conjectures that every integer n ≥50 has a canonical partition with at most three terms and he states that this conjecture was verified by computer up to 5,000,000.
Lau, Chu, Suhadak, Foo, and Ng introduced SD-prime cordial labelings as follows. Given a finite, simple graph G with n vertices and a bijection f : V (G) → {1, 2, . . . , n}, for each edge uv let S = f(u) + f(v) and D = |f(u) −f(v)|. For each edge uv define f ′ induced by f by assigning f ′(uv) = 1 if gcd(S, D) = 1 and f ′(uv) = 0 otherwise. Then f ′ is said to be SD-prime cordial if f ′(uv) = 1 for all edges uv. They provide results about paths, complete bipartite graphs, stars, double stars, wheels, fans, double fans, ladders, and grids. They conjecture that Pm × Pn is SD-prime for all m ≥2 and n ≥2. Lau, Shiu, Ng, and Jeyanthi give sufficient conditions for a theta graph to have an SD-prime cordial labeling, provide a way to construct new SD-prime cordial graphs from existing ones, and investigate SD-prime cordialness of some general graphs.
Lourdusamy and Patrick provide a way to construct SD-prime cordial graphs from an existing graph G with an SD-prime cordial labeling by identifying a vertex of G having a particular label with a vertex of maximum degree of a star or fan or with an endpoint of a path. In Lourdusamy and Patrick investigated SD-prime cordial labelings of subdivision graphs, splitting graphs, shadow graphs of stars and bistars, T(Pn), T(Cn), the graph obtained by duplication of each vertex of a path and a cycle by an edge, Qn, A(Tn), the electronic journal of combinatorics (2023), #DS6 283 triangular ladders, Pn ⊙K1, Cn ⊙K1, and jewel graphs. Shiu and Lau provided SD-prime labelings for some one-point unions of gear graphs.
The following definitions appear in , , , and . A double trian-gular snake DTn consists of two triangular snakes that have a common path; a double quadrilateral snake DQn consists of two quadrilateral snakes that have a common path; an alternate triangular snake A (Tn) is the graph obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertex vi (that is, every alternate edge of a path is replaced by C3); a double alternate triangular snake DA (Tn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to two new vertices vi and wi; an alternate quadrilateral snake A (Qn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertices vi and wi respectively and then joining vi and wi (that is, every alternate edge of a path is replaced by a cycle C4); a double alternate quadrilateral snake DA (Qn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertices vi, xi and wi and yi respectively and then joining vi and wi and xi and yi.
Prajapati and Vantiya proved that the following snake graphs have SD-prime cordial labelings: triangular (except for n = 3), alternate triangular, quadrilateral, alter-nate quadrilateral, double triangular, double alternate triangular, double quadrilateral, and double alternate quadrilateral. Lourdusamy, Wency, and Patrick proved that the union of stars and paths, subdivision of combs, subdivision of ladders, and the graph obtained by attaching a star at one end of a path are SD-prime graphs. They proved that the union of two SD-prime cordial graphs need not be SD-prime cordial graphs. Also, they proved that given a positive integer n, there is SD-prime cordial graph with n ver-tices. In Prajapati and Vantiya investigated SD-prime cordial labelings of various kinds of snakes involving K4. In they investigate the SD-prime cordial labeling of alternate k-polygonal snake graphs of type-1, type-2 and type-3.
Vaidya and Prajapati have introduced the notion of k-prime labeling. A k-prime labeling of a graph G is an injective function f : V (G) →{k+1, k+2, k+3, . . . , k+ |V (G)| −1} for some positive integer k that induces a function f + on the edges of G defined by f +(uv) = gcd(f(u), f(v)) such that gcd(f(u), f(v)) = 1 for all edges uv. A graph that admits a k-prime labeling is called a k-prime graph. They prove the following are prime graphs: a tadpole (that is, a graph obtained by identifying a vertex of a cycle to an end vertex of a path); the union of a prime graph of order n and a (n + 1)-prime graph; the graph obtained by identifying the vertex labeled with n in an n-prime graph with either of the vertices labeled with 1 or n in a prime graph of order n. Arockiamary and Vijayalakshmi proved that shell graphs, uniform shell butterfly graphs, and shell-flower graphs admit k-prime total labelings.
For connected graphs G1(p1, q1) and G2(p2, q2), G1 ˆ o G2 is the graph obtained by superimposing any selected vertex of G2 on any selected vertex of G1. Then for any graph k-prime graph G, Vijayalashmi and Arockiamary proved the following graphs have k-prime labelings: C2n∪C2n, G∪tPm(m > 1), G∪K1,n, G ˆ o Pn, Gˆ o K1,n and G ˆ o (Pn+K1).
Chaurasiya, Limda, Parmar, and Rupani proved that Pn ⊙3K1, Pn ⊙K1, ladders, new and quadrilateral snakes admit k-prime labelings.
the electronic journal of combinatorics (2023), #DS6 284 A dual of prime labelings has been introduced by Deretsky, Lee, and Mitchem .
They say a graph with edge set E has a vertex prime labeling if its edges can be labeled with distinct integers 1, . . . , |E| such that for each vertex of degree at least 2 the greatest common divisor of the labels on its incident edges is 1. Deretsky, Lee, and Mitchem show the following graphs have vertex prime labelings: forests; all connected graphs; C2k ∪Cn; C2m ∪C2n ∪C2k+1; C2m ∪C2n ∪C2t ∪Ck; and 5C2m. They further prove that a graph with exactly two components, one of which is not an odd cycle, has a vertex prime labeling and a 2-regular graph with at least two odd cycles does not have a vertex prime labeling. They conjecture that a 2-regular graph has a vertex prime labeling if and only if it does not have two odd cycles. Let G = St i=1 C2ni and N = Pt i=1 ni. In Borosh, Hensley and Hobbs proved that there is a positive constant n0 such that the conjecture of Deretsky et al. is true for the following cases: G is the disjoint union of at most seven cycles; G is a union of cycles all of the same even length 2n where n ≤150 000 or where n ≥n0; ni ≥(log N)4 log log log n for all i = 1, . . . , t; and when each C2ni is repeated at most ni times. They end their paper with a discussion of graphs whose components are all even cycles, and of graphs with some components that are not cycles and some components that are odd cycles. In Lavanya and Ganesa provided vertex prime labelings for graphs related to Cn ⊙K1.
In Bapat proved the following graphs have vertex prime labelings: kayak paddles KP(k, m, l); books; irregular books not necessarily with pages of the same size; triangular snakes; m-fold triangular snakes of length n obtained from a path v1, v2, . . . , vn, vn+1 by joining vi and vi+1 to new m vertices wi 1, wi 2, . . . , wi m, for i = 1, 2, . . . , n giving edges viwi j and wi jvi+1), for j = 1, . . . , m, i = 1, 2, . . . , n; m-fold petal sunflowers obtained from a cycle v1, v2, . . . , vn by joining vi and vi+1 to new m vertices wi 1, wi 2, . . . , wi m, for i = 1, 2, . . . , n giving edges viwi j and wi jvi+1) for j = 1, . . . , m, i = 1, 2, . . . , n(vn+1 = v1); and one-point unions of cycles not necessarily of the same length.
A bijection f from V (G to {1, 2, . . . , |V | + |E|} is said to be a total prime if for each edge uv the labels assigned to u and v are relatively prime and for each vertex of degree at least 2, the labels on the incident edges are relatively prime. A graph that admits a total prime labeling is called a total prime graph. In Prajapati and Gajjar defined a braided star graph as follows. Let a0 be the apex vertex and a1, a2, . . . , an−1, an be consecutive n rim vertices of Wn, (n ≥3). Let b1, b2, b3, . . . , b2n−1, b2n be consecutive 2n vertices of C2n (n > 1); and let c1, c2, c3, . . . , c2n−1, c2n be consecutive 2n vertices of a second copy of C2n. Join each ai to b2i−1 by an edge and b2i to c2i by an edge. For each i, join a new vertices di to each c2i−1 and c2i+1 by an edge taking the subscripts modulo n. They proved that braided stars are prime, total prime, and vertex prime.
In Arockiamary, Baskar Babujee, and Vijayalakshmi say a graph G has a k-prime total labeling if k is a positive integer and there is a bijection f : V (G) ∪E(G) → {k, k+1, k+2, . . . , k+V (G)∪E(G)−1} such that for every edge uv the labels f(u), f(v), and f(uv) are pairwise relatively prime. A graph that admits a k-prime total labeling is called a k-prime total graph. They proved that Pn is k-prime total for odd k, K1,n is k-prime total for odd k and n ≥2, the bistar B(m, n) is a k−prime total for odd k and m, n ≥2, and various other tree-related graphs are k-prime total.
the electronic journal of combinatorics (2023), #DS6 285 Jothi calls a graph G highly vertex prime if its edges can be labeled with distinct integers {1, 2, . . . , |E|} such that the labels assigned to any two adjacent edges are relatively prime. Such labeling is called a highly vertex prime labeling. He proves: if G is highly vertex prime then the line graph of G is prime; cycles are highly vertex prime; paths are highly vertex prime; Kn is highly vertex prime if and only if n ≤3; K1,n is highly vertex prime if and only if n ≤2; even cycles with a chord are highly vertex prime; Cp ∪Cq is not highly vertex prime when both p and q are odd; and crowns Cn ⊙K1 are highly vertex prime.
For a finite simple graph G(V, E) with n vertices and v ∈V let N(v) denote the open neighborhood of v. Patel and Shrimali say a bijective function f :→{1, 2, 3, . . . , n} is a neighborhood-prime labeling of G, if for every vertex v ∈V with deg(v) > 1, gcd {f(u) : u ∈N(v)} = 1. A graph that admits a neighborhood-prime labeling is called a neighborhood-prime graph. In , , and they prove the following graphs have a prime-neighborhood labeling: graphs with a vertex of degree |V | −1; paths; Cn if and only if n ̸≡2 (mod 4); helms; closed helms; flowers; graphs obtained by the du-plication of an arbitrary vertex of cycle or path; G1 + G2 where each of G1 and G2 have at least 2 vertices; Cn ∪Cm is a neighborhood-prime graph if and only if n ≡0 (mod 4) and m ≡0 (mod 4), or n ≡0 (mod 4) and m ≡1 (mod 2); Wm ∪Wn; the union of a finite number of paths; Pm × Pn; and the tensor product of two paths of the same order.
They also prove that if G is neighborhood-prime graph and v is a vertex in G that is not adjacent to any pendent vertices, then the graph obtained by duplicating the vertex v is neighborhood-prime .
Patel showed that the generalized Petersen graph P(n, k) is neighborhood-prime when the greatest common divisor of n and k is 1, 2, or 4 and that P(n, 8) is neighborhood-prime for all n. Patel and Kansagara proved that the snake graphs of the type Cm k are neighborhood-prime if and only if either k ̸≡2 (mod 4) or m ̸≡1 (mod 4) and that Cm k,2 and Cm k,3 are neighborhood-prime for all m ≥2. Rozario Raj and Sheriff gave neighborhood-prime labelings for books with triangular and rectangle pages. Shrimali, Rathod, and Vihol proved the following graphs are neighborhood-prime: graphs obtained from the helm Hn by identifying each vertex of degree 1 with a vertex of Wn, graphs obtained from the helm Hn by identifying each vertex of degree 1 with a vertex of the fan Fn graphs obtained by identifying each pendent vertex of Hn with a vertex of outer cycle of closed helm of Hn, and graphs obtained by identifying each pendent vertex of Hn by vertex of outer cycle of the Petersen graph.
Let G(V, E) be a graph with p vertices and Q edges. Rajesh Kumar and Mathew Varkey call a bijection from V (G) ∪E(G) to {1, 2, . . . , p + q} an total neighborhood prime labeling if for for each vertex of degree at least two, the gcd of labeling on its neighborhood vertices is 1 and for each vertex of degree at least two, the gcd of labeling on the induced edges is 1. They proved that paths, combs, and C4n+2 are total neighborhood-prime graphs. Shrimali and Pandya proved that the following graphs have total neighborhood-prime labelings: combs Pn ⊙K1, Pm ∪Pn, (Pm ⊙K1)∪(Pn ⊙K1), Wm ∪Wn, graphs obtained from a copy of Pn and n copies K1,n by joining the ith vertex of Pn with an edge to the center vertex of the ith copy of K1,m, Cn⊙mK1, and subdivisions of bistars the electronic journal of combinatorics (2023), #DS6 286 Bm,n. Shrimali, Rathod, and Vihol proved that following graphs are neighborhood-prime graphs: the graph obtained by identifying each pendent vertex of a helm Hn with a rim vertex of the wheel Wn; the graph obtained by identifying each pendent vertex of a helm Hn with a vertex of maximum degree of the fan Pn +K1; and the graph obtained by identifying each pendent vertex of Hn with a vertex of outer cycle of closed helm graph Hn. Rajesh Kumar proved that double stars, spiders, caterpillars, and firecrackers admit total neighborhood-prime labelings.
In Rajesh Kumar and Mathew Varkey extend the neighborhood-prime labeling concept to Gaussian integers. Using the spiral order on the Gaussian integers, they showed the following graphs have Gaussian neighborhood-prime labelings: graphs obtained by connecting the centers of two stars with a path, combs Pn ⊙K1, spiders, Cn where n ̸= 2 mod 4, C4 (n ≥4) with a cord, graphs obtained by switching of any vertex Cn, and graphs obtained by duplicating arbitrary vertex of Cn. Rajesh Kumar, Jerome, and Santhosh Kumar extended the Gaussian neighborhood-prime labeling concept to Gaussian total neighborhood-prime labeling. Among the graphs they prove to have a Gaussian neighborhood-prime labeling are: paths, stars, trees with one vertex of degree at least 3 and all other vertices having degree 1 or 2, and caterpillars. In Shrimali and Trivedi provided results depending upon the Hamiltonicity of a graph that guarantee that the graph is Gaussian neighborhood-prime graph under the spiral order and proved that generalized Petersen graphs are Gaussian neighborhood-prime graphs under the spiral order.
In Pandya and Shrimali introduce a vertex-edge neighborhood-prime label-ing of graphs as follows.
For a graph G an injective function f : V (G) ∪E(G) → {1, 2, . . . , |V (G)| + |E(G|} is said to be a vertex-edge neighborhood-prime labeling if it has the following properties: If u has degree 1, then gcd{f(w), f(uw)} = 1 taken over all vertices w adjacent to u; if u has degree greater than 1, then gcd{f(w)} = 1 taken over all vertices w adjacent to u and gcd{f(wu)} = 1 taken over all vertices w adjacent to u. A graph that admits vertex-edge neighborhood-prime labeling is called a vertex-edge neighborhood-prime graph. They give vertex-edge neighborhood-prime labelings for paths, helms, Cn ⊕K1, bistars, the central edge subdivision of bistars, and subdivisions of edges of bistars. They observe that every vertex-edge neighborhood-prime graph is a total neighborhood-prime graph and that a total neighborhood-prime graph that does not have a vertex of degree 1 is vertex-edge neighborhood-prime. Shrimali and Rathod proved that coconut trees, double coconut trees, spiders with legs of equal length, olive trees, combs, and F(n, 2)-firecrackers have vertex-edge neighborhood-prime labelings.
Let G(V, E) be a graph with p verticies and q edges. For s ∈V , let NV [s] = {s, t ∈ V | t is adjacent to s} and NE[s] = {s ∈V, e ∈E | e is incident with s}. A one-one mapping φ : V ∪E →{1, 2, . . . , p + q} is called a VECN prime labeling if, for each vertex s with d(s) ≥1, where d(s) is the degree of s, gcd(φ(s), φ(t)) | t ∈NV [s] = 1 and gcd(φ(s), φ(e) | e ∈N(E[s]) = 1. A graph that admits VECN prime labeling is called a VECN prime graph. In Sunitha and Revathi investigate VECN prime labelings of new m-fold quadrilateral snakes, m-fold alternate quadrilateral snakes, and m-fold irregular triangular snakes.
the electronic journal of combinatorics (2023), #DS6 287 Patel and Ghodasara proved the following graphs are neighborhood-prime: one point union C(k) n (k ≥2, n ≥3) of k copies of cycle Cn, the barycentric subdivision of wheels and gears, the middle and total graph of crowns Cn ⊙K1 (n ≥3), the square of crowns, tadpoles T(n, l) (n ≥3, l ≥1), cycles, and umbrellas.
In Delman, Koilraj, and Raj gave neighborhood-prime labelings of arbitrary super subdivision of helms, tadpoles, and triangular snakes.
Let G(V, E) be a graph with p verticies and q edges. For s ∈V , let NV [s] = {s, t ∈ V | t is adjacent to s} and NE[s] = {s ∈V, e ∈E | e is incident with s}. A one-one mapping φ : V ∪E →{1, 2, . . . , p + q} is called a VECN prime labeling if, for each vertex s with d(s) ≥1, where d(s) is the degree of s, gcd(φ(s), φ(t)) | t ∈NV [s] = 1 and gcd(φ(s), φ(e) | e ∈N(E[s]) = 1. A graph that admits VECN prime labeling is called a VECN prime graph. In Sunitha and Revathi investigate VECN prime labelings of new m-fold quadrilateral snakes, m-fold alternate quadrilateral snakes, and m-fold irregular triangular snakes.
In Prajapati and Shah introduce an odd prime labeling as follows. Let G(V, E) be a graph. A bijection f from V to {1, 3, . . . , 2|V | −1} is called an odd prime labeling if for each edge uv, gcd(f(u), f(v)) = 1. A graph that admits odd prime labeling is called an odd prime graph. They prove paths, ladders, complete bipartite graphs, wheels, gears, flowers, helms, closed helms, and generalized Petersen graphs P(n, 2) are odd prime and conjecture that generalized Petersen graphs P(n, k) and every prime graph is an odd prime graph. In they proved the following graphs are odd prime graphs: graphs obtained by duplication of a vertex of paths, stars, and wheels, and graphs obtained by duplication of an edge of cycles, stars, and wheels. Carter and Fox prove that the disjoint union of cycles; cycle chains; snakes; books; triangular, pentagonal, and hexagonal stacked prisms; spiders; perfect binary trees; special cases of caterpillars; and firecrackers are odd prime. They characterize when odd prime labelings exist for powers of paths and cycles and make progress towards proving the conjecture that all prime graphs are also odd prime. Meena and Gajalakshmi proved that the corona product of Cn × P2 and K1, K2, and K3 are odd prime graphs.
For a graph G(V, E) with p vertices and q edges Shiu, Lau, and Lee call a bijection f from E to {1, 2, . . . , q} an edge-prime labeling if for each edge uv in E, we have gcd(f +(u), f +(v)) = 1, where f +(u) = Σf(uw) over all uw ∈E. A graph that admits an edge-prime labeling is called an edge-prime graph. A bijection f from E to {1, 2, . . . , q} is an semi-edge-prime labeling if for each edge uv in E, we have gcd(f +(u), f +(v)) = 1 or f +(u) = f +(v). They obtained a necessary and sufficient condition for the disjoint union of paths to be edge-prime, proved that all 2-regular graphs are edge-prime, proved that many bipartite and tripartite graphs are edge-prime (or not edge-prime), and showed that certain bipartite and tripartite graphs are semi-edge-prime graphs. In Lau, Lee, and Shiu proved that if G is a cubic graph and every component is of order 4, 6 or 8, then G is edge-prime if and only if G ̸≈K4 or nK3,3 and n = 2 or 3 (mod 4). They conjectured that a connected cubic graph G is not edge-prime if and only if G ≈K4.
In Jagadesh and Baskar Babujee introduced an edge vertex prime labeling of a graph G as an injection f from V (G) ∪E(G) to {1, 2, . . . , |V (G)| + |E(G)|} such that the electronic journal of combinatorics (2023), #DS6 288 for every edge uv, the labels f(u), f(v), and f(uv) are pairwise relatively prime. A graph that admits such a labeling is called an edge vertex prime graph.
They proved that paths, cycles, and stars are are edge vertex prime. In and Parmar proved that wheels, fans, friendship graphs, and K2,n are edge vertex prime.
Simaringa and Muthukumaran proved that following graphs have edge vertex prime labelings: triangular and rectangular books, butterfly graphs, Kn ∪K1,m, K1,m + K1, Km ∪Kn, Jahangir graphs Jn,3 and Jn,4. In Simaringa and Muthukumaran investigated the existence of edge vertex prime labelings for crowns, unions of cycles, and wheel relate graphs. Shrimali and Parmar proved that the following graphs have edge vertex prime labelings: bistars B(m, n), n-centipede trees, coconut trees obtained from the path Pn by appending m new pendent edges at an end vertex of Pn), double coconut trees (graphs obtained by attaching n > 1 pendent vertices to one end of the path Pr and m > 1 pendent vertices to the other end of path Pr), and special classes of banana trees and firecrackers. In Shrimali and Parmar showed that helms and graphs obtained by joining a fan (n ≥2) and C2n by sharing a common vertex are edge vertex prime graphs and that Pn1 ∪Pn2 ∪· · · ∪Pnk and K1,n1 ∪K1,n2 ∪· · · ∪K1,nk are not an edge vertex prime graphs. They also provided a necessary condition for being edge vertex prime graph.
Rilwan and Radha investigated edge vertex prime labelings and super edge vertex prime labelings of Cayley graphs and Cayley digraphs of finite groups. Baskar Babujee and Jagadesh proved generalized stars K1,n1,n2,n3,...,nm and generalized cycle stars Cn ∗(Pn1 ∪Pn2m ∪. . . ∪Pnm)) that are obtained by joining one of the pendant vertices of each of the paths with an edge to any vertex of Cn, admit edge vertex prime labelings.
Simaringa and Muthukumaran proved that the following graphs are edge vertex prime graphs: Cm ∪K1,n, Cm ∪Pn, K2,m ∪Cn, Cn ∪Cn when n = 0 or 2 mod n, and the one point union of wheel and cycle-related graphs. They also proved that for graphs G with |V (G)| + |E(G)| even G ∪K1,n and G ∪Pn are edge vertex prime graphs.
In Youssef and Almoreed gave a new variation of the prime labeling as follows: A graph G(V, E) has an odd prime labeling if its vertices can be labeled with distinct odd integers from 1 to 2|V (G)| −1 such that for every edge xy in E the labels assigned to the vertices of x and y are relatively prime. A graph that admits an odd prime labeling is called an odd prime graph. They provided some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. They conjectured that every prime graph is odd prime graph.
A coprime labeling of vertices of a graph G with distinct labels from the set {1, . . . , m} for some integer m ≥n such that adjacent labels are relatively prime. The minimum value m for which G has a coprime labeling is defined as the minimum coprime number , denoted by pr(G), and a coprime labeling of G with largest label being pr(G) is called a minimum coprime labeling of G. Obviously, if G is a prime graph of order n, then pr(G) = n.
In Asplund and Fox focus on the problem of determining the minimum coprime number for graphs that have been shown to not be prime. Amomg them are Kn (n ≥4), W2n+1, and the union of odd cycles. C. Lee determined the minimum coprime number of coronas of complete graphs with empty graphs, the joins of two paths, and prisms. She also proved that gears are prime, double wheels DWn are prime if and only the electronic journal of combinatorics (2023), #DS6 289 if n is even, and the graph that obtained by attaching P2 to each vertex of Cn followed by attaching the star Sm at its center to each pendent vertex is prime.
In Periasamy, Venugopal, and Rozario Raj introduced the kth Fibonacci prime labeling of graphs as follows. A kth Fibonacci prime labeling of a graph G(V, E) with |V (G)| = n is an injective function g from V (G) to {fk, fk+1, . . . , fk+n−1} where fk is the kth Fibonacci number, that induces a function g∗from E(G) to the non-negative integers defined by g(uv) = gcd(g(u), g(v)) = 1 for all uv ∈E(G). A graph that admits a kth Fibonacci prime labeling and is called a kth Fibonacci prime graph. They proved that paths, cycles, Pn ⊙K1, triangular snakes, quadrilateral snakes, and tadpoles are kth Fibonacci prime graphs.
The tables following summarize the state of knowledge about prime labelings and vertex prime labelings.
In the table, P means prime labeling exists, and VP means vertex prime labeling exists. A question mark following an abbreviation indicates that the graph is conjectured to have the corresponding property.
the electronic journal of combinatorics (2023), #DS6 290 Table 21: Summary of Prime Labelings Graph Types Notes Pn P stars P complete binary trees P spiders P trees P?
Cn P Cn ∪C2m P Kn P iff n ≤3 Wn P iff n is even helms P fans P flowers P K2,n P K3,n P n ̸= 3, 7 Pn + Km not P n ≥3 Pn + K2 P iff n = 2 or n is odd books P Cm + Cn not P C2 n not P n ≥4 P 2 n not P n ≥6, n ̸= 7 Continued on next page the electronic journal of combinatorics (2023), #DS6 291 Table 21 – Continued from previous page Graph Types Notes Mn (Möbius ladders) not P n even Sm ∪Sn P Cm ∪Sn P Km ∪Sn P iff no. of primes ≤m + n + 1 is at least m Kn ⊙K1 P iff n ≤7 Pm × Pn (grids) P m ≤3, n prime m prime > 2, p < n ≤p2 Cn ⊙Ki (crowns) P Pn ⊙K2 P iff n ̸= 2 Cm-snakes (see §2.2) P unicyclic P?
Table 22: Summary of Vertex Prime Labelings Graph Types Notes Cm + Cn not P C2 n not P n ≥4 Pn not P n = 6, n ≥8 M2n (Möbius ladders) not P connected graphs VP forests VP Continued on next page the electronic journal of combinatorics (2023), #DS6 292 Table 22 – Continued from previous page Graph Types Notes C2m ∪Cn VP C2m ∪C2n ∪C2k+1 VP C2m ∪C2n ∪C2t ∪Ck VP 5C2m VP G ∪H VP if G, H are connected and one is not an odd cycle 2-regular graph G not VP G has at least 2 odd cycles VP?
iff G has at most 1 odd cycle 7.3 Edge-graceful Labelings In 1985, Lo introduced the notion of edge-graceful graphs. A graph G(V, E) is said to be edge-graceful if there exists a bijection f from E to {1, 2, . . . , |E|} such that the induced mapping f + from V to {0, 1, . . . , |V |−1} given by f +(x) = (P f(xy)) (mod |V |) taken over all edges xy is a bijection. Note that an edge-graceful graph is antimagic (see §6.1). A necessary condition for a graph with p vertices and q edges to be edge-graceful is that q(q + 1) ≡p(p + 1)/2 (mod p). Lee notes that this necessary condition extends to any multigraph with p vertices and q edges. It was conjectured by Lee that any connected simple (p, q)-graph with q(q + 1) ≡p(p −1)/2 (mod p) vertices is edge-graceful. Lee, Kitagaki, Young, and Kocay prove that the conjecture is true for maximal outerplanar graphs. Lee and Murthy proved that Kn is edge-graceful if and only if n ̸≡2 (mod 4).
(An edge-graceful labeling given in for Kn for n ̸≡2 (mod 4) is incorrect.) Lee notes that a multigraph with p ≡2 (mod 4) vertices is not edge-graceful and conjectures that this condition is sufficient for the edge-gracefulness of connected graphs. Lee has conjectured that all trees of odd order are edge-graceful. Small has proved that spiders for which every vertex has odd degree with the property that the distance from the vertex of degree greater than 2 to each end vertex is the same are edge-graceful. Keene and Simoson proved that all spiders of odd order with exactly three end vertices are edge-graceful. Cabaniss, Low, and Mitchem have shown that regular spiders of odd order are edge-graceful. For a (p, q) the electronic journal of combinatorics (2023), #DS6 293 connected edge-graceful graph G with q = kp + r, where kis aninteger and 0 ≤r < p.
Kayathri and Amutha proved that every edge-graceful labeling f of G induces ((k + 1)!)r(k!)p−r edge-graceful labelings of G.
Lee and Seah have shown that Kn,n,...,n is edge-graceful if and only if n is odd and the number of partite sets is either odd or a multiple of 4. Lee and Seah have also proved that Ck n (the kth power of Cn) is edge-graceful for k < ⌊n/2⌋if and only if n is odd and Ck n is edge-graceful for k ≥⌊n/2⌋if and only if n ̸≡2 (mod 4) (see also ). Lee, Seah, and Wang gave a complete characterization of edge-graceful P k n graphs. Shiu, Lam, and Cheng proved that the composition of the path P3 and any null graph of odd order is edge-graceful. Uma and Mazuda Shanofer prove that C2n+1 ⊙K2 is edge graceful and the graphs obtained by starting with Cn and, for each edge of Cn, adjoining a copy of Cn that shares an edge with the starting copy (the flower graph FLn) is not edge-graceful.
Lo proved that all odd cycles are edge-graceful and Wilson and Riskin proved the Cartesian product of any number of odd cycles is edge-graceful. Lee, Ma, Valdes, and Tong investigated the edge-gracefulness of grids Pm×Pn. The necessity condition of Lo that a (p, q) graph must satisfy q(q+1) ≡0 or p/2 (mod p) severely limits the possibilities. Lee et al. prove the following: P2 × Pn is not edge-graceful for all n > 1; P3 × Pn is edge-graceful if and only if n = 1 or n = 4; P4 × Pn is edge-graceful if and only if n = 3 or n = 4; P5 × Pn is edge-graceful if and only if n = 1; P2m × P2n is edge-graceful if and only if m = n = 2. They conjecture that for all m, n ≥10 of the form m = (2k + 1)(4k + 1), n = (2k + 1)(4k + 3), the grids Pm × Pn are edge-graceful. Riskin and Weidman proved: if G is an edge-graceful 2r-regular graph with p vertices and q edges and (r, kp) = 1, then kG is edge-graceful when k is odd; when n and k are odd, kCr n is edge-graceful; and if G is the Cartesian product of an odd number of odd cycles and k is odd, then kG is edge-graceful. They conjecture that the disjoint union of an odd number of copies of a 2r-regular edge-graceful graph is edge-graceful.
Shiu, Lee, and Schaffer investigated the edge-gracefulness of multigraphs derived from paths, combs, and spiders obtained by replacing each edge by k parallel edges. Lee, Ng, Ho, and Saba construct edge-graceful multigraphs starting with paths and spiders by adding certain edges to the original graphs. Lee and Seah have also investigated edge-gracefulness of various multigraphs.
Lee and Seah (see ) define a sunflower graph SF(n) as the graph obtained by starting with an n-cycle with consecutive vertices v1, v2, . . . , vn and creating new vertices w1, w2, . . . , wn with wi connected to vi and vi+1 (vn+1 is v1). In they prove that SF(n) is edge-graceful if and only if n is even. In the same paper they prove that C3 is the only triangular snake that is edge-graceful. Lee and Seah prove that for k ≤n/2, Ck n is edge-graceful if and only if n is odd, and for k ≥n/2, Ck n is edge-graceful if and only if n ̸≡2 (mod 4). Lee, Seah, and Lo (see ) have proved that for n odd, C2n ∪C2n+1, Cn ∪C2n+2, and Cn ∪C4n are edge-graceful. They also show that for odd k and odd n, kCn is edge-graceful. Lee and Seah (see ) prove that the generalized Petersen graph P(n, k) (see Section 2.7 for the definition) is edge-graceful if and only if n is even and k < n/2. In particular, P(n, 1) = Cn × P2 is edge-graceful if and only if n the electronic journal of combinatorics (2023), #DS6 294 is even.
Schaffer and Lee proved that Cm × Cn (m > 2, n > 2) is edge-graceful if and only if m and n are odd. They also showed that if G and H are edge-graceful regular graphs of odd order then G × H is edge-graceful and that if G and H are edge-graceful graphs where G is c-regular of odd order m and H is d-regular of odd order n, then G×H is edge-magic if gcd(c, n) = gcd(d, m) = 1. They further show that if H has odd order, is 2d-regular and edge-graceful with gcd(d, m) = 1, then C2m × H is edge-magic, and if G is odd-regular, edge-graceful of even order m that is not divisible by 3, and G can be partitioned into 1-factors, then G × Cm is edge-graceful.
In 1987 Lee (see ) conjectured that C2m ∪C2n+1 is edge-graceful for all m and n except for C4 ∪C3. Lee, Seah, and Lo have proved this for the case that m = n and m is odd. They also prove: the disjoint union of an odd number copies of Cm is edge-graceful when m is odd; Cn ∪C2n+2 is edge-graceful; and Cn ∪C4n is edge-graceful for n odd. Bu gave necessary and sufficient conditions for graphs of the form mCn ∪Pn−1 to be edge-graceful.
Kendrick and Lee (see ) proved that there are only finitely many n for which Km,n is edge-graceful and they completely solve the problem for m = 2 and m = 3. Ho, Lee, and Seah use S(n; a1, a2, . . . , ak) where n is odd and 1 ≤a1 ≤a2 ≤· · · ≤ ak < n/2 to denote the (n, nk)-multigraph with vertices v0, v1, . . . , vn−1 and edge set {vivj| i ̸= j, i −j ≡at (mod n) for t = 1, 2, . . . , k}. They prove that all such multigraphs are edge-graceful. Lee and Pritikin (see ) prove that the Möbius ladders (see §2.2 for definition) of order 4n are edge-graceful. Lee, Tong, and Seah have conjectured that the total graph of a (p, p)-graph is edge-graceful if and only if p is even. They have proved this conjecture for cycles. In Khodkar and Vinhage proved that there exists a super edge-graceful labeling of the total graph of K1,n and the total graph of Cn. Wang and Zang proved that a regular graph of odd degree is edge-graceful if it contains either a quasi-prism factor or a claw factor.
Kuang, Lee, Mitchem, and Wang have conjectured that unicyclic graphs of odd order are edge-graceful. They have verified this conjecture in the following cases: graphs obtained by identifying an endpoint of a path Pm with a vertex of Cn when m + n is even; crowns with one pendent edge deleted; graphs obtained from crowns by identifying an endpoint of Pm, m odd, with a vertex of degree 1; amalgamations of a cycle and a star obtained by identifying the center of the star with a cycle vertex where the resulting graph has odd order; graphs obtained from Cn by joining a pendent edge to n −1 of the cycle vertices and two pendent edges to the remaining cycle vertex.
In Wang and Zhang introduced the notion called edge-graceful deficiency, which is a parameter to measure how close a graph is away from being an edge-graceful graph.
The edge-graceful deficiency of a graph G is the minimum value of k such that the edge labeling f E →{1, 2, . . . , q+k} is edge-graceful. They proved that an odd regular graph is edge-graceful if it contains a quasi-prism factor or a claw factor and completely determine the edge-graceful deficiency of Hamiltonian regular graphs of even degree.
Gayathri and Subbiah say a graph G(V, E) has a strong edge-graceful labeling if there is an injection f from the E to {1, 2, 3, . . . , ⟨3|E|/2⟩} such that the induced the electronic journal of combinatorics (2023), #DS6 295 mapping f + from V defined by f +(u) = (Σf(uv)) (mod 2|V |) taken all edges uv is an injection. They proved the following graphs have strong edge graceful labelings: Pn(n ≥ 3), Cn, K1,n(n ≥2), crowns Cn ⊙K1, and fans Pn + K1(n ≥2). In his Ph.D. thesis Subbiah provided edge-graceful and strong edge-graceful labelings for a large variety of graphs. Among them are bistars, twigs, y-trees, spiders, flags, kites, friendship graphs, mirror of paths, flowers, sunflowers, graphs obtained by identifying a vertex of a cycle with an endpoint of a star, and K2 ⊙Cn, and various disjoint unions of path, cycles, and stars.
Hefetz has shown that a graph G(V, E) of the form G = H ∪f1 ∪f2 ∪· · · ∪fr where H = (V, E′) is edge-graceful and the fi’s are 2-factors is also edge-graceful and that a regular graph of even degree that has a 2-factor consisting of k cycles each of length t where k and t are odd is edge-graceful.
Bača and Holländer investigated a generalization of edge-graceful labeling called (a, b)-consecutive labelings.
A connected graph G(V, E) is said to have an (a, b)-consecutive labeling where a is a nonnegative integer and b is a positive proper divisor of |V |, if there is a bijection from E to {1, 2, . . . , |E|} such that if each vertex v is assigned the sum of all edges incident to v the vertex labels are distinct and they can be partitioned into |V |/b intervals Wj = [wmin = (j −1)b + (j −1)a, wmin + jb + (j −1)a −1], where 1 ≤j ≤p/b and wmin is the minimum value of the vertices. They present necessary conditions for (a, b)-consecutive labelings and describe (a, b)-consecutive labelings of the generalized Petersen graphs for some values of a and b.
A graph with p vertices and q edges is said to be k-edge-graceful if its edges can be labeled with k, k+1, . . . , k+q−1 such that the sums of the edges incident to each vertex are distinct modulo p. In Lee and Wang show that for each k ̸= 1 there are only finitely many trees that are k-edge graceful (there are infinitely many 1-edge graceful trees).
They describe completely the k-edge-graceful trees for k = 0, 2, 3, 4, and 5. Gayathri and Sarada Devi obtained some necessary conditions and characterizations for k-edge-gracefulness of trees. They also proved that specific families of trees are edge-graceful and k-edge-graceful and conjecture that all odd trees are k-edge-graceful.
Gayathri and Sarada Devi defined a k-even edge-graceful labeling of a (p, q) graph G(V, E) as an injection f from E to {2k −1, 2k, 2k + 1, . . . , 2k + 2q −2} such that the induced mapping f + of V defined by f +(x) = P f(xy) (mod 2s) taken over all edges xy, are distinct and even, where s = max{p, q} and k is a positive integer. A graph G that admits a k-even-edge-graceful labeling is called a k-even-edge-graceful graph.
In , , , and Gayathri and Sarada Devi investigate the k-even edge-gracefulness of a wide variety of graphs. Among them are: paths; stars; bistars; cycles with a pendent edge; cycles with a cord; crowns Cn ⊙K1; graphs obtained from Pn by replacing each edge by a fixed number of parallel edges; and sparklers (paths with a star appended at an endpoint of the path).
In 1991 Lee defined the edge-graceful spectrum of a graph G as the set of all nonnegative integers k such that G has a k-edge graceful labeling. In Lee, Wang, Ng, and Wang determine the edge-graceful spectrum of the following graphs: G ⊙K1 the electronic journal of combinatorics (2023), #DS6 296 where G is an even cycle with one chord; two even cycles of the same order joined by an edge; and two even cycles of the same order sharing a common vertex with an arbitrary number of pendent edges attached at the common vertex (butterfly graph). Lee, Chen, and Wang have determined the edge-graceful spectra for various cases of cycles with a chord and for certain cases of graphs obtained by joining two disjoint cycles with an edge (i.e., dumbbell graphs). More generally, Shiu, Ling, and Low call a connected with p vertices and p + 1 edges bicyclic.
In particular, the family of bicyclic graphs includes the one-point union of two cycles, two cycles joined by a path and cycles with one cord. In they determine the edge-graceful spectra of bicyclic graphs that do not have pendent edges. Kang, Lee, and Wang determined the edge-graceful spectra of wheels and Wang, Hsiao, and Lee determined the edge-graceful spectra of the square of Pn for odd n. Results about the edge-graceful spectra of three types of (p, p+1)-graphs are given by Chen, Lee, and Wang . In Wang and Lee determine the edge-graceful spectra of the one-point union of two cycles, the corona product of the one-point union of two cycles with K1, and the cycles with one chord.
Lee, Levesque, Lo, and Schaffer investigate the edge-graceful spectra of cylin-ders. They prove: for odd n ≥3 and m ≡2 (mod) 4, the spectra of Cn × Pm is ∅; for m = 3 and m ≡0, 1 or 3 (mod 4), the spectra of C4 ×Pm is ∅; for even n ≥4, the spectra of Cn × P2 is all natural numbers; the spectra of Cn × P4 is all odd positive integers if and only if n ≡3 (mod) 4; and Cn × P4 is all even positive integers if and only if n ≡1 (mod) 4. They conjecture that C4 × Pm is k-edge-graceful for some k if and only if m ≡2 (mod) 4. Shiu, Ling, and Low determine the edge-graceful spectra of all connected bicyclic graphs without pendent edges.
A graph G(V, E) is called super edge-graceful if there is a bijection f from E to {0, ±1, ±2, . . . , ±(|E|−1)/2} when |E| is odd and from E to {±1, ±2, . . . , ±|E|/2} when |E| is even such that the induced vertex labeling f ∗defined by f ∗(u) = Σf(uv) over all edges uv is a bijection from V to {0, ±1, ±2, . . . , ±(p −1)/2} when p is odd and from V to {±1, ±2, . . . , ±p/2} when p is even. Lee, Wang, Nowak, and Wei proved the following: K1,n is super-edge-magic if and only if n is even; the double star DS(m, n) (that is, the graph obtained by joining the centers of K1,m and K1,n with an edge) is super edge-graceful if and only if m and n are both odd. They conjecture that all trees of odd order are super edge-graceful. In Lee, Su, and Wei exhibit a family of trees of odd orders which are super edge-graceful. Chung, Lee, Gao and Schaffer posed the problems of characterizing the paths and tress of diameter 4 that are super edge-graceful.
In Chung, Lee, and Gao prove various classes of caterpillars, combs, and amal-gamations of combs and stars of even order are super edge-graceful. Lee, Sun, Wei, Wen, and Yiu proved that trees obtained by starting with the paths the P2n+2 or P2n+3 and identifying each internal vertex with an endpoint of a path of length 2 are super edge-graceful.
Shiu has shown that Cn×P2 is super-edge-graceful for all n ≥2. More generally, he defines a family of graphs that includes Cn × P2 and generalized Petersen graphs are follows. For any permutation θ on n symbols without a fixed point the θ-Petersen graph P(n; θ) is the graph with vertex set {u1, u2, . . . , un} ∪{v1, v2, . . . , vn} and edge set the electronic journal of combinatorics (2023), #DS6 297 {uiui+1, uiwi, wiwθ(i) | 1 ≤i ≤n} where addition of subscripts is done modulo n. (The graph P(n; θ) need not be simple.) Shiu proves that P(n; θ) is super-edge-graceful for all n ≥2. He also shows that certain other families of connected cubic multigraphs are super-edge-graceful and conjectures that every connected cubic of multigraph except K4 and the graph with 2 vertices and 3 edges is super-edge-graceful.
In Shiu and Lam investigated the super-edge-gracefulness of fans and wheel-like graphs. They showed that fans F2n and wheels W2n are super-edge-graceful. Although F3 and W3 are not super-edge-graceful the general cases F2n+1 and W2n+1 are open. For a positive integer n1 and even positive integers n2, n3, . . . , nm they define an m-level wheel as follows. A wheel is a 1-level wheel and the cycle of the wheel is the 1-level cycle. An i-level wheel is obtained from an (i−1)-level wheel by appending ni/2 pairs of edges from any number of vertices of the i −1-level cycle to ni new vertices that form the vertices in the i-level cycle. They prove that all m-level wheels are super-edge-graceful. They also prove that for n odd Cm ⊙Kn is super-edge-graceful, for odd m ≥3 and even n ≥2 Cm ⊙Kn is edge-graceful, and for m ≥3 and n ≥1 Cm ⊙Kn is super-edge-graceful. For a cycle Cm with consecutive vertices v1, v2, . . . , vm and nonnegative integers n1, n2, . . . , nm they define the graph A(m; n1, n2, . . . , nm) as the graph obtained from Cm by attaching ni edges to the vertex vi for 1 ≤i ≤m. They prove A(m; n1, n2, . . . , nm) is super-edge-graceful if m is odd and A(m; n1, n2, . . . , nm) is super-edge-graceful if m is even and all the ni are positive and have the same parity. Chung, Lee, Gao, and Schaffer provide super edge-graceful labelings for various even order paths, spiders and disjoint unions of two stars. In Chung and Lee characterize spiders of even orders that are not super-edge-graceful and exhibit some spiders of even order of diameter at most four that are super-edge-graceful. They raised the question of which paths are super edge-graceful.
This was answered by Cichacz, Fronček, and Xu who showed that the only paths that are not super edge-graceful are P2 and P4. Cichacz et al. also proved that the only cycles that are not super edge-graceful are C4 and C6. Gao and Zhang proved that some cases of caterpillars are super edge-graceful.
In Chung, Lee, Gao, and Schaffer asked for a characterize trees of diameter 4 that are super edge-graceful. Krop, Mutiso, and Raridan provide a super edge-graceful labelings for all caterpillars and even size lobsters of diameter 4 that permit such labelings.
They also provide super edge-graceful labelings for several families of odd size lobsters of diameter 4. They were unable to find general methods that describe super edge-graceful labelings for a few families of odd size lobsters of diameter 4, although they are able to show that certain lobsters in these families are super-edge graceful. They conclude with three conjectures about rooted trees of height 2 and diameter 4.
Although it is not the case that a super edge-graceful graph is edge-graceful, Lee, Chen, Yera, and Wang proved that if G is a super edge-graceful with p vertices and q edges and q ≡−1 (mod p) when q is even, or q ≡0 (mod p) when q is odd, then G is also edge-graceful. They also prove: the graph obtained from a connected super edge-graceful unicyclic graph of even order by joining any two nonadjacent vertices by an edge is super edge-graceful; the graph obtained from a super edge-graceful graph with p vertices and p + 1 edges by appending two edges to any vertex is super edge-graceful; the electronic journal of combinatorics (2023), #DS6 298 and the one-point union of two identical cycles is super edge-graceful. Collins, Magnant, and Wang present a stronger concept of “tight” super-edge-graceful labeling. Such a super-edge graceful labeling has an additional constraint on the edge and vertices with the largest and smallest labels. They use this concept to recursively construct tight super-edge graceful trees of any order.
Gayathri, Duraisamy, and Tamilselvi calls a (p, q)-graph with q ≥p even edge-graceful if there is an injection f from the set of edges to {1, 2, 3, . . . , 2q} such that the values of the induced mapping f + from the vertex set to {0, 1, 2, . . . , 2q −1} given by f +(x) = (Σf(xy))(mod 2q) over all edges xy are distinct and even. In and Gayathri et al. prove the following: cycles are even edge-graceful if and only if the cycles are odd; even cycles with one pendent edge are even edge-graceful; wheels are even edge-graceful; gears (see §2.2 for the definition) are not even edge-graceful; fans Pn + K1 are even edge-graceful; C4 ∪Pm for all m are even edge-graceful; C2n+1 ∪P2n+1 are even edge-graceful; crowns Cn⊙K1 are even edge-graceful; C(m) n (see §2.2 for the definition) are even edge-graceful; sunflowers (see §3.7 for the definition) are even edge-graceful; triangular snakes (see §2.2 for the definition) are even edge-graceful; closed helms (see §2.2 for the definition) with the center vertex removed are even edge-graceful; graphs decomposable into two odd Hamiltonian cycles are even edge-graceful; and odd order graphs that are decomposable into three Hamiltonian cycles are even edge-graceful.
In Gayathri and Duraisamy generalized the definition of even edge-graceful to include (p, q)-graphs with q < p by changing the modulus from 2q the maximum of 2q and 2p. With this version of the definition, they have shown that trees of even order are not even edge-graceful whereas, for odd order graphs, the following are even edge-graceful: banana trees (see §2.1 for the definition); graphs obtained joining the centers of two stars by a path; Pn⊙K1,m; graphs obtained by identifying an endpoint from each of any number of copies of P3 and P2; bistars (that is, graphs obtained by joining the centers of two stars with an edge); and graphs obtained by appending the endpoint of a path to the center of a star. They define odd edge-graceful graphs in the analogous way and provide a few results about such graphs. Kathiresan, Muthumari, and Ramalakshmi proved that Kc n ∨2K2 and the flower graph FLn (n ≥3) odd edge-graceful labelings. They further prove that Pa,b (the graph obtained by identifying the end points of b internally disjoint paths each of length a) admit odd edge-graceful labelings when a and b are odd, whereas Pa,2 is not odd edge-graceful when a ≥2. In Aljohanis and Daoud provided edge- new odd graceful labelings five new families of wheel-related graph and proved that Sn are not edge-odd graceful when n is odd.
Paley graphs are dense undirected graphs raised from the vertices as elements of an appropriate finite field by joining pairs of vertices that differ by a quadratic residue.
In , Kamaraj and Thangakani study the construction of edge even (odd) graceful labeling for Paley graphs and prove that Paley graphs of prime order are edge even (odd) graceful. Lee, Pan, and Tsai call a graph G with p vertices and q edges vertex-graceful if there exists a labeling f V (G) →{1, 2, . . . , p} such that the induced labeling f + from E(G) to Zq defined by f +(uv) = f(u) + f(v) (mod q) is a bijection. Vertex-graceful graphs can be viewed the dual of edge-graceful graphs. They call a vertex-graceful the electronic journal of combinatorics (2023), #DS6 299 graph strong vertex-graceful if the values of f +(E(G) are consecutive.
They observe that the class of vertex-graceful graphs properly contains the super edge-magic graphs and strong vertex-graceful graphs are super edge-magic. They provide vertex-graceful and strong vertex-graceful labelings for various (p, p + 1)-graphs of small order and their amalgamations.
Shiu and Wong proved the one-point union of an m-cycle and an n-cycle is vertex-graceful only if m+n ≡0 (mod 4); for k ≥2, C(3, 4k−3) is strong vertex-graceful; C(2n + 3, 2n + 1) is strong vertex-graceful for n ≥1; and if the one-point union of two cycles is vertex-graceful, then it is also strong vertex-graceful. In Somashekara and Veena found the number of (n, 2n −3) strong vertex graceful graphs. Gao, Zhang, and Xu proved that Cn, Cn ⊙K1 and Cn ⊙K1,t are vertex-graceful if n is odd; Cn is super vertex-graceful if n ̸= 4, 6; and Cn ⊙K1 is super vertex-graceful if n is even. They proposed two conjectures on (super)vertex-graceful labelings.
As a dual to super edge-graceful graphs Lee and Wei define a graph G(V, E) to be super vertex-graceful if there is a bijection f from V to {±1, ±2, . . . , ±(|V |−1)/2} when |V | is odd and from V to {±1, ±2, . . . , ±|V |/2} when |V | is even such that the induced edge labeling f ∗defined by f +(uv) = f(u)+f(v) over all edges uv is a bijection from E to {0, ±1, ±2, . . . , ±(|E|−1)/2} when |E| is odd and from E to {±1, ±2, . . . , ±|E|/2} when |E| is even. They show: for m and n1, n2, . . . , nm each at least 3, Pn1 × Pn2 × · · · × Pnm is not super vertex-graceful; for n odd, books K1,n × P2 are not super vertex-graceful; for n ≥3, P 2 n × P2 is super vertex-graceful if and only if n = 3, 4, or 5; and Cm × Cn is not super vertex-graceful. They conjecture that Pn × Pn is super vertex-graceful for n ≥3.
In Lee and Wong generalize super edge-vertex graphs by defining a graph G(V, E) to be P(a)Q(1)-super vertex-graceful if there is a bijection f from V to {0, ±a, ±(a + 1), . . . , ±(a −1 + (|V | −1)/2)} when |V | is odd and from V to {±a, ±(a + 1), . . . , ±(a−1+|V |/2)} when |V | is even such that the induced edge labeling f ∗defined by f +(uv) = f(u) + f(v) over all edges uv is a bijection from E to {0, ±1, ±2, . . . , ±(|E| − 1)/2} when |E| is odd and from E to {±1, ±2, . . . , ±|E|/2} when |E| is even.
They show various classes of unicyclic graphs are P(a)Q(1)-super vertex-graceful. In Lee, Leung, and Ng more simply refer to P(1)Q(1)-super vertex-graceful graphs as super vertex-graceful and show how to construct a variety of unicyclic graphs that are super vertex-graceful. They conjecture that every unicyclic graph is an induced subgraph of a super vertex-graceful unicyclic graph. Lee and Leung determine which trees of di-ameter at most 6 are super vertex-graceful graphs and propose two conjectures. Lee, Ng, and Sun found many classes of caterpillars that are super vertex-graceful. In Gao shows that the generalized butterfly graph Bt n is super vertex-graceful when t > 0 is even, B0 n is super vertex-graceful when n ≡0 or 3 (mod 4), and C(t) 3 is super vertex-graceful if and only if t = 1, 2, 3, 5, or 7.
In Chopra and Lee define a graph G(V, E) to be Q(a)P(b)-super edge-graceful if there is a bijection f from E to {±a, ±(a+1), . . . , ±(a+(|E|−2)/2)} when |E| is even and from E to {0, ±a, ±(a+1), . . . , ±(a+(|E|−3)/2)} when |E| is odd and f +(u) is equal to the sum of f(uv) over all edges uv is a bijection from V to {±b, ±(b+1), . . . , (|V |−2)/2} when |V | is even and from V to {0, ±b, ±(b + 1), . . . , ±(|V | −3)/2} when |V | is odd.
the electronic journal of combinatorics (2023), #DS6 300 They say a graph is strongly super edge-graceful if it is Q(a)P(b)-super edge-graceful for all a ≥1. Among their results are: a star with n pendent edges is strongly super edge-graceful if and only if n is even; wheels with n spokes are strongly super edge-graceful if and only if n is even; coronas Cn ⊙K1 are strongly super edge-graceful for all n ≥3; and double stars DS(m, n) are strongly super edge-graceful in the case that m is odd and at least 3 and n is even and at least 2 and in the case that both m and n are odd and one of them is at least 3. Lee, Song, and Valdés investigate the Q(a)P(b)-super edge-gracefulness of wheels Wn for n = 3, 4, 5, and 6.
In Lee, Wang, and Yera proved that some Eulerian graphs are super edge-graceful, but not edge-graceful, and that some are edge-graceful, but not super edge-graceful. They also showed that a Rosa-type condition for Eulerian super edge-graceful graphs does not exist and pose some conjectures, one of which was: For which n, is Kn is super edge-graceful? It was known that the complete graphs Kn for n = 3, 5, 6, 7, 8 are super edge-graceful and K4 is not super edge-graceful. Khodkar, Rasi, and Sheikholeslami, answered this question by proving that all complete graphs of order n ≥3, except 4, are super edge-graceful.
In 1997 Yilmaz and Cahit introduced a weaker version of edge-graceful called E-cordial. Let G be a graph with vertex set V and edge set E and let f a function from E to {0, 1}. Define f on V by f(v) = P{f(uv)|uv ∈E} (mod 2). The function f is called an E-cordial labeling of G if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1 and the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1.
A graph that admits an E-cordial labeling is called E-cordial. Yilmaz and Cahit prove the following graphs are E-cordial: trees with n vertices if and only if n ̸≡2 (mod 4); Kn if and only if n ̸≡2 (mod 4); Km,n if and only if m + n ̸≡2 (mod 4); Cn if and only if n ̸≡2 (mod 4); regular graphs of degree 1 on 2n vertices if and only if n is even; friendship graphs C(n) 3 for all n (see §2.2 for the definition); fans Fn if and only if n ̸≡1 (mod 4); and wheels Wn if and only if n ̸≡1 (mod 4). They observe that graphs with n ≡2 (mod 4) vertices can not be E-cordial.
They generalized E-cordial labelings to Ek-cordial (k > 1) labelings by replacing {0, 1} by {0, 1, 2, . . . , k −1}. Of course, E2-cordial is the same as E-cordial (see §3.7). Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of an E-cordial graph thereby ruling out any possibility of obtaining any forbidden subgraph characterization for E-cordial graphs. They also proved that a connected graph can be embedded as an induced subgraph of an E-cordial connected graph and every planar graph can be embedded as an induced subgraph of an E-cordial planar graph.
Liu, liu, and Wu provide two necessary conditions for a graph to be Ek-cordial and prove that Pn (n ≥3) is Ep-cordial for odd p. They also discuss the E2-cordiality of graphs that have a subgraph that is a 1-factor. Devaraj has shown that M(m, n), the mirror graph of K(m, n), is E-cordial when m+n is even and the generalized Petersen graph P(n, k) is E-cordial when n is even. (Recall that P(n, 1) is Cn × P2.) In Vaidya and Vyas prove that the following graphs are E-cordial: the mirror graphs (see §2.3 for the definition) even paths, even cycles, and the hypercube are E-cordial. In they show that the middle graph, the total graph, and the splitting the electronic journal of combinatorics (2023), #DS6 301 graph of a path are E-cordial and the composition of P2n with P2. (See §2.7 for the definitions of middle, total and splitting graphs.) In Vaidya and Lekha prove the following graphs are E-cordial: the graph obtained by duplication of a vertex (see §2.7 for the definition) of a cycle; the graph obtained by duplication of an edge (see §2.7 for the definition) of a cycle; the graph obtained by joining of two copies of even cycle by an edge; the splitting graph of an even cycle; and the shadow graph (see §3.8 for the definition) of a path of even order.
Vaidya and Vyas proved the following graphs have E-cordial labelings: K2n×P2; P2n × P2; Wn × P2 for odd n; and K1,n × P2 for odd n. Vaidya and Vyas proved that the Möbius ladders, the middle graph of Cn, and crowns Cn ⊙K1 are E-cordial graphs for even n while bistars Bn,n and its square graph B2 n,n are E-cordial graphs for odd n. In and Vaidya and Vyas proved the following graphs are E-cordial: flowers, closed helms, double triangular snakes, gears, graphs obtained by switching of an arbitrary vertex in Cn except n ≡2 (mod 4), switching of rim vertex in wheel Wn except n ≡1 (mod 4), switching of an apex vertex in helms, and switching of an apex vertex in closed helms. Sugumaran and Vishnu Prakash proved that the following graphs are E-cordial: theta graphs, duplication of any vertex in theta graphs, switching of any vertex in theta graphs, the fusion of any two vertices in theta graphs, and the open star of n copies of a fixed theta graph (that is, the graph obtained by replacing each endpoint vertex of K1,n by copies of the theta graph).
In her PhD thesis Vanitha defines a (p, q) graph G to be directed edge-graceful if there exists an orientation of G and a labeling of the arcs of G with {1, 2, . . . , q} such that the induced mapping g on V defined by g(v) = |f +(v) −f −(v)| (mod p) is a bijection where, f +(v) is the sum of the labels of all arcs with head v and f −(v) is the sum of the labels of all arcs with tail v. She proves that a necessary condition for a graph with p vertices to be directed edge-graceful is that p is odd. Among the numerous graphs that she proved to be directed edge-graceful are: odd paths, odd cycles, fans F2n (n ≥2), wheels W2n, nC3-snakes, butterfly graphs Bn (two even cycles of the same order sharing a common vertex with an arbitrary number of pendent edges attached at the common vertex), K1,2n (n ≥2), odd order y-trees with at least 5 vertices, flags Fl2n (the cycle C2n with one pendent edge), festoon graphs Pn ⊙mK1, the graphs Tm,n,t obtained from a path Pt (t ≥2) by appending m edges at one endpoint of Pt and n edges at the other endpoint of Pt, Cn 3 , P3 ∪K1,2n+1, P5 ∪K1,2n+1, and K1,2n ∪K1,2m+1.
In Boonklurb, Ruamkaew, and Singhun use C(c×a) to denote the graph obtained by identifying a vertex of c cycles Ca to a single vertex. They provide directed edge-graceful labelings for C(c × a) in the case when a ≥3 is an odd integer and c ≥2 and in the case where a and c are even with a ≥4 and c ≥2.
The table following summarizes the state of knowledge about edge-graceful labelings.
In the table EG means edge-graceful labeling exists. A question mark following an ab-breviation indicates that the graph is conjectured to have the corresponding property.
the electronic journal of combinatorics (2023), #DS6 302 Table 23: Summary of Edge-graceful Labelings Graph Types Notes Kn EG iff n ̸≡2 (mod 4) odd order trees EG?
Kn,n,...,n (k terms) EG iff n is odd or k ̸≡2 (mod 4) Ck n, k < ⌊n/2⌋ EG iff n is odd Ck n, k ≥⌊n/2⌋ EG iff n ̸≡2 (mod 4) P3[Kn] EG n is odd M4n (Möbius ladders) EG odd order dragons EG odd order unicyclic graphs EG?
P2m × P2n EG iff m = n = 2 Cn ∪P2 EG n even C2n ∪C2n+1 EG n odd Cn ∪C2n+2 EG Cn ∪C4n EG n odd C2m ∪C2n+1 EG?
(m, n) ̸= (4, 3) odd P(n, k) generalized Petersen EG n even, k < n/2 graph Cm × Cn EG?
(m, n) ̸= (4, 3) the electronic journal of combinatorics (2023), #DS6 303 7.4 Radio Labelings In 2001 Chartrand, Erwin, Zhang, and Harary were motivated by regulations for channel assignments of FM radio stations to introduce radio labelings of graphs. A radio labeling of a connected graph G is an injection c from the vertices of G to the natural numbers such that d(u, v) + |c(u) −c(v)| ≥1 + diam(G) for every two distinct vertices u and v of G. The radio number of c, rn(c), is the maximum number assigned to any vertex of G. The radio number of G, rn(G), is the minimum value of rn(c) taken over all radio labelings c of G. Chartrand et al. and Zhang gave bounds for the radio numbers of cycles. The exact values for the radio numbers for paths and cycles were reported by Liu and Zhu as follows: for odd n ≥3, rn(Pn) = (n −1)2/2 + 2; for even n ≥4, rn(Pn) = n2/2 −n + 1; rn(C4k) = (k + 2)(k −2)/2 + 1; rn(C4k+1) = (k + 1)(k−1)/2; rn(C4k+2) = (k+2)(k−2)/2+1; and rn(C4k+3) = (k+2)(k−1)/2. However, Chartrand, Erwin, and Zhang obtained different values than Liu and Zhu for P4 and P5. Chartrand, Erwin, and Zhang proved: rn(Pn) ≤(n −1)(n −2)/2 + n/2 + 1 when n is even; rn(Pn) ≤n(n −1)/2 + 1 when n is odd; rn(Pn) < rn(Pn+1) (n > 1); for a connected graph G of diameter d, rn(G) ≥(d + 1)2/4 + 1 when d is odd; and rn(G) ≥d(d + 2)/4 + 1 when d is even. Karst, Langowitz, Oehrlein, and Troxell provide general lower bounds for rck(Cn) for all cycles Cn when k ≥diam(Cn) and show that these bounds are exact values when k = diam(Cn) + 1. In Bloomfield, Liu, and Ramirez combine a lower bound approach with the cyclic group structure to determine the value of rnk(Cn) for k ≥n−3. For d ≤k < n−3, they obtain the values of rnk(Cn) when n and k have the same parity, and prove partial results when n and k and have different parities. These results extend the known values of rnd(Cn) and rnd+1(Cn) shown by Liu and Zhu , and by Karst, Langowitz, Oehrlein, and Troxell in , respectively.
Benson, Porter, and Tomova determined the radio numbers of all graphs of order n and diameter n −2.
In Liu obtained lower bounds for the radio number of trees and the radio number of spiders (trees with at most one vertex of degree greater than 2) and characterized the graphs that achieve these bounds. Bantva, Vaidya, and Zhou and give a lower bound for the radio number of trees and a necessary and sufficient condition for their bound to be achieved. They determine the radio number for symmetric trees (that is, trees whose non-leaf vertices all have the same degree and whose leaf vertices all have the same eccentricity), banana trees, and firecracker trees. In Kola and Panigrahi provide the radio number for a class of caterpillars. In Bantva and Der-Fen Liu give a lower bound for the radio number of the Cartesian product of two trees. Moreover, they present three necessary and sufficient conditions, and three sufficient conditions for the product of two trees to achieve this bound. Applying these results, they determine the radio number of the Cartesian product of two stars as well as a path and a star. Nazeer, Khan, Kousar, and Nazeer investigated the radio number for some families of generalized caterpillar graphs. Chakraborty, Nandi, Sen, and Supraja gave the exact values of rck(G) for powers of paths where the diameter of the graph is strictly less than k. Their proof provides a linear time algorithm for providing such labelings.
the electronic journal of combinatorics (2023), #DS6 304 Chartrand, Erwin, Zhang, and Harary proved: rn(Kn1,n2,...,nk) = n1 + n2 + · · · + nk +k −1; if G is a connected graph of order n and diameter 2, then n ≤rn(G) ≤2n−2; and for every pair of integers k and n with n ≤k ≤2n−2, there exists a connected graph of order n and diameter 2 with rn(G) = k. They further provide a characterization of connected graphs of order n and diameter 2 with prescribed radio number.
Fernandez, Flores, Tomova, and Wyels proved rn(Kn) = n; rn(Wn) = n + 2; and the radio number of the gear graph obtained from Wn by inserting a vertex between each vertex of the rim is 4n + 2.
Morris-Rivera, Tomova, Wyels, and Yeager determine the radio number of Cn × Cn. Martinez, Ortiz, Tomova, and Wyels define generalized prisms, denoted Zn,s, s ≥1, n ≥s, as the graphs with vertex set {(i, j) | i = 1, 2 and j = 1, . . . , n} and edge set {((i, j), (i, j ±1))}∪{((1, i), (2, i+σ)) | σ = − s−1 2 . . . , 0, . . . , s 2 }. They determine the radio number of Zn,s for s = 1, 2 and 3.
In and Bantva determines the radio number for three families of trees obtained by taking a graph operation on a given tree or a family of trees and the radio number for the middle graph of paths. Sun determined the radio number for the middle graph of a dandelion (the one-point union of a star and a path), which generalized Bantva’s results in . Zhang, Nazeer, Habib, Zia, and Ren determined the radio number for the generalized Petersen graphs P(4k + 2, 2) and provided a lower bound for P(4k, 2).
In Vasoya and Bantva gave a lower bound for the radio number for the Cartesian product of the generalized Petersen graph and a tree.
They give two necessary and sufficient conditions and three other sufficient conditions to achieve the lower bound.
They use these results to determine the radio number for the Cartesian product of the Petersen graph and stars.
Elrokh, Badr, Al-Shamiri, and Ramadhan provided upper bounds for the radio number of triangular snakes and double triangular snakes that improved on those given by Badr and Moussa and by Saha and Panigrahi in and .
DeVito, Niedzialomski, and Warren determined the radio number of all diameter 3 Hamming graphs and showed that an infinite subset of them have radio graceful labelings.
In Alkasasbeh, · Badr, · Attiya, and · Shabana provided radio numbers of the new friendship graphs F3,k, F4,k, F5,k, and F6,k. Kang, Nazeer, Nazeer, Kousar and Jung new determined the radio number for a family of caterpillars In Mari and Jeyaraj new determined the radio number of the supersub−division of Pn (n ≥3) with K2,m.
Gomathi and Venugopal determined lower and upper bounds of the radio new antipodal numbers of triangular grids and the upper bound of the ladder-related graphs.
In Gomathi, Venugopal, and Jose determined bounds of the radio numbers of new triangular grid graphs, triangular ladders, and pagoda graphs. In Basunia, Saha, new and Tiwary determined the radio numbers of some special type of path related graphs.
William and Kenneth also determined the radio numbers of some special type of new path related graphs.
In Ramyal and Sooryanarayana generalized the notion of radio labeling as fol-lows.
Let M be a subset of non-negative integers and (M, ⋆) be a monoid with the identity e.
A radio ⋆-labeling of graph G(V, E) is a mapping f : V →M such that |f(u)f(v)| ⋆d(u, v) ≥diam(G) + 1 −e, for all u, v ∈V . The radio ⋆-number rn ⋆(f) of the electronic journal of combinatorics (2023), #DS6 305 a radio ⋆-labeling f of G is the maximum label assigned to a vertex of G. The radio ⋆-number of G, denoted by rn ⋆(G), is the minimum of rn ⋆(f) taken over all radio ⋆-labeling f of G. They completely determine rn ⋆(G) of some transformation graphs of paths and cycles where ⋆is the usual multiplication of integers.
Sooryanarayana and Ranghunath define a radio labeling f of a graph G to be a consecutive radio labeling of G if f(V (G)) = {1, 2, . . . , |V (G)|}. They call a graph for which a consecutive radio labeling exists radio graceful. In her Ph.D. thesis Niedzialomski (see also ) investigated the existence of radio graceful labelings of Cartesian products of graphs. Among her results are: for n ≥3 and 1 ≤t ≤n −1 the Cartesian product of t copies of Kn is radio graceful; for 2 ≤p ≤n2 the Cartesian product of p · ⌈n/p⌉copies of Kn is radio graceful; the Cartesian product Kn1 × Kn2, . . . , ×Kns is radio graceful when n1, n2, . . . , ns are relatively prime; certain families of generalized Petersen graphs are radio graceful; and the Cartesian product of t ≥1 + n(n2 −1)/6 copies of Kn is not radio graceful. Locke and Niedzialomski proved that Kn × P is radio graceful where P is the Peterson graph. Wyels and Tomova proved that that P × P is radio graceful. Sooryanarayana and Raghunath determine the values of n for which C3 n is radio graceful.
For a simple connected graph G with at least 3 vertices the triameter of G, denoted by tr(G), is the smallest positive integer M such that d(u, v) + d(v, w) + d(w, u) ≤M for every triplet u, v, and w in V (G). Saha and Basunia proved that a graph with diameter 2 is radio graceful if and only if it contains a Hamiltonian path. They also prove that if both a graph G and its complement G are connected and tr(G) > 9, then G is radio graceful only if G contains a Hamiltonian path and tr(G) = 5 or 6.
The generalized gear graph Jt,n is obtained from a wheel Wn by introducing t-vertices between every pair (vi, vi+1) of adjacent vertices on the n-cycle of wheel. Ali, Rahim, Ali, and Farooq gave an upper bound for the radio number of generalized gear graph, which coincided with the lower bound found in and . They proved for t < n−1 and n ≥7, rn(Jt,n) = (nt2+4nt+3n+4)/2. They pose the determination of the radio number of Jt,n when n ≤7 and t > n −1 as an open problem. In Sunitha determine the new radio number of double triangular snake graphs.
Saha and Panigrahi determined the radio number of the toroidal grid Cm × Cn when at least one of m and n is an even integer and gave a lower bound for the radio number when both m and n are odd integers. Liu and Xie determined the radio numbers of squares of cycles for most values of n. In Liu and Xie proved that rn(P 2 n) is ⌊n/2⌋+ 2 if n ≡1 (mod 4) and n ≥9 and rn(P 2 n) is ⌊n/2 + 1⌋otherwise. In Liu found a lower bound for the radio number of trees and characterizes the trees that achieve the bound. She also provides a lower bound for the radio number of spiders in terms of the lengths of their legs and characterizes the spiders that achieve this bound. Sweetly and Joseph prove that the radio number of the graph obtained from the wheel Wn by subdividing each edge of the rim exactly twice is 5n −3. Marinescu-Ghemeci determined the radio number of the caterpillar obtained from a path by attaching a new terminal vertex to each non-terminal vertex of the path and the graph obtained from a star by attaching k new terminal vertices to each terminal vertex of the star. Ahmad and the electronic journal of combinatorics (2023), #DS6 306 Marinescu-Ghemeci determined the radio numbers of Mongolian tents, diamonds, fans, and double fans.
Sooryanarayana and Raghunath determined the radio number of C3 n, for n ≤20 and for n ≡0 or 2 or 4 (mod 6). Sooryanarayana, Vishu Kumar, Manjula determine the radio number of P 3 n, for n ≥4. Lo and Alegria completely determine the radio number for the fourth-power of Pn for n ≥6, except when n ≡1 (mod 8). Saha and Panigrahi prove that for an n-vertex simple connected graph G, the difference between the upper and lower bounds of the radio number of G2 is at most ⌊(n −1)/2⌋.
They also determine the radio number for square of graphs belonging to some specific class and apply this to find the radio number for square of hypercube Q2 n (n ̸≡0 (mod 4)), the square of toroidal grid T 2 m,n (m+n ≡1, 2, 3, 4, 6 (mod 8)), and the square of some generalized prism graphs. Wang, Xu, Yang, Zhang, Luo, and Wang determine the radio number of ladder graphs. Jiang completely determined the radio number of the grid graph Pm × Pn (m, n > 2). In Vaidya and Vihol determined upper bounds on radio numbers of cycles with chords and determined the exact radio numbers for the splitting graph and the middle graph of Cn.
In Li, Mak, and Zhou determine the radio number of complete m-ary trees. Kim, Hwang, and Song determine the radio numbers of Pn with n ≥4 and Km with m ≥3. Bantva improved the lower bound for the radio number of graphs given by Das et al. in and gave necessary and sufficient condition to achieve the lower bound. He also determined the radio number for Cartesian product of paths Pn and the Peterson graph P and provided a short proof for the radio number of Cartesian product of paths Pn and complete graphs Km given by Kim . Bantva determined the radio number of the Cartesian product of a path and a wheel.
A radio k-coloring of G when k = diam(G)−1 is called a radio antipodal labeling. The minimum span of a radio antipodal labeling of G is called the radio antipodal number of G and is denoted by an(G). Khennoufa and Togni determined the radio number and the radio antipodal number of the hypercube by using a generalization of binary Gray codes. They proved that rn(Qn) = (2n−1−1)⌈n+3 2 ⌉+1 and an(Qn) = (2n−1−1)⌈n 2⌉+ε(n), with ε(n) = 1 if n ≡0 mod 4, and ε(n) = 0 otherwise. In Gomathi and Venu-gopal provided bounds for the radio antipodal number of honeycombs derived networks— triangular and rhombic honeycombs. These bounds give the optimum number of channels (bandwidth) needed for these honeycomb derived networks for effective communication without interference.
In Xavier and Thivyarathi introduced the notion of radio antipodal mean new labeling as follows. A radio antipodal mean labeling of a graph G is a function f that assigns to each vertex a non-negative integer such that for distinct vertices u and v of G, d(u, v) < diam(G) and d(u, v) + ⌈(f(u) + f(v))/2⌉≥d. The radio antipodal mean number of f, denoted by ramn(f), is the maximum number assigned to any vertex of G.
The radio antipodal mean number of G, denoted by ramn(G), is the minimum value of ramn(f) taken over all radio antipodal mean labelings of G. They determined an upper bound for the antipodal mean number of paths, cycles, wheels, meshes (Pm × Pn), and enhanced meshes. Jose, Prabakaran, and Gomath determined the radio antipodal new the electronic journal of combinatorics (2023), #DS6 307 mean number of triangular grids and torus grids. Yenoke, Jose, and Venugopal new obtained the radio antipodal mean number of mongolian tents, triangular ladders, and Pagoda graphs.
In Nazeer, Kousar, and Nazeer give radio and radio antipodal labelings for certain circulant graphs.
In Nazeer, Kousar, and Munir determined the radio number and radio antipodal number of non-bipartite cubic graphs of order 2k. Shen, Dong, Zheng, and Guo use C(m, t) to denote the caterpillar consisting of a path x1x2 · · · xm with t pendent edges at each inner vertex. They determine the exact value of the radio number of C(m, t) for all integers m ≥4 and t ≥2, and explicitly construct an optimal radio labeling. They also show that the radio number and the construction of optimal radio labelings of paths are the special cases of C(m, t) with t = 2. An edge-joint graph G is a 1-edge connected graph having an edge uv such that eccentricity of u equals the eccentricity of v and deletion of uv disconnects G.
Niranjan and Rao Kola determined rn(Pn ⊙Cm) when n is even and m ≥5 and gave an upper bound for the same when n is odd. For m ≥4 they determined the radio number of Pn ⊙Pm when n is even, and gave both upper and lower bounds for rn(Pn ⊙Pm) when n is odd.
In Canales, Tomova, and Wyels investigated the question of which radio numbers of graphs of order n are achievable. They proved that the achievable radio numbers of graphs of order n must lie in the interval [n, rn(Pn)], and that these bounds are the best possible. They also show that for odd n, the integer rn(Pn) −1 = (n−1)2 2 + 2 is an unachievable radio number for any graph of order n. In Sokolowsky settled the question of exactly which radio numbers are achievable for a graph of order n.
Adefokun and Ajayi investigated the radio number of K1,m × P2n for the case m ≥4. They also obtained new lower and upper bounds of the radio number for the case that m = 3 that improve similar their results in . For a graph H, Naseem, Shabbir, and Shaker use JuH to denote the graph obtained by joining two disjoint copies of a graph H by an edge between the vertices labeled with u in H. They provided a lower bound for the radio number of such graphs and show that their lower bound is optimal (i.e., equal to the radio number) for certain a subfamily of JuH. They obtained similar results for graphs obtained by contracting the edge uv of JuH.
For any connected graph G and positive integer k Chartrand, Erwin, and Zhang, define a radio k-coloring as an injection f from the vertices of G to the natural numbers such that d(u, v) + |f(u) −f(v)| ≥1 + k for every two distinct vertices u and v of G.
Using rck(f) to denote the maximum number assigned to any vertex of G by f, the radio k-chromatic number of G, rck(G), is the minimum value of rck(f) taken over all radio k-colorings of G. Note that rc1(G) is χ(G), the chromatic number of G, and when k = diam(G), rck(G) is rn(G), the radio number of G. Chartrand, Nebesky, and Zang gave upper and lower bounds for rck(Pn) for 1 ≤k ≤n −1.
Kchikech, Khennoufa, and Togni improved Chartrand et al.’s lower bound for rck(Pn) and Kola and Panigrahi improved the upper bound for certain special cases of n. The exact value of rcn−2(Pn) for n ≥5 was given by Khennoufa and Togni in and the exact value of rcn−3(Pn) for n ≥8 was given by Kola and Panigrahi in . Kola and the electronic journal of combinatorics (2023), #DS6 308 Panigrahi gave the exact value of rcn−4(Pn) when n is odd and n ≥11 and an upper bound for rcn−4(Pn) when n is even and n ≥12. In Saha and Panigrahi provided an upper and a lower bound for rck(Cr n) for all possible values of n, k and r and showed that these bounds are sharp for antipodal number of Cr n for several values of n and r. Kchikech, Khennoufa, and Togni gave upper and lower bounds for rck(G × H) and rck(Qn). In the same authors proved that rck(K1,n) = n(k −1) + 2 and for any tree T and k ≥2, rck(T) ≤(n −1)(k −1).
Karst, Langowitz, Oehrlein, and Troxell provide general lower bounds for rck(Cn) for all cycles Cn when k ≥diam(Cn) and show that these bounds are exact values when k = diam(Cn) + 1.
In Čada, Ekstein, Holub, and Togni defined a k-labeling of a connected graph as an assignment c of non-negative integers to the vertices of the graph so that for every pair of vertices x and y, |c(x) −c(y)| ≥k + 1 −d(x, y). The radio k-number of a graph is the largest number that has to be used in a k-labeling of the graph. A distance graph is a graph with integer vertices, where two vertices are adjacent if the absolute value of their difference is in some chosen set D. They established some lower and upper bounds for the radio k-number of graphs with distance sets D(1, 2, . . . , t), D(1, t), and D(t −1, t) for a positive t. Korze, Shao, and Vesel improved some lower and upper bounds for the radio k-labeling number of the same three families studied by Čada et al. and conjectured that some of these upper bounds for the radio k-labeling number of D(1, 2, . . . , t) are exact radio k-labeling numbers. They obtained their main results using theoretical constructions and the mathematical optimization methods of integer linear programming. Arockiamary and Vijayalakshmi investigated the existence of vertex k-labeling of various cyclic snakes and mCn ⊙K1.
For a positive integer k, Arockiamary and Vijayalakshmi define a vertex k-prime labeling f of a graph G(V, E) as a bijective function from V to {k, k + 1, . . . , k + |V | −1} such that gcd(f(u), f(v) = 1 for every edge uv. A graph G that admits vertex k-prime labeling is called a vertex k-prime graph. They proved that the class of planar graphs containing the maximum number of edges possible in a graph with n vertices is vertex k-prime if and only if n is odd, G ∪K1,n is vertex k-prime, and if G and H are k-prime labelings, then so is G ∪H. They also prove that Kn (n ≥4) is not vertex k-prime.
The Dd-distance was introduced by Anto Kinsley and Siva Ananthi as follows.
For a connected graph G, the Dd-length of a connected uv path is defined as Dd(u, v) = D(u, v) + deg(u) + deg(v). Letting DDd(u, v) denote the Dd-distance between u and v and diamDd(G) denote the Dd-diameter of G, Viola and Nicholas define a radio mean Dd-distance labeling of a connected graph G as an injective map f from V (G) to the positive integers such that for two distinct vertices u and v of G, DDd(u, v)+⌈(f(u)+ f(v))/2⌉≥1 + diamDd(G).
The radio mean Dd-distance number of f, rmnDd(f), is the maximum label assigned to any vertex of G. The radio mean Dd-distance number of G, rmnDd(G), is the minimum value of f of G. They determined the radio mean Dd-distance number of complete graphs, stars, bistars, subdivisions of stars, paths, fans, and freindship graphs.
In Jose and Giridaran introduced the notion a radial radio mean labeling of new the electronic journal of combinatorics (2023), #DS6 309 graphs as follows. A radial radio mean labeling of a graph is a function f from V (G) to the non-negative integers that satisfies the condition d(u, v) + ⌈(f(u) + f(v))/2⌉≥1 + r(G), where u and v are vertices of G and r(G) is the radius of G. The span of a radial radio mean labeling is the largest integer in the range of f. The radial radio mean number of a graph G is the minimum span taken over all radial radio mean labelings of G. They determined the radial radio mean number of Mongolian tents and diamond graphs. In Paramasivam, Yenoke, and Muralidharan investigated the upper bounds for radial new radio number of chess board graphs and king’s graph. Kaabar and Yenoke studied new the relationship between the radio number and radial radio number for any connected graph. They provided upper bounds of the radio numbers and radial radio numbers for certain sunflower graphs.
The survey article by Panigrahi includes background information and further results about radio k-colorings.
7.5 Product and Divisor Cordial Labelings Sundaram, Ponraj, and Somasundaram introduced the notion of product cordial labelings.
A product cordial labeling of a graph G with vertex set V is a function f from V to {0, 1} such that if each edge uv is assigned the label f(u)f(v), the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a product cordial labeling is called a product cordial graph.
In and Sundaram, Ponraj, and Somasundaram prove the following graphs are product cordial: trees; unicyclic graphs of odd order; triangular snakes; dragons; helms; Pm ∪Pn; Cm ∪Pn; Pm ∪K1,n; Wm ∪Fn (Fn is the fan Pn + K1); K1,m ∪K1,n; Wm ∪ K1,n; Wm ∪Pn; Wm ∪Cn; the total graph of Pn (the total graph of Pn has vertex set V (Pn) ∪E(Pn) with two vertices adjacent whenever they are neighbors in Pn); Cn if and only if n is odd; C(t) n , the one-point union of t copies of Cn, provided t is even or both t and n are even; K2+mK1 if and only if m is odd; Cm∪Pn if and only if m+n is odd; Km,n∪Ps if s > mn; Cn+2 ∪K1,n; Kn ∪Kn,(n−1)/2 when n is odd; Kn ∪Kn−1,n/2 when n is even; and P 2 n if and only if n is odd. They also prove that Km,n (m, n > 2), Pm × Pn (m, n > 2) and wheels are not product cordial and if a (p, q)-graph is product cordial graph, then q ≤(p −1)(p + 1)/4 + 1.
The bicyclic graph B[m, n] is the one-point union of Cm and Cn. Meena and Usharani provided product cordial labelings for some bicyclic related graphs. Among them are B[n, n], the graphs obtained by adjoining the center of K1,m to each vertex of B[n, n], B[n, n] ⊙K2, and B[n, n] ⊙K3.
In Seoud and Helmi obtained the following results: Kn is not product cordial for all n ≥4; Cm is product cordial if and only if m is odd; the gear graph Gm is product cordial if and only if m is odd; all web graphs are product cordial; the corona of a triangular snake with at least two triangles is product cordial; the C4-snake is product cordial if and only if the number of 4-cycles is odd; Cm ⊙Kn is product cordial; and they determine all graphs of order less than 7 that are not product cordial. Seoud and the electronic journal of combinatorics (2023), #DS6 310 Helmi define the conjunction G1^G2 of graphs G1 and G2 as the graph with vertex set V (G1) × V (G2) and edge set {(u1, v1)(u2, v2)| u1u2 ∈E(G1), v1v2 ∈E(G2)}. They prove: Pm^Pn (m, n ≥2) and Pm^Sn (m, n ≥2) are product cordial. Nada, Diab, Elrokh, and Sabra proved that Pn ⊙Cm is product cordial if and only if (n, m) ̸= (1, 3) (mod 4). Gao, Lau, and Lee investigated the friendly index and product-cordial index sets of a family of Möbius-like cubic graphs. Rokad proved the following graphs are product cordial: double wheels DWn = 2Cn + K1, path unions of finite number of copies of double wheels, the graphs obtained by joining two copies of double wheels by a path of arbitrary length, DWn ⊕K1,n, and DFn ⊕K1,n (DFn = Pn + K2).
Vaidya and Kanani prove the following graphs are product cordial: the path union of k copies of Cn except when k is odd and n is even; the graph obtained by joining two copies of a cycle by path; the path union of an odd number copies of the shadow of a cycle (see §3.8 for the definition); and the graph obtained by joining two copies of the shadow of a cycle by a path of arbitrary length. In Vaidya and Kanani prove the following graphs are product cordial: the path union of an even number of copies of Cn(Cn); the graph obtained by joining two copies of Cn(Cn) by a path of arbitrary length; the path union of any number of copies of the Petersen graph; and the graph obtained by joining two copies of the Petersen graph by a path of arbitrary length.
Vaidya and Barasara prove that the following graphs are product cordial: friend-ship graphs; the middle graph of a path; odd cycles with one chord except when the chord joins the vertices at a diameter distance apart; and odd cycles with two chords that share a common vertex and form a triangle with an edge of the cycle and neither chord joins vertices at a diameter apart. In Vaidya and Barasara investigated the product cordial labeling of the line graph of the middle graphs of paths, triangular snakes, armed crowns, the square of paths, the splitting graphs of paths, and the total graph of paths.
In Vaidya and Dani prove the following graphs are product cordial: < S(1) n : S(2) n : . . . : S(k) n > except when k odd and n even; < K(1) 1,n : K(2) 1,n : . . . : K(k) 1,n >; and < W (1) n : W (2) n : . . . : W (k) n > if and only if k is even or k is odd and n is even with k > n.
(See §3.7 for the definitions.) Vaidya and Barasara proved the following graphs are product cordial: closed helms, web graphs, flower graphs, double triangular snakes obtained from the path Pn if and only if n is odd, and gear graphs obtained from the wheel Wn if and only if n is odd.
Vaidya and Barasara proved that the graphs obtained by the duplication of an edge of a cycle, the mutual duplication of pair of edges of a cycle, and mutual duplication of pair of vertices between two copies of Cn admit product cordial labelings. Moreover, if G and G′ are the graphs such that their orders or sizes differ at most by 1 then the new graph obtained by joining G and G′ by a path Pk of arbitrary length admits product cordial labeling.
Vaidya and Barasara define the duplication of a vertex v of a graph G by a new edge u′v′ as the graph G′ obtained from G by adding the edges u′v′, vu′ and vv′ to G.
They define the duplication of an edge uv of a graph G by a new vertex v′ as the graph G′ obtained from G by adding the edges uv′ and vv′ to G. They proved the following graphs have product cordial labelings: the graph obtained by duplication of an arbitrary vertex the electronic journal of combinatorics (2023), #DS6 311 by a new edge in Cn or Pn (n > 2); the graph obtained by duplication of an arbitrary edge by a new vertex in Cn (n > 3) or Pn (n > 3); and the graph obtained by duplicating all the vertices by edges in path Pn. They also proved that the graph obtained by duplicating all the vertices by edges in Cn (n > 3) and the graph obtained by duplicating all the edges by vertices in Cn are not product cordial.
Recall (see , , , ) a double triangular snake DTn consists of two triangular snakes that have a common path; a double quadrilateral snake DQn consists of two quadrilateral snakes that have a common path; an alternate triangular snake A (Tn) is the graph obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertex vi (that is, every alternate edge of a path is replaced by C3); a double alternate triangular snake DA (Tn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to two new vertices vi and wi; an alternate quadrilateral snake A (Qn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertices vi and wi respectively and then joining vi and wi (that is, every alternate edge of a path is replaced by a cycle C4); a double alternate quadrilateral snake DA (Qn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertices vi, xi and wi and yi respectively and then joining vi and wi and xi and yi.
Vaidya and Barasara prove that the shell graph Sn is product cordial for odd n and not product cordial for even n. They also show that D2(Cn); D2(Pn); C2 n; M(Cn); S′(Cn); circular ladder CLn; Möbius ladder Mn; step ladder S(Tn) and Hn,n does not admit product cordial labeling.
Vaidya and Vyas prove the following graphs are product cordial: alternate triangular snakes A(Tn) except n ≡3 (mod4); alternate quadrilateral snakes A(QSn) except except n ≡2 (mod4); double alternate triangular snakes DA(Tn) and double alternate quadrilateral snakes DA(QSn).
Vaidya and Vyas prove the following graphs are product cordial: the splitting graph of bistar S′(Bn,n); duplicating each edge by a vertex in bistar Bn,n and duplicating each vertex by an edge in bistar Bn,n. They also proved that D2(Bn,n) is not product cordial.
Ghodasara and Vaghasiya prove the following graphs admit product cordial labelings: the path union of an odd number of copies of Cn with a chord except for n = 4, the path union of an odd number of copies of Cn with twin chords except when n = 6, the path union of Cn (n > 6) with three cords that form two triangles and a cycle of length n −3, the graph obtained by joining two copies of the same cycle that has one chord by a path, the graph obtained by joining two copies of same cycle that has twin chords by a path, and the graph obtained by joining two copies of Cn (n ≥7) with three cords that form two triangles and a cycle of length n−3 by a path. Ghodasara and Vaghasiya prove the following graphs are product cordial: the path union of helms, the path union of closed helms, the path union of gear graphs Gn for odd n, the graph obtained by joining two copies of the same helm by a path, the graph obtained by joining two copies of the same closed helm by a path, and the graph obtained by joining two copies of the same gear graph by a path.
In Bapat proves the following graphs are product cordial: graphs obtained by identifying an endpoint of Pn with each vertex of C3, graphs obtained by identifying an the electronic journal of combinatorics (2023), #DS6 312 endpoint of Pn with each vertex of C4, graphs obtained by identifying the degree m vertex of K1,m with each vertex of C3, and graphs obtained by identifying the degree m vertex of K1,m with each vertex of the shell Cn,n−3) (Cn with n −3 chords that share a common endpoint) if and only n is even or n is odd and m is even. In Bapat proves K5 ⊙Cn and kayak paddles are product cordial, the one-point union of n copies of Km is product cordial if and only if n is even, and graphs obtained by identifying one edge of K5 with each edge of Pn is product cordial if and only if n is even.
Prajapati and Raval investigated product cordial labelings of the graphs ob-tained by duplication of vertices and edges of gears and graphs obtained by the vertex switching operation of gears. In Prajapati and Raval proved that the book Bm,n is a product cordial graph if and only if m and n both are odd and m ≥3. They showed that graphs obtained from books by duplicating or deleting vertices or edges are product cordial. For graphs with an even number of vertices they proved that the duplication of each of the vertices of a product cordial graph with an edge is a product cordial graph and that for graphs that have an odd number of vertices and even number of edges the duplication of each of the vertices of a product cordial graph with an edge is a product cordial graph.
Kwong, Lee, and Ng determine the product-cordial index sets of Möbius ladders and the graphs obtained by subdividing an edge of K4 and an edge of a Möbius ladder that is not a rung and joining the two new vertices by an edge. They show that no Möbius ladder is product cordial.
Gao, Sun, Zhang, Meng, and Lau provide sufficient conditions for a graph to admit (or not admit) a product cordial labeling. Gao, Lau, and Lee investigated the friendly index and product-cordial index sets of a family of cubic graphs known as Möbius-like graphs. Prajapati and Raval proved that windmills, barbells, the one point union at the apex of copies of a wheel (generalized wheel), and the one point union of copies of a wheel connected at one common rim vertex of the wheel are product cordial graphs. They also showed that duplicating all rim edges with a vertex and duplicating all the vertices with an edge of generalized wheels, and the graphs obtained by switching an apex vertex in a generalized wheel are product cordial graphs. Patel, Prajapati, and Kansagara proved that graphs obtained from the barbell graph by duplicating all vertices by edges and duplicating all edges by vertices in the path joining complete graphs are product cordial and the graphs obtained by switching a vertex of path in the barbell graph are product cordial.
In Salehi called the set {|ef(0) −ef(1)| : f is a friendly labeling of G} the product-cordial set of G. He determines the product-cordial sets for paths, cycles, wheels, complete graphs, bipartite complete graphs, double stars, and complete graphs with an edge deleted. Salehi and Mukhin say a graph G of size q is fully product-cordial if its product cordial set is {q −2k : 0 ≤k ≤⌊q/2⌋}. They proved: Pn (n ≥2) is fully product-cordial; trees with a perfect matching are fully product-cordial; and P2×Pn is not fully product-cordial. They determine the product-cordial sets of P2 × Pn, Pn × P2m, and Pn × P2m+1, where m ≥n. Because the product-cordial set is the multiplicative version of the friendly index set, Kwong, Lee, and Ng called it the product-cordial index set of G. They determined the exact values of the product-cordial index set of Cm and the electronic journal of combinatorics (2023), #DS6 313 Cm×Pn and that Pm×Pn has the maximum product cordial-index 2mn−m−n. In Kwong, Lee, and Ng determined the friendly index sets and product-cordial index sets of 2-regular graphs and the graphs obtained by identifying the centers of any number of wheels. In z Salehi, Churchman, Hill, and Jordan determine the product-cordial index sets of certain classes of trees.
In Shiu and Kwong define the full product-cordial index of G under f as FPCI(G) = {i∗ f(G) | f is a friendly labeling of G}. They provide a relation between the friendly index and the product-cordial index of a regular graph. As applications, they determine the full product-cordial index sets of Cm and Cm × Cn, which was asked by Kwong, Lee, and Ng in . Shiu determined the product-cordial index sets of grids Pm × Pn. Recall the twisted cylinder graph is the permutation graph on 4n (n ≥2) vertices, P(2n; σ), where σ = (1, 2)(3, 4) · · · (2n−1, 2n) (the product of n transpositions).
Shiu and Lee determined the full friendly index sets and the full product-cordial index sets of twisted cylinders.
Jeyanthi and Maheswari define a mapping f : V (G) →{0, 1, 2} to be a 3-product cordial labeling if |vf(i) −vf(j)| ≤1 and |ef(i) −ef(j)| ≤1 for any i, j ∈{0, 1, 2}, where vf(i) denotes the number of vertices labeled with i, ef(i) denotes the number of edges xy with f(x)f(y) ≡i (mod 3). A graph with a 3-product cordial labeling is called a 3-product cordial graph. In they prove that for a (p, q) 3-product cordial graph: p ≡0 (mod 3) implies q ≤p2−3p+6 3 ; p ≡1 (mod 3) implies q ≤p2−2p+7 3 ; and p ≡2 (mod 3) implies q ≤ p2−p+4 3 . They prove the following graphs are 3-product cordial: paths; stars; Cn if and only if n ≡1, 2 (mod 3); Cn ∪Pn, Cm ⊙Kn; Pm ⊙Kn for m ≥3 and n ≥1; Wn when n ≡1 (mod 3); and the graph obtained by joining the centers of two identical stars to a new vertex. They also prove that Kn is not 3-product cordial for n ≥3 and if G1 is a 3-product cordial graph with 3m vertices and 3n edges and G2 is any 3-product cordial graph, then G1 ∪G2 is a 3-product cordial graph. In they prove that ladders, < W (1) n : W (2) n : . . . : W (k) n > (see §3.7 for the definition), graphs obtained by duplicating an arbitrary edge of a wheel, graphs obtained by duplicating an arbitrary vertex of a cycle or a wheel are 3-product cordial. They also prove that the graphs obtained by from the ladders Ln = Pn × P2 (n ≥2) by adding the edges uivi+1 for 1 ≤i ≤n −1, where the consecutive vertices of two copies of Pn are u1, u2, . . . , un and v1, v2, . . . , vn and the edges are uivi. They call these graphs triangular ladders . The graph B∗ n,n is obtained from the bistar Bn,n with V (Bn,n) = {u, v, ui, vi | 1 ≤i ≤n} and E(Bn,n) = {uv, uui, vvi, vui, uvi | 1 ≤i ≤n} by joining u with vi and v with ui for 1 ≤i ≤4. Jeyanthi and Maheswari proved: the splitting graphs S′(K1,n) and S′(Bn,n) are 3-product cordial graphs; B∗ n,n is a 3-product cordial graph if and only if n ≡0, 1 (mod 3); and the shadow graph D2(Bn,n) is a 3-product cordial graph if and only if n ≡0, 1 mod 3. Jeyanthi, Maheswari, and Vijaya Laksmi prove the following: graphs obtained by switching an apex vertex in a closed helm are 3-product cordial; Wn are 3-product cordial if and only if n ≡2 (mod 3); double fans are 3-product cordial if and only if n ≡0 (mod 3); books are 3-product cordial; and permutation graphs P(K2 + mK1; T) are 3-product cordial if and only if m ≡2 (mod 3). In Jeyanthi, Maheswari, and Vijayalaksmi investigated the 3-product cordial behavior of alternate the electronic journal of combinatorics (2023), #DS6 314 triangular snakes, double alternate triangular snakes, and triangular snakes.
A k-product cordial labeling of a graph G is a map f from V (G) to {0, 1, . . . , k −1} where k is a positive integer at most |V (G)| such that when each edge uv is assigned the label f(u)f(v) (mod k), the number of vertices (edges) labeled with i and the number of edges (vertices) labeled with j differ by at most 1. In and Jeya Daisy, Sabibha, Jeyanthi, and Youssef showed that the following graphs admit k-product cordial labelings: fans and double fans when k = 4 and 5; cones (K1 + Cm) and double cones (K2 + Cm) for k = 5. They also proved that double cones are not 4-product cordial.
In they investigated the k-product cordial behavior of G + Kt, where G is a k-product cordial graph and found an upper bound of the size of k-product cordial graphs.
In they investigated the k-product cordial behavior of union of graphs. In Jeya Daisy, Santrin Sabibha, Jeyanthi, and Youssef proved that the 1-cone Cn + K1 and double cone Cm+K2 admit 5-product cordial labelings. They also showed that the double cone does not admit 4-product cordial labeling. In they define the Napier bridge, Pn(t), as the graph obtained from the path Pn by joining the pairs of vertices u, v of Pn with d(u, v) = t. They provided necessary and significant conditions for Pn(3), Pn(4) and Pn(5) to have 3-product and 4-product cordial labelings. In Jeya Daisy, Sabibha, Jeyanthi and Youssef proved that fans Fn and double fans DFn admit 4-product cordial labelings and 5-product cordial labelings. In they investigated the existence of k-product cordial labelings of direct products, Cartesian products, strong products, and lexicographic products of graphs. Jeya Daisya, Sabibhab, Jeyanthic, and Youssef new showed that the splitting graph of star graphs admit k-product cordial labeling.
Sundaram and Somasundaram introduced the notion of total product cordial labelings. A total product cordial labeling of a graph G with vertex set V is a function f from V to {0, 1} such that if each edge uv is assigned the label f(u)f(v) the number of vertices and edges labeled with 0 and the number of vertices and edges labeled with 1 differ by at most 1. A graph with a total product cordial labeling is called a total product cordial graph.
In and Sundaram, Ponraj, and Somasundaram prove the following graphs are total product cordial: every product cordial graph of even order or odd order and even size; trees; all cycles except C4; Kn,2n−1; Cn with m edges appended at each vertex; fans; double fans; wheels; helms; Cn × P2; K2,n if and only if n ≡2 (mod 4); Pm × Pn if and only if (m, n) ̸= (2, 2); Cn + 2K1 if and only if n is even or n ≡1 (mod 3); Kn × 2K2 if n is odd, or n ≡0 or 2 (mod 6), or n ≡2 (mod 8). Y.-L. Lai, the reviewer for MathSciNet , called attention to some errors in . Pedrano and Rulete determined the total product cordial labeling of Pm × Cn, Cm × Cn and the generalized Petersen graph P(m, n). In Pedrano and Rulete determined the total product cordial labeling of Pm ⊙Cn, Pm ⊙Pn, Cm ⊙Pn, Pm ⊙Fn, Pm ⊙Wn, and Pm ⊙Kn.
Villar proved Pn ⊙Cm (n ≥2, m ≥3) is product cordial, Pn ⊙Pm (n, m ≥2) is product cordial except when n and m are both even, and P2n+1 ⊙Km (n ≥1, m ≥4) is not product cordial. Gao, Sun, Zhang, Meng, and Lau proved that P m n+1 is total product cordial.
Ramanjaneyulu, Venkaiah, and Kothapalli give total product cordial labeling for a family of planar graphs for which each face is a 4-cycle.
Vaidya and Vihol prove the following graphs have total product cordial labelings: the electronic journal of combinatorics (2023), #DS6 315 a split graph; the total graph of Cn; the star of Cn (recall that the star of a graph G is the graph obtained from G by replacing each vertex of star K1,n by a graph G); the friendship graph Fn; the one point union of k copies of a cycle; and the graph obtained by the switching of an arbitrary vertex in Cn.
Sundaram, Ponraj, and Somasundaram introduced the notion of EP-cordial labeling (extended product cordial) labeling of a graph G as a function f from the vertices of a graph to {−1, 0, 1} such that if each edge uv is assigned the label f(u)f(v), then |vf(i) −vf(j)| ≤1 and |ef(i) −ef(j)| ≤1 where i, j ∈{−1, 0, 1} and vf(k) and ef(k) denote the number of vertices and edges respectively labeled with k.
An EP-cordial graph is one that admits an EP-cordial labeling. In Sundaram, Ponraj, and Somasundaram prove the following: every graph is an induced subgraph of an EP-cordial graph, Kn is EP-cordial if and only if n ≤3; Cn is EP-cordial if and only if n ≡1, 2 (mod 3), Wn is EP-cordial if and only if n ≡1 (mod 3); and caterpillars are EP-cordial.
They prove that all K2,n, paths, stars and the graphs obtained by subdividing each edge of a star exactly once are EP-cordial. They also prove that if a (p, q) graph is EP-cordial, then q ≤1 + p/3 + p2/3. They conjecture that every tree is EP-cordial.
Ponraj, Sivakumar, and Sundaram introduced the notion of k-product cordial labeling of graphs. Let f be a map from V (G) to {0, 1, 2, . . . , k −1}, where 2 ≤k ≤|V |.
For each edge uv assign the label f(u)f(v) (mod k).
f is called a k-product cordial labeling if |vf(i)−vf(j)| ≤1 and |ef(i)−ef(j)| ≤1, i, j ∈{0, 1, 2, . . . , k −1}, where vf(x) and ef(x) denote the number of vertices and edges labeled with x. A graph with a k-product cordial labeling is called a k-product cordial graph. Observe that 2-product cordial labeling is simply a product cordial labeling and 3-product cordial labeling is an EP-cordial labeling. In and Ponraj et al. prove the following are 4-product cordial: Pn if and only n ≤11, Cn if and only if n = 5, 6, 7, 8, 9, or 10, Kn if and only if n ≤2, Pn ⊙K1, Pn ⊙2K1, K2,n if and only if n ≡0, 3 (mod 4), Wn if and only if n = 5 or 9, Kn + 2K2 iff n ≤2, and the subdivision graph of K1,n. Sivakumar proved the following coronas are 4-total product cordial: Pn⊙K1, Pn⊙2K1, S(Pn⊙K1), S(Pn⊙2K1), S(Cn⊙K1) and S(Cn ⊙2K1). Jeyanthi, Maheswari, and Vijayalakshmi investigated the 3-product cordial behavior of alternate triangular snakes, double alternate triangular snakes, and triangular snake graphs. In they establish that vertex switching graphs of wheels, gears, and degree splitting of bistars are 3-product cordial graphs.
Let f be a map from V (G) to {0, 1, 2, . . . , k −1} where 2 ≤k ≤|V |. For each edge uv assign the label f(u)f(v) (mod k). Ponraj, Sivakumar, and Sundaram define f to be a k-total product cordial labeling if |f(i) −f(j)| ≤1, i, j ∈{0, 1, 2, . . . , k −1}, where f(x) denote the number of vertices and edges labeled with x. A graph with a k-total product cordial labeling is called a k-total product cordial graph . A 2-total product cordial labeling is simply a total product cordial labeling. In , , , and , Ponraj et al. proved the following graphs are 3-total product cordial: Pn, Cn if and only if n ̸= 3 or 6, K1,n if and only if n ≡0, 2 (mod 3), Pn ⊙K1, Pn ⊙2K1, K2 + mK1 if and only if m ≡2 (mod 3), helms, wheels, Cn ⊙2K1, Cn ⊙K2, dragons Cm@Pn (obtained by identifying an endpoint of Pn with a vertex of Cm), Cn ⊙K1, bistars Bm,n, and the subdivision graphs of K1,n, Cn ⊙K1, K2,n, Pn ⊙K1, Pn ⊙2K1, Cn ⊙K2, the electronic journal of combinatorics (2023), #DS6 316 wheels and helms. They also proved that every graph is a subgraph of a connected k-total product cordial graph, Bm,n is (n + 2)-total product cordial, and Km,n is (n + 2)-total product cordial. Ahmad, Hasni, Irfan, Naseem, and Siddiqui proved that Pm × Pn for m, n ≥2 is 3-total edge product cordial. Sharon Philomena and Thirusangu proved that the flower graph is 3-total product cordial. Ahmada, Bača, Naseemc, and Semaničová-Feňovčíková described a method for obtaining a 3-total edge product cordial labeling of the hexagonal grid from a smaller hexagonal grid.
In Ahmad proved that the generalized Petersen graphs P(n, m) are 3-total edge product cordial.
In Azaizeh, Hasni, Lau, and Ahmad proved that complete graphs, bipartite graphs and generalised friendship graphs have 3-total edge product cordial labelings. Ahmad, Ali, Bilal, Zafar, and Zahid prove that webs, helms, gears, and Ln ⊙2K1 (Ln is the ladder with 2n vertices) have 3-total edge product cordial labelings. Ivančo characterized graphs admitting a 2-total edge product cordial labeling and proved that dense graphs and regular graphs of degree 2(k−1) admit a k-total edge product cordial labeling. Javed and Jamil proved that the rhombic grid graphs Rm 1 , Rm 2 and Rm 3 are 3-total edge product cordial for m ≥1 and that the rhombic grid graph Rm n is 3-total edge product cordial for m, n ≥1. Ullah, Rahmat, Numan, Anoh Yannick, and Aslam proved that the stellation of Pn×Pn (that is, the graph obtained from Pn×Pn by adding a vertex in each square of Pn×Pn and then joining this vertex to each vertex of that square) admits a 3-total edge product cordial labeling. Bala, Krithika Devi, and Thirusangu proved the existence of 3-total cordial edge magic labeling, 3-total sum cordial labeling, and total product cordial labeling for the square graph of a comb. Jeba Jesintha, Devakirubanithi, and Monisha provided edge product cordial labelings for the duplication of prism new graphs.
In Kumari and Mehra call a vertex labeling f of a graph G with 0 and 1 with the induced edge labeling f given by f(uv) = f(u)f(v) a vertex product cordial labeling if the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. They prove the following graphs have vertex product cordial labelings: P 2 n if and only if n is odd, the path unions of k copies of P 2 n, Pn ⊙K1, helms, gears Gn for odd n, graphs obtained from Cn after switching of a vertex, Cn ⊙K1, Cn ⊙¯ Km, and certain banana trees.
A labeling f of a graph G(V, E) from V ∪E to {0, 1} is said to be a total product cordial labeling of G if f(xy) = f(x)f(y) for all x, y ∈V and xy ∈E and the difference of the total number vertices and edges labeled 0 and the total number vertices and edges labeled 1 differ by at most 1. Cyrile, Valdehueza, and Pedrano determined the total edge product cordial labeling of the corona graphs Pn ⊙Pm and Pn ⊙Cm. Gondalia proved that the graphs obtained by vertex switching of the following graphs are Fibonacci product cordial: cycles, cycles with one chord, cycles with twin chords where the chords form two triangles and one cycle Cn−2, and cycles with twin chords where chords form two triangles and one cycle Cn(1, 1, n −5).
In Gajjar and Raval define an E-sum cordial labeling of a graph G as an edge labeling f ∗: E(G) →{0, 1}, where the induced vertex labeling defined by f(u) = P f(uv) the electronic journal of combinatorics (2023), #DS6 317 mod 2 over all edges incident to u, has the property that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1.
They introduce two new graphs insprited by origamai models that they call the north star graph and the lotus star (see article for the definitions) and investigate the existence of cordial labelings, sum cordial labelings, signed product cordial labelings, and E-sum cordial labelings for these new graph.
In Soundar Rajan and Baskar Babujee investigated the existence of signed new product cordial labeling for splitting graphs of bull graphs, the splitting graphs of stars, P 2 n, Cn ⊙3K1, and the graphs obtained by joining two copies of the helm H4 by a path of arbitrary length.
In 2019 the concept of a Fibonacci product cordial labeling was introduced by Abra-ham and Jose as follows. An injective function f from the vertices of a graph to the set {F0, F1, F2, . . . , Fn+1} where Fj is the jth Fibonacci number is said to be a Fibonacci cordial labeling if the induced function f ∗from the edges s to the set {0, 1} defined by f ∗(uv) = (f(u)fv))(mod 2) has the property that the number of edges with label 0 and the number of edges with label 1 differ by at most 1. A graph which admits a Fibonacci cordial labeling is called a Fibonacci cordial graph. They proved that the following graphs are Fibonacci product cordial: paths, cycles, wheels Wn except when n = 2 mod 3, and the Petersen graph. For n ≥2, Sulayman and Pedrano determined Fibonacci cordial labelings of alternate triangular snakes A(Tn), quadrilateral snakes Qn, cycle quadrilateral snakes CQn, and for n ≥3, double alternate quadrilateral snakes.
For a graph G Sundaram, Ponraj, and Somasundaram defined the index of product cordiality, ip(G), of G as the minimum of {|ef(0) −ef(1)|} taken over all the 0-1 binary labelings f of G with |vf(i) −vf(j)| ≤1 and f(uv) = f(u)f(v), where ef(k) and vf(k) denote the number of edges and the number of vertices labeled with k. They established that ip(Kn) = ⌊n/2⌋2; ip(Cn) = 2 if n is even; ip(Wn) = 2 or 4 according as n is even or odd; ip(K2,n) = 4 or 2 according as n is even or odd; ip(K2 + nK1) = 3 if n is even; ip(G × P2) ≤2ip(G); ip(G1 ∪G2) ≤ip(G1) + ip(G2) + 2 min{∆(G1), ∆(G2)} where G1 and G2 are graphs of odd order; and ip(G1 ⊙G2) ≤ip(G1)+ip(G2)+2δ(G2)+3 where G1 and G2 have odd order.
In Tenguria and Verma called a mapping f from V (G) to {0, 1, 2} such that each edge uv is labeled (f(u)+f(v)) mod 3 a 3-total super sum cordial labeling if |f(i)−f(j)| ≤1 for i, j ∈{0, 1, 2}, where f(x) denotes the total number of vertices and edges labeled with x and for each edge uv, |f(u) −f(v)| ≤1. A graph that has a 3-total super sum cordial labeling is called 3-total super sum cordial graph. They proved Pm ∪Pn, Cm ∪Cn, and K1,m ∪K1,n are 3-total super sum cordial graphs. (These results also appeared in and ).
Sridevi, Nagarajan, Nellaimurugan, and Navaneethakrishnan introduced the notion of Fibonacci divisor cordial labeling of a graph G(V, E) as a bijection f : V → {F1, F2, F3, . . . , Fp}, where Fi is the ith Fibonacci number such that if each edge uv is assigned the label 1 if f(u) divides f(v) or f(v) divides f(u) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most the electronic journal of combinatorics (2023), #DS6 318 1. If a graph has a Fibonacci divisor cordial labeling, then it is called a Fibonacci divisor cordial graph.
They proved Pn, Cn, K2,n, and the subdivision of the bistar Bn,n are Fibonacci divisor cordial graphs. They further proved that Kn (n ≥3) is not a Fibonacci divisor cordial graph.
Ghosh and Pal proved that the graphs obtained by the switching of an arbitrary vertex of cycles and wheels, the duplication of an arbitrary vertex of cycles, the degree splitting graphs of paths, umbrella graphs, and bistars Bn,n are also Fibonacci divisor cordial graphs.
Sridevi, Palani, Navanaeethakrishnan, and Nagarajan proved that K1,n for n ≤9 and n = 11, Wn for n = 7, 8, 9, 10 and Cm@Pn are Fibonacci divisor cordial graphs. They also we prove that K1,n for n ≥12 and n = 10, and Wn for n = 4, 5, 6 and n ≥11 are not Fibonacci divisor cordial graphs.
Vaidya and Vyas define the tensor product G1(Tp)G2 of graphs G1 and G2 as the graph with vertex set V (G1) × V (G2) and edge set {(u1, v1)(u2, v2)| u1u2 ∈E(G1), v1v2 ∈V (G2)}. They proved the following graphs are product cordial: Pm(Tp)Pn; C2m(Tp)P2n; C2m(Tp)C2n; the graph obtained by joining two components of Pm(Tp)Pn an by arbitrary path; the graph obtained by joining two com-ponents of C2m(Tp)P2n by an arbitrary path; and and the graph obtained by joining two components of C2m(Tp)C2n by an arbitrary path.
In Ponraj introduced the notion of an (α1, α2, . . . , αk)-cordial labeling of a graph.
Let S = {α1, α2, . . . , αk} be a finite set of distinct integers and f be a function from a vertex set V (G) to S. For each edge uv of G assign the label f(u)f(v). He calls f an (α1, α2, . . . , αk)-cordial labeling of G if |vf(αi)−vf(αj)| ≤1 for all i, j ∈{1, 2, . . . , k} and |ef(αiαj) −ef(αrαs)| ≤1 for all i, j, r, s ∈{1, 2, . . . , k}, where vf(t) and ef(t) denote the number of vertices labeled with t and the number of edges labeled with t, respectively. A graph that admits an (α1, α2, . . . , αk)-cordial labeling is called an (α1, α2, . . . , αk)-cordial graph.
Note that an (−α, α)-cordial graph is simply a cordial graph and a (0, α)-cordial graph is a product cordial graph. Ponraj proved that K1,n is (α1, α2, . . . , αk)-cordial if and only if n ≤k and for α1 ̸= 0, α2 ̸= 0, α1 + α2 ̸= 0 proved the following: Kn is (α1, α2)-cordial if and only if n ≤2; Pn is (α1, α2)-cordial; Cn is (α1, α2)-cordial if and only if n > 3; Km,n (m, n > 2) is not (α1, α2)-cordial; the bistar Bn,n+1 is (α1, α2)-cordial; Bn+2,n is (α1, α2)-cordial if and only if n ≡1, 2 (mod 3); Bn+3,n is (α1, α2)-cordial if and only if n ≡0, 2 (mod 3); and Bn+r,n, r > 3 is not (α1, α2)-cordial. He also proved that if G is an (α1, α2)-cordial graph with p vertices and q edges, then q ≤3p2/8 −p/2 + 9/8. In Ponraj proved that combs Pn ⊙K1 are (α1, α2)-cordial; coronas Cn ⊙K1 are (α1, α2)-cordial for n ≡0, 2, 4, 5 (mod 6); C(t) 3 is not (α1, α2)-cordial; Wn is not (α1, α2)-cordial; and Kn + 2K2 is (α1, α2)-cordial if and only if n = 2.
In Varatharajan, Navanaeethakrishnan Nagarajan define a divisor cordial label-ing of a graph G with vertex set V as a bijection f from V to {1, 2, . . . , |V |} such that an edge uv is assigned the label 1 if one f(u) or f(v) divides the other and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. If graph that has a divisor cordial labeling, it is called a divisor cordial graph.
They proved the standard graphs such as paths, cycles, wheels, stars and some complete bipartite graphs are divisor cordial. They also proved that complete graphs are not divisor cordial. In they proved dragons, coronas, wheels, and complete binary trees are the electronic journal of combinatorics (2023), #DS6 319 divisor cordial. For t copies S1, S2, . . . , St of an n-star K1,n they define ⟨S1, S2, . . . , St⟩as the graph obtained by starting with S1, S2, . . . , St and joining the central vertices of Sk−1 and Sk to a new vertex xk−1. They prove that ⟨S1, S2⟩and ⟨S1, S2, S3⟩are divisor cordial.
In Barasara and Thakkar gave e some characterizations of various divisor cordial new type labelings.
Vaidya and Shah proved that the splitting graphs of stars and bistars are divisor cordial and the shadow graphs and the squares of bistars are divisor cordial. In they proved that helms, flower graphs, and gears are divisor cordial graphs. They also proved that graphs obtained by switching of a vertex in a cycle, switching of a rim vertex in a wheel, and switching of an apex vertex in a helm admit divisor cordial labelings. Raj and Valli proved the following graphs divisor cordial: the duplication of a vertex of a cycle; graphs obtained by joining two wheels of the same size by a path of length at least 3; Gv ⊙K1, where Gv is a graph obtained by switching any vertex of a cycle of size at least 4; graphs obtained by joining the apex vertices of two shells of the same size to an isolated vertex; graphs obtained by joining the centers of two wheels of the same size to an isolated vertex; and a class of graphs obtained by removing certain edges from complete graphs. Bosmia and Kanani proved that the graphs of the form G ⊙K1 where G any of the following admits a divisor cordial labeling: K1,n, K2,n, K3,n, a wheel, a helm, a flower, a fan, a double fan, and a barycentric subdivision of a star. Bosmia and Kanani prove that the following graphs admit divisor cordial labelings: bistars, the splitting graph of bistars, the degree splitting graph of bistars, the shadow graph of bistars, the restricted square graph of bistars, the barycentric subdivision of bistars, and the corona product of a bistar with K1. Thirusangu and Madhu proved that the extended duplicate graph of star and bistar graphs are divisor cordial graphs. Krithika and Stanley proved that subdivided shell graphs, disjoint union of two subdivided new shell graphs and uniform shell bow graphs are sum divisor cordial.
In Sugumaran and Mohan proved that the following graphs are divisor cordial graphs: degree splitting graph of K1,n,n), the splitting graph of the graph obtained from two isolated vertices are joined by n paths of length 2, the W−graph (the graph obtained from two copies of K1,n and identifying the last pendent vertex of the first copy with the first pendent edge of the second copy), B(n)⊙um K1, where B(n) = 2Pn +K1 (bow graph, the Herschel graph Hs and switching of an apex vertex in the Herschel graph (see [615, p.
53]. In Sugumaran and Suresh proved that the following graphs are divisor cordial graphs: fans, Petersen graphs, Cm ⊙K1, friendship graphs Fn, and the switching of a end vertex in path Pn, switching of any one of the inner vertices of Petersen graph Pe.
Barasara and Thakkar investigated divisor cordial labelings of armed crowns, closed helms, webs, and the one point union of t copies of Cn). In they investigated divisor cordial labeling of the combs, P 2 n, C2 n for n ̸= 4, and middle graphs of paths and cycles. Barasara and Thakkar proved that ladders, circular ladders, Möbius ladders, total graphs of paths, and total graphs of cycles are divisor cordial graphs.
Barasara and Thakkar proved that the triangular snake Tn and the quadrilateral new snake Qn are divisor cordial graphs.
Moreover, the graphs DS(Pn), DS(Sn)(n ≥5), DS(Cn ⊙K1), DS(Pn ⊙K1) (n ≥4), and the degree splitting graph of cycle Cn with the electronic journal of combinatorics (2023), #DS6 320 one chord are divisor cordial graphs. In , Barasara and Thakkar showed that the Cn-quadrilateral snake is a divisor cordial graph.
Also, the graphs DS(Tn) (n ≥4), DS(Qn) (n ≥4), DS(Ln) (n ≥3), and DS(Hn) are divisor cordial graphs. Moreover, the graph obtained by duplication of an arbitrary edge by a new vertex in Pn, the graph obtained by duplication of an arbitrary vertex by a new edge in Pn, the graph obtained by duplication of an arbitrary vertex by a new vertex in Pn, and the graph obtained by duplication of an arbitrary edge by a new edge in Pn are divisor cordial graphs.
Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of a product cordial graph, thereby ruling out any possibility of obtaining any forbidden subgraph characterization for product cordial graphs. They also proved that every connected graph can be embedded as an induced subgraph of a product cordial connected graph and every planar graph can be embedded as an induced subgraph of a product cordial planar graph. Similar results for total product cordial labeling were also obtained. In they prove that every graph can be embedded as an induced subgraph of a divisor cordial graph, thereby ruling out any possibility of obtaining any forbidden subgraph characterization for divisor cordial graphs. They also proved that every connected graph can be embedded as an induced subgraph of a divisor cordial connected graph and every planar graph can be embedded as an induced subgraph of a divisor cordial planar graph.
Gondalia proved that the Herschel graph, the fusion of any two adjacent vertices of degree 3 in a Herschel graph, the duplication of any vertex of degree 3 in a Herschel graph, the switching of a central vertex in the Herschel graph, the joint sum of two copies of a Herschel graph, and the degree splitting of Herschel graph are divisor cordial graphs.
Motivated by the concept of divisor cordial labeling, Lourdusamy and Patrick introduced a new concept of divisor cordial labeling called sum divisor cordial labeling. Let G = (V (G), E(G)) be a simple graph and f be a bijection from V (G) to {1, 2, . . . , |V (G)|}.
For each edge uv, assign the label 1 if 2 divides f(u)+f(v) and the label 0 otherwise. The function f is called a sum divisor cordial labeling if the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph which admits a sum divisor cordial labeling is called a sum divisor cordial . They prove that paths, combs, stars, complete bipartite, K2+mK1, bistars, jewels, crowns, flowers, gears, subdivisions of stars, the graph obtained from K1,3 by attaching the center of K1,n at each pendent vertex of K1,3, and the square Bn,n are sum divisor cordial graphs. In Lourdusamy and Patrick proved that every transformed tree admits a sum divisor cordial labeling. They also investigated the sum divisor cordial labelings of the graphs obtained by identifying a vertex of graphs of paths, middle graphs of paths, and the splitting graphs of cycles.
In , , , and Sugumaran and Rajesh proved that the following graphs are sum divisor cordial: swastiks, path unions of finite number of copies of swastiks, cycles of k copies of swastiks, when k is odd, jelly fish, Petersen graphs, theta graphs, the fusion of any two vertices in the cycle of swastiks, duplication of any vertex in the cycle of swastiks, the switchings of a central vertex in swastiks, the path unions of two copies of a swastik, the star graph of the theta graphs, the Herschel graph, the fusion of any two adjacent vertices of degree 3 in Herschel graphs, the duplication of any vertex the electronic journal of combinatorics (2023), #DS6 321 of degree 3 in the Herschel graph, the switching of central vertex in Herschel graph, the path union of two copies of the Herschel graph, H-graph Hn, when n is odd, C3@K1,n (obtained by identifying the center of K1,n with a vertex of C3), < F 1 n△F 2 n > (the graph obtained by joining the apex vertices of F 1 n and F 2 n by an edge and by joining the two apex vertices to a new vertex v′) and open star of swastik graphs S(t.Swn), when t is odd.
In , and Sugumaran and Rajesh proved that the following graphs are sum divisor cordial graphs: H-graph Hn, when n is even, duplication of all edges of the H-graph Hn, when n is even, Hn ⊙K1, P(r.Hn), C(r.Hn), plus graphs, umbrella graphs, path unions of odd cycles, kites, complete binary trees, drums graph (two copies of Cn that share exactly one vertex v and two copies of Pn that have an end point at v, twigs (graphs obtained from a path by attaching exactly two pendent edges to each internal vertices of the path), fire crackers of the form Pn ⊙Sn, where n is even, and the double arrow graph DAn m, where |m −n| ≤1 and n is even (obtained from Pm × Pn by adding two new vertices u and v such that each of the top row vertices of Pm × Pn are connected to u by an edge and the bottom row vertices of Pm × Pn are connected to v by an edge).
Sugumaran and Rajesh proved that the following graphs are sum divisor cordial: Pn + Pn (n is odd), Pn@K1,m (obtained by identifying an endpoint of Pn with the center of K1,n), Cn@K1,m (n is odd), the graph obtained from W2n by attaching the apex vertex of a copy of K1,m to each rim vertex, the graph obtained by joining the central vertices of two copies of K1,n,n by an edge and to a new vertex, the graph obtained by starting with Cn and, for each edge of Cn, adjoining a copy of Cn that shares an edge with the starting copy (the flower graph FLn). In Adalja and Ghodasara provide sum divisor cordial labelings for the graphs resulting from the duplication of graph elements in stars, cycles, and paths. In Raheem, Javaid, Numan, and Hasni investigated the existence sum divisor cordial labelings for the disjoint union of paths and subdivided stars. In new Arockiams, Wencys, and Patrick investigate sum divisor cordial labelings of transformed tree related graphs.
The zero divisor graph of a commutative ring R, Γ(R), is the graph whose vertex set is set of zero divisors in R in which two distinct vertices u, v are adjacent if uv = 0.
In Lourdusamy, Jenifer Wency, and Joy Beaula investigate the existence of sum divisor cordial labelings of graphs of the form Γ(Zn), Γ(Zm) + Γ(Zn), and Γ(Zm) + Γ(Zn).
Murugesan introduced a square divisor cordial labeling. Let G be a simple graph and f :→{1, 2, . . . , |V (G)|} a bijection. For each edge uv, assign the label 1 if either (f(u))2 divides f(v) or (f(v))2 divides f(u) and the label 0 otherwise. Call f a square divisor cordial labeling if |ef(0) −ef(1)| ≤1. A graph with a square divisor cordial labeling is called a square divisor cordial graph. Murugesan proved that the following are square divisor cordial graphs: Pn (n ≤12), Cn (3 ≤n ≤11), wheels, some stars, some complete bipartite graphs, and some complete graphs. Vaidya and Shah proved that the following are square divisor cordial graphs: flowers, bistars, shadow graphs of stars, splitting graphs of stars and bistars, degree splitting graphs of paths and bistars.
Kanani and Bosmia define a cube divisor cordial labeling f of a simple graph G as a bijection from V (G) to {1, 2, . . . , |V (G)|} such that, when each edge uv is assigned the label 1 if (f(u))3 divides f(v) or (f(v))3 divides f(u) and the label 0 otherwise, it the electronic journal of combinatorics (2023), #DS6 322 holds that |ef(0) −ef(1)| ≤1. A graph with a cube divisor cordial labeling is called a cube divisor cordial graph. They proved that the following graphs admit cube divisor cordial labelings: Kn if and only if n = 1, 2, 3; K1,n if and only if n = 1, 2, 3; K2,n for all n; K3,n if and only if n = 1, 2; bistars Bn,n for all n ; and the graph obtained by joining leaves of one star of a bistar with the center of the opposite star of the bistar. Kanani and Bosmia prove: the edge deleted graph of a cube divisor cordial graph is also a cube divisor cordial graph; Pn is a cube divisor cordial graph if and only if n = 1, 2, 3, 4, 5, 6, 8; Cn is a cube divisor cordial graph if and only if n = 3, 4, 5; and wheels, flowers and fans are cube divisor cordial, The Lucas sequence of numbers is a linear recurrence relation satisfying the conditions: l1 = 1, l2 = 3 and ln = ln−1 + ln−2, n ≥3.
Let G = (V, E) be a simple graph and f : V (G) →{l1, l2, . . . , l|V (G)|} be a bijection such that each edge uv, assign the label 1 if either f(u) divides f(v) or f(v) divides f(u) and label 0 otherwise. In Sugumaran and Rajesh call such an f a Lucas divisor cordial labeling if |ef(0) −ef(1)| ≤1. A graph with a Lucas divisor cordial labeling is called a Lucas divisor cordial graph. In Sugumaran and Rajesh proved that the following graphs are Lucas divisor cordial graphs: bistars, jelly fish, square graphs of bistars, switching of a vertex in cycles, and switching of a pendent vertex in paths.
A variation of divisor cordial labeling called vertex odd divisor cordial labeling was introduced by Muthaiyan and Pugalenthi (see ) as follows. Let G be a graph with p vertices and a bijection f from V (G) to {1, 3, 5, . . . , 2p −1} such that if each edge uv is assigned the label 1 if f(u) divides f(v) or f(v) divides f(u), and the label 0 otherwise.
The function f is called a vertex odd divisor cordial labeling if |ef(0) −ef(1)| ≤1. A graph with vertex odd divisor cordial labeling is called a vertex odd divisor cordial graph.
Muthaiyan and Pugalenthi (see ) proved paths, cycles, K2,n, K1,n ∪K1,m, helms, flowers, < K(1) 1,n, K(2) 1,n >, the switching of the apex vertex in helms, and the splitting graph of stars are vertex odd divisor cordial graphs under some conditions. In Muthaiyan and Pugalenthi proved the following graphs have vertex odd divisor cordial labelings: wheels, the switching of a pendent vertex in paths and cycles, bistars Bn,n, the subdivision graph of K1,n, B2 n,n, DS(Bn,n), the splitting graph of Bn,n, and < K(1) 1,n, K(2) 1,n, K(3) 1,n >.
Let G1 and G2 be two copies of any graph G that has an apex vertex. The graph obtained by joining the apex vertices of G1 and G2 by an edge and the graph obtained by joining the apex vertices of G1 and G2 by an edge and by joining the two apex vertices to a new vertex v′, is denoted G1△G2. For any vertex u of Km,n the graph obtained by joining u to a new pendent vertex is denoted by Km,n ⊙u(K1). In Sugumaran and Suresh proved that the following graphs are vertex odd divisor cordial graphs: the shadow graph of K1,n, K2,n ⊙u(K1), K1,n△K1,n, the subdivision of the edge between the apex vertices of Bn,n, and the graph K1,n ∗Pn+2 (the graph obtained by identifying an end vertex of Pn+2 with the apex vertex of K1,n). In they showed that the graphs Fn△Fn, K1,n△K1,n△K1,n, K1,n△K1,n△K1,n△K1,n, theta graphs, and switching of a vertex in a Petersen graph are vertex odd divisor cordial graphs. In they proved that gears, switching of an apex vertex in S(K1,n), P2 + mK1, C(n, n −3), and C(n, n −4) are vertex odd divisor cordial graphs. In they showed that the globe the electronic journal of combinatorics (2023), #DS6 323 Gl(n), jewels, G ∗Wn (appending the central vertex of wheel Wn with any one of the vertices of G), G ∗C(n, n −3), and wheels are vertex odd divisor cordial graphs.
Let f : V (G) →{1, 2, . . . , |V (G)|} be an injective map. For each edge uv assign the label r where r is the remainder when f(u) is divided by f(v) or f(v) is divided by f(u) according as f(u) ≥f(v) or f(v) ≥f(u). The function f is called a remainder cordial labeling of G if |ef(0)−ef(1)| ≤1 where ef(0) and ef(1) respectively denote the number of edges labelled with even integers and number of edges labelled with odd integers. A graph G with admits a remainder cordial labeling is called a remainder cordial graph. In Ponraj, Annathurai, and Kala investigated the remainder cordial behavior of S(K1,n), S(Bn,n), S(Wn) and union of some star related graphs. Ponraj, Gayathri, and Somasun-daram investigated the 4-remainder cordial labeling behavior of parachutes, twigs, and kayak paddles graphs.
A double divisor cordial labeling of a graph G(V, E) is a bijective function ψ from V to {1, 2, 3, . . . , |V |} such that each edge ab is given label 1 if 2ψ(a)|ψ(b) or 2ψ(b)|ψ(a) and 0 otherwise, then the number of edges labeled 0 and the number of edges labeled 1 differ by at most 1. An graph that admits a double divisor cordial labeling is said to be a double divisor cordial graph. In Parthiban and Sharma proved that the following graphs admit double divisor cordial labelings: full binary trees, Pn (n ≥3), Cn, Wn (n odd), helms, fans, stars, bistars, jelly fish, and friendship graphs. They also prove that Kn (n ≥5) does not admit a double divisor cordial labeling. Sharma and Parthiban proved that the following graphs admit double divisor cordial labelings: the barycentric subdivision of stars and bistars B(n, n); the splitting graph of K1,n (n ≥2); the splitting graph of B(m, n) (m, n ≥2); ⟨K(1) 1,n, K(2) 1,n⟩; ⟨K(1) 1,n, K(2) 1,n, K(3) 1,n⟩; the graph constructed by connecting an apex vertex of one copy of Bn,n to a new vertex r an apex vertex of another copy of Bn,n to another vertices s; and Ki,n ⊙K1 for i = 1, 2, and 3.
7.6 Other Cordial Labelings In Barasara and Thakkar introduced a new concept called power cordial labeling which is defined as follows: a power cordial labeling of a graph G = (V, E) is a bijection f : V (G) →{1, 2, . . . , |V (G)|} such that when an edge e = uv is assigned the label 1 if f(u) = (f(v))n or f(v) = (f(u))n, for some n ∈N ∪{0} and the label 0 otherwise, the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph that admits a power cordial labeling is called a power cordial graph. They proved that the helm Hn is a power cordial graph for n ≤9 and n = 13 and not a power cordial graph for n = 10, 11, 12 and n ≥14, the flower Fln is a power cordial graph, the gear graph Gn is a power cordial graph for n ≤9 and n = 13 and not a power cordial graph for n = 10, 11, 12 and n ≥14, the fan Fn is a power cordial graph, the jewel graph Jn is a power cordial graph, the graph S(K1,n) is a power cordial graph, the graph 2K1,n is a power cordial graph, the graph S′(K1,n) is a power cordial graph for n ≤5 and n = 7 and not a power cordial graph for n = 6 and n ≥8, the graph G = ⟨K(1) 1,n, K(2) 1,n⟩is a power cordial graph and the graph S′(Bn,n) is a power cordial graph for n ≤8 and not a power cordial graph for n > 8. In Barasara and Thakkar proved that any graph G can be the electronic journal of combinatorics (2023), #DS6 324 embedded as an induced subgraph of a power cordial graph. This rules out the possibility of forbidden subgraph characterization for power cordial labeling. Also, they showed that for given positive integer n, there is a power cordial graph G that has n vertices. If G is a power cordial graph of even size, then G −e is also a power cordial graph for all e ∈E(G), If G is a power cordial graph of odd size, then G −e is also a power cordial graph for some e ∈E(G), for G be a power cordial graph of even size and G ∗Kn, then adding an edge between any two nonadjacent vertices of G is also a power cordial graph.
Moreover they proved that for given graph G with size n and G has a vertex v such that jn 2 k ≤d(v) ≤ ln 2 m then G is a power cordial graph.
In Ponraj, Sathish Narayanan, and Ramasamy introduced a new graph labeling called parity combination cordial labeling. Let G be a (p, q)-graph. Let f be an injective map from V (G) to {1, 2, . . . , p}. For each edge xy, assign the label x y or y x according as x > y or y > x. Call f a parity combination cordial labeling if f is a one to one map and |ef(0) −ef(1)| ≤1, where ef(0) and ef(1) denote the number of edges labeled with an even number and odd number, respectively. A graph with a parity combination cordial labeling is called a parity combination cordial graph.
They proved that the following are parity combination cordial graphs: paths, cycles, stars, triangular snakes, alternate triangular snakes, olive trees, combs, crowns, fans, umbrellas, P 2 n, helms, dragons, bistars, butterfly graphs, and graphs obtained from Cn and K1,m by unifying a vertex of Cn and a pendent vertex of K1,m. They also proved that Wn admits a parity combination cordial labeling if and only if n ≥4 and conjectured that for n ≥4, Kn is not a parity combination cordial graph. In , Ponraj, Rajpal Singh, and Sathish Narayanan proved that if G is a parity combination cordial graph, then G ∪Pn is also parity combination cordial if n ̸= 2, 4. In Seoud and Aboshady surveyed all graphs of order at most six with regard to whether they have a parity combination cordial labeling or not. They obtained an upper bound for the number of edges of any graph that satisfies a certain condition, and described the parity combination cordial labelings for two families of graphs.
In Naduvath defines the notion of the set-cordial labeling of a graph as follows.
For a non-empty set X, a function f from the vertices of a graph G to the power set of X is said to be a set-cordial labeling of G if |f(vi)||f(vj)| = ±1 for all edges vivj of G.
A graph that admits a set-cordial labeling is called a set-cordial graph . He proves that paths are set-cordial and that a graph G admits a set-cordial labeling if and only if G is bipartite. He defines the glutting number of a graph G as the minimum number of edges of G that can be removed so that the resulting graph admits a set-cordial labeling and provides the glutting number of wheels, helms, and complete graphs.
In 2011 Murugan and Selvaraj introduced the concept of V -cordial labeling of a graph G with vertex set V as a bijective function φ : V →{0, 1} such that the induced function φ∗: E →{0, 1} is defined by φ(uv) = 0 if φ(u) = φ(v) = 0 and 1 otherwise with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and likewise for the edges. A graph that the admits a V -cordial labeling is called a V -cordial graph. In 2015 Murugan and Mathubala introduced the concept of homo-cordial labeling as follows. A homo-cordial labeling of a graph G with vertex set V is a bijection φ : V →{0, 1} such that the induced function φ∗: E →{0, 1} the electronic journal of combinatorics (2023), #DS6 325 given by φ∗(uv) = 1, if φ(u) = φ(v) and 0 otherwise with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and likewise for the edges. A graph that the admits a homo-cordial labeling is called a homo-cordial graph. In Murugan and Vidhya say that a hetro-cordial labeling of a graph G with vertex set V is a bijection φ : V →{0, 1} such that the induced function φ∗: E →{0, 1} given by φ∗(uv) = 0, if φ(u) = φ(v) and 1, otherwise with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and likewise for the edges. A graph that the admits a hetro-cordial labeling is called a hetro-cordial graph. Murugan, and Mathubala proved that odd cycles, shadow graphs of cycles, graphs obtained by identifying the end points of n paths of length 2 (globe graphs), and double triangular snakes are homo-cordial graphs. Murugan, and Selva Vidhya proved that paths, combs, fans, double fans, and ladders are hetro-cordial graphs. Murugan and Selvaraj proved that paths, fans double fans, and combs are V cordial graphs. Bala, Sundarraj, and Thirusangu proved existence of a homo-cordial labeling, a hetro-cordial labeling, and a V -cordial labeling for the extended triplicate graph of a comb.
In Nicholas and Maya introduced the concept of an integer edge cordial label-ing of a graph G with edge set E as an injective map f from E to [−q/2, . . . , q/2] or [⌊−q/2⌋, . . . , ⌊q/2⌋] as q is even or odd, which induces a vertex labeling f ∗: V →{0, 1} such that, a vertex u is assigned the label 1 if P(f ∗(ei) ≥0 taken over all i, and 0 otherwise, and the number of vertices labeled with 1 and the number of vertices labeled with 0 differ by at most by 1. A graph that has integer edge cordial labeling is called an integer edge cordial graph. They proved that Pn (n ≥3), cycles, Wn (n > 3), helms, closed helms, K1,2n, and flower graphs are integer edge cordial graphs. They also proved that Kn,n is not integer edge cordial and that Kn,n \M is integer edge cordial if n is even, where M is a perfect matching of Kn,n.
For a planar graph G(p, q) suppose that g : E(G) →[ −q 2 , . . . , q 2] or [−⌊q 2⌋, . . . , ⌊q 2⌋] as q is even or odd, is an injective map that induces a vertex labeling g∗: V (G) →{0, 1} defined by g∗(v) = 1 if the sum of all g(e) over all edges e of G adjacent to v is nonnegative and g∗(v) = 0 otherwise, and the face labeling function g∗∗from the faces of G to {0, 1} defined by g∗∗(f) = 1, if the sum of g(e) over all edges e of f is nonnegative, and g∗∗(f) = 0 otherwise. Then g is called a face integer edge cordial labeling if the number of vertices labeled 0 and the number of vertices labeled 1 differ by at most 1 and the number of faces labeled 0 and the number of faces labeled 1 differ by at most 1. A graph that admits a face integer edge cordial labeling is called face integer edge cordial. Sheriff and Abbas gave face integer edge cordial labelings of the duplication of each vertex by an edge in nontrivial fans, the planar graph G′ obtained from joining the outer vertex of the two cop For a simple, finite and planar graph G of order p and size q. Sumathi and J. Suresh Kumar introduced the concept of fuzzy quotient-3 cordial labeling as follows. Let σ : V (G) →[0, 1] be a function defined by σ(v) = r 10, r ∈Z4 −{0}. For each edge uv define µ : E(G) →[0, 1] by µ(uv) = 1 10 3σ(u) σ(v) where σ(u) ≤σ(v). The function σ is called fuzzy quotient-3 cordial labeling of G if the number of vertices labeled with i and the number of vertices labeled with j differ by at most 1, the number of edges labeled with i the electronic journal of combinatorics (2023), #DS6 326 and the number of edges labeled with j differ by at most 1 where i ̸= j have the form r 10, where r ∈{1, 2, 3}. They proved that stars and star related graphs are fuzzy quotient-3 cordial.
The concept of multiply divisor cordial labeling was introduced in 2019 by Gondalia and Rokad as follows. A multiply divisor cordial labeling of a graph G having vertex set V is a bijective h from V to {1, 2, . . . , |V |} such that an edge xy is assigned the label 1 if 2 divides h(x) · h(y) and 0 otherwise, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph having multiply divisor cordial labeling is said to be a multiply divisor cordial graph. In and Gondalia new new Rokad proved that cycles, cycles with one chord, cycles with a twin chord, cycles with a triangle, paths, stars, jelly fish, and coconut trees are multiply divisor cordial graph.
Further they proved that ring sum of star with a cycle, the ring sum of star with a cycle having one chord, the ring sum of star with a cycle having twin chords, the ring sum of star with a cycle having triangle, the ring sum of star with double fan, the ring sum of star with double wheel, and the ring sum of star with helm graph are multiply divisor cordial labeling.
In Gondalia proved the following graphs admit multiply divisor cordial label- new ings: graphs obtained by joining two copies of Petersen graph by a path of arbitrary length; graphs obtained by joining two copies of Cn with one chord by a path of arbitrary length; graphs obtained by joining two copies of Cn with twin chords by a path of arbitrary length; graphs obtained by joining two copies of Cn with a triangle by a path of arbitrary length; graphs obtained by joining two copies of a fan Fn by a path of arbitrary length; and graphs obtained by joining two copies of flower graph Fln by a path of arbitrary length.
7.7 Edge Product Cordial Labelings Vaidya and Barasara introduced the concept of edge product cordial labeling as edge analogue of product cordial labeling. An edge product cordial labeling of graph G is an edge labeling function f : E(G) →{0, 1} that induces a vertex labeling function f ∗: V (G) →{0, 1} defined as f ∗(u) = Q{f(uv) | uv ∈E(G)} such that the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1 and the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1. A graph with an edge product cordial labeling is called an edge product cordial graph.
In , , , , and Vaidya and Barasara proved the following graphs are edge product cordial: Cn for n odd; trees with order greater than 2; unicyclic graphs of odd order; C(t) n , the one point union of t copies of Cn for t even or t and n both odd; Cn ⊙K1; armed crowns Cm ⊙Pn ; helms; closed helms; webs; flowers; gears; shells Sn for odd n; tadpoles Cn@Pm for m + n even or m + n odd and m > n while not edge product cordial for m + n odd and m < n; triangular snakes; for odd n, double triangular snakes DTn, quadrilateral snakes Qn and double quadrilateral snakes DQn; P 2 n for odd n; M(Pn), T(Pn); S′(Pn) for even n; the tensor product of Pm and Pn; and the tensor the electronic journal of combinatorics (2023), #DS6 327 product of Cn and Cm if m and n are even. In Vaidya and Barasara investigate product and edge product cordial labelings of the degree splitting graphs of paths, shells, bistars, and gear graphs. They proved the following graphs are not edge product cordial: Cn for n even; Kn for n ≥4; Km,n for m, n ≥2; wheels; the one point union of t copies of Cn for t odd and n even; shells Sn for even n; tadpoles Cn@Pm for m + n odd and m < n; for n even double triangular snake DTn, quadrilateral snake Qn and double quadrilateral snake DQn; double fans; C2 n for n > 3; P 2 n for even n; D2(Cn), D2(Pn); M(Cn); T(Cn); S′(Cn); S′(Pn) for odd n; Pm × Pn and Cm × Cn; the tensor product of Cn and Cm if m or n odd; and Pn[P2] and Cn[P2].
Barasara proved that Cn with one chord except when n is even and the chord joins vertices that are at diameter distance, Cn with twin chords except when n is even and the chords join vertices that are at diameter distance, and Pn ⊙K1 are edge product cordial graphs, whereas triangular ladders are not edge product cordial.
In , Barasara proved that Cn with one chord, Cn with twin chords, Pn ⊙K1, and triangular ladders are total edge product cordial graphs. Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of an edge product cordial graph, thereby ruling out any possibility of obtaining any forbidden subgraph characterization for edge product cordial graphs. They also proved that every connected graph can be embedded as an induced subgraph of an edge product cordial connected graph and every planar graph can be embedded as an induced subgraph of an edge product cordial planar graph. In , Barasara proved that the shadow graph of Wn and the splitting graph of Wn are not edge product cordial graphs, whereas the middle graph of Wn is edge product cordial for n > 5, and the total graph of Wn is edge product cordial for n > 8. He also investigated total edge product cordial labeling of the shadow graph, the splitting graph, the middle graph, and the total graph of Wn.
Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of a total edge product cordial graph thereby ruling out any possibility of ob-taining any forbidden subgraph characterization for total edge product cordial graphs.
They also proved that a connected graph can be embedded as an induced subgraph of a total edge product cordial connected graph and every planar graph can be embedded as an induced subgraph of a total edge product cordial planar graph.
Prajapati and Shah proved the following graphs are edge product cordial: graphs obtained from a crown by duplication of a vertex, duplication of a vertex by an edge, or duplication of an edge by a vertex; graphs obtained from a gear graph by duplication of each of the vertices of degree three by an edge; and the graph obtained from a helm by duplication of each of the pendent vertices by a new vertex. In Prajapati and Patel provided results about the existence of edge product cordial labelings closed webs, lotus inside a circle, and sunflower graphs.
Vaidya and Barasara introduced the concept of a total edge product cordial labeling as edge analogue of total product cordial labeling. An total edge product cordial labeling of graph G is an edge labeling function f : E(G) →{0, 1} that induces a vertex labeling function f ∗: V (G) →{0, 1} defined as f ∗(u) = Q{f(uv) | uv ∈E(G)} such that the number of edges and vertices labeled with 0 and the number of edges and vertices the electronic journal of combinatorics (2023), #DS6 328 labeled with 1 differ by at most 1. A graph with total edge product cordial labeling is called a total edge product cordial graph.
In and Vaidya and Barasara proved the following graphs are total edge product cordial: Cn for n ̸= 4; Kn for n > 2; Wn; Km,n except K1,1 and K2,2; gears; C(t) n , the one point union of t copies of Cn; fans; double fans; C2 n; M(Cn); D2(Cn); T(Cn); S′(Cn); P 2 n for n > 2; M(Cn); D2(Cn) for n > 2; T(Cn); S′(Cn). Moreover, they prove that every edge product cordial graph of either even order or even size admits total edge product cordial labeling. Bača, Irfan, Javad, and Semaničová-Feňočová investigated the existence of total edge product cordial labeling of toroidal fullerenes and for Klein-bottle fullerenes. Prajapati and Patel proved that the one point union of t copies of a wheel with a rim vertex in common is edge product cordial if and only is t is even; all pentagonal snakes (obtained from the path by replacing every edge of a path by C5) are edge product cordial; and a double pentagonal snakes (two pentagonal snakes that have a common path) is edge product cordial if and only is t is odd.
In Prasad and Maheswari developed a technique for coding a secret messages using sunflower graphs by subdividing edges and applying edge product cordial labeling.
7.8 Difference Cordial Labelings Ponraj, Sathish Narayanan, and Kala introduced the notion of difference cordial labelings. A difference cordial labeling of a graph G is an injective function f from V (G) to {1, . . . , |V (G)|} such that if each edge uv is assigned the label |f(u)−f(v)|, the number of edges labeled with 1 and the number of edges not labeled with 1 differ by at most 1. A graph with a difference cordial labeling is called a difference cordial graph. Jeba Jesintha, Devakirubanithi, and Jasma Pershi proved that the graphs obtained by switching new a central vertex in the path is difference cordial and the crown graph with r copies of fan graph attached to it is also difference cordial.
The following definitions appear in , , , and . A double trian-gular snake DTn consists of two triangular snakes that have a common path; a double quadrilateral snake DQn consists of two quadrilateral snakes that have a common path; an alternate triangular snake A (Tn) is the graph obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertex vi (that is, every alternate edge of a path is replaced by C3); a double alternate triangular snake DA (Tn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to two new vertices vi and wi; an alternate quadrilateral snake A (Qn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertices vi and wi respectively and then joining vi and wi (that is, every alternate edge of a path is replaced by a cycle C4); a double alternate quadrilateral snake DA (Qn) is obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to new vertices vi, xi and wi and yi respectively and then joining vi and wi and xi and yi.
In and Ponraj and Sathish Narayanan define the irregular triangular snake ITn as the graph obtained from the path Pn : u1, u2, . . . , un with vertex set V (ITn) = V (Pn) ∪{vi : 1 ≤i ≤n ≤2} and the edge set E(ITn) = E(Pn) ∪{uivi, viui+2 : the electronic journal of combinatorics (2023), #DS6 329 1 ≤i ≤n −2}. The irregular quadrilateral snake IQn is obtained from the path Pn : u1, u2, . . . , un with vertex set V (IQn) = V (Pn) ∪{vi, wi : 1 ≤i ≤n −2} and edge set E (IQn) = E (Pn)∪{uivi, wiui+2, viwi : 1 ≤i ≤n −2}. They proved the following graphs are difference cordial: triangular snakes Tn, quadrilateral snakes, alternate triangular snakes, alternate quadrilateral snakes, irregular triangular snakes, irregular quadrilateral snakes, double triangular snakes DTn if and only if n ≤6, double quadrilateral snakes, double alternate triangular snakes DA (Tn), and double alternate quadrilateral snakes.
In , , , and Ponraj, Sathish Narayanan, and Kala proved the following graphs have difference cordial labelings: paths; cycles; wheels; fans; gears; helms; K1,n if and only if n ≤5; Kn if and only if n ≤4; K2,n if and only if n ≤4; K3,n if and only if n ≤4; bistar B1,n if and only if n ≤5; B2,n if and only if n ≤6; B3,n if and only if n ≤5; DTn⊙K1; DTn⊙2K1; DTn⊙K2; DQn⊙K1; DQn⊙2K1; DQn⊙K2; DA (Tn)⊙K1; DA (Tn) ⊙2K1; DA (Tn) ⊙K2; DA (Qn) ⊙K1; DA (Qn) ⊙2K1; and DA (Qn) ⊙K2. They also proved: if G is a (p, q) difference cordial graph, then q ≤2p −1; if G is a r-regular graph with r ≥4, then G is not difference cordial; if m ≥4 and n ≥4, then Km,n is not difference cordial; if m + n > 8 then the bistar Bm,n is not difference cordial; and every graph is a subgraph of a connected difference cordial graph. If G is a book, sunflower, lotus inside a circle, or square of a path, they prove that G⊙mK1 (m = 1, 2) and G⊙K2 is difference cordial.
In , , and Ponraj, Sathish Narayanan, and Kala proved that the following graphs are difference cordial: crowns Cn ⊙K1; combs Pn ⊙K1; Pn ⊙Cm; Cn ⊙ Cm; Wn ⊙K2; Wn ⊙2K1; Gn ⊙K1 where Gn is the gear graph; Gn ⊙2K1; Gn ⊙K2; (Cn × P2)⊙K1; (Cn × P2)⊙2K1; (Cn × P2)⊙K2; Ln⊙K1; Ln⊙2K1; and Ln⊙K2. Ponraj, Sathish Narayanan and Kala proved that the following subdivision graphs are difference cordial: S (Tn); S (Qn); S (DTn); S (DQn); S (A (Tn)); S (DA (Tn)); S (AQn); S (DAQn); S (K1,n); S (K2,n); S (Wn); S (Pn ⊙K1); S (Pn ⊙2K1); S (LCn); S (P 2 n); S (K2 + mK1); subdivision graphs of sunflowers S (SFn); subdivisions graphs flowers S (Fln); S (Bm) (Bm is a book with m pages); S (Cn × P2); S (Bm,n); subdivisions n-cubes; S (J (m, n)); S (W (t, n)); subdivisions of Young tableaus S (Yn,n); and if S (G) is difference cordial, then S (G ⊙mK1) is difference cordial. For graphs G that are a tree, a unicycle, or when |E(G)| = |V (G)| + 1, they proved that G ⊙Pn and G ⊙mK1 (m = 1, 2, 3) are difference cordial.
In Prajapati and Gajjar define a holiday star as follows. Let v1, v2, . . . , v4n−1, v4n be the consecutive 4n vertices of C4n (n ≥3).
Let u0 be the central vertex and u1, u2, . . . , u2n−1, u2n be end vertices of K1,2n. Join u0 to v4i−3 by an edge; for each i from 1 to n. In they define a Kusadama flower graph as follows. Let v0 be the apex vertex and v1, v2, v3, . . . , v2n−1, v2n be 2n consecutive rim vertices of the wheel W2n (n ≥3).
Subdivide the spoke edge v0v2i−1 by a vertex wi and at each wi join two copies of path of length 2; P ℓ 2 = v0, u2i−1, wi and P r 2 = v0, u2i, wi, for each i ∈[n]. In and Prajapati and Gajjar proved that the holiday star graph and the Kusudama flower graph admit cordial, E-cordial, difference cordial, prime, vertex prime, and total prime label-ings. In Prajapati and Gajjar define a braided star graph as follows: Let a0 be the apex vertex and a1, a2, . . . , an−1, an be consecutive n rim vertices of Wn (n ≥3); let the electronic journal of combinatorics (2023), #DS6 330 b1, b2, b3, . . . , b2n−1, b2n be 2n consecutive vertices of the cycle C2n; let c1, c2, . . . , c2n−1, c2n be consecutive 2n vertices of C2n. Join each ai to b2i−1 by an edge and b2i to c2i by an edge. Take a new vertex di and join each di to c2i−1 and c2i+1 by an edge for each i ∈[n] where subscripts are taken modulo n. Prajapati and Gajjar proved that braided star graphs are cordial, E-cordial and difference cordial. In Prajapati and Gajjar investigated the existence of cordial, E-cordial, vertex prime, and difference cordial label-ings of graphs that are inspired by the origami model called an aboreale star (see their paper for the definition).
Recall the splitting graph of G, S′(G), is obtained from G by adding for each vertex v of G a new vertex v′ so that v′ is adjacent to every vertex that is adjacent to v and the shadow graph D2(G) of a connected graph G is constructed by taking two copies of G, G′ and G′′, and joining each vertex u′ in G′ to the neighbors of the corresponding vertex u′ in G′′.
Ponraj and Sathish Narayanan , proved the following graphs are difference cordial: S ′(Pn); S ′ (Cn); S ′ (Pn ⊙K1); and S ′ (K1,n) if and only if n ≤3. They proved following are not difference cordial: S ′ (Wn); S ′ (Kn); S ′ (Cn × P2); the splitting graph of a flower graph; DS (SFn); DS (LCn); DS (Fln); D2 (G) where G is a (p, q) graph with q ≥p; and DS (Bm,n) (m ̸= n) with m + n > 8.
Let G (V, E) be a graph with V = S1 ∪S2 ∪· · · ∪St ∪T where each Si is a set of vertices having at least two vertices and having the same degree. Panraj and Sathish Narayanan , define the degree splitting graph of G denoted by DS (G) as the graph obtained from G by adding vertices w1, w2, . . . , wt and joining wi to each vertex of Si (1 ≤i ≤t). They proved the following graphs are difference cordial: DS (Pn); Wn; DS (Cn); DS (Kn) if and only if n ≤3; DS (K1,n) if and only if n ≤4; DS (Wn) if and only if n = 3; DS (Kc n + 2K2) if and only if n = 1; DS (K2 + mK1) if and only if n ≤3; DS (Kn,n) if and only if n ≤2; DS (Tn) if and only if n ≤5; DS (Qn) if and only if n ≤5; DS (Ln) if and only if n ≤5; DS (Bn,n) if and only if n ≤2; DS (B1,n) if and only n ≤4; DS (B2,n) if and only n ≤4; D2 (Pn); D2 (Kn) if and only if n ≤2; and D2 (K1,m) if and only if m ≤2.
In , Ponraj and Sathish Narayanan proved the following graphs are difference cordial: Tn ⊙K1, Tn ⊙2K1, Tn ⊙K2, A(Tn) ⊙K1, A(Tn) ⊙2K1 and A(Tn) ⊙K2 where Tn and A(Tn) are triangular snake and alternate triangular snake respectively. In [2575,2576] Ponraj, Sathish Narayanan, and Kala proved the following graphs are difference cordial: Cn × P2; M¨ obius ladders; the n-cube; sunflower graphs; lotuses inside a circle; pyra-mids; books with n pentagonal pages; mongolian tents; graphs obtained from a lad-der by subdividing each step exactly once; permutation graphs P(P2k, f) where f = (1 2)(3 4) · · · (k k+1) · · · (2k−1 2k); and P(Pn, I), P(Cn, I), P(Pn⊙K1, I), P(Pn⊙2K1, I) where I is the identity permutation. Ponraj, Sathish Narayanan, and Kala proved the following graphs are not difference cordial: G1(p1, q1)×G2(p2, q2) with q1 ≥p1 and q2 ≥p2; Cm × Cn; G × Kn where G connected graph and n ≥5, G + K1 where |E(G) > |V (G) + 1; G1 + G2 where G1 and G2 are connected and |E(G1)| > 1 and E(G2)| > 3; permutation graphs P(G × K2, f) where |E(G)| ≥|V (G)| and f is any permutation; P(Wn, f) for any permutation f; P(S ′(G), f) where S ′(G) is the splitting the electronic journal of combinatorics (2023), #DS6 331 graph of G, |E(G)| ≥|V (G)|, and f is any permutation; and P(Fln, f) where Fln is a flower graph and f is any permutation. They also obtained the following necessary and sufficient conditions for difference cordiality: Km×P2 if and only if m ≤3; for a connected graph G, G × Wn if and only if G = K1; books Bm if and only if m ≤6; G + G if and only if |V (G)| ≤3 and |E(G)| ≤1; K2 + mK1 if and only if m ≤4; Kn + 2K2 if and only if n ≤2; the double fan DFn if and only if n ≤4; the t-fold wheel Wn + Kt if and only if t ≤2 and n = 3; cocktail party graphs Hn,n if and only n ≤6; P(Kn, I) if and only if n ≤3; P(K2 + mK1, I) if and only if m ≤3; and P(Km,n, I) (m, n > 1) if and only if m = n = 2 and n = 3, 4, 5.
Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of a difference cordial graph, thereby ruling out any possibility of obtaining any forbidden subgraph characterization for difference cordial graphs. They also proved that every connected graph can be embedded as an induced subgraph of a difference cordial connected graph and every planar graph can be embedded as an induced subgraph of a difference cordial planar graph.
In , Ponraj, Maria Adaickalam, and Kala introduced a new graph labeling called a k-difference cordial labeling. Let G be a (p, q)-graph and 2 ≤k ≤|V (G)|. Let f : V (G) →{1, 2, . . . , k} be a map. For each edge uv, assign the label |f(u) −f(v)|. They say f is a k-difference cordial labeling of G if |vf(i) −vf(j)| ≤1 and |ef(0) −ef(1)| ≤1, where vf(x) denotes the number of vertices labeled with x, ef(1) denotes the number of edges labeled with 1, and ef(0) denotes the number of edges that are not labeled with 1.
A graph with a k-difference cordial labeling is called a k-difference cordial graph.
They proved the following: every graph is a subgraph of a connected k-difference cordial graph; if k is even, then k-copies of K1,p is k-difference cordial; and if n ≡0 (mod k) and k ≥6, then K1,n is not k-difference cordial. They further prove the following are 3-difference cordial graphs: paths; Cn where n ≡0, 3 (mod 4); Km,n (m ≤n) and m is even; combs; double combs; quadrilateral snakes; bistars; subdivisions of a star; subdivisions of a bistar; C(t) 4 ; Kn if and only if n ∈{1, 2, 3, 4, 6, 7, 9, 10}; and K1,n if and only if n ∈{1, 2, 3, 4, 5, 6, 7, 9}.
In , , , Ponraj and Maria Adaickalam proved the following are 3-difference cordial graphs: K1,n ⊙K2, Pn ⊙3K1, Cn ⊙K2, mC4, splitting graph of a star, fan, double fan, Wn where n ≡0, 1 (mod 3), helms, flower, sunflower graph, lotus inside a circle, closed helm, double wheel DWn where V (DWn) = V (Wn) ∪{vi : 1 ≤i ≤n} and edge set E(DWn) = E(Wn) ∪{uvi : 1 ≤i ≤n} ∪{vivi+1 : 1 ≤i ≤n −1} ∪{v1vn}, degree splitting graph of a bistar, spl(K1,n) ∪K1,n, spl(K1,n) ∪Pn, K3,n ∪spl(K1,n), DFn ∪spl(K1,n), S(K1,n) ∪S(Bn,n), K2,n ∪S(K1,n), Fn ∪S(K1,n), Wn ∪S(K1,n), Bn,n ∪ S(Bn,n), K2,n ∪Bn,n, (Cn ⊙K1) ∪(Pn ⊙K1), Fn ∪Fn, jelly fish, Pn ∪K1,n, K1,n ∪K2,n, K1,n ∪S(K1,n), are Let Cn be the cycle u1u2 . . . unu1. If G is (p, q) 3-difference cordial graph with p ≡0 (mod 2) and q ≡0 (mod 3), then G ∪G also 3-difference cordial.
Let G be the graph obtained from Cn with V (G) = V (Cn) ∪{vi : 1 ≤i ≤ n 2 } and E(G) = {uivi, ui+1vi : 1 ≤i ≤n}. Then G is 3-difference cordial. The graph Gn with the vertex set V (Gn) = {ui, vi, wi : 1 ≤i ≤n} and E(Gn) = {uiui+1, vivi+1 : 1 ≤i ≤ n −1} ∪{unu1, v1u1} ∪{uivi, viwi : 1 ≤i ≤n} is 3-difference cordial. Let C3 be the cycle the electronic journal of combinatorics (2023), #DS6 332 u1u2u3u1. Let G be a graph obtained from C3 with V (G) = V (C3)∪{vi, wi, zi : 1 ≤i ≤n} and E(G) = E(C3) = {u1vi, u2wi, u3zi : 1 ≤i ≤n}. Then G is 3-difference cordial if n ≡0, 2, 3 (mod 4).
If n ≡0, 1 (mod 3), then K1,n ∪K1,n is 3-difference cordial.
Ponraj, Adaickalam, and Kala proved the following graphs have 3-difference cordial labelings: DA(Tn)⊙K1, DA(Tn)⊙2K1, DA(Tn)⊙K2, DA(Qn)⊙K1, and DA(Qn)⊙2K1 (Tn is a triangular snake.) In Ponraj, Adaickalam, Maria Adaickalam, and Kala investigated the 3-difference cordial labeling behavior of ladders, books, dumbbell graphs, and umbrella graphs. Ponraj, Subbulakshmi, and Somasundarami investigated the 4-difference cordial labeling behavior of jelly fish, jewel graphs, combs, subdivisions of stars, subdivisions of bistars, and books with triangle pages.
For graphs G and H and a vertex v of G the graph G ⊙v H is obtained by joining any particular vertex of H to vertex v. In Sugumaran and Mohan proved that the following graphs are difference cordial graphs: the path union of r copies of P 2 n (that is, P(r.P 2 n))–see Section 2.7 for the definition), the cycle union of r copies of C2 n, (that is, C(r.C2 n)), the open star of r copies the square graph P 2 n (that is, S(r.P 2 n)), the graph C2 n ⊙vn Pk, and the graph C2 n ⊙vn P 2 k . In they proved that the plus graph Pln, the path union of plus graph P(r.Pln), the cycle union of plus graph C(r.Pln), the barycentric subdivision of Pln, the hanging pyramid HPyn graph, and the path union of hanging pyramid P(r.HPyn). In they proved that switching of a pendent vertex in path Pn, switching of an apex vertex in CHn, the graph obtained by duplication of each vertex of path Pn by an edge, the barycentric subdivision of Cn ⊙K1, the path union of r copies of fan P(r.Fn), the cycle union of r copies of fan C(r.Fn), and the open star of r copies of fan S(r.Fn) are difference cordial graphs.
7.9 Prime Cordial Labelings Sundaram, Ponraj, and Somasundaram have introduced the notion of prime cordial labelings. A prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, . . . , |V |} such that if each edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 and 0 if gcd(f(u), f(v)) > 1, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. In Sundaram, Ponraj, and Somasundram prove the following graphs are prime cordial: Cn if and only if n ≥6; Pn if and only if n ̸= 3 or 5; K1,n (n odd); the graph obtained by subdividing each edge of K1,n if and only if n ≥3; bistars; dragons; crowns; triangular snakes if and only if the snake has at least three triangles; ladders; K1,n if n is even and there exists a prime p such that 2p < n + 1 < 3p; K2,n if n is even and if there exists a prime p such that 3p < n + 2 < 4p; and K3,n if n is odd and if there exists a prime p such that 5p < n + 3 < 6p. They also prove that if G is a prime cordial graph of even size, then the graph obtained by identifying the central vertex of K1,n with the vertex of G labeled with 2 is prime cordial, and if G is a prime cordial graph of odd size, then the graph obtained by identifying the central vertex of K1,2n with the vertex of G labeled with 2 is prime cordial. They further prove that Km,n is not prime cordial for a number of special cases of m and n. Sundaram and Somasundaram and Youssef observed that for n ≥3, Kn is not prime the electronic journal of combinatorics (2023), #DS6 333 cordial provided that the inequality φ(2) + φ(3) + · · · + φ(n) ≥n(n −1)/4 + 1 is valid for n ≥3 (φ is the Euler phi-function). This inequality was proved by Yufei Zhao .
Haque, Lin, Yang, and Zhao show that with the exception of P(4, 1), all generalized Petersen graphs are prime cordial. Haque, Lin, Yang, and Zhang show that the flower snark and related graphs are prime cordial. In Ghosh, Mohanty, and Pal gave an algorithmic approach to find cordial labelings of Cartesian product of two balanced bipartite graphs.
A signed product cordial labeling of graph G(V, E) is a function f from V to {−1, 1} such that when each edge uv is assigned the label f(u)f(v), the number of vertices labeled −1 and the number of vertices labeled 1 differ by at most 1, and likewise for edges. A graph that admits signed product cordial labeling is called a signed product cordial graph.
Signed product cordial labelings for fractal graphs given by Santhi and Albert in and by Raval and Prajapati in . Ghosh and Pal proved that Km,m × Pn admits signed product cordial labelings and total signed product cordial labelings. El-hay, Elmshtaye, and Elrokh provided necessary and sufficient conditions for which there exist a signed product cordial labeling of corona product of paths and third power of lemniscate graph.
Seoud and Salim give an upper bound for the number of edges of a graph with a prime cordial labeling as a function of the number of vertices. For bipartite graphs they give a stronger bound. They prove that Kn does not have a prime cordial labeling for 2 < n < 500 and conjecture that Kn is not prime cordial for all n > 2. They determine all prime cordial graphs of order at most 6. For a graph with n vertices to admit a prime cordial labeling, Seoud and Salim proved that the number of edges must be less than n(n −1) −6n2/π2 + 3. As a corollary they get that Kn (n > 2) is not prime cordial thereby proving their earlier conjecture.
In Ghodasara and Jena prove that the following graphs are prime cordial: Cn with one chord, Cn with twin chords (that is, two cords that form a triangle with an edge of the cycle), Cn with three cords that form two triangles and a cycle of length n −3 (n ≥7), the graph obtained by joining two copies of Cn with one chord by a path, and the graph obtained by joining two copies of the same cycle with twin chords by a path is prime cordial.
In Baskar Babujee and Shobana proved sun graphs Cn ⊙K1; Cn with a path of length n −3 attached to a vertex; and Pn (n ≥6) with n −3 pendent edges attached to a pendent vertex of Pn have prime cordial labelings. Additional results on prime cordial labelings are given in .
In and Vaidya and Vihol prove following graphs are prime cordial: the total graph of Pn and the total graph of Cn for n ≥5 (see §2.7 for the definition); P2[Pm] for all m ≥5; the graph obtained by joining two copies of a fixed cycle by a path; and the graph obtained by switching of a vertex of Cn except for n = 5 (see §3.6 for the definition); the graph obtained by duplicating each edge by a vertex in Cn except for n = 4 (see §2.7 for the definition); the graph obtained by duplicating a vertex by an edge in cycle Cn (see §2.7 for the definition); the path union of any number of copies of a fixed cycle (see §3.7 for the definition); and the friendship graph Fn for n ≥3. Vaidya and Shah prove the electronic journal of combinatorics (2023), #DS6 334 following results: P 2 n is prime cordial for n = 6 and n ≥8; C2 n is prime cordial for n ≥10; the shadow graphs of K1,n (see §3.8 for the definition) for n ≥4 and the bistar Bn,n are prime cordial graphs.
Let Gn be a simple nontrival connected cubic graph with vertex set V (Gn) = {ai, bi, ci, di : 0 ≤i ≤n −1}, and edge set E(Gn) = {aiai+1, bibi+1, cici+1, diai, dibi, dici : 0 ≤i ≤n −1}, where the edge labels are taken modulo n. Let Hn be a graph obtained from Gn by replacing the edges bn−1b0 and cn−1c0 with bn−1c0 and cn−1b0 respectively. For odd n ≥5, Hn is called a flower snark whereas Gn, H3 and all Hn with even n ≥4, are called the related graphs of a flower snark. Mominul Haque, Lin, Yang, and Zhang proved that flower snarks and related graphs are prime cordial for all n ≥3.
In Vaidya and Shah prove that the following graphs are prime cordial: split graphs of K1,n and Bn,n; the square graph of Bn,n; the middle graph of Pn for n ≥4; and Wn if and only if n ≥8. Vaidya and Shah prove following graphs are prime cordial: the splitting graphs of K1,n and Bn,n; the square of Bn,n; the middle graph of Pn for n ≥4; and wheels Wn for n ≥8. Gajjar proved that the generalized web graph W(t, n) (n ≥3) is prime cordial for t = 1, 2, 3, and 4 and that W(t, 2p) is prime cordial for all t and odd primes p.
In Vaidya and Shah proved following graphs are prime cordial: gear graphs Gn for n ≥4; helms; closed helms CHn for n ≥5; flower graphs Fln for n ≥4; degree splitting graphs of Pn and the bistar Bn,n; double fans Dfn for n = 8 and n ≥10; the graphs obtained by duplication of an arbitrary rim edge by an edge in Wn where n ≥6; and the graphs obtained by duplication of an arbitrary spoke edge by an edge in wheel Wn where n = 7 and n ≥9.
Let G(p, q) with p ≥4 be a prime cordial graph and K2,n be a bipartite graph with bipartition V = V1 ∪V2 with V1 = {v1, v2} and V2 = {u1, u2, . . . , un}. If G1 is the graph obtained by identifying the vertices v1 and v2 of K2,n with the vertices of G having labels 2 and 4 respectively, Vaidya and Prajapati proved that G1 admits a prime cordial labeling if n is even; if n, p, q are odd and with ef(0) = ⌊q/2⌋; and if n is odd, p is even and q is odd with ef(0) = ⌈q/2⌉. Prajapati and Gajjar proved the following graphs are prime cordial: Cn × P2 except for n = 1, 2 and 4, Cn × P4 (n ≥3), C3 × Pn (n > 1), C5 × Pn (n > 1), C6 × Pn (n > 1), C2p × Pn where p is an odd prime and n > 1, and C4 × Pn (n > 2).
In Sugumaran and Mohan proved the following graphs are prime cordial: the cycle butterfly graph Bn,m (two copies of Cn that share a common vertex with m pendent vertices attached to the common vertex), W−graph (obtained by starting with the two copies of K1,n and merging the last pendent vertex in the first copy of K1,n with the initial pendent vertex in the second copy of K1,n), Hn graph (the graph obtained from two paths u1, u2, . . . , un and v1, v2, . . . , vn by joining the vertices u(n+1)/2 and v(n+1)/2 if n is odd and joining un/2 and vn/2+1 if n is even), and duplication of all edges of an Hn graph. In Sugumaran and Mohan proved that the following graphs are prime cordial: Hn ⊙K1, the path union of r copies of Hn, the cycle union of r copies of an Hn, the open star of r copies of an Hn−graph (obtained by replacing each pendent vertex of K1,n by a copy of Hn.
the electronic journal of combinatorics (2023), #DS6 335 In Sugumaran and Suresh proved that the following graphs are prime cordial graphs: the duplication of each vertex by an edge of paths, stars, jelly fish, bistars, and Cn ⊙K1. Sugumaran and Vishnu Prakash proved that the following graphs are prime cordial graphs: duplication of any vertex of degree 3 in theta graph , switching of any vertex of degree 3 in theta graph, fusion of any two vertices in theta graph, the path union of two copies of theta graph, and two copies of theta graph joined by a path of any length. They further proved that the theta graph is not a prime cordial labeling. In they showed that one point union of path graph P t n(tn.Tα), the open star of theta graph, and any path union of even number of theta graphs are prime cordial graphs. Also they proved the subdivision of bistar Bn,n, Pn J K1,n−1, (that is, each ith vertex of path Pn is append with the apex vertex of ith copy of K(1, n −1)), the disconnected graph Pn ∪Pm are prime cordial graphs.
Vaidya and Barasara proved that every graph can be embedded as an induced subgraph of a prime cordial graph thereby ruling out any possibility of obtaining any forbidden subgraph characterization for prime cordial graphs. They also proved that a connected graph can be embedded as an induced subgraph of a prime cordial connected graph and every planar graph can be embedded as an induced subgraph of a prime cordial planar graph. In Gayathri, Thanjavur, Maniammai, and Sakar proved that 8-polygonal snakes containing n 8-polygons, splitting graphs of Cn for n ≥5, and armed crowns C2k ⊙Pm for all k ≥3 and m ≥2 admit prime cordial labelings. Barasara and Prajapati showed that switching of a vertex of degree 1 in the path Pn is a prime cordial graph for n ̸= 2, 3, 4, 5, 7 and not a prime cordial graph for n = 2, 3, 4, 5, 7, the armed crown ACn is a prime cordial graph for all n, all web graphs are prime cordial, and C(t) n , one-point union of cycles, are prime cordial for t ̸= 2 and n ̸= 3, 4 and not a prime cordial graph for t = 2 and n = 3, 4. Babitha and Baskar Babujee proved that if G is prime cordial, then so is G with an edge deleted, K1,m ⊙Pn (m, n > 2) is prime cordial, and in certain cases, the one-point union of K1,m and Pn and the one-point union of a prime cordial graph G and K1,n are prime cordial. They further provided some characterization results.
For a graph G(V, E) and the group S3 of all permutations of {1, 2, 3} Chandra and Kala define a function g : V (G) →S3 such that xy ∈E if g(x) and g(y) have relatively prime orders. Let nj(g) denote the number of vertices of G having label j under g. Then g is called a group S3 cordial prime labeling if |ni(g) −nj(g)| ≤1 for every i, j ∈S3. A graph that admits a group S3 cordial prime labeling is called a group S3 cordial prime cordial prime graph. Chandra and Kala prove that all paths, cycles, gears, ladders, and fans are group S3 cordial prime and characterize wheels that are group S3 cordial prime.
In Parthiban and Sharma gave a comprehensive survey on prime cordial and divisor cordial labeling of graphs.
Vaidya and Prajapati call a graph strongly prime cordial if for any vertex v there is a prime labeling f of G such that f(v) = 1.
They prove the following: the graphs obtained by identifying any two vertices of K1,n are prime cordial; the graphs obtained by identifying any two vertices of Pn are prime cordial; Cn, Pn, and K1,n are the electronic journal of combinatorics (2023), #DS6 336 strongly prime cordial; and Wn is a strongly prime cordial for every even integer n ≥4.
Prajapati and Gajjar proved that generalized prism graphs Yn,2 is prime cordial except for n = 1, 2 and 4; Yn,4 is prime cordial for n ≥3; Y3,n, Y5,n, Y6,n and Y2p,n (for odd prime p) are prime cordial for n > 1; and Y4,n is prime cordial for n > 2. They also proved the following graphs are prime cordial: Cn × P2 except for n = 1, 2 and 4, Cn × P4 (n ≥3), C3 × Pn (n > 1), C5 × Pn (n > 1), C6 × Pn (n > 1), C2p × Pn where p is an odd prime and n > 1, and C4 × Pn (n > 2).
In Ponraj, Singh, Kala, and Sathish Narayanan introduced a new graph labeling called k-prime cordial labeling. Let G be a (p, q)-graph and 2 ≤p ≤k and let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). They say that f is a k-prime cordial labeling of G if |vf(i) −vf(j)| ≤1 for i, j ∈{1, 2, . . . , k} and |ef(0) −ef(1)| ≤1, where vf(x) denotes the number of vertices labeled with x, and ef(1) and ef(0), respectively, denote the number of edges labeled with 1 and not labeled with 1. A graph with a k-prime cordial labeling is a k-prime cordial graph. They proved that every graph is a subgraph of a connected k-prime cordial graph; if k is even, then Pn, n ̸= 3, is k-prime cordial; Cn, n ̸= 3, is k-prime cordial when k is even; and the bistar Bn,n is k-prime cordial for all even k. They studied 3-prime cordiality of paths, cycles, and olive trees. They also proved that if T is a 3-prime cordial tree, then T ⊙K1 is 3-prime cordial; K1,n is 3-prime cordial if and only if n ≤3; Kn is 3-prime cordial if and only if n < 3; combs Pn ⊙K1 are 3-prime cordial; and Cn ⊙K1 is 3-prime cordial if and only if n ̸= 3. They proved that K2 + mK1, K2,n, and wheels are not 3-prime cordial graphs.
In Ponraj, Singh, and Sathish Narayana proved if G is 3-prime cordial, then G∪Pn is a 3-prime cordial for n > 12, the splitting graph of a star is not a 3-prime cordial graph, and the jelly fish J(m, n) is 3-prime cordial if 10m ≥n + 2.
For a 4-prime cordial graph G Ponraj and Singh proved G ∪Pn (n ≥5), G ∪ 2mKn,n, and G ∪2mK1,n are 4-prime cordial. For a (4t, q) 4-prime cordial graph G they prove that G + K1 and G + 2K1 are 4-prime cordial. Ponraj, Singh, and Kala prove that Pm × Pn and subdivisions of wheels and helms are 4-prime cordial. They also show that if G is bipartite then G ∪G is 4-prime cordial; and if G is 4-prime cordial then G ⊙K1 is 4-prime cordial. Ponraj, Singh, and Kala proved the following graphs are 4-prime cordial: 2m(Kn,n), 2m(Pn × P2), m(Cn ⊕K1), mBn,n, and 2W2n+1.
Murugesan, Jayaraman, and Shiama (see ) defined a 3-equitable prime cordial labeling of a graph G as a bijection f from V (G) to {1, 2, . . . , |V (G)|} such that if an edge uv is assigned the label 1 when gcd (f(u), f(v)) = 1 and gcd (f(u)+f(v), f(u)−f(v)) = 1, the label 2 when gcd (f(u), f(v)) = 1 and gcd(f(u) + f(v), f(u) −f(v)) = 2, and the label 0 otherwise, then the number of edges labeled with i and the number of edges labeled with j differ by at most 1 for 0 ≤i ≤2 and 0 ≤j ≤2. A graph that has a 3-equitable prime cordial labeling is called a 3-equitable prime cordial graph. Sugumaran and Vishnu Prakash proved the following graphs are 3-equitable prime cordial graphs: bistars, combs, ladders, kites, and slanting ladders. In they showed that theta graphs, the duplication of any vertex in theta graphs, switching of any vertex in theta graphs, the fusion of any two vertices in theta graphs, path unions of two copies of theta graphs, open star graphs of copies of a fixed theta graph are 3-equitable prime the electronic journal of combinatorics (2023), #DS6 337 cordial graphs. Seoud and Jaber proved that the butterfly BFn,m, helms Hn, the graph ⟨Wn : Wm⟩obtained by joining apex vertices of two wheels with a new vertex are prime cordial, and determine the prime cordial graphs of order 7. They also gave an algorithm to calculate the maximum number of edges in a 3-equitable prime cordial graph.
Murugesan, Jayaraman, and Shiama proved the following graphs are 3-equitable prime cordial: Pn, Cn, (n ≥4), K1,n if and only if n = 2 (mod 3), and Kn if and only if n ≤2. Ghodasaram and Sonchhatra proved the following graphs admit 3-equitable labeling: wheels, helms, gears, cycles with one pendant edge, and the graphs obtained by joining two copies of a fan by a path of arbitrary length.
For a simple, finite and planar graph G of order p and size q. Sumathi and J. Suresh Kumar introduced the concept of fuzzy quotient-3 cordial labeling as follows. Let σ : V (G) →[0, 1] be a function defined by σ(v) = r 10, r ∈Z4 −{0}. For each edge uv define µ : E(G) →[0, 1] by µ(uv) = 1 10 3σ(u) σ(v) where σ(u) ≤σ(v). The function σ is called fuzzy quotient-3 cordial labeling of G if the number of vertices labeled with i and the number of vertices labeled with j differ by at most 1, the number of edges labeled with i and the number of edges labeled with j differ by at most 1 where i ̸= j have the form r 10, where r ∈{1, 2, 3}. They proved that stars and star related graphs are fuzzy quotient-3 cordial.
7.10 Mean Labelings Somasundaram and Ponraj introduced the notion of mean labelings of graphs. A graph G with p vertices and q edges is called a mean graph if there is an injective function f from the vertices of G to {0, 1, 2, . . . , q} such that when each edge uv is labeled with (f(u)+f(v))/2 if f(u)+f(v) is even, and (f(u)+f(v)+1)/2 if f(u)+f(v) is odd, then the resulting edge labels are distinct. In , , , , , and they prove the following graphs are mean graphs: Pn, Cn, K2,n, K2 + mK1, Kn + 2K2, Cm ∪ Pn, Pm ×Pn, Pm ×Cn, Cm ⊙K1, Pm ⊙K1, triangular snakes, quadrilateral snakes, Kn if and only if n < 3, K1,n if and only if n < 3, bistars Bm,n (m > n) if and only if m < n+2, the subdivision graph of the star K1,n if and only if n < 4, the friendship graph C(t) 3 if and only if t < 2, the one point union of two copies a fixed cycle, dragons (the one point union of Cm and Pn, where the chosen vertex of the path is an end vertex), the one point union of a cycle and K1,n for small values of n, and the arbitrary super subdivision of a path, which is obtained by replacing each edge of a path by K2,m. They also prove that Wn is not a mean graph for n > 3 and enumerate all mean graphs of order less than 5.
Gayathri and Gopi prove the following are mean graphs: double triangular snakes; double quadrilateral snakes; generalized antiprisms; graphs obtained by joining the 2 vertices of K2,n of degree n with an edge; and graphs obtained from Cn with consecutive vertices v1, v2, . . . , vn by adding the chords joining vi and vn−i+2 for 2 ≤i ≤ ⌊n/2⌋. In Gayathri and Gopi gave various necessary conditions for mean labelings.
Kannan, Manivannan, Loganathan, and Gyeltshen investigated the existence of mean labelings for the graphs obtained by duplicating an edge or vertex of paths, cycles, combs, and the splitting graph of paths.
the electronic journal of combinatorics (2023), #DS6 338 Lourdusamy and Seenivasan prove that kCn-snakes are means graphs and every cycle has a super subdivision that is a mean graph. They define a generalized kCn-snake in the same way as a Cn-snake except that the sizes of the cycle blocks can vary (see Section 2.2). They prove that generalized kCn-snakes are mean graphs. Recall that Pa,b denotes the graph obtained by identifying the endpoints of b internally disjoint paths each of length a. Vasuki and Nagarajan proved that the following graphs admit mean labelings: Pr,2m+1 for all r and m; Pr,2m for all m and 2 ≤r ≤6; P 2m+1 r for all r and m; and P 2m r for all m and 2 ≤r ≤6. Anusa, Sandhya, and Somasundaram proved that triangular ladders, triangular snakes, double triangular snakes, quadrilateral snakes, and double quadrilateral snakes are mean graphs.
Lourdusamy and Seenivasan define an edge linked cyclic snake, EL(kCn), as the connected graph obtained from k copies of Cn (n ≥4) by identifying an edge of the (i + 1)th copy to an edge of the ith copy for i = 1, 2, . . . , k −1 in such a way that the consecutive edges so chosen are not adjacent. They proved that all EL(kC2n) are mean graphs and some cases of EL(C2n−1) are mean graphs. They also define a generalized edge linked cyclic snake in the same way but allow the cycle lengths (at least 4) to vary.
They prove that certain cases of generalized edge linked cyclic snakes are mean graphs.
Barrientos and Krop proved that there exist n! graphs of size n that admit mean labelings. They give two necessary conditions for the existence of a mean labeling of a graph G with m vertices and n edges: if G is a mean graph, then n + 1 ≥m; if G is a mean graph with n edges and maximum degree ∆(G), then ∆(G) ≤n+3 2 when n is odd and ∆(G) ≤ n+2 2 when n is even. They proved that the disjoint union of n copies of C3 is a mean graph and if a mean r-regular graph has n vertices, then r < n −2. They established a connection between α-labelings and mean labelings by proving that every tree that admits an α-labeling is a mean graph when the size of its stable sets differ by at most one. When the tree is a caterpillar, this difference can be up to two. Barrientos and Krop call a mean labeling of a bipartite graph an α-mean labeling if the labels assigned to vertices of the same color have the same parity. They show that the complementary labeling of a α-mean labeling is also an α-mean labeling.
They use graphs with α-mean labelings to construct new mean graphs. One construction consists of connecting a pair of corresponding vertices of two copies of an α-mean graph by an edge. The other construction identifies a pair of suitable vertices from two α-mean graphs. Barrientos and Krop also proved that every quadrilateral snake admits an α-mean labeling. They conjecture that all trees of size n and maximum degree at most ⌈(n + 1)/2⌉are mean graphs and state some open problems. In Barrientos proves that all trees with up to four end-vertices except K1,4 are mean graphs. Bailey and Barrientos prove the following are mean graphs: Cn ∪Cm, Cn ∪Pm, K2 + nK1, 2K2 + nK1, Cn × K2.
In Bailey and Barrientos study several operations with mean graphs. They prove that the coronas G ⊙K1 and G ⊙K2 are mean graphs when G is an α-mean graph. Also, if G and H are mean graphs with n vertices and n −1 edges and H is an α-mean graph, then G × H is a mean graph. They prove that given two mean graphs G and H, there exists a mean graph obtained by identifying an edge from G with an edge from H and uses this result to prove that the graphs Rn (n ≥2) of order 2n and size 4n−3 with vertex the electronic journal of combinatorics (2023), #DS6 339 set V (Rn) = {v1, v2, . . . , v2n} and edge set E(Rn) = {vivi+1 | 1 ≤i ≤n −1 and n + 1 ≤ i ≤2n −1} ∪{vivn+i | 1 ≤i ≤n} ∪{vivn+i−1 | 2 ≤i ≤n} (rigid ladders) are mean graphs.
Barrientos, Abdel-Aal, Minion, and Williams use An to denote the set of all α-mean labeled graphs of size n such that the difference of the cardinalities of the bipartite sets of the vertices of the graphs is at most one.They prove that the class An is equivalent to the class of α-labeled graphs of size n with bipartite sets that differ by at most one.
They also prove that when G ∈An, the coronas G ⊙mK1, G ⊙P2, and G ⊙P3 admit mean labelings.
In Vaidya and Bijukumar define two methods of creating new graphs from cycles as follows. For two copies of a cycle Cn the mutual duplication of a pair of vertices vk and v′ k respectively from each copy of Cn is the new graph G such that N(vk) = N(v′ k).
For two copies of a cycle Cn and an edge ek = vkvk+1 from one copy of Cn with incident edges ek−1 = vk−1vk and ek+1 = vk+1vk+2 and an edge e′ m = umum+1 in the second copy of Cn with incident edges e′ m−1 = um−1um and e′ m+1 = um+1um+2, the mutual duplication of a pair of edges ek and e′ m respectively from two copies of Cn is the new graph G such that N(vk) −vk+1 = N(um) −um+1 = {vk−1, um−1} and N(vk+1) −vk = N(um+1) −um = {vk+2, um+2}. They proved that the graph obtained by mutual duplication of a pair of vertices each from each copy of a cycle and the mutual duplication of a pair of edges from each copy of a cycle are mean graphs. Moreover, they proved that the shadow graphs of the stars K1,n and bistars Bn,n are mean graphs.
Vasuki and Nagarajan proved the following graphs are admit mean labelings: the splitting graphs of paths and even cycles; Cm ⊙Pn; Cm ⊙2Pn; Cn ∪Cn; disjoint unions of any number of copies of the hypercube Q3; and the graphs obtained from by starting with m copies of Cn and identifying one vertex of one copy of Cn with the corresponding vertex in the next copy of Cn.) Jeyanthi and Ramya define the jewel graph Jn as the graph with vertex set {u, x, v, y, ui : 1 ≤i ≤n} and edge set {ux, vx, uy, vy, xy, uui, vui : 1 ≤i ≤n}. They proved that the jewel graphs, jelly fish graphs, and the graph obtained by joining any number of isolated vertices to the two endpoints of P3 are mean graphs. Ramya and Jeyanthi proved several families of graphs constructed from Tp-tree are mean graphs. Ahmad, Imran, and Semaničová-Feňovčiková studied the relation between mean labelings and (a, d)-edge-antimagic vertex labelings. They show that two classes of caterpillars admit mean labelings. Revathi proved that the shadow graphs of bistars, combs, and the splitting graph of combs have mean labelings.
Recall from Section 2.7 that given connected graphs G1, G2, . . . , Gn, Kaneria, Makadia, and Jariya define a cycle of graphs C(G1, G2, . . . , Gn) as the graph obtained by adding an edge joining Gi to Gi+1 for i = 1, . . . , n−1 and an edge joining Gn to G1. (The resulting graph can vary depending on which vertices of the Gi are chosen.) When the n graphs are isomorphic to G the notation C(n · G) is used. Also recall Kanneria and Makadia define a step grid graph Stn as the graph obtained by starting with paths Pn, Pn, Pn−1, . . . , P2 (n ≥3) arranged vertically parallel with the vertices in the paths forming horizontal rows and edges joining the vertices of the rows. In , , and the electronic journal of combinatorics (2023), #DS6 340 , Kaneria, Viradia, and Makadia proved the following graphs are mean graphs: the path union of any number of copies of a mean graph; C(2t·Pn); C(2t·Cn); C(2t·Pn×Pm); C(2r · B2 n,n) (B2 n,n is the square of the bistar Bn,n); C(2r · M(Cn)) (M(Cn) is the middle graph of Cn); C(2r · (P2n + 2K1)); step grid graphs; the path union of finitely copies of the step grid graphs; cycles of step grid graphs C(2r · Stn); and C(2t · K2,m).
For a fixed vertex v of Cm Avadayappan and Vasuki use (Pm; Cn) to denote the graph obtained from m copies of Cn and the path Pm : u1u2 · · · um by joining ui with v of the ith copy of Cn with an edge for 1 ≤i ≤m. They define (Pm; Q3), (P2n; Sm) , (Pn; S1) and (Pn; S2), where v is a fixed vertex of the cube Q3 and v is the center of the star Sk, in an analogous way. For Cn : v1v2 . . . vnv1 they use [Pm; Cn] to denote the graph obtained from m copies of Cn with vertices v11, v12, . . . , v1n, v21, . . . , v2n, . . . , vm1, . . . , vmn by joining vij and v(i+1)j with an edge, for some j and 1 ≤i ≤m −1. They define [Pm; Q3] and [Pm; C(2) m ], where C(2) m is the friendship graph, similarly. In they prove these families are mean graphs.
Maheswari, Hariprabakaran, and Balaji introduced a coding technique for con-verting a text message using a subsuper mean labeling on two and three star graphs to a picture coding message. For more on coding a text message using a graph labelings see , , and .
Ramya, Ponraj, and Jeyanthi called a mean graph super mean if vertex labels and the edge labels are {1, 2, . . . , p + q}. They prove following graphs are super mean: paths, combs, odd cycles, P 2 n, Ln⊙K1, Cm∪Pn (n ≥2), the bistars Bn,n and Bn+1,n. They also prove that unions of super mean graphs are super mean and Kn and K1,n are not super mean when n > 3. In Jeyanthi, Ramya, and Thangavelu prove the following are super mean: nK1,4; the graphs obtained by identifying an endpoint of Pm (m ≥2) with each vertex of Cn; the graphs obtained by identifying an endpoint of two copies of Pm (m ≥2) with each vertex of Cn; the graphs obtained by identifying an endpoint of three copies of Pm (m ≥2); and the graphs obtained by identifying an endpoint of four copies of Pm (m ≥2). In Jeyanthi and Ramya prove the following graphs have super mean labelings: the graph obtained by identifying the endpoints of two or more copies of P5; the graph obtained from Cn (n ≥4) by joining two vertices of Cn distance 2 apart with a path of length 2 or 3; Jeyanthi and Rama use S(G) to denote the graph obtained from a graph G by subdividing each edge of G by inserting a vertex. They prove the following graphs have super mean labelings: S(Pn ⊙K1), S(Bn,n), Cn ⊙K2; the graphs obtained by joining the central vertices of two copies of K1,m by a path Pn (denoted by ⟨Bm,m : Pn⟩); generalized antiprisms (see §6.2 for the definition), and the graphs obtained from the paths v1, v2, v3, . . . , vn by joining each vi and vi+1 to two new vertices ui and wi (double triangular snakes).
Lourdusamy and Seenivasan introduced the notion of super vertex mean la-beling as follows.
For a (p, q)-graph and an injective function f from the edges to the set {1, 2, 3, . . . , p + q} that induces for each vertex v the label defined by f ∗(v) = (Round P e∈Ev f(e))/d(v), where Ev denotes the set of edges in G that are incident to the vertex v, d(v) is the degree of v, and Round(x) is the integer nearest to x, such that the set of all edge labels and the induced vertex labels is {1, 2, 3, . . . , p + q} is called a the electronic journal of combinatorics (2023), #DS6 341 super vertex mean labeling of G and G is called a super vertex mean graph. In they investigated the all graphs of order up to 5 and regular graphs of order up to 7 for the property of being super vertex mean and proved that all linear triangular snakes are super vertex mean. Lourdusamy, George, and Seenivasan proved that all cycles except C4 are super vertex mean and Lourdusamy and George proved that linear Cn snakes with at least 2 blocks are super vertex mean graphs for the following cases: n = 4, 5, 6, and 7; n ≥8 even; n ≥9 and n ≡1 mod 4; and n ≥11 and n ≡3 mod 4.
Inayah, Sudarsana, Musdalifah, and Mangesa have showed that the total graphs of paths and cycles are super mean graphs.
In Sudarsana, Suryanto, Lusianti, and Putri show how super mean graph label-ings can be used to increase the security level of encrypted text on social medias.
A graph G with q edges is called a k-mean graph if there is an injective function f from the vertices of G to {0, 1, 2, . . . , k + q −1} such that when each edge uv is labeled with (f(u) + f(v))/2 if f(u) + f(v) is even, and (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd, the resulting edge labels are {k, k + 1, k + 2, . . . , k + q −1}. A graph G with q edges is said to have a restricted k-mean labeling if there is an injective function f from the vertices of G to {k −1, k, k +1, . . . , k +q −1} such that when each edge uv is labeled with {k, k + 1, k + 2, . . . , k + q −1}, the resulting edge labels {k, k + 1, k + 2, . . . , k + q −1} are distinct where k is a positive integer. A graph that admits a restricted k-mean labeling is called a restricted k-mean graph. Gayathri and Gopi proved some properties of k-mean labelings in . In they proved that if G1 and G2 are restricted k-mean graphs for all k, then G1∪G2 is restricted k-mean for all k, and if G1 is a restricted k-mean graph for all k ≥k1 and G2 is a restricted k-mean graphs for all k, then G1 ∪G2 is restricted k-mean for all k ≥k1.
A mean graph is called k-super mean if vertex labels and the edge labels are {k, k + 1, k +2, . . . , p+q +k −1}. Jeyanthi, Ramya, Thangavelu give super mean labelings for Cm∪Cn and k-super mean labelings for a variety of graphs. Tamilselvi, Akilandeswari, and Suguna proved that the following graphs admit k-super mean labelings: the graph obtained by subdividing the central edge of the bistar Bn,n (n ≥2), the subdivision graph of Bn,n, and the corona product of a triangular snake and K1. Tamilselvi, and Akilandeswari proved that the following graphs admit k-super Heronian mean new labelings: (Pn × P2) ⊙K1 (n > 1), subdivision graphs of triangular snakes Tn (n > 1), subdivsion graphs of Tn ⊙K1, graphs obtained from a Pn by fusing one edge of C4 to each vertex of Pn, and graphs obtained from Pn by fusing one vertex two copies of C6 to each vertex of Pn.
Vasuki and Nagarajan define Hn, called the H-graph of a path Pn, as the graph obtained from two copies of Pn with vertices v1, v2, . . . , vn and u1, u2, . . . , un by joining the vertices v(n+1)/2 and u(n+1)/2 if n is odd, and the vertices v n 2 +1 and u n 2 if n is even, and a cyclic snake mCn as the graph obtained from m copies of Cn by identifying the vertex v(k+2)j in the jth copy of the vertex v1j+1 in the (j + 1)th copy if n = 2k + 1 and identifying the vertex v(k+1)j in the jth copy with the vertex v1j+1 in the (j + 1)th copy if n = 2k. They establish the super meanness of even cycles, H-graphs, the coronas of H-graphs, 2-coronas of H-graphs, coronas of cycles, mCn-snakes (n ̸= 4), dragons Pn(Cm) the electronic journal of combinatorics (2023), #DS6 342 for m ̸= 4, and Cm × Pn for m = 3 and 5. Vasuki, Sugirtha, and Venkateswari proved that the subdivision of the following graphs are super mean graphs: Hn, Hn ⊙K1, Hn with two pendent edges attached to each vertex, Cn ⊙K1 (n ≥3), slanting ladders, triangular snakes with a pendent edge at each vertex, and Cm@Cn (the graph obtained by attaching paths Pn to Cm by identifying the endpoints of the paths with each successive pairs of vertices of Cm).
Let G(V, E) be a simple graph of order p and size q. Then G is said to be a relaxed mean graph if it is possible to label the vertices x ∈V with distinct elements f(x) from {0, 1, 2 . . . , q−1, q+1} in such a way that when each edge uv is labeled with (f(u)+f(v))/2 if f(u)+f(v) is even and (f(u)+f(v)+1)/2 if f(u)+f(v) is odd, then the resulting edge labels {1, 2, 3, . . . , q} are distinct. Such an f is called a relaxed mean labeling of G. Balaji, Ramesh, and Sudhaker prove that the disjoint union of any path with n −1 edges joining the pendent vertices of distinct paths is a relaxed mean graph and K1,m is not a relaxed mean graph for m ≥5. They also prove that the graph consisting of two stars K1,m and Kn,1 with an edge in common is a relaxed mean graph if and only if |m−n| ≤5. Balaji and Maheswari proved the following graphs are relaxed mean graphs Cn (n > 4); K2,n; triangular snakes; quadrilateral snakes; P 2 n ; (Pn × P2) ⊙K1; Kn + 2K2; K2 + mK1; C(t) 3 ; the union of any two trees; Pm × Pn (m > 1, n > 1); and Pm × Cn (m > 1). They also prove that Kn is not a relaxed mean graph. Maheswari, Ramesh, and Balaji proved the following graphs are relaxed mean graphs: Pn (n > 5); bistars Bm,n if and only if |m −n| ≤3; combs; and C3 ∪Pn (n > 1). They also proved that K1,n is not a relaxed mean graph for n > 5. Francis and Balaji proved that Wn (n > 4) are relaxed mean graphs.
In Arockiaraj, Rajesh Kannan, and Durai Baskar introduced the F-centroidal mean labeling of graphs by defining a function f to be an F-centroidal mean labeling of a graph G(V, E) with q edges if f : V (G) →{1, 2, 3, . . . , q + 1} is injective and the induced function f ∗: E(G) →{1, 2, 3, . . . , q} defined as f ∗(uv) = j 2 [f(u)2+f(u)f(v)+f(v)2] 3 [f(u)+f(v)] k for all uv ∈E(G) is bijective. A graph that admits an F-centroidal mean labeling is called an F-centroidal mean graph. They discussed the F-centroidal meanness of the tree Pn(X1, X2, . . . Xn) obtained from a path on n vertices by attaching Xi pendent vertices at each ith vertex of the path for 1 ≤i ≤n, the twig graph TW(Pn), the graph Pn ◦Sm for m ≤4, Pm×Pn for m ≤3, ladders, Pn◦K2, P b a for a ≥2 and b ≤3, the middle graphs and splitting graphs of paths, the total graphs of paths, P 2 n, and P(1, 2, . . . , n −1) the graph obtained by replacing each ith edge of Pn by identifying its end vertices with the vertices of the two element component of K2,i. In the same article they introduced super F-centroidal mean graphs as follows. Let G be a graph and f : V (G) →{1, 2, 3, . . . , p + q} be an injection. With f ∗defined as for the F-centroidal case f is called a super F-centroidal mean labeling if f(V (G)) ∪{f ∗(uv) : uv ∈E(G)} = {1, 2, 3, . . . , p + q}. A graph that admits a super F-centroidal mean labeling is called a super F-centroidal mean graph. They proved that the following graphs are super F-centroidal mean graphs: paths, cycles, the union of any number of paths, the mirror graph of Pn, Pn ◦Sm, TW(Pn), Pn ∪Cm, P 2 n, and dragons Pn(Cm) the graph obtained from Cm by identifying an end vertex of Pn at a the electronic journal of combinatorics (2023), #DS6 343 vertex of Cm.
In Gopi and Kumar proved the following graphs are are F-centroidal mean graphs: the mirror graph M(Pn) (n ≥3), Spl(Pn) (n ≥3), P 2 n (n ≥3), V D(G), the tortoise graph Tn (n ≥4) (obtained from a path v1, v2, . . . , vn by attaching an edge between vi and vn−i+1 for 1 ≤i ≤⌊n/2⌋.
(n = 1, 3 (mod 4)), and PCn (n ≥5) (obtained from Cn = v1, v2, . . . , vn by adding chords joining vi and vn−i+2 for 2 ≤i ≤t, where t = ⌊n/2⌋), In they proved D(Tn) (n ≥3), AD(Tn) (n ≥4), Qn (n ≥2), A(Qn) (n ≥2) AD(Qn) (n ≥2), and SLn (n ≥2) are F-centrodial mean graphs.
Arockiaraj, Kannan, Manivannan, and Durai Baskar investigated the F-centroidal mean property for paths, cycles, stars, Kn, Pn ◦S1, the triangular snakes, arbitrary subdivisions of K1,3, and some line graphs.
In and Balaji, Ramesh, and Subramanian use the term “Skolem mean” labeling for super mean labeling. They prove: Pn is Skolem mean; K1,m is not Skolem mean if m ≥4; K1,m ∪K1,n is Skolem mean if and only if |m −n| ≤4; K1,l ∪K1,m ∪K1,n is Skolem mean if |m−n| = 4+l for l = 1, 2, 3, . . . , m = 1, 2, 3, . . . , and l ≤m < n; K1,l ∪ K1,m ∪K1,n is not Skolem mean if |m −n| > 4 + l for l = 1, 2, 3, . . . , m = 1, 2, 3, . . . , n ≥ l + m + 5 and l ≤m < n; K1,l ∪K1,l ∪K1,m ∪K1,n is Skolem mean if |m −n| = 4 + 2l for l = 2, . . . , m = 2, 3, 4 . . . , n = 2l + m + 4 and l ≤m < n; K1,l ∪K1,l ∪K1,m ∪K1,n is not Skolem mean if |m −n| > 4 + l for l = 1, 2, 3, . . . , m = 1, 2, 3, . . . , n ≥l + m + 5 and l ≤m < n; K1,l ∪K1,l ∪K1,m ∪K1,n is not Skolem mean if |m −n| > 4 + 2l for l = 2, . . . , m = 2, 3, 4 . . . , n ≥2l + m + 5 and l ≤m < n; K1,l ∪K1,l ∪K1,m ∪K1,n is Skolem mean if |m −n| = 7 for m = 1, 2, 3, . . . , n = m + 7 and 1 ≤m < n; and K1,l ∪K1,l ∪K1,m ∪K1,n is not Skolem mean if |m −n| > 7 for m = 1, 2, 3, . . . , n ≥m + 8 and 1 ≤m < n. Balaji proved that K1,l∪K1,m∪K1,n is Skolem mean if |m−n| < 4+l for integers 1, m ≥1 and l ≤m < n. In Shainy and Balaji determined necessary and sufficient conditions for the disjoint union of three stars to be Skolem mean.
In Prakash, Gopi, and Shalini introduced the following graph labeling. A graph new G(V, E) with p vertices and q edges where p < q + 1 is said to be an anti skolem mean graph if it is possible to label the vertices with distinct labels from {0, 1, 2, . . . , q + 1} in such a way that if each edge uv is labeled with f ∗(uv) = ⌈(f(u) + f(v))/rceil, the resulting edge labels are distinct labels from the set {2, 3, . . . , q + 1}. They proved that quadrilateral snake related graphs have anti skolem mean labelings.
In Shendra Shainy, Hariprabakaran, Swathy, and Balaji provided a technique for coding a secret messages by applying a Skolem mean-like labeling on a graph obtained from the two stars of the form K1,n and K1,n+1 that are joined by an edge from one vertex of degree 1 in K1,n to one vertex of degree 1 in K1,n+1.
In Jeyanthi, Selvi, and Ramya prove that Cm ∪Cn, (Pn + K1) ∪(n −2)K2 (n > 2), (Pn + K2) ∪(2n −3)K2 (n ≥2) and Wn ∪(n −1)K2 (n ≥3) are Skolem difference mean graphs. In they show that the union of any finite number of paths, the union of any finite number of stars, G ∪nK2 where G is Skolem difference mean and all the vertex labels are odd, Cm ∪Pm (m ≥2), Km,n ∪(m−1)(n−1)K2, and K1,1,n ∪(n−1)K2.
are skolem difference mean graphs.
In Jeyanthi, Ramya, and Thangavelu proved the following graphs have super the electronic journal of combinatorics (2023), #DS6 344 mean labelings: the one point union of any two cycles, graphs obtained by joining any two cycles by an edge (dumbbell graphs), C2n+1 ⊙C2m+1, graphs obtained by identifying a copy of an odd cycle Cm with each vertex of Cn, the quadrilateral snake Qn, where n is odd, and the graphs obtained from an odd cycle u1, u2, . . . , un by joining the vertices ui and ui+1 by the path Pm (m is odd) for 1 ≤i ≤n−1 and joining vertices un and u1 by the path Pm. Jeyanthi, Ramya, Thangavelu, and Aditanar give super mean labelings of Cm ∪Cn and Tp-trees. Vasuki and Arockiaraj proved that nC4, n > 1, triangular grid graphs, the edge mCn-snakes, and the braid graphs are super mean graphs. They further proved that the graphs obtained by identifying an edge of two cycles Cm and Cn is a super mean graph.
In Jeyanthi and Ramya define Sm,n as the graph obtained by identifying one endpoint of each of n copies of Pm and < Sm,n : Pm > as a graph obtained by identifying one end point of a path Pm with the vertex of degree n of a copy of Sm,n and the other endpoint of the same path to the vertex of degree n of another copy of Sm,n. They prove the following graphs have super mean labelings: caterpillars, < Sm,n : Pm+1 >, and the graphs obtained from P2m and 2m copies of K1,n by identifying a leaf of ith copy of K1,n with ith vertex of P2m. They further establish that if T is a Tp-tree, then T ⊙K1, T ⊙K2, and, when T has an even number of vertices, T ⊙Kn (n ≥3) are super mean graphs.
In Dhanalakshmi and Parvathi define a mean square cordial labeling of a graph G(V, E) with p vertices and q edges as one for which there is a bijection from V to {0, 1} such that when each edge uv is assigned the label ⌈(f(u)2 + f(v)2)/2⌉the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled 0 and the number of edges labeled with 1 differ by at most 1 is satisfied. In Dhanalakshmi and Parvathi proved that helms, closed helms, gears, sunlet graphs, fans, and Cp ⊙nK1 admit mean square cordial labelings. In Dhanalakshmi and Parvathi showed that the paths, combs, n-centipedes, and stars admit mean square cordial labelings. They also proved that the induced subgraph obtained by the upper approximation of any subgraph H of the above acyclic graphs admits a mean square cordial labeling. In Dhanalakshmi and Parvathi prove that bistars, subdivisions of stars, coconut trees, banana trees, the one-point union of Cn and K1,n, and Pm ⊙K1,n admit mean square cordial labelings.
Dhanalakshmim proved that shell-butterfly graphs with shell orders m > 2 and n > 2 admit mean square cordial labelings. Dhanalakshmi and Thirunavukkarasu z proved pentagonal snakes, subdivision of a pentagonal snakes, double pentagonal snakes, and alternate pentagonal snakes admit mean square cordial labelings.
Arockiaraj, Durai Baskar, and Rajesh Kannan calls a graph G with p vertices and q edges a F-root square mean graph if there is an injective function f from the vertices of V (G) to {1, 2, . . . , q + 1} such that for each edge uv the induced function f ∗(uv) = ⌊ p (f(u)2 + f(v)2)/2⌋is bijective. They proved that the following are F-root square mean graphs: paths, the graph Pn ◦Sm obtained from the path Pn by attaching m pendant vertices to each vertex of Pn, twigs TW(Pn), the graph [Pn; Sm] obtained from n copies of Sm and the path Pn by joining ui with the central vertex v(i) 1 of the ith copy of Sm with of an edge for i ≤n, the mirror graph of Pn, the total graph of Pn, P 2 n, ladders, and the electronic journal of combinatorics (2023), #DS6 345 slanting ladders. In , Arockiaraj, Durai Baskar, and Rajesh Kannan analyzed that the line graph operation preserves the F-root square meanness of line graph of the path, cycle, star, Pn ◦S1, Pn ◦S2, [Pn; S1], S(Pn ◦S1), ladder, slanting ladder, the crown graph Cn ◦S1 and the arbitrary subdivision of S3. Gopi proved that triangular snakes Tn (n ≥2), A(Tn) (n ≥3), D(Tn) (n ≥2), quadrilateral snakes, A(Qn), D(Qn) (n ≥3) are F-root square mean graphs.
Rajesh Kannan, Vikrama Prasad, and Gopi call a graph G with p vertices and q edges a super root mean graph if there is an injective function f from the ver-tices of G to {1, 2, . . . , p + q} such that for each edge uv the induced function f ∗(uv) = ⌊ p (f(u)2 + f(v)2)/2⌋or f ∗(uv) = ⌈ p (f(u)2 + f(v)2)/2⌉yields the set of vertex labels and edge labels {1, 2, . . . , p + q}. They proved the following are super root square mean graphs: Pm∪Pm (m, n ≥3); Pm∪(Pn⊙K1) (m, n ≥3); (Pm⊙K1)∪(Pn⊙K1) (m, n ≥3); the union of a path and a triangular snake; and the union of Pn ⊙K1 and a triangular snake. Gopi and Kalaiyarasi prove that the following graphs have a super root square mean labeling: P 2 n (n ≥4), slanting ladders SLn(n ≥3), triangular snakes with a pendent edge attached to each vertex, and quadrilateral snakes with a pendent edge attached to each vertex.
Chitra devi and Saravana Kumar proved that nPm, nK3, Pn ⊙K2, the middle graph of path Pn (graphs obtained by starting with a path and joining every consecutive pair of vertices excluding the two end vertices of the path to a new isolated vertex), and Cm ⊙Pn admits super root square mean labelings. Venkatesan and Thirugnanasamban-dam proved that the following graphs admit super root square mean labelings: P 2 n (n ≥4), slanting ladders, Tn ⊙K1 (n ≥3) ( Tn is a triangular snake), and Qn ⊙K1 (Qn is a quadrilateral snake). Orias and Pedrano determined the super root square mean labelings of Cn ⊙K1, middle cycles (graphs obtained by starting with a cycle and joining every consecutive pair of vertices of the cycle to a new isolated vertex), polygonal chains, alternate polygonal chains, and kayak paddles.
In Akilandeswari calls a graph is a k-super root square mean graph if it is possible to label the vertices with distinct elements from {k, k + 1, k + 2, . . . , p + q + k −1} in such a way that when each edge uv is labeled with ⌊ p (f(u)2 + f(v)2)/2⌋or ⌈ p (f(u)2 + f(v)2)/2⌉, the union of the vertex and edge labels is the set {k, k + 1, k + 2, . . . , p + q + k −1}. He proved that Pn ⊙K1,2, Pn ⊙K1,3, the corona product of the quadrilateral snake and K1, double triangular snakes, and the corona product of the double triangular snake and K1 admit k-super root square mean labelings.
A radio mean square labeling of a connected graph is an injective map h from the set of vertices of the graph G to the set of positive integers N, such that for any two distinct vertices x, y, the inequality d(x, y) + ⌈(h(x))2 + (h(y))2/2⌉≥dim(G) + 1 holds. For a particular radio mean square labeling h, the maximum number of h(v) taken over all vertices of G is called its spam, denoted by rmsn(h), and the minimum value of rmsn(h) taking over all radio mean square labeling h of G is called the radio mean square number of G, denoted by rmsn(G). In Badr, Nada, Al-Shamiri, Abdel-Hay, and Elrokh investigated the radio mean square numbers for paths and cycles. For a graph G, they present an approximate algorithm to determine rmsn(G). Finally, they introduced a new the electronic journal of combinatorics (2023), #DS6 346 mathematical model to find the upper bound of rmsn(G) for graph G.
In Sandhya, Somasundaram, and Anusa say a graph with q edges is a root square mean graph if it is possible to label the vertices with distinct elements from {1, 2, . . . , q+1} in such a way that when each edge uv is labeled with ⌊ p (f(u)2 + f(v)2)/2⌋ or ⌈ p (f(u)2 + f(v)2)/2⌉, the resulting edge labels are distinct. They prove that paths, cycles, combs, ladders, triangular snakes, quadrilateral snakes, K1,n) (1 ≤6), and Kn for n = 1, 2 and 3 are root square mean graphs. In they proved that the following graphs admit root square mean labelings: double triangular snakes, alternate double tri-angular snakes, double quadrilateral snakes, alternate double quadrilateral snakes, and polygonal chains.
In Kulandhai Therese and Romila introduced the notion of cube root cube mean labeling as follows. For a graph G(V, E) with |(V (G)| = p and |E(G)| = q, let f : V (G) →{1, 2, . . . , q + 1} be an injective function. The induced edge labeling f ∗for a vertex labeling f is defined by f ∗(e) = ⌊3 p (f(u)3 + f(v)3)/2⌋or ⌈3 p (f(u)3 + f(v)3)/2⌉ for all e = uv ∈E(G). If f is a bijection it is called a cube root cube mean labeling. If such labeling exists, G is said to be a k-cube root cube mean graph. They proved that paths, combs, ladders, and quadrilateral snakes admit cube root cube mean labelings.
In Princy Kala introduced the notion of k-super cube root cube mean la-beling as follows.
For a graph G(V, E) with |(V (G)| = p and |E(G)| = q let f : V (G) →{k, k + 1, k + 2, . . . , p + q + k −1} be an injective function.
The induced edge labeling f ∗for a vertex labeling f is defined by f ∗(e) = ⌊3 p (f(u)3 + f(v)3)/2⌋or ⌈3 p (f(u)3 + f(v)3)/2⌉for all e = uv ∈E(G). If f is a bijection and f(V (G))∪f ∗(E(G)) = {k, k+1, k+2, . . . , p+q+k−1}, f is called a k-super cube root cube mean labeling. If such labeling exists, then G is said to be a k-super cube root cube mean graph. He proved the existence k-super cube root mean labelings for triangular snakes, double triangular snakes, quadrilateral snakes, double quadrilateral snakes, alternate triangular snakes, alternate double triangular snakes, alternate quadrilateral snakes, and alternate double quadrilat-eral snakes. In Princy Kala proved that the corona products of a triangular snake and K1, an alternate triangular snake and K1, an alternate triangular snake and 2K1, an alternate quadrilateral snake and K1, Pn ⊙K1,2, and Pn ⊙K1,3 are k-super cube root cube mean graphs.
Let G be a graph and let f : V (G) →{1, 2, . . . , n} be a function such that the label of the edge uv is (f(u) + f(v))/2 or (f(u) + f(v) + 1)/2 according as f(u) + f(v) is even or odd and f(V (G)) ∪{f ∗(e) : e ∈E(G)} ⊆{1, 2, . . . , n}. If n is the smallest positive integer satisfying these conditions together with the condition that all the vertex and edge labels are distinct and there is no common vertex and edge labels, then n is called the super mean number of a graph G and it is denoted by Sm(G). Nagarajan, Vasuki, and Arockiaraj proved that for any graph of order p, Sm(G) ≤2p −2 and provided an upper bound of the super mean number of the graphs: K1,n n ≥7; tK1,n, n ≥5, t > 1; the bistar B(p, n), p > n; the graphs obtained by identifying a vertex of Cm and the center of K1,n, n ≥5; and the graphs obtained by identifying a vertex of Cm and the vertex of degree 1 of K1,n. They also gave the super mean number for the graphs Cn, tK1,4, and B(p, n) for p = n and p = n + 1.
the electronic journal of combinatorics (2023), #DS6 347 Manickam and Marudai defined a graph G with q edges to be an odd mean graph if there is an injective function f from the vertices of G to {1, 3, 5, . . . , 2q −1} such that when each edge uv is labeled with (f(u) + f(v))/2 if f(u) + f(v) is even, and (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd, then the resulting edge labels are distinct.
Such a function is called a odd mean labeling. For integers a and b at least 2, Vasuki and Nagarajan use P b a to denote the graph obtained by starting with vertices y1, y2, . . . , ya and connecting yi to yi+1 with b internally disjoint paths of length i + 1 for i = 1, 2, . . . , a−1 and j = 1, 2, . . . , b. For integers a ≥1 and b ≥2 they use P b ⟨2a⟩to denote the graph obtained by starting with vertices y1, y2, . . . , ya+1 and connecting yi to yi+1 with b internally disjoint paths of length 2i for i = 1, 2, . . . , a and j = 1, 2, . . . , b. They proved that the graphs P2r,m, P2r+1,2m+1, and P m ⟨2r⟩are odd mean graphs for all values of r and m. Jeyanthi and Gomathi proved the edge linked cyclic snake EL(kCn) (n ≥6) is an odd mean graph.
For a Tp-tree T with m vertices T@Pn is the graph obtained from T and m copies of Pn by identifying one pendent vertex of ith copy of Pn with ith vertex of T. For a Tp-tree T with m vertices T@2Pn is the graph obtained from T by identifying the pendent vertices of two vertex disjoint paths of equal lengths n −1 at each vertex of T. Ramya, Selvi and Jeyanthi prove that Pm ⊙Kn (m ≥2, n ≥1) is an odd mean graph, Tp trees are odd mean graphs, and, for any Tp tree T, the graphs T@Pn, T@2Pn, ⟨T ˜ oK1,n⟩ are odd mean graphs.
For a Tp-tree T with m vertices let T ˆ oCn denote the graph obtained from T and m copies of Cn by identifying a vertex of ith copy of Cn with ith vertex of T and T ˜ oCn denote the graph obtained from T and m copies of Cn by joining a vertex of ith copy of Cn with ith vertex of T by an edge. In Selvi, Ramya, and Jeyanthi prove that for a Tp tree T the graphs T ˆ oCn (n > 3, n ̸= 6) and T ˜ oCn, (n > 3, n ̸= 6) are odd mean graphs.
Ramya, Selvi, and Jeyanthi prove that for a Tp-tree T the following graphs are odd mean graphs: T@Pn, T@2Pn, Pm⊙Kn, and the graph obtained from T and m copies of K1,n by joining the central vertex of ith copy of K1,n with ith vertex of T by an edge.
For a graph G and some fixed vertex v of G, Pooranam, Vaski, and Suganthi proved the following graphs have odd mean labelings: graphs obtained from a path Pm : u1u2 · · · um and G by joining ui to v to the ith copy of G. Their results include the cases where G = C4n, a star, or the cube Q3. For a graph G and some fixed vertex v of G they also proved the existence of odd mean labelings for graphs obtained from a path Pm : u1u2 · · · um and G by identifying ui with v in the ith copy of G, where G is Q3 or C(2) 4n and v is the vertex of C(2) 4n of degree 4.
A graph G is said to be vertex odd mean graphvertex odd mean if there exist an injective function f : V (G) to {1, 3, 5, . . . , 2|E(G)|−1} such that the induced mapping f ∗: E(G) to the set of positive integers defined by f ∗(uv) = (f(u)+f(v))/2 is injective. Such a function is called a vertex odd mean labeling. A graph G is called a vertex even mean graph if there exist an injective function f : V (G) to {2, 4, 6, . . . , 2|E(G)|} such that the induced mapping f ∗: E(G) to the set of positive integers defined by f ∗(uv) = (f(u) + f(v))/2 is injective. Such a function is called a vertex even mean labeling. Revathi gave vertex odd and even mean labelings for nontrivial umbrella graphs, mongolian tents, and the electronic journal of combinatorics (2023), #DS6 348 K1 + Cn.
Anitha, Selvam, and Thirusangu proved that the extended duplicate graph of kite graph admits mean, even mean, and odd mean labelings, Prajapati and Raval and proved that cyclic snakes for n = 3, 4, 5 are vertex even mean and vertex odd mean graphs. They have proved that double cyclic snake and alternating cyclic snake for n = 3, 4, 5 are vertex odd mean and vertex even mean graph. Prajapati and Raval proved that every vertex even mean graph is vertex odd mean graph and vice versa. They also proved that triangular snakes TSn, TSn ⊙P2, quadrilateral snakes QSn, QSn⊙P2, double quadrilateral snakes, alternating double quadrilateral snakes, jelly fish, crowns, shells, and fans are vertex even mean and vertex odd mean graphs. Raval and Prajapati proved that cyclic snakes (obtained by replacing each vertex of a path by a cycle), quadrilateral snakes, pentagonal snakes, and alternating quadrilateral snakes admit vertex even and odd mean labelings. They also proved that even vertex odd mean graphs are even mean graphs. In Basher obtained sufficient conditions for new certain uniform theta graphs to be even vertex odd mean graphs.
Maheswari, Azhagarasi, and Samuvel prove that the path union Pm (m ≥2) new of even cycles C2n (n ≥3) with parallel P3 chords and the path union Pm (m ≥2) of odd cycles C2n+1 (n ≥3) with parallel P3 chords are vertex even mean and vertex odd mean graph. They also prove that crowns C2n ⊙K1 (n ≥3) with parallel P3 chords and crowns C2n+1 ⊙K1 (n ≥3) with parallel P3 chords possess vertex even mean labeling and vertex odd mean labeling.
In Amuthavalli and S. Dineshkumar say a (p, q) graph G has a k-odd edge mean labeling if there exists an injection f from the edges of G to {0, 1, 2, 3, . . . , 2k+2p−3} such that the induced map f ∗defined on V (G) by f ∗(v) = ⌈P(f(vu)/deg(v)⌉is a bijection from V to {2k−1, 2k+1, 2k+3, . . . , 2k+2p−3}. A graph that admits a k-odd edge mean labeling is called a k-odd edge mean graph. They proved that Pn (n ̸= 4), K1,2n (n ≥2), and Cn (̸= 6, 7) are k-odd edge mean graphs for all k.
Thirugnanasambandam, Chitra, and Vishnupriya introduced a notion of prime odd mean graphs as follows. A graph G with p vertices and q edges for which there exists an injective function f : V (G) →{1, 3, 5, . . . , 2q + 1} such that gcd (f(u), f(v)) = 1 and the induced edge labeling f ∗: E(G) →{2, 4, 6, . . . , 2p −2} defined by f ∗(uv) = ⌈(f(u) + f(v))/2⌉is injective is called a prime odd mean graph. They proved that caterpillars, cycles, kites, ladders, spiders, and stars are prime odd mean graphs.
In Chitra, Priya, and Vishnupriya introduced the notion of odd Fibonacci mean labeling of graph G as an injective f from the vertices of G to the odd Fibonacci numbers such that the induced edge labeling f ∗defined by f ∗(uv) = (f(u) + f(v))/2 is injective. A graph that admits a odd Fibonacci mean labeling is called an odd Fibonacci mean graph.
They proved the following graphs admit odd Fibonacci mean labelings: wheels, helms, gears, cycles, banana trees, tadpoles, fire crackers, fans, stars, ladders, complete tripartite graphs, and Dutch windmills.
Gayathri and Amuthavalli (see also ) say a (p, q)-graph G has a (k, d)-odd mean labeling if there exists an injection f from the vertices of G to {0, 1, 2, . . . , 2k − 1 + 2(q −1)d} such that the induced map f ∗defined on the edges of G by f ∗(uv) = ⌈(f(u)+f(v))/2⌉is a bijection from edges of G to {2k−1, 2k−1+2d, 2k−1+4d, . . . , 2k− the electronic journal of combinatorics (2023), #DS6 349 1 + 2(q −1)d}. When d = 1 a (k, d)-odd mean labeling is called k-odd mean. For n ≥2 they prove the following graphs are k-odd mean for all k: Pn; combs Pn ⊙K1; crowns Cn ⊙K1 (n ≥4); bistars Bn,n; Pm ⊙Kn (m ≥2); Cm ⊙Kn; K2,n; Cn except for n = 3 or 6; the one-point union of Cn (n ≥4) and an endpoint of any path; grids Pm × Pn (m ≥2); (Pn × P2) ⊙K1; arbitrary unions of paths; arbitrary unions of stars; arbitrary unions of cycles; the graphs obtained by joining two copies of Cn (n ≥4) by any path; and the graph obtained from Pm × Pn by replacing each edge by a path of length 2. They prove the following graphs are not k-odd mean for any k: Kn; Kn with an edge deleted; K3,n (n ≥3); wheels; fans; friendship graphs; triangular snakes; Möbius ladders; books K1,m × P2 (m ≥4); and webs. For n ≥3 they prove K1,n is k-odd mean if and only if k ≥n −1. Gayathri and Amuthavalli prove that the graph obtained by joining the centers of stars K1,m and K1,n are k-odd mean for m = n, n + 1, n + 2 and not k-odd mean for m > n + 2. For n ≥2 the following graphs have a (k, d)-mean labeling : Cm ∪Pn (m ≥4) for all k; arbitrary unions of cycles for all k; P2m; P2m+1 for k ≥d; (P2m+1 is not (k, d)-mean when k < d); combs Pn ⊙K1 for all k; K1,n for k ≥d; K2,n for k ≥d; bistars for all k; nC4 for all k; and quadrilateral snakes for k ≥d.
In Seoud and Salim proved that a graph has a k-odd mean labeling if and only if it has a mean labeling. In Seoud and Salim give upper bounds of the number of edges of graphs with a (k, d)-odd mean labeling Pricilla defines an even mean labeling of a graph G as an injective function f from the vertices of G to {2, 4, . . . , 2|E(G)|} such that the edge labels given by (f(u)+f(v))/2 are distinct. Vaidya and Vyas proved that D2(Pn), M(Pn), T(Pn), S′(Pn), P 2 n, P 3 n, switching of pendent vertex in Pn, S′(Bn,n), double fans, and duplicating each vertex by an edge in paths are even mean graphs.
Gayathri and Gopi defined a graph G with q edges to be an k-even mean graph if there is an injective function f from the vertices of G to {0, 1, 2, . . . , 2k + 2(q −1)} such that when each edge uv is labeled with (f(u) + f(v))/2 if f(u) + f(v) is even, and (f(u)+f(v)+1)/2 if f(u)+f(v) is odd, then the resulting edge labels are {2k, 2k+2, 2k+ 4, . . . , 2k + 2(q −1)}. Such a function is called a k-even mean labeling. In they proved that the graphs obtained by joining two copies of Cn with a path Pm are k-even mean for all k and all m, n ≥3 when n ≡0, 1 (mod 4) and for all k ≥1, m ≥7, and n ≥3. In Gayathri and Gopi proved that various graphs obtained by joining two copies of stars K1,m and K1,n with a path by identifying the one endpoint of the path with the center of one star and the other endpoint of the path with the center of the other star are k-even mean. In they proved that various graphs obtained by appending a path to a vertex of a cycle are k-even mean. In they proved that Cn ∪Pm, n ≥4, m ≥2, is k-even mean for all k. Gayathri and Gopi proved the following are k-even mean graphs: shadow graphs of stars with at least 3 vertices; edge duplication graphs of cycles with at least 4 vertices; and vertex duplication graphs of paths and cycles with at least 4 vertices.
Gayathri and Gopi say graph G with q edges has a (k, d)-even mean labeling if there exists an injection f from the vertices of G to {0, 1, 2, . . . , 2k + 2(q −1)d} such that the induced map f ∗defined on the edges of G by f ∗(uv) = (f(u) + f(v))/2 if f(u) + f(v) the electronic journal of combinatorics (2023), #DS6 350 is even and f ∗(uv) = (f(u) + f(v) + 1)2 if f(u) + f(v) is odd is a bijection from edges of G to {2k, 2k + 2d, 2k + 4d, . . . , 2k + 2(q −1)d}. A graph that has a (k, d)-even mean labeling is called a (k, d)-even mean graph. They proved that Pm ⊕nK1(m ≥3, n ≥2) has a (k, d)-even mean labeling in the following cases: all (k, d) when m is even; all (k, d) when m is odd and n is odd; and m is odd, n is even and k ≥d.
Kalaimathy investigated conditions under which a mean labeling for a graph G will yield a (k, d)-even mean labeling for G and vice versa. He also gave conditions under which two graphs that have (1, 1)-mean labelings can be joined by an single edge to obtain a new graph that has a (1, 1)-even mean labeling. Gopi’s Ph. D. thesis has a large number of results about mean, k-mean, k-odd mean, k-even mean, (k, d)-odd mean , and (k, d)-mean labelings.
In Vasuki, Nagarajan, and Arockiaraj introduced the notion of even vertex odd mean graphs as follows. A (p, q)-graph is said to have an even vertex odd mean labeling if there exists an injective function f from V (G) to {0, 2, 4, . . . , 2q −2, 2q} such that the induced map f ∗: E(G) to {1, 3, 5, . . . , 2q −1} defined by f ∗(uv) = (f(u) + f(v))/2 is a bijection. A graph that admits an even vertex odd mean labeling is called an even vertex odd mean graph. They proved that paths, C4n, K2,n, bistars Bm,n for n = m, m + 1, quadrilateral snakes Qn, combs, Pm × Pn, C4m × Pn, ladders, and dragons are even vertex odd mean graphs. Rajesh Kannan, Vikrama Prasad, Gopi proved the following graphs have an even vertex odd mean labeling: slanting ladders SLn(n ≥3); double triangular snakes; alternative double triangular snakes; graphs obtained by starting with a tree G with at least 3 vertices and a mean labeling and a copy G′ of G by joining each vertex of G to its corresponding vertex in G′ with an edge; graphs obtained by starting with a path v1v2 · · · vn (n ≥4) and joining v1 and v3 to an isolated vertex; graphs obtained by starting with a path v1v2 · · · vn (n ≥4) and appending two edges to each of v2, v3, . . . , vn−1; and graphs obtained from a quadrilateral snake and appending an edge at each vertex. The H-graph of a path Pn is the graph obtained from two copies of Pn with vertices v1, v2, . . . , vn and u1, u2, . . . , un by joining the vertices vn/2+1 and un/2+1 by an edge if n is odd and the vertices v(n+1)/2 and u(n/2 by an edge if n is even. Rajesh Kannan, Vikrama Prasad, and Gopi prove that the H-graph of Pn (n ≥3) and the graph H ⊙K1 have even vertex odd mean labelings where H is the H graph of Pn (n ≥3).
In and Rajesh Kannan, Vikrama Prasad, and Gopi proved the following are even vertex odd mean graphs: graphs obtained by joining the centers of two stars K1,m and K1,n by a path Pt (m, n, t ≥2), graphs obtained by duplicating an edge of Cn (n ≥4), graphs obtained by joining each endpoint of P3 to n isolated vertices, shadow graphs of stars, shadow graphs of bistars B(n, n), mirror graphs of paths, and the graphs obtained taking two copies of Pn × P2 and joining each vertex of one with the matching vertex in the other with an edge. Kannan, Vikrama Prasad, and Gopi proved that the following graphs admit even vertex odd mean labelings: slanting ladders SLn (n ≥3), twigs TW(n) (n ≥4, double triangular snakes, alternative double quadrilateral snakes, and Qn ·K1. Prasad, Rejesh Kannan, and Gopi proved that C4m ⊙K1,4n, Pm ⊙Pn, and K2 + Kn have even vertex odd mean labelings. In Prajapati and Raval proved that quadrilateral snakes, pentagonal snakes, and alternating quadrilateral snakes are the electronic journal of combinatorics (2023), #DS6 351 vertex even and odd mean graphs. They also proved that even vertex odd mean graphs are even mean graphs. Jeyanthi, Ramya, and Selvi prove that TP-trees (transformed trees), T@Pn, T@2Pn, and ⟨T ˆ oK1,n⟩(where T is a TP tree) are even vertex odd mean graphs.
Basher investigated an even vertex labeling for the calendula graphs and intro-duced an arbitrary calendula graph as one in which if every edge from Cm is attached by an edge from arbitrary Cni, where ni may vary for each 1 ≤i ≤m. He proved that these graphs are also even vertex odd mean graphs. In Basher proved several cycle related graphs are even vertex odd mean graphs. Basher and Kamran Siddiqui proved that paths, cycles, combs, crowns, and planar grid super subdivisions are even vertex odd mean graphs.
For a graph G(V, E) a bijection f from V (G) ∪E(G) onto {1, 2, . . . , |V (G)| + |V (E)|} is said to be a total mean labeling if the values of f ∗(uv) = ⌈(f(u) + f(v) + f(uv))/3⌉ taken over all edges are distinct. A graph G is said to be a total mean labeling graph if it admits a total mean labeling.
Karuppasamy and Kaleeswari proved that Pn, P + n , K1,n, K2,n, Cn, Bm,n, triangular snakes, and alternate triangular snakes are total mean labeling graphs.
Murugan and Subramanian say a (p, q)-graph G has a Skolem difference mean labeling if there exists an injection f from the vertices of G to {1, 2, . . . , p + q} such that the induced map f ∗defined on the edges of G by f ∗(uv) = (|f(u)−f(v)|)/2 if |f(u)−f(v)| is even and f ∗(uv) = (|f(u) −f(v)| + 1)/2 if |f(u) + f(v)| is odd is a bijection from edges of G to {1, 2, . . . , q}. A graph that has a Skolem difference mean labeling is called a Skolem difference mean graph. They show that the graphs obtained by starting with two copies of Pn with vertices v1, v2, . . . , vn and u1, u2, . . . , un and joining the vertices v(n+1)/2 and u(n+1)/2 if n is odd and the vertices vn/2+1 and un/2 if n is even are Skolem difference mean. Parmar and Vaghela proved the following graphs have Skolem difference mean labelings: brooms Bn,d (n ≥4, d ≥2) (the graph with n vertices obtained from Pd by appending n−d edges at an endpoint), combs Pn⊙K1 (n ≥2), K1,m∪K1,n (m, n ≥2), and K1,3∗K1,n (n ≥2) obtained from K1,3 by attaching root of a star K1,n at each pendent vertex of K1,3. Jeyanthi proved ∪Pni (ni ≥2); ∪K1,ni (ni ≥2); Cn ∪Pm (n ≥ 3, m ≥2); Km,n ∪(m −1)(n −1)K2; and K1,1,n ∪(n −1)K2 are Skolem difference mean graphs. She also proved that if G is a Skolem difference mean graph, then G ∪nK2 is a Skolem difference mean graph.
Let L0, L1, . . . denote the sequence of Lucas numbers. In Ponmoni, Navaneetha Krishnan, and Nagarajan introduce the following graph labeling method. A graph G with p vertices and q edges is said to have a Skolem difference Lucas mean labeling if there is an injective function f from the vertices to {1, 2, . . . , Lp+q} such that when the edge uv is labeled with |f(u) −f(v)|/2 if |f(u) −f(v)| is even, and (|f(u) −f(v)| + 1)/2 if |f(u) − f(v)| is odd, then the resulting edge labels are distinct and belong to {L1, L2, . . . Lq}. A graph that admits a Skolem difference Lucas mean labeling is called a Skolem difference Lucas mean graph. They proved the graphs obtained from K1,m by identifying the center of K1,n with the endpoint of each non-center vextex of K1,m, bistars, K1,m ⊙2Pn and ⟨K(1) 1,n, K(2) 1,n, . . . , K(m) 1,n ⟩are Skolem difference Lucas mean graphs.
the electronic journal of combinatorics (2023), #DS6 352 Selvi, Ramya, and Jeyanthi prove that Cn@Pn (n ≥3, m ≥1), Kn(n ≤3), the shrub St(n1, n2, · · · , nm), and the banana tree Bt(n, n, . . . , n) are Skolem difference mean graphs. They show that if G is a (p, q) graph with q > p then G is not a Skolem difference mean graph and prove that Kn (n ≥4) is not a Skolem difference mean graph.
A skolem difference mean labeling for which all the labels are odd is called an extra Skolem difference mean labeling. They also prove that the graph T ⟨K1,n1 : K1,n2 : · · · : K1,nm⟩, obtained from the stars K1,n1, K1,n2, . . ., K1,nm by joining the central vertex of K1,nj and K1,nj+1 to a new vertex wj for 1 ≤j ≤m−1 and the graph T ⟨K1,n1 ◦K1,n2 ◦· · · ◦K1,nm⟩, obtained from K1,n1, K1,n2, . . ., K1,nm by joining a leaf of K1,nj+1 to a new vertex wj for 1 ≤j ≤m −1 by an edge are extra Skolem difference mean graphs. Jeyanthi, Selvi, and Ramya proved that the union of any number of paths, any number of stars, G ∪nK2 where G is an extra Skolem difference mean tree, Cn ∪Pm (n ≥3, m ≥2), Km,n ∪(m −1)(n −1)K2, and K1,1,n ∪(n −1)K2 have Skolem difference mean labelings.
Vaghela and Parmar provided an extra Skolem difference mean labeling for combs, twigs of Pn, H graphs of Pn, graphs obtained from K1,2 by attaching the root of K1,n at each pendant vertex of K1,2, graphs obtained from K1,3 by attaching the root of K1,n at each pendant vertex of K1,3, and related graphs. Gross and Yellen proved that Tp-trees and caterpillars are extra Skolem difference mean graphs.
Let G(V, E) be a graph with p vertices and q edges. Ramya, Kalaiyarasi, and Jeyanthi say G is a Skolem odd difference mean if there exists an injective function f : V (G) →{0, 1, 2, 3, . . . , p+3q−3} such that the induced map f ∗: E(G) →{1, 3, 5, . . . , 2q− 1} denoted by f ∗(uv) = ⌈|f(u) −f(v)|/2⌉is a bijection. A graph that admits a Skolem odd difference mean labeling is called a odd difference mean graph. They prove that Pn, Cn (n ≥4), K1,n, Pn ⊙K1,n, coconut trees T(n, m) obtained by identifying the central vertex of the star K1,m with a pendent vertex of Pn, Bm,n, caterpillars S(n1, n2, . . . , nm), Pm@Pn and Pm@2Pn are Skolem odd difference mean graphs. (Pm@Pn is obtained from Pm and m copies of Pn by identifying one pendent vertex of the i-th copy of Pn with the i-th vertex of Pm; Pm@2Pn is defined analogously.) They establish that Kn, n > 3 and K2,n (n ≥3) are not Skolem odd difference mean graphs. They also prove that K2,n is a Skolem odd difference mean graph if n ≤2. In Jeyanthi, Kalaiyarasi, Ramya, and Saratha Devi prove that bistars B(m, n), mPn, mPn ∪tPs, mK1,n ∪tK1,s and the graph ⟨Pm˜ oSn⟩obtained from Pm and m copies of K1,n by joining the central vertex of ith copy of K1,n with ith vertex of Pm by an edge admit Skolem odd difference mean labelings.
They also prove that if G(p, q) is a Skolem odd differences mean graph then p ≥q and that wheels, umbrellas, books, and ladders are not Skolem odd difference mean graphs.
They call a Skolem odd difference mean labeling a Skolem even vertex odd difference mean labeling if all the vertex labels are even. They prove that Pn, K1,n, Pn ⊙K1, the coconut tree T(n, m) obtained by identifying the central vertex of K1,m with a pendent vertex of a path Pn, B(m, n), caterpillars S(n1, n2, . . . , nm), Pm@Pn are Pm@2Pn are even vertex odd difference mean and Cn is not a Skolem even vertex odd difference mean graph. In Kalaiyarasi, Ramya, and Jeyanthi prove the following graphs have Skolem odd difference mean labelings: graphs obtained from a Tp tree with m vertices and m copies of K1,n by identifying the central vertex of ith copy of K1,n, with ith vertex of T; graphs obtained by the electronic journal of combinatorics (2023), #DS6 353 connecting an isolated vertex to central vertex of each of a number of stars; the banana trees obtained by connecting an isolated vertex to one leaf of each of any number of K1,n; graphs obtained from K1,n1, K1,n2, . . . , K1,nm by joining the central vertices of K1,nj and K1,nj+1 to a new vertex wj for 1 ≤j ≤m−1; graphs obtained from K1,n1, K1,n2, . . . , K1,nm by joining a leaf of K1,nj and a leaf of K1,nj+1 to a new vertex wj for 1 ≤j ≤m −1.
Lau, Jeyanthi, Ramya, and Kalaiyarasi say a (p, q)-graph G(V, E) is a Skolem even difference mean if there exists an injective function f : V (G) →{0, 1, 2, 3, . . . , p + 3q −1} such that the induced map f ∗: E(G) →{2, 4, . . . , 2q} defined by f ∗(uv) = ⌈|f(u) −f(v)|/2⌉is a bijection. A graph that admits a Skolem even difference mean labeling is called a even difference mean. They prove: the disjoint union of paths of length at least 2 and K2,n∪(n−1)K2 (n ≥2) are Skolem even vertex odd difference mean graphs; if G is a Skolem even vertex odd difference mean (q + 1, q)-graph, then G ∪nK2, G ∪Pn, and G ∪K1,n are Skolem odd difference mean graphs; Cm ∪Pn (n ≥2) is a Skolem odd difference mean graph for m = 4 and 6; the caterpillar S(n1, n2, . . . , nm) is a Skolem even vertex even difference mean graph; Pm@Pn, mPn, Km,n ∪(m −1)(n −1)K2 (m, n ≥2), K1,n∪nK2, and K1,1,n∪nK2 are Skolem even difference mean graphs; and if G is a Skolem even vertex even difference mean (q+1, q)-graph, then G∪nK2 is a Skolem even difference mean graph. They conclude with the following open problem: Establish that G ∪nK2 where G is a (complete) multipartite graph is a Skolem even difference mean graph.
Kalaiyarasi, Ramya, and Jeyanthi say a graph G(V, E) with p vertices and q edges has a centered triangular mean labeling if it is possible to label the vertices with distinct elements f(x) from S, where S is a set of non-negative integers in such a way that for each edge e = uv, f ∗(e) = ⌈(f(u) + f(v))/2⌉and the resulting edge labels are the first q centered triangular numbers. A graph that admits a centered triangular mean labeling is called a centered triangular mean graph. They prove that Pn, K1,n, bistars Bm,n, coconut trees, caterpillars S(n1, n2, n3, . . . , nm), St(n1, n2, n3, . . . , nm), banana trees Bt(n, n, . . . , n) and Pm@Pn are centered triangular mean graphs.
Selvi, Ramya, and Jeyanthi define a triangular difference mean labeling of a graph G(p, q) as an injection f : V − →Z+, such that when the edge labels are defined as f ∗(uv) = ⌈|f(u) −f(v)|/2⌉the values of the edges are the first q triangular numbers.
A graph that admits a triangular difference mean labeling is called a triangular differ-ence mean graph. They prove that the following are triangular difference mean graphs: Pn, K1,n, Pn ⊙K1, bistars Bm,n, graphs obtained by joining the roots of different stars to the new vertex, trees T(n, m) obtained by identifying a central vertex of a star with a pendent vertex of a path, the caterpillar S(n1, n2, . . . , nm) and the graph Cn@Pm.
A graph G(V, E) with p vertices and q edges is said to have centered triangular dif-ference mean labeling if there is an injective mapping f from V to Z+ such that the edge labels induced by f ∗(uv) = ⌈|f(u) −f(v)|/2⌉are the first q centered triangu-lar numbers.
A graph that admits a centered triangular difference mean labeling is called a centered triangular difference mean graph. Ramya, Selvi, and Jeyanthi prove that Pn, K1,n, Cn ⊙K1, bistars Bm,n, Cn (n > 4), coconut trees, caterpillars S(n1, n2, n3, . . . , nm), Cn@Pm (n > 4) and Sm,n are centered triangular difference mean graphs.
the electronic journal of combinatorics (2023), #DS6 354 Gayathri and Tamilselvi say a (p, q)-graph G has a (k, d)-super mean labeling if there exists an injection f from the vertices of G to {k, k+d, . . . , k+(p+q)d} such that the induced map f ∗defined on the edges of G by f ∗(uv) = ⌈(f(u)+f(v))/2⌉has the property that the vertex labels and the edge labels together are the integers from k to k + (p + q)d.
When d = 1 a (k, d)-super mean labeling is called k-super mean. For n ≥2 they prove the following graphs are k-super mean for all k: odd cycles; Pn; Cm ∪Pn; the one-point union of a cycle and the endpoint of Pn; the union of any two cycles excluding C4; and triangular snakes. For n ≥2 they prove the following graphs are (k, d)-super mean for all k and d: Pn; odd cycles; combs Pn ⊙K1; and bistars. In Jeyanthi, Ramya, and Thangavelu proved the following graphs have k-super mean labelings: C2n, C2n+1 ×Pm, grids Pm ×Pn with one arbitrary crossing edge in every square, and antiprisms on 2n vertices (n > 4).
(Recall an antiprism on 2n vertices has vertex set {x1,1, . . . , x1,n, x2,1, . . . , x2,n} and edge set {xj,i, xj,i+1}∪{x1,i, x2,i}∪{x1,i, x2,i−1} where subscripts are taken modulo n). Jeyanthi, Ramya, Thangavelu give k-super mean labelings for a variety of graphs. Jeyanthi, Ramya, Thangavelu, and Aditanar show how to construct k-super mean graphs from existing ones. For n ≥3 Gayathri and Tamilselvi proved the following graphs are k-super edge mean for all k: paths; cycles; combs Pn ⊙K1; triangular snakes; crowns Cn ⊙K1; the one-point union of C3 and an endpoint of Pn; and Pn ⊙K2.
In Sandhya, Somasundaram, and Ponraj call a graph with q edges a harmonic mean graph if there is an injective function f from the vertices of the graph to the integers from 1 to q + 1 such that when each edge uv is labeled with ⌈2f(u)f(v)/(f(u) + f(v))⌉ or ⌊2f(u)f(v)/(f(u) + f(v))⌋the edge labels are distinct.
They prove the following graphs have such a labeling: paths, ladders, triangular snakes, quadrilateral snakes, Cm ∪ Pn (n > 1); Cm ∪Cn; nK3; mK3 ∪Pn (n > 1); mC4; mC4 ∪Pn; mK3 ∪nC4; and Cn ⊙K1 (crowns). They also prove that wheels, prisms, and Kn (n > 4) with an edge deleted are not harmonic mean graphs. In Sandhya, Somasundaram, and Ponraj investigated the harmonic mean labeling for a polygonal chain, square of the path and dragon and enumerate all harmonic mean graph of order at most 5. In Jayasekaran and David Raj prove that some disconnected graphs are harmonic mean graphs. In Raj, Jayasekaran, and Sandhya investigate some new families of harmonic mean graphs.
Seoud and Salim provided upper bounds of the number of edges of graphs of given orders with harmonic mean labelings and showed that all graphs of order at most 9 have have harmonic mean labelings using the floor function portion of the definition.
Meena and Sivasakth prove that subdivision graphs of Pn ⊙K1, Pn ⊙K2, H-graphs, Cn ⊙K1, Cn ⊙K2, quadrilateral snakes, and triangular snakes are harmonic mean.
Sandhya, Somasundaram, and Ponraj proved that the following graphs have harmonic mean labelings: graphs obtained by duplicating an arbitrary vertex or an arbi-trary edge of a cycle; graphs obtained by joining two copies of a fixed cycle by an edge; the one-point union of two copies of a fixed cycle; and the graphs obtained by starting with a path and replacing every other edge by a triangle or replacing every other edge by a quadrilateral.
Vaidya and Barasara proved that the following graphs have harmonic mean labelings: graphs obtained by the duplication of an arbitrary vertex or arbitrary edge of the electronic journal of combinatorics (2023), #DS6 355 a path or a cycle; the graphs obtained by the duplication of an arbitrary vertex of a path or cycle by a new edge; and the graphs obtained by the duplication of an arbitrary edge of a path or cycle by a new vertex.
In Narasimhan and Sampathkumar called a graph with p vertices a contra harmonic mean graph if there is an injective function f from the vertices of the graph to the integers from 1 to p such that when each edge uv is labeled with f(uv) = ⌈(f(u))2 + (f(v))2/(f(u) + f(v))⌉or f(uv) = ⌊(f(u))2 + (f(v))2/(f(u) + f(v))⌋the edge labels are distinct.
They prove the following graphs have such a labeling: paths, cycles, Cm ∪ Pn, Cm ∪Cn, nK3, nK3 ∪Pm, and nK3 ∪Cm. Gopi called a graph with q edges a k-contra harmonic mean graph if there is an bijective function f from the edges of the graph to the integers from k −1 to k + q + 1 such that each edge uv is labeled with f(uv) = ⌈(f(u))2+(f(v))2/(f(u)+f(v))⌉or f(uv) = ⌊(f(u))2+(f(v))2/(f(u)+f(v))⌋. He proves that triangular snakes, double triangular snakes, quadrilateral snakes, and double quadrilateral snakes have k-contra harmonic mean labelings.
Gopi and Suba say a graph G with p vertices and q edges is a super Lehmer-3 mean graph if there is an injective function f from the vertices of G to {1, 2, . . . , q+1} such that for each edge uv the induced function f ∗(uv) = ⌊(f(u)3 + f(v)3)/(f(u)2 + f(v)2)⌋or f ∗(uv) = ⌈(f(u)3 + f(v)3)/(f(u)2 + f(v2))⌉yields that the set of vertex labels and edge labels is {1, 2, . . . , p}. They prove that Pm ⊙K1,n and the graph obtained by identifying each endpoint of a path with an endpoint of the star K1,n have a super Lehmer-3 labeling.
In Gopi and Nirmala provide Lehmer-3 mean labelings for Pm ⊙Cn (m, n ≥3) and Pm ⊙K1 ⊙Cn (m, n ≥3). In Gopi and Prakash investigated the k-super Lehmer-3 new meanness of triangular snakes, double triangular snakes, alternative triangular snakes, quadrilateral snakes, double quadrilateral snakea, and alternative quadrilateral snakes.
In Somasundaram, Sandhya, and Pavithra proved that paths, combs, ladders, and new crowns are super Lehmer-3 mean graphs.
In Shalini and Meena introduced the notion of Lehmer-4 mean labelings as follows. A graph G(V, E) with p vertices and q edges is called a Lehmer-4 mean graph, if there is an injection from V to {2, 4, 6, . . . , 2p} and the induced function f ∗(uv) = ⌊(f(u)4 + f(v)4)/(f(u)2 + f(v)2)⌋or f ∗(uv) = ⌈(f(u)4 + f(v)4)/(f(u)2 + f(v2))⌉in an injection. In this case, f is called a Lehmer-4 mean labeling mean of G. They proved paths, combs, Pn ⊙K1,2 and Pn ⊙K1,3 are Lehmer-4 mean graphs.
An F-geometric mean labeling of a graph G with q edges, is an injective function from the vertex set of G to {1, 2, . . . , q + 1} such that the edge labels obtained from the floor function of geometric mean of the vertex labels of the end vertices of each edge, are all distinct and the set of edge labels is {1, 2, . . . , q}.
Durai Baskar, Arockiaraj, and Rajendran proved that the following graphs are F-geometric mean: graphs obtained by identifying a vertex of consecutive cycles (not necessarily of the same length) in a particular way; graphs obtained by identifying an edge of consecutive cycles (not necessarily of the same length) in a particular way; graphs obtained by joining consecutive cycles (not necessarily of the same length) by paths (not necessarily of the same length) in a particular way; Cn ⊙K1; Pn ⊙K1; Ln ⊙K1; G ⊙K1 where G is the graph obtained by joining two copies of Pn by an edge in a particular way; graphs obtained by appending the electronic journal of combinatorics (2023), #DS6 356 two edges at each vertex of graphs obtained by joining two copies of Pn by an edge in a particular way; graphs obtained from Cn by appending two edges at each vertex of Cn; graphs obtained from ladders by appending two edges at each vertex of the ladders; graphs obtained from Pn by appending an end point of the star S2 to each vertex of Pn; and graphs obtained from Pn by appending an end point of the star S3 to each vertex of Pn.
A C-geometric mean labeling of a graph G with q edges, is an injective function from the vertex set of G to {1, 2, 3, . . . , q + 1} such that the edge labels obtained from the ceiling function of the geometric mean of the vertex labels of the end vertices of each edge are all distinct and the set of edge labels is {2, 3, 4, . . . , q + 1}. A graph is said to be a C-geometric mean graph if it admits a C-geometric mean labeling. In Durai Baskar and Arockiaraj study the C-geometric meanness of some cycle related graphs such as cycle, union of a path and a cycle, unions of two cycles, the graphs C3 × Pn, corona of cycle, the graphs Pa,b, P a b and some chain graphs.
A geometric mean labeling f of G(V, E) is called a super geometric mean labeling if f(V ) ∪f(E) = {1, 2, . . . , |V | + |E|}. Sandhya, Merly, and Shiny prove that the subdivision graphs of the following graphs have super geometric mean labelings: alternate quadrilateral snakes, double quadrilateral snakes, alternate double quadrilat-eral snakes, triple quadrilateral snakes, and subdivisions of alternate triple quadrilateral snakes. In they prove that the following graphs have super geometric mean la-belings: triangular ladders, triangular snakes, alternate triangular snakes, quadrilateral snakes, and alternate quadrilateral snakes. Hemalatha and Selvi prove that fol-lowing graphs have super geometric mean labelings: flags, kayak paddles, dumbbells, polygonal snakes, and graphs obtained by connecting any number of copies of Cn where each joined to the next with an edge.
Durai Basker and Arockiaraj study the F-geometric meanness of cycles, stars, complete graphs, combs, ladders, triangular ladders, middle graphs of paths, graphs ob-tained from duplicating arbitrary vertex by a vertex as well as arbitrary edge by an edge in cycles, and subdivisions of combs and stars.
In 2017 Hemalatha, Mohanaselvi, and Amuthavalli defined a radio geometric mean labeling of a graph G as a mapping from the vertex set to the set of natural numbers such that for two distinct vertices u and v of G, d(u, v) + ⌈f(u)f(v)⌉≥1 + diam(G).
The radio geometric mean number of f is the maximum number assigned to any vertex of G. The radio geometric mean number of G is the minimum value taken over all radio geometric mean labeling f of G. A radio antipodal geometric mean labeling of a graph G is a mapping f from the vertex set of G to the set of natural numbers such that for two distinct vertices u and v of G, d(u, v) + ⌈ p f(u)f(v)⌉≥diam(G). If d(u, v) = diam(G), the vertices u and v can be given the same label and if d(u, v) ̸= diam(G), the vertices u and v are assigned different labels. The radio antipodal geometric mean number of f is the maximum number assigned to any vertex of G. The radio antipodal geometric mean number of G is the minimum value taken over all radio geometric mean labeling f of G. Hemalatha, Mohanaselvi, and Amuthavalli provided the radio geometric mean numbers for stars, bistars, complete k-ary trees with 3 levels, and the graph lotus inside a the electronic journal of combinatorics (2023), #DS6 357 circle. In Hemalatha and Amuthavalli determined the radio geometric mean number of splitting graphs of stars and bistars. Hemalatha and Mohanaselvi found the radio geometric mean numbers of the subdivision graphs of complete graphs, complete bipartite graphs, books, and windmills. In Giridaran, Jose, and Jeony provided upper bounds for the radio antipodal geometric mean number of ladders, triangular ladders, circular ladders (Cn×K2), and pagoda graphs (graphs obtained ny joining two adjacent endpoints of a ladder to new vertex.
Arockiaraj and Meena Kumari introduced the F-face magic mean labeling of graphs in . This motivated Meena Kumari and Arockiaraj to introduce the (1,0,0)-F-face magic mean labeling of graphs as follows. A bijection φ from V (G) to {1, 2, . . . , |V (G)|} is called a (1,0,0)-F-face magic mean labeling of G if the induced face labeling φ∗(fi) = ⌊(sum of the labels of the vertices in the boundary of fi)/deg(fi)⌋is a constant for each face fi, including the exterior face of G, where deg fi is the number of edges that bound the face. A graph that admits an (1,0,0)-F-face magic mean labeling is called (1,0,0)-F-face magic mean. In Arockiaraj and Meena Kumar showed that Pn + K1 (n ≥2), cycles with certain cords, Cm × Pn where m and n are even, and graphs obtained by duplicating every edge of a cycle by a vertex admit (1,0,0)-F-face magic mean labelings.
In Sundaram, Ponraj, and Somasundaram introduced a new labeling parameter called the mean number of a graph. Let f be a function from the vertices of a graph to the set {0, 1, 2, . . . , n} such that the label of any edge uv is (f(u) + f(v))/2 if f(u) + f(v) is even and (f(u)+f(v)+1)/2 if f(u)+f(v) is odd. The smallest integer n for which the edge labels are distinct is called the mean number of a graph G and is denoted by m(G).
They proved that for a graph G with p vertices m(tK1,n) ≤t(n + 1) + n −4; m(G) ≤ 2p−1 −1; m(K1,n) = 2n −3 if n > 3; m(B(p, n)) = 2p −1 if p > n + 2 where B(p, n) is a bistar; m(kT) = kp −1 for a mean tree T, m(Wn) ≤3n −1, and m(C(t) 3 ) ≤4t −1.
Let f be a function from V (G) to {0, 1, 2}. For each edge uv of G, assign the label ⌈(f(u) + f(v))/2⌉. Ponraj, Sivakumar, and Sundaram say that f is a mean cordial labeling of G if |vf(i) −vf(j)| ≤1 for i and j in {0, 1, 2} where vf(x) and ef(x) denote the number of vertices and edges labeled with x, respectively. A graph with a mean cordial labeling is called a mean cordial graph. Observe that if the range set of f is restricted to {0, 1}, a mean cordial labeling coincides with that of a product cordial labeling. Ponraj, Sivakumar, and Sundaram prove the following: every graph is a subgraph of a connected mean cordial graph; K1,n is mean cordial if and only n ≤2; Cn is mean cordial if and only n ≡1, 2 (mod 3); Kn is mean cordial if and only n ≤2; Wn is not mean cordial for all n ≥3; the subdivision graph of K1,n is mean cordial; the comb Pn ⊙K1 is mean cordial; Pn ⊙2K1 is mean cordial; and K2,n is a mean cordial if and only n ≤2.
Seoud and Salim provided upper bounds of the number of edges of graphs of given orders with mean cordial labelings and proved that P2t × P2 is mean cordial if and only if t ≡2 mod 3 and Cn ⊙K1 is mean cordial if and only if n ≡1 or 2 mod 3.
In Ponraj and Sivakumar proved the following graphs are mean cordial: mG where m ≡0 (mod 3); Cm ∪Pn; Pm ∪Pn; K1,n ∪Pm; S(Pn ⊙K1); S(Pn ⊙2K1); P 2 n if and only if n ≡1 (mod 3) and n ≥7; and the triangular snake Tn (n > 1) if and only if n ≡0 (mod 3). They also proved that if G is mean cordial then mG, m ≡1 (mod 3) is the electronic journal of combinatorics (2023), #DS6 358 mean cordial. Deshmukh and Shaikh prove the graph ⟨K1,n : 2⟩and the path union of n copies of K1,m are mean cordial graphs.
In Ponraj and Sathish Narayanan proved double triangular snakes D(Tn) are mean cordial if and only if n > 3 and obtained partial results on mean cordial label-ings of alternate triangular snakes, double alternate triangular snakes. Shendra shainy, Vinothkumar, and Balaji proved that for β1 < β2 < β3, the three star graph K1,β1 ∧K1,β2 ∧K1,β3 is a mean cordial graph if and only if |β2 −β3| ≤3β1 + 7. Further, in Maheshwari, Vinothkumar and Balaji proved that for β1 = β2 < β3 the three star graph K1,β1 ∧K1,β2 ∧K1,β3 is a mean cordial graph if and only if |β2 −β3| ≤3β1 + 6.
In Ponraj, Sathish Narayanan, and Ramasamy introduced the notion of total mean cordial labeling. A total mean cordial labeling of a graph G(V, E) is a function f : V (G) →{0, 1, 2} such that when each edge xy is assigned the label ⌈(f(x) + f(y))/2⌉ we have |evf(i) −evf(j)| ≤1, i, j ∈{0, 1, 2}, where evf(x) denotes the total number of vertices and edges labeled with x. A graph with a total mean cordial labeling is called total mean cordial. In , , and , Ponraj, Sathish Narayanan, and Ramasamy determined the total mean cordiality of the following graphs: Pn; Cn; K1,n; Wn; K2 + mK1; combs Pn ⊙K1; double combs Pn ⊙2K1; crowns; flowers; lotuses inside a circle; bistars; quadrilateral snakes; K2,n; olive trees; S(Pn ⊙K1); S(K1,n) (S(G) denotes the subdivision of G); triangular snakes; P 2 n; fans Fn; umbrellas; butterflies; and dumbbells.
In , , and , Ponraj and Sathish Narayanan determined the total mean cordiality of Kc n +2K2; prisms; gears; helms; P1 ∪P2 ∪· · ·∪Pn; Ln ⊙K1; S(Wn); S(Pn ⊙ 2K1); and graphs obtained by subdividing each step of a ladder exactly once.
Let G be a (p, q)-graph. Ponraj and Sathish Narayanan and proved the following. If G satisfies any one of the following three conditions then G ⊙2K is total mean cordial: (1) G is a tree, (2) G is a unicycle, (3) q = p + 1. If G satisfies any one of the following three conditions then the shadow graph of G is total mean cordial: (1) G is a tree, (2) G is a unicycle, (3) q = p + 1. They also proved that the following are total mean cordial graphs: Cn ⊙K2, C(2) n , dragons, splitting graphs of stars, splitting graphs of combs, books, ladders, Pn ⊙K2 if and only if n ̸= 1, and G ∪Pn (n ̸= 3).
In Lourdusamy and Joy Beaula define the notion of S3 (all permutations of {1, 2, 3}) mean cordial graphs as follows. Let G(V, E) be a graph and g : V (G) →S3 be a function. For each edge xy assign the label 1 if ⌈(|g(x)| + |g(y)|)/2⌉is odd, and 0 otherwise. Such a g is said to be a group S3 mean cordial labeling if the number of vertices labeled with i and j differ by at most 1 where i, j are the elements in S3 and the number of edges labeled with 0 and 1 differ by at most 1. A graph with a group S3 mean cordial labeling is called a group S3 mean cordial graph. They prove that K2,n, subdivisions of fans, double fans, wheels, and helms are S3 mean cordial graphs.
Ponraj, Sathish Narayanan, and Kala introduced the concept of radio mean labeling in . A radio mean labeling of a connected graph G is a one-to-one map f from V (G) to the set of natural numbers such that for each pair of distinct vertices u and v of G, d (u, v) + l f(u)+f(v) 2 m ≥1 + diam (G). The radio mean number of f, rmn (f), is the maximum number assigned to any vertex of G. The radio mean number of G, rmn (G), the electronic journal of combinatorics (2023), #DS6 359 is the minimum value of rmn (f) taken over all radio mean labelings f of G. They proved rmn (G) ≥|V (G)|; if G is a (p, q)-graph with diameter d ≥2, then rmn (G) ≤p + d −2; and if G is a (p, q)-connected graph with diameter 2 or 3, then rmn (G) = p. They also determine the radio mean number of Kn, Km,n, sunflowers, helms, gears, lotuses inside a circle, and graphs obtained by identifying any two vertices of two wheels of the same size, Aasi, Asif, Iqbal, and Ibrahim determine the radio number and the radio mean number for the lexicographic product of a path with a path, a path with a cycle, and a cycle with a cycle. Moreover, they present computer programs for finding such radio labelings of these families of graphs. Sunitha, Raj, and Subramanian investigated new radio mean labeling of path and cycle related graphs. Lavanya, Dhanyashree, and Meera determined the radio mean number of subdivision graphs of complete graphs, new Mongolian tents, subdivision of friendship graphs, and diamond graphs. If the minimum value of the radio mean number taken over all radio mean labelings of G is |V (G)|, the graph is said to be radio mean graceful. They proved that the graphs they studied are radio mean graceful.
In and Ponraj, Sathish Narayanan, and Kala determine the radio mean numbers of S(Km,n) (m > 1, n > 1); Km,n ⊙Pt; C(t) 6 ; Wn ⊙Pm; graphs obtained by joining the rim vertices of the two wheels with an edge; and graphs obtained from a wheel by subdividing each spoke by a vertex. In Ponraj, Sathish Narayanan, and Kala give the radio mean number of graphs with diameter three, lotuses inside a circle, helms, and sunflower graphs.
In and Ponraj and Sathish Narayanan give the radio mean number of the following graphs: subdivisions of stars, subdivisions of wheels, subdivisions of K2 + mK1, subdivisions of bistars, jelly fish, subdivisions of jelly fish, books with pentagonal pages, graphs obtained by taking m disjoint copies of K1,n and joining a new vertex to the centers of the m copies of K1,n.
A radio mean D-distance labeling of a connected graph G is an injective map f from V (G) to the natural numbers such that for two distinct vertices u and v of G, dD(u, v) + ⌈(f(u) + f(v))/2⌉≥1 + diamD(G), where dD(u, v) denotes the distance D between u and v and diamD(G) denotes the D-diameter of G. The radio mean D-distance number of f, rmnD(f), is the maximum label assigned to any vertex of G. The radio mean D-distance number of G, rmnD(G), is the minimum value of rmnD(f) taken over all radio mean D-distance labeling f of G. Nicholas and Bosco determined the radio mean D-distance number of cycles, wheels, gears, helms, fans, and friendship graphs.
In Ponraj and Sathish Narayanan proved that the following graphs are not mean cordial: K2 + Km; Kn + 2K2; Pn × K2; flower graphs; sunflower graphs; Cn ⊙K2.
Also they proved the following: the Mongolian tent MTm,n is mean cordial if and only if m ≡0 (mod 3) or n ≡0 (mod 3) (MTm,n is the graph obtained from Pm × Pn, n odd, by adding one extra vertex above the grid and joining every other vertex of the top row of Pm × Pn to the new vertex); the book Bm is mean cordial if and only if m = 1; books with n pentagonal pages are mean cordial if and only if n ≡1 (mod 3); Pn ⊙K2 is mean cordial if and only if n ≡0 (mod 3); quadrilateral snakes are mean cordial; alternate quadrilateral snakes A(Qn) are mean cordial if and only if the square starts from second the electronic journal of combinatorics (2023), #DS6 360 vertex of the path Pn, ends with (n −1)th vertex and n ≡0, 2 (mod 3), or the square starts from first vertex, ends with nth vertex and n ≡0, 2 (mod 3), or the square starts from second vertex, ends with nth vertex and n ≡0, 1 (mod 3).
Let G be a graph and let f : V (G) →{0, 1, . . . , k −1} k > 1.
For each edge uv, assign the label f(uv) = ⌈f(u)+f(v) 2 ⌉.
Ponraj, Subbulakshmi, and Somasundaram say that f is a k-total mean cordial labeling of G if |tmf(i) −tmf(j)| ≤1, for all i, j ∈{0, 1, . . . , k −1}, where tmf(x) denotes the total number of vertices and edges labeled with x ∈{0, 1, . . . , k −1}. A graph with admit a k-total mean cordial labeling is called k-total mean cordial graph. Ponraj, Subbulakshmi, and Somasundaram investigated the 4-total mean cordial labeling behavior of fans, wheels, jelly fish, jewel graphs, ladders, and triangular snakes.
A balanced mean cordial labeling is a mean cordial labeling f with the property that the number of vertices labeled with i is the same as the number labeled with j for i, j ∈ {0, 1, 2} and likewise for the edges. Kaneria, Khoda, and Karavadiya prove: the path union of n copies of a graph G is a mean cordial when n ≡0 (mod 3); if G is balanced mean cordial, then Pn × G and Cn × G are balanced mean cordial; and if f : V (G) − →{0, 1, 2} is a balanced mean cordial labeling for G, then G∗is also a balanced mean cordial graph.
In Jeyanthi and Maheswari define a one modulo three mean labeling of a graph G with q edges as an injective function φ from the vertices of G to {a | 0 ≤a ≤3q −2 where a ≡0 (mod 3) or a ≡1 (mod 3)} and φ induces a bijection φ∗from the edges of G to {a | 1 ≤a ≤3q −2 where a ≡1 (mod 3)} given by φ∗(uv) = ⌈(φ(u) + φ(v))/2⌉. They proved that P2n, combs, bistars Bn,n, Tp-trees with an even number of vertices, C4n+1, ladders, K1,2n × P2 are one modulo three mean graphs. They also proved that bistars Bm,n (m ̸= n), K1,n (n > 3), and Kn, (n > 3) are not one modulo three mean graphs.
In Jeyanthi, Maheswari, and Pandiaraj proved that DA(Qn), DA(Q2) ⊙ nK1, DA(Qm) ⊙nK1, DA(T2) ⊙nK1, DA(Tm) ⊙nK1, S(DA(Tn)), S(DA(Qn)), and mPn are one modulo three mean graphs.
Jeyanthi, Maheswari, and Pandiaraj prove that following graphs have one mod-ulo three mean labelings: books K1,2n × P2; splitting graphs S′(P2n); vertex duplica-tion graphs D(G, v′); edge duplication graphs D(G, e′); nth alternate quadrilateral snake graphs NA(Qm); graphs obtained by joining the endpoints of paths P4m to n isolated vertices; and extended jewel graphs EJn with vertex set {u, v, x, y, w, z, ui : 1 ≤i ≤n} and edge set {uv, ux, xy, yz, vw, wz, vui, zui : 1 ≤i ≤n}. For graphs G1 and G2, G1b ◦G2 is the graph obtained from G1 and |V (G1)| copies of G2 by joining a vertex of ith copy of G2 with the ith vertex of G1 by an edge. Jeyanthi, Maheswari, and Pandiaraj prove that the graphs T ⊙Kn, T b ◦K1,n, T b ◦Pn, and T b ◦2Pn are one modulo three mean graphs. Sudarvizhi and Balasangu provide one modulo three mean labelings for triangular books, duplication subdivisions of the central edge of bistars, and slanting ladders (graphs obtained from two paths u1, u2, u3, . . . , un and v1, v2, v3, . . . , vn by joining each ui with vi+1 for 1 ≤i ≤n1).
In and Gayathri and Prakash proved the following graphs admit one modulo three mean graphs: the mirror graph of Pn, Spl(Pn), Pm × Pn, (Pm × P2) ⊙K1,n, the electronic journal of combinatorics (2023), #DS6 361 bistars, Cn@Pm, Cn⊙Pm, and H⊙Km, where H is the graph is obtained from two copies of Pn with vertices u1, u2, . . . , un and v1, v2, . . . , vn by joining u(n+1)/2 to v(n+1)/2 with an edge when n > 1 is odd, and un/2 to vn/2+1 with an edge when n is even. In Gayathri and Prakash investigated the existence of one modulo three mean labelings of disconnected graphs. In they provided a necessary condition for a graph to admit a one modulo three mean labeling. In Gayathri and Prakash provided one modulo three mean labelings for various trees. In and they proved the following graphs admit are k-one modulo three mean labelings: Pn, K1,n, combs, Pm ⊙K1,n, Bm,n, Ln = Pn × P2, K1,n × P2, A(Qn), D(Qn), A(D(Qn)), A(D(Qn)) ⊙Km, A(Qn) ⊙Km, and the graphs obtained by joining the end points of Pm to n isolated points.
A graph G is said to be one modulo three root square mean graph if there is an injective function φ from the vertex set of G to the set {0, 1, 3, . . . , 3q−2, 3q} where q is the number of edges of G and φ induces a bijection φ∗from the edge set of G to {1, 4, . . . , 3q −2} given by φ∗(uv) = q [φ(u)]2+[φ(v)]2 2 or q [φ(u)]2+[φ(v)]2 2 . The function φ is called a one modulo three root square mean labeling of G. In Jayasekaran and Jaslin Melbha investigated some path related graphs that have one modulo three root square mean labelings.
Somasundaram, Vidhyarani, and Ponraj introduced the concept of a geometric mean labeling of a graph G with p vertices and q edges as an injective function f : V (G) → {1, 2, . . . , q + 1} such that the induced edge labeling f ∗: E(G) →{1, 2, . . . , q} defined as f ∗(uv) = lp f(u)f(v) m or jp f(u)f(v) k is bijective. Among their results are: paths, cycles, combs, ladders are geometric mean graphs and Kn (n > 4) and K1,n (n > 5) are not geometric mean graphs. Somasundaram, Vidhyarani, and Sandhya proved Cm ∪Pn, Cm ∪Cn, nK3, nK3 ∪Pn, nK3 ∪Cm, P 2 n, and crowns are geometric mean graphs. Vaidya and Barasara investigated geometric mean labelings in context of duplication of graph elements in cycle Cn and path Pn. Durai Baskar, Arockiyaraj, and Rajendran investigate the geometric meanness of some graphs obtained from paths.
In Jeyanthi, Maheswari, and Pandiaraj define a graph G to be a one modulo three geometric mean graph if there is an injective function φ from the vertex set of G to the set {a|1 ≤a ≤3q −2 and either a ≡0 (mod 3) or a ≡1 (mod 3)} where q is the number of edges of G and φ induces a bijection φ∗from the edge set of G to {a|1 ≤a ≤3q −2 and a ≡1 (mod 3)} given by φ∗(uv) = lp φ(u)φ(v) m or jp φ(u)φ(v) k the function φ is called one modulo three geometric mean labeling of G. They proved paths, cycles with length at least 5, ladders, Pn ⊙K1, Pn ⊙P2, Pn ⊙P2, subdivision graphs S(Pn ⊙K1), and subdivision graphs S(Pn ⊙K2) are one modulo three geometric graphs.
They also prove that K1,n (n ≥3) and graphs in which every edge lies on a triangle are not one modulo three geometric mean graph.
In Jeyanthi, Selvi and Ramya introduced a new graph labeling as follows. A graph G(p, q) is said to be a one modulo N-difference mean graph if there is an injection f from the vertex set of G to the set {0 ≤a ≤2(q −1)N + 1} and either a = 0 (mod N) or a = 1 (mod N), where N is a positive integer and f induces a bijection f ∗from the edge set the electronic journal of combinatorics (2023), #DS6 362 of G to {1 ≤a ≤(q−1)N +1} and a = 1 (mod N) given by f ∗(uv) = ⌈|f(u)f(v)|/2⌉. Such a function f is called a one modulo N-difference mean labeling of G. They establish that Bm,n, Sm,n, Pn@Pm, B(l, m, n), T(n, m), shrubs, caterpillars, and K1,n are one modulo N-difference mean graph. In addition, they show that C3 is not a one modulo N-difference mean graph.
Jeyanthi, Selvi, and Ramya define a restricted triangular difference mean label-ing of a graph G with p vertices and q edges as an injection f : V →{1, 2, 3, . . . , pq} such that for each edge uv, the edge labels defined by f ∗(uv) = ⌈|f(u)−f(v)|/2⌉are the first q triangular numbers. A graph that admits a restricted triangular difference mean labeling is called a restricted triangular difference mean graph. Jeyanthi, Selvi, and Ramya investigate the restricted triangular difference mean behaviors of the paths, combs, Kn, bistars Bm,n, caterpillars S(n1, n2, . . . , nm), Km,n, wheels, and graphs obtained by joining the centers of different stars to the new vertex. They also give a necessary condition for a graph to be a restricted triangular difference mean graph.
A (p, q) graph G(V, E) is said to be an analytic mean graph if it is possible to injectively label the vertices with {0, 1, 2, . . . , p −1} in such a way that when each edge uv is labeled with |(f(v)2 −(f(u))2|/2 when |(f(v)2 −(f(u))2| is even and |(f(v)2 −(f(u))2 +1|/2 when |(f(v)2 −(f(u))2| + 1 is odd and the edge labels are distinct. In this case, f is called an analytic mean labeling of G. Raj and Vivek proved that Pm ∪Cn ∪K1,s (m, s ≥ 2, n ≥3), (Pm + K1) ∪K1,n (m, n ≥2), graphs obtained by identifying the apex vertices of K1,m and K1,n and one vertex of two copies of Cs where m, n ≥2, c ≥3 are analytic mean graphs.
Let G = (V, E) be a graph with p vertices and q edges. A graph G is analytic odd mean if there exist an injective function f : V →{0, 1, 3, 5 . . . , 2q −1} with an induce edge labeling f ∗: E →Z such that for each edge uv with f(u) < f(v), f ∗(uv) = l f(v)2−(f(u)+1)2 2 m if f(u) ̸= 0, and f ∗(uv) = l f(v)2 2 m if f(u) = 0 is injective. In this case we say that f is an analytic odd mean labeling of G. Jeyanthi, Gomathi, and Lau proved that fans, double fans, double wheels, closed helms, total graphs of cycles, total graphs of paths, armed crowns CnΘPm, generalized Petersen graphs GP(n, 2) are analytic odd mean graphs. In they prove that wheels, flower graphs, some splitting graphs, and multiples of graphs are analytic odd mean graphs. In they prove that quadrilateral snakes, double quadrilateral snakes, coconut trees, fire cracker graphs, some star graphs, splitting graphs, complete bipartite graphs, unicyclic graphs, and the graphs obtained from a path of vertices v1, v2, v3, . . . , vn by joining i pendent vertices at each of ith vertex 1 ≤i ≤n (denoted Pn(1, 2, . . . , n)) are analytic odd mean graphs. Jeyanthi, Gomathi, and Lau proved the square graphs of Pn, Cn, Bn,n H-graphs, and H ⊙mK1 admit analytic odd mean labelings.
In Jeyanthi, Selvi, and Rama proved that the following graphs admit ana-lytic odd mean labeling: arbitrary union of paths, M2(Pn)(n ≥2), ladders, slanting slanting ladders, diamond snakes, quadrilateral snakes, alternately quadrilateral snakes, Jln(P3)(n ≥1), C4 ⊙K1,n(n ≥1), DUP2(K1,n), DUP2(Bn,n), friendship graphs, and nC4(n ≥1). Jeyanthi, Gomathi, Lau, and Shiu proved that the following graphs admit analytic odd mean labeling: Pn + Cm, tadpoles T(m, n), Cm1 + Cm2, JC(m1, m2), the electronic journal of combinatorics (2023), #DS6 363 edge linked cycle snakes EL(kCn), triangular snakes Tn, double triangular snakes D(Tn), Pn(Cm), nCm, Pn(T(m, 1)), shell graphs Shn. They propose the conjecture that a (p, q)-graph is analytic odd mean when p ≤q + 1.
In Jeyanthi and Gomathi proved that the subdivision and super subdivision of the following graphs are analytic odd mean: cycles, stars, combs, and graphs obtained from Pn ⊙K1 by subdividing the each edge of Pn. Jeyanthi, Gomathi, and Lau proved that quadrilateral snakes, double quadrilateral snakes, coconut trees, fire cracker graphs, and Pn(1, 2, . . . , n) are analytic odd mean graphs. They also proved that Cn⊙K1, prisms, helms, banana trees, perfect binary trees, unicyclic graphs, certain caterpillars, and spiders are analytic odd mean graphs. Jeyanthi and Gomathi proved that the graphs TLn, (the triangular ladder obtained from Ln), TLn⊙K1, Tn⊙K1, and Qn⊙K1 ad-mit an analytic odd mean labelings. Simaringa and Thirunavukkarasu proved that triangular books, double triangular books, triangular snakes, double triangular snakes, butterflies, drums, Cc J Pc, Fc J K1,d, 1 ≤d ≤2c −1, Wc ⊙K1,d, 1 ≤d ≤2c, P 2 c ⊙K1,d, c ≥3 and 1 ≤d ≤2c −3 are analytic odd mean graphs.
Let G be a (p, q) graph and f a injective function from V (G) to {k, k+1, . . . , p+q+k− 1} For each edge uv, let f ∗= ⌈(2f(u)f(v)/(f(u)+f(v)⌉or ⌊(2f(u)f(v)/(f(u)+f(v)⌋. We say f is a k-super harmonic mean if f(V )∪{f ∗(uv) | uv ∈E(G)} = {k, k+1, . . . , p+q+k− 1}. A graph that admits a k-super harmonic mean labeling is called a k-super harmonic mean graph. In the case that k = 1 the labeling is called a super harmonic mean labeling.
For all n > 1 Tamilselvi and Revathi prove that the following graphs have k-super harmonic mean labelings: Pn, nPm (m > 1), Pn ⊙K1, Pn ⊙K2, Pn ⊙K3, P 2 n (n ≥4), the subdivision graph of Pn ⊙K1, and the middle graph of Pn.
In Sandya, Merly, and Deepa introduced a Heronian mean labeling of graphs as new follows. A graph G(V, E) with p vertices and q edges is said to be a Heronian mean graph if there exists an injective function f from V to 1, 2, 3, . . . , q+1 such that when each edge uv is labeled with f ∗(uv) = ⌊(f(u)+f(v)+ p f(u)f(v))/3⌋or ⌈(f(u)+f(v)+ p f(u)f(v))/3⌉, the resulting edge labels are distinct. In this case f is called a Heronian mean labeling of G.
Sampath, Narasimhan, and Nagaraja proved cycles, K1,n if and only if n ≤4, Cm ∪Pn, Cm ∪Cn, nK3, nK3 ∪Pm, nK3 ∪Cm, mC4, crowns Cn ⊙K1, dragons Cn@Pm, and P 2 n admit Heronian mean labelings. Anitha, Selvam, and Thirusangu provide Heronian mean labelings for the extended duplicate graph of the kite graph.
A graph G = (V, E) with p vertices and q edges is said to be a (k, d)-Heronian mean graph if it is possible to label the vertices x ∈V with distinct labels f(x) from k, k + d, k + 2d, . . . , k + qd in such a way that when each edge uv is labeled with f ∗(uv) = ⌊(f(u) + f(v) + p f(u)f(v))/3⌋or ⌈(f(u) + f(v) + p f(u)f(v))/3⌉, then the resulting edge labels are distinct. In this case f is called a (k, d)-Heronian mean labeling of G.
Note that the case k = 1 and d = 1 is a Heronian mean labeling. Akilandeswari and Tamilselvi proved that paths, ladders, and Pn ⊙mK1 for n ≥2, 1 ≤m ≤4, are k-Heronian mean graphs. In Akilandeswari and Tamilselvi proved that the following graphs have (k, d)-Heronian mean labelings: paths, (Pn × P2) ⊙K1, Tn ⊙K1 (Tn is the triangular snake obtained from Pn), Qn ⊙K1, TLn ⊙K1 (TLn is the triangular ladder obtained from Ln), Peterson graphs, and the graphs obtained from two copies of Pn with the electronic journal of combinatorics (2023), #DS6 364 vertices v1, v2, . . . , vn and u1, u2, . . . , un by joining the vertices u(n+1)/2 and v(n+1)/2 if n is odd and un/2+1 and vn/2+1 if n is even.
A pronic number is one of the form n(n + 1), where n is a positive integer. In Porchelvi and Devi defined a pronic Heron mean labeling of a graph G with n vertices as a bijection f : V (G) →{0, 2, 6, 12, . . . , n(n + 1)} such that the resulting edge labels obtained by f ∗(uv) = ⌈(f(u)+f(v)+ p f(u)f(v)/3⌉for every edge uv are distinct. In Devi and Porchevi gave pronic Heron mean labelings for the generalized Peterson graphs P(6, 2), P(8, 3),P(10, 2), P(10, 3), and P(12, 5).
They also investigated the existence of pronic Heron labelings for mK3, the unions of cycles and paths, and the union of combs and paths. They described an algorithm to label the vertices for the pronic Heron mean labeling for certain disconnected graphs. They further investigated the existence of proniceron labelings for mK3, the unions of cycles and paths, and the union of combs and paths.
Arockiaraj and Meena say a planar graph has an F-face magic mean labeling it there exists an assignment of labels to the edges that induces an assignment of labels to the faces of the graph such that the mean weight of each face is constant. They proved that the following graphs have F-face magic mean labelings: P2n + K1, the one-point union of m copies of Cn, mCn-snakes, and graphs obtained identifying the endpoints of any number of copies Pn. In and Arockiaraj and Meena Kumari investigated the existence of (1, 0, 0) F-face magic mean labelings for paths, butterflies, and middle graphs of cycles, Pn+K1, Cm×Pn, latitude graphs, cyclic ladders, graphs obtained by duplicating every edge of Pn (n ≥2), even cycles, the middle graphs of cycles, and the middle graphs of butterflies. In they investigated (0, 1, 0), (1, 1, 0)-F-face magic mean labelings of slanting ladders, Pm×Pn, K2,n, and certain graphs obtained by applying graph operations.
Vani Shree and Dhanalakshmi investigated the existence of (1, 0, 0)-F-face magic mean labelings of ladders, tortoises, and the middle graph of paths. They also provided (1,0,0)- and (1,1,0)-F-face magic mean labelings for ortho chain square cactus graphs, para chain square cactus graphs, triangular snakes, and quadrilateral snakes.
Amara Jothi, Baskar Babujee, and David investigated face magic labeling of planar graphs of types (1, 0, 1), (1, 1, 0), (0, 1, 1) and (1, 1, 1) on duplication graphs.
For a graph G(V, E), Rajesh Kannan, Manivannan, and Durai Basker de-fined an exponential labeling as injective function χ from V to {1, 2, 3, . . . , |E| + 1} such that the induced function χ∗from E to {1, 2, . . . , |E|} given by χ(uv)∗ = ⌊(1/e)(χ(v)χ(v)/χ(u)χ(u))(1/χ(v)−χ(u))⌋for all uv ∈E is a bijection. A graph that allows an exponential mean labeling is called an exponential. They investigated the exponential mean labeling of graphs obtained by duplication of an edge or vertex of paths and path related graphs.
In Baskaran and Ganapathy introduced a new variant of mean labeling as follows.
A function h is said to be a C-exponential mean labeling of a graph G(V, E) if h : V → {1, 2, . . . , |E| + 1} is injective and the induced function h∗: E →{2, 3, . . . , |E| + 1} defined by h∗(uv) = ⌈1/((h(v)h(v)/(h(u)h(u)))1/(h(v)−h(u))⌉is a bijection. They investigate the exponential meanness of paths, triangular trees (see Section 3.1), Cm ⊙Pn, Pm × Pn, one-sided step graphs (the graph obtained by starting with the path Pn with consecutive the electronic journal of combinatorics (2023), #DS6 365 edges e1, e2, . . . , en−1 and erecting a ladder that has that has n −i −1 steps with ei at the bottom) double-sided step graphs (the graph obtained by starting with the path P2n with 2n −1 edges e1, e2, . . . e2n−1, where on every edge ei we erect a ladder that has i + 1 steps for i = 1, 2, . . . n, and on every edge and for i = n + 1, n + 2, . . . , 2n −1 erect a ladder on edge ei that has 2n + 1 −i steps), one-sided arrow graphs (the graphs obtained from joining the vertices of the bottom row of Pm × Pn to a new vertex), double-sided arrow graphs, graphs (the graphs obtained from joining the vertices of the bottom row of Pm × Pn to a new vertex and by joining the vertices of the right column of Pm × Pn to a new vertex), and subdivision of ladders.
A function f is said to be an absolute mean graceful of a graph G, if f : V (G) →{0, ±1, ±2, . . . , ±|E|} is injective and the edge labeling function f ∗: E(G) → {1, 2, . . . , |E|} defined as f ∗(uv) = ⌈(f(u) −f(v))/2⌉is bijective. A graph that admits such labeling is called absolute mean graceful graph. In and Kaneria, Chu-dasama, and Andharia proved that the path unions of a finite number of copies of trees, paths, cycles, complete bipartite graphs, grid graphs, step grid graphs, and double step grid graphs are absolute mean graceful graphs. In Akbari1, Kaneria, and Parmar prove that jewel graphs, jelly fish graphs, and the extended jewel graphs admit absolute mean graceful labelings.
In Palani and Shunmugapriya defined the concept of near mean labeling in digraphs as follows. Let D(V, A) be a digraph where V is the vertex is set and A is the arc set. Let f : V →{0, 1, 2, . . . , q} be a 1-1 map. Define f ∗: A →{1, 2, . . . , q} by f ∗(e = ⃗ uv) = l f(u)+f(v) 2 m . Let f ∗(v) = P w∈V f ∗( ⃗ vw) −P w∈V f ∗( ⃗ wv) . If f ∗(v) ≤2 for all v ∈A(D), f is said to be a near mean labeling of D and D is said to be a near mean digraph. They investigated the existence of near mean labeling in dragon digraphs (the one-point union of cycle and a path where the edges of a cycle are directed clockwise or anti-clockwise and the edges of the path edges are directed towards the cycle). In they investigated the existence of near mean labelings for various dicyclic snakes.
7.11 Pair Sum and Pair Mean Graphs For a (p, q) graph G Ponraj and Parthipan define an injective map f from V (G) to {±1, ±2, . . . , ±p} to be a pair sum labeling if the induced edge function fem from E(G) to the nonzero integers defined by fe(uv) = f(u) + f(v) is one-one and fe(E(G)) is either of the form {±k1, ±k2, . . . , ±k q 2} or {±k1, ±k2, . . . , ±k q−1 2 }∪{k q+1 2 }, according as q is even or odd. A graph with a pair sum labeling is called pair sum graph. In and they proved the following are pair sum graphs: Pn, Cn, Kn if and only if n ≤4, K1,n, K2,n, bistars Bm,n, combs Pn ⊙K1, Pn ⊙2K1, and all trees of order up to 9. Also they proved that Km,n is not pair sum graph if m, n ≥8 and enumerated all pair sum graphs of order at most 5.
In , , , and Ponraj, Parthipan, and Kala proved the following are pair sum graphs: K1,n ∪K1,m, Cn ∪Cn, mKn if n ≤4, (Pn × K1) ⊙K1, Cn ⊙K2, dragons Dm,n for n even, Kn + 2K2 for n even, Pn × Pn for n even, Cn × P2 for n even, (Pn×P2)⊙K1, Cn⊙K2 and the subdivision graphs of Pn×P2, Cn⊙K1, Pn⊙K1, triangular the electronic journal of combinatorics (2023), #DS6 366 snakes, and quadrilateral snakes. Ponraj, Parthipan, and Kala prove the following graphs are pair sum graphs: the union of two stars, the union of a path and a star, ladders, Cn ⊙K1, Cm ∪Cm, mKn if and only if n ≤4, quadrilateral snake Qn(n odd), and triangular snakes.
A (p, q)-graph G is said to be a super pair sum if there exists a bijection f from V (G) ∪E(G) to {0, ±1, ±2, . . . , ±( p+q−1 2 )} when p + q is odd and from V (G) ∪E(G) to {±1, ±2, . . . , ±( p+q 2 )} when p + q is even such that f(uv) = f(u) + f(v). A graph that admits a super pair sum labeling is called a super pair sum graph. Vasuki, Velmurugan, and Sugirtha prove that the graphs Hn ⊙mK1, (Hn is obtained from two copies of Pn (n ≥3) with vertices v1, v2, . . . , vn and u1, u2, . . . , un by joining v(n+1)/2 and u(n+1)/2 if n is odd and vn/2 and u(n+2)/2 if n is even); (P2n; Sm), S′(P2n), < Bm,n : Pk > for m ≥1, n ≥1, k ≡2 (mod 4), < B(m) : Pk > for m ≥1 k ≡0, 2 (mod 4) and 2Bm,n (m ≥1, n ≥1) are super pair sum graphs.
Jeyanthi and Sarada Devi define an injective map f from E(G) to {±1, ±2, . . . , ±q} as an edge pair sum labeling of a graph G(p, q) if the induced func-tion of f ∗from V (G) to Z −{0} defined by f ∗(v) = P f(e) taken over all edges e incident to v is one-one and f ∗(V (G)) is either of the form {±k1, ±k2, . . . , ±kp/2} or {±k1, ±k2, . . . , ±k(p−1)/2} ∪{kp/2} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. They proved that Pn, Cn, triangular snakes, Pm ∪K1,n, and Cn ⊙Km are edge pair sum graphs.
Jeyanthi, Sarada Devi, and Lau proved that the following graphs have edge pair sum labelings: triangular snakes Tn, Cn ∪Cn, K1,n ∪K1,m, and bistars Bm,n. They also proved that every graph is a subgraph of a connected edge pair sum graph. Jeyanthi and Sarada Devi showed that P2n × P2 and the graphs Pn(+)Nm obtained from a path Pn by joining its endpoints to m isolated vertices are edge pair sum graphs. Jeyanthi and Sarada Devi proved that the following graphs have edge pair sum labeling: shadow graphs S2(Pn), S2(K1,n), total graphs T(C2n) and T(Pn), the one-point union of any number of copies of Cn, the one-point union of Cm and Cn, P 2 2n−1, and full binary trees in which all leaves are at the same level and every parent has two children. Jeyanthi and Sarada Devi proved the spiders SP(1m, 2t), SP(1m, 2t, 3), SP(1m, 2t, 4), and for t even SP(1m, 3t, 3) are edge pair sum graphs. In Jeyanthi and Sarada Devi prove some cycle related graphs are edge pair sum graphs. In they prove that the one point union of cycles, perfect binary trees, shadow graphs, total graphs, and P 2 n admit edge pair sum graph. In Jeyanthi and Sarada provide edge pair sum labelings for jewel graphs, gears, triangular ladders, balanced lobsters, and double wheels 2Cn + K1.
The tree WT(n) is obtained from K1,n+2 with central vertex c1 and end vertices xi : 1 ≤i ≤n+2 and another K1,n+2 with central vertex c2 and end vertices yj : 1 ≤j ≤n+2 by identifying vertex xn+2 and yn+2 and denoting the identified vertices by w. A w-tree WT(n : k) is obtained from k copies of WT(n) by joining a new vertex a to vertex w of each copy of WT(n). Jeyanthi, Sarada Devi, and Lau proved that the graphs WT(n : k) trees have edge pair sum labelings (see also ).
In , , , Jeyanthi and Sarada Devi prove the following graphs are edge pair sum graphs: shell graphs; some butterfly graphs; jelly fish; Y -trees; theta the electronic journal of combinatorics (2023), #DS6 367 graphs; wheels with subdivided spokes, Pm + 2K1; C4 × Pm; Pn ⊙Km; (P2 × Pm) ⊙Kn; Pm × C3; books; graphs obtained from the path Pn having an even fixed even number quadrilaterals on each edge of the path; K2+mK1; graphs obtained by identifying one end point from each of m copies of Pn; closed helms; graphs that are two copies of generalized Petersen graphs joined by a path Pn, n ≥5; and graphs that two copies of fan Pn ⊙K1 joined by a path Pn, n ≥5.
In Jeyanthi and Sarada Devi prove the following graphs admit edge pair sum labelings: K2,n, double triangular snakes, wheels, flowers, ⟨Cm, K1,n⟩(m ≥4, n odd) obtained from Cm and K1,n by identifying any vertex of Cm with the central vertex of K1,n, and ⟨Cm ∗K1⟩(m ≥4) the graphs obtained from Cm and K1,n by identifying any vertex of Cm with an endpoint vertex of K1,n. In they prove that the subdivision of graph of bistars Bm,n, Pn ⊙K1, triangular snakes when the path has an odd num-ber of vertices, double triangular snakes, double quadrilateral snakes, double alternative triangular snakes, and double alternative quadrilateral snakes are edge pair sum graph.
In Amudha and Jayapriya introduced the notion of labeling pair sum modulo labelings as follows. An undirected simple graph G with p vertices and q edges is said to be a pair sum modulo graph if there is an injective function f from V (G) to {±1, ±2, . . . , ±p} such that the induced edge labeling g from E(G) to {0, 1, 2, . . . , q −1} defined by g(uv) = (f(u) + f(v))(mod q) is injective. They proved the Petersen graphs, a certain family of coconut trees, K2,n, and bistars have pair sum modulo labelings. Jayapriya proved spider graphs with at most five legs admit pair sum modulo labelings. Ponraj, Parthipan, and Kala prove the following graphs are pair sum graphs: the union of two stars, the union of a path and a star, ladders, Cn ⊙K1, Cm ∪Cm, mKn if and only if n ≤4, quadrilateral snake Qn(n odd), and triangular snakes.
For a (p, q) graph G Ponraj and Parthipan define an injective map f from V (G) to {±1, ±2, . . . , ±p} to be a pair mean labeling if the induced edge function fem from E(G) to the nonzero integers defined by fem(uv) = (f(u) + f(v))/2 if f(u) + f(v) is even and fem(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is one-one and fem(E(G)) = {±k1, ±k2, . . . , ±kq/2} or fem(E(G)) = {±k1, ±k2, . . . , ±k(q−1)/2} ∪{k(q+1)/2}, according as q is even or odd. A graph with a pair mean labeling is called a pair mean graph. They proved the following graphs have pair mean labelings: Pn, Cn if and only if n ≤3, Kn if and only if n ≤2, K2,n, bistars Bm,n, Pn⊙K1, Pn⊙2K1, and the subdivision graph of K1,n. Also they found the relation between pair sum labelings and pair mean labelings.
The graph G@Pn is obtained by identifying an end vertex of a path Pn with any vertex of G. A graph G(V, E) with q edges is called a (k + 1)-equitable mean graph if there is a function f from V to {0, 1, 2, . . . , k} (1 ≤k ≤q) such that the induced edge that labeling f ∗from E to {0, 1, 2, . . . , k} given by f ∗(uv)−⌈(f(u)+f(v))/2⌉has the properties |vf(i) −vf(j)| ≤1 and |ef∗(i) −ef∗(j)| ≤1 for i, j = 0, 1, 2, . . . , k where vf(x) and ef∗(x) are the number of vertices and edges of G respectively with the label x. In Jeyanthi proved the following: a connected graph with q edges is a (q + 1)-equitable mean graph if and only if it is a mean graph; a graph is 2-equitable mean graph if and only if it is a product cordial graph; for every graph G, the graph 3mG is a 3-equitable mean graph; the electronic journal of combinatorics (2023), #DS6 368 for every 3-equitable mean graph G, the graph (3m + 1)G is a 3-equitable mean graph; Cn is a 3-equitable mean graph if and only if n ̸≡0 (mod 3); Pn is a 3-equitable mean graph for all n ≥2; if G is a 3-equitable mean graph then G@Pn is a 3-equitable mean graph for n ≡1 (mod 3); the bistar B(m, n) with m ≥n is a 3-equitable mean graph if and only if n ≥⌊q/3⌋; K1,n is a 3-equitable mean graph if and only if n ≤2; and for any graph H and 3m copies H1, H2, . . . , H3m of H, the graph obtained by identifying a vertex of Hi with a vertex of Hi+1 for 1 ≤i ≤3m −1 is a 3-equitable mean graph.
In Lakshmi and Nagarajan introduced the notion of geometric mean cordial labeling of graphs as follows. Let G = (V, E) be a graph and f be a mapping from V (G) to {0, 1, 2}. The graph G is called geometric mean cordial if each edge uv can be assigned the label ⌈ p f(u)f(v)⌉in such a way that and |vf(i)vf(j)| ≤1 and |ef(i)ef(j)| ≤1, where vf(x) and ef(x) denote the number of vertices and edges labeled with x and x ∈ {0, 1, 2} They proved that Pn, Cn (n ≡1, 2 (mod 3)) and K1,n are geometric mean cordial graphs and Kn(n > 2), K2,n (n > 2), Kn,n (n > 2) and wheels are not geometric mean cordial graphs. In Kaneria, Meera, and Maulik call these graphs geometric mean 3-equitable. They proved: Kmn (m, n ≥4) is not a geometric mean 3-equitable graph, caterpillars S(x1, x2, . . . , xt) and Cn ⊙K1 (t ≥2) are geometric mean 3-equitable graphs, and Cn ⊙K1 is a geometric mean 3-equitable graph if and only if n ≡1, 2 (mod 3).
7.12 Irregular Total Labelings In 1988 Chartrand, Jacobson, Lehel, Oellermann, Ruiz, and Saba defined an irregular labeling of a graph G with no isolated vertices as an assignment of positive integer weights to the edges of G in such a way that the sums of the weights of the edges at each vertex are distinct.
The minimum of the largest weight of an edge over all irregular labelings is called the irregularity strength s(G) of G. If no such weight exists, s(G) = ∞.
Chartrand et al.
gave a lower bound for s(mKn).
Faudree and Lehel obtained s(G) ≤⌈n/2⌉+ 9 for regular graphs.
For every graph G of order n ≥4 and with finite irregularity strength, Nierhoff proved s(G) ≤n −1. Using deterministic and mostly probabilistic techniques, Frieze, Gould, Karonski, and Pfender obtained the following bounds: s(G) ≤c1 · n/δ) if ∆≤n1/2; and s(G) ≤c2 · (log n)n/δ if ∆> n1/2 , where c1 and c2 are explicit constants. Bohman and Kravitz found an infinite sequence of trees with strength converging to (11 − √ 5)/8. Faudree, Jacobson, and Lehel gave an upper bound for s(mKn) when n ≥5 and proved that for graphs G with δ(G) ≥ n −2 ≥1, s(G) ≤3.
They also proved that if G has order n and δ(G) = n −t and 1 ≤t ≤ p n/18, s(G) ≤3.
Aigner and Triesch proved s(G) ≤n + 1 for any graph G with n ≥4 vertices for which s(G) is finite. In Przybylo proved that s(G) < 112n/δ + 28, where δ is the minimum degree of G and G has n vertices. The best bound of this form is currently due to Kalkowski, Karońki, and Pfender, who showed in that s(G) ≤6⌈n/δ⌉< 6n/δ + 6. In Przybylo showed that for d-regular graphs s(G) < 16n/d + 6. In 1991 Cammack, Schelp, and Schrag proved that the irregularity strength of a full d-ary tree (d = 2, 3) is its number of pendent vertices and conjectures that the irregularity strength of a tree with no vertices of degree two is its the electronic journal of combinatorics (2023), #DS6 369 number of pendent vertices. This conjecture was proved by Amar and Togni in 1998. Muthu Guru Packiam, Manimaran, and Thuraiswamy prove the following: s(Cn ⊙mK1) = mn, s(Pn ⊙K2) = n + 1, s(Cn ⊙K2) = n + 1, s(Pn ⊙K3) = n + 1, and sCn ⊙K3) = n + 1. In Jinnah and Kumar determined the irregularity strength of triangular snakes and double triangular snakes. Hasni, Tarawneh, Siddiqui, Raheem, and Asim determined the exact value of edge irregularity strength of disjoint union of cycles and generalized prisms. These results provide an upper bound Hashi, Tarawneh, and Husin determined the edge irregularity strength of Pn ⊙Pm for n ≥2 and m = 3, 4, and 5. Imran, Aslam, Zafar, and Nazeer determined the exact value of the edge irregularity strength of caterpillars, n-star graphs, (n, t)-kite graphs, cycle chains, and friendship graphs. Tarawneh, Hasni, Ahmad, and Asim determined the exact value of the edge irregularity strength for some classes of plane graphs. In Ahmad, Khan, Ahmad, Nadeem, and Siddiqui determined the exact value of the edge irregularity strength for categorical product of two paths. In Imran, Cancan, and new Ali determined the edge irregularity strength of certain splitting graphs, shadow graphs, jewel graphs, jellyfish graphs, and m copies of 4-pan graphs. Uma Devi, Kamaraj, and Arockiaraj determined the odd Fibonacci edge irregularity strength for Pn (n ≥ 2), K1,n, Pn ⊙K1 (n ≥2), bistars B(m,n) and proved the nonexistence of an odd Fibonacci edge irregular labeling for Kn (n ≤3) and Km,n (m ≥2, n ≥4). In Devi, Kamaraj, and Arockiaraj determined the odd Fibonacci edge irregularity strength for Pn (n ≥2), stars, subdivision of stars, subdivision of fans, Pn ⊙mK1 (m, n ≥2), and the graphs obtained by joining i pendant vertices to the ith vertex of Pn (n ≥2).
Motivated by the notion of the irregularity strength of a graph and various kinds of other total labelings, Bača, Jendroľ, Miller, and Ryan introduced the total edge irregularity strength of a graph as follows.
For a graph G(V, E) a labeling ∂: V ∪ E →{1, 2, . . . , k} is called an edge irregular total k-labeling if for every pair of distinct edges uv and xy, ∂(u) + ∂(uv) + ∂(v) ̸= ∂(x) + ∂(xy) + ∂(y).
Similarly, ∂is called an vertex irregular total k-labeling if for every pair of distinct vertices u and v, ∂(u) + P ∂(e) over all edges e incident to u ̸= ∂(v) + P ∂(e) over all edges e incident to v. The minimum k for which G has an edge (vertex) irregular total k-labeling is called the total edge (vertex) irregularity strength of G. The total edge (vertex) irregular strength of G is denoted by tes(G) (tvs(G)). They prove: for G(V, E), E not empty, ⌈(|E| + 2)/3⌉≤ tes(G)≤|E|; tes(G)≥⌈(∆(G) + 1)/2⌉and tes(G)≤|E| −∆(G), if ∆(G) ≤(|E| −1)/2; tes(Pn) = tes(Cn)= ⌈(n + 2)/3⌉; tes(Wn)= ⌈(2n + 2)/3⌉; tes(Cn 3 ) (friendship graph) = ⌈(3n + 2)/3⌉; tvs(Cn) = ⌈(n + 2)/3⌉; for n ≥2, tvs(Kn)= 2; tvs(K1,n) = ⌈(n + 1)/2⌉; and tvs(Cn×P2)= ⌈(2n+3)/4⌉. In Salma1, Nagesh1, and Narahari determined the new exact values of the edge irregular k-labeling for the one-point union of Ck and K1,n−k (n > k) for 3 ≤k ≤7. ⌈((|E(G)| + 2)/3)⌉given in is a lower bound for the total edge irregularity strength of a graph G then this bound is also tight for the Cartesian product of G with any path.
Ahmad, Nurdin, and Baskoro determined the exact value of the total edge (vertex) irregularity strength of generalized Halin graphs. Al-Mushayt, Ahmad, and Siddiqui determined the exact values of the total edge irregular strength of hexagonal grid graphs. The (m, n)-lollipop graph denoted by Lm,n is a graph obtained the electronic journal of combinatorics (2023), #DS6 370 by joining a complete graph Km to a path graph Pn with a bridge.
Ni’mah and Indriati determined tvs(Lm,n) for m ≥3 and n ≥1.
Aftiana and Indriati proved that for n ≥3 the total edge irregularity strength of the graph obtained by joining two copies of Kn (barbell graph) with an edge is ⌈(n2 −n + 3)/3⌉.
In Nurdin and Hye consider the splitting graph of stars as a land transportation system and give the exact value of their total vertex irregularity strength. For m, n ≥ 3 Indriati, Widodo, and Sugeng determined the exact value of the total vertex irregularity strength for generalized helm graphs Hm n (obtained from Wn by attaching Pm vertices at each vertex of the n-cycle) and for prisms with outer pendent edges.
Nurdin determined the total vertex irregularity strength of banana trees and quad trees. Susanti, Puspitasari and Khotimah determined the exact value of total edge irregularity strength for staircase graphs, double staircase graphs, and mirror-staircase graphs. In Susanti, Wahyuni, Sutjijana, Sutopo, and Ernanto provide the total edge irregularity strength of generalized arithmetic staircase graphs and generalized double-staircase graphs.
For R be a commutative ring R with the set of all zero divisor set Z(R) of R the zero divisor graph of R, Γ(R), has vertex set Z(R) and (x, y) is an edge if and only if xy = 0. Ahmad determined the total vertex irregularity strength of zero divisor graphs associated with the commutative rings Zp2 ×Zq where p and q are distinct primes.
In Ahmad, Ibrahim, and Siddiqui determined the total irregularity strength of generalized Petersen graphs.
In Al-Mushayt, Ahmad, and Siddiqui de-termined the total edge (vertex) irregularity strength for convex polytope graphs hav-ing the same diameter.
In Siddiqui determined the irregularity strength of six classes of convex polytope graphs with pendent edges. Ramdani, Salman, Assiyatum, and Semaničová-Feňovčíková establish upper bounds for the total vertex (edge) irregularity strength and total irregularity strength for disjoint union of arbitrary graphs.
Naeem and Siddiqui determined the total irregularity strength of disjoint union of isomorphic copies of the generalized Petersen graph. In Yang, Siddiqui, Ibrahim, Ahmad, and Ahmad determined the exact value of the total irregularity strength of three planar graphs. In Salama introduced three kinds of snake graphs and investigated their total irregularity strength. Ibrahim, Khan, Asim, and Waseem determined the exact value of the total irregularity strength of cubic graphs. Tilukay, Tomasouw, and Rumlawang determined the total irregularity strength of complete graphs and complete bipartite graphs.
Jendroľ, Miškul, and Soták (see also ) proved: tes(K5) = 5; for n ≥6, tes(Kn)= ⌈(n2 −n + 4)/6⌉; and that tes(Km,n)= ⌈(mn + 2)/3⌉. They conjecture that for any graph G other than K5, tes(G) = max{⌈(∆(G) + 1)/2⌉, ⌈(|E| + 2)/3⌉}. Ivančo and Jendroľ proved that this conjecture is true for all trees. Jendroľ, Mis̆kuf, and Soták prove the conjecture for complete graphs and complete bipartite graphs. The conjecture has been proven for the categorical product of two paths , the categorical product of a cycle and a path , the categorical product of two cycles , the Cartesian product of a cycle and a path , the subdivision of a star , and the toroidal polyhexes . In Ahmad, Siddiqui, and Afzal proved the conjecture is true the electronic journal of combinatorics (2023), #DS6 371 for graphs obtained by starting with m vertex disjoint copies of Pn (m, n ≥2) arranged in m horizontal rows with the jth vertex of row i + 1 directly below the jth vertex row i for 1 = 1, 2, . . . , m−1 and joining the jth vertex of row i to the j +1th vertex of row i+1 for 1 = 1, 2, . . . , m −1 and j = 1, 2, . . . , n −1 (the zigzag graph). Siddiqui, Ahmad, Nadeem, and Bashir proved the conjecture for the disjoint union of p isomorphic sun graphs (i. e., Cn ⊙K1) and the disjoint union of p sun graphs in which the orders of the n-cycles are consecutive integers. They pose as an open problem the determination of the total edge irregularity strength of disjoint union of any number of sun graphs. Brandt, Misškuf, and Rautenbach proved the conjecture for large graphs whose maximum degree is not too large relative to its order and size. In particular, using the probabilistic method they prove that if G(V, E) is a multigraph without loops and with nonzero maximum degree less than |E|/103p 8|V |, then tes(G) = (⌈|E| + 2)/3⌉. As corollaries they have: if G(V, E) satisfies |E| ≥3·103|V |3/2, then tes(G) = ⌈(|E|+2)/3⌉; if G(V, E) has minimum degree δ > 0 and maximum degree ∆such that ∆< δ p |V |/103 · 4 √ 2 then tes(G) = ⌈(|E| + 2)/3⌉; and for every positive integer ∆there is some n(∆) such that every graph G(V, E) without isolated vertices with |V | ≥n(∆) and maximum degree at most ∆satisfies tes(G) = ⌈(|E| + 2)/3⌉. Notice that this last result includes d-regular graphs of large order. They also prove that if G(V, E) has maximum degree ∆≥2|E|/3, then G has an edge irregular total k-labeling with k = ⌈(∆+ 1)/2⌉. Pfender proved the conjecture for graphs with at least 7 × 1010 edges and proved for graphs G(V, E) with ∆(G) ≤E(G)/4350 we have tes(G) = (⌈|E| + 2)/3⌉. Susanti and Haq determined the minimum k such that an odd staircase graph can be labeled by an edge irregular total k-labeling. Murhu Guru Packiam, Manimaran, and Thuraiswamy investigate how the addition of a new edge affects the total edge irregularity strength of a graph.
Laurence and Kathiresan determined the total edge irregular strength of path union of cycles. Sivakumar, Vidyanandini, Sreedevi, Nayak, and Bhoi determined the total edge irregularity strength of complete tripartite graphs.
In Jeyanthi and Sudha investigated the total edge irregularity strength of the disjoint union of wheels. They proved the following: tes(2Wn) = ⌈(4n + 2)/3⌉, n ≥3; for n ≥3 and p ≥3 the total edge irregularity strength of the disjoint union of p isomorphic wheels is ⌈(2(pn + 1)/3⌉; for n1 ≥3 and n2 = n1 + 1, tes(Wn1 ∪Wn2) = ⌈(2(n1 + n2 + 1)/3)⌉; for n1, n2, n3 where n1 ≥3 and ni+1 = n1 + i for i = 1, 2, tes(Wn1 ∪ Wn2∪Wn3) = ⌈(2(n1 + n2 + n3 + 1)/3)⌉; the total edge irregularity strength of the disjoint union of p ≥4 wheels Wn1 ∪Wn2 ∪· · · ∪Wnp with ni+1 = n1 + i and N = Pp j=1 nj + 1 is ⌈2N/3)⌉; and the total edge irregularity strength of p ≥3 disjoint union of wheels Wn1∪Wn2∪· · ·∪Wnp and N = Pp j=1 nj+1 is ⌈(2N/3⌉if max{ni | 1 ≤i ≤p} ≤1 2 ⌈(2N/3⌉.
In Mughal and Jamil determined the tight lower bounds for the total face irregular strength of type (1, 1, 0) of grids and wheels. Mughal, Jamil, and Virk new determined the exact tight lower bounds for the face irregular strength of generalized plane graphs under a ρ−labeling of class (α1, β1, γ1) for the vertex ρ-(1, 0, 0), the edge ρ-(0, 1, 0), the face ρ-(0, 0, 1), the vertex-face ρ-(1, 0, 1), the edge-face ρ-(0, 1, 1) and the ρ-(1, 1, 1) for grid graphs.
A k-labeling φ of a planar graph G is defined to be a face irregular k-labeling of type the electronic journal of combinatorics (2023), #DS6 372 (α, β, γ) if for every two different faces f and g of G we have wtφ(α,β,γ)(f) ̸= wtφ(α,β,γ)(g).
The face irregularity strength of type (α, β, γ) of a planar graph G, denoted fs(α, β, γ)(G), is the smallest integer k such that G admits a face irregular k-labeling of type (α, β, γ).
In Bača, Ovais, Semaničová-Feňovčíková, and Nengah Suparta, estimated the lower bounds and the upper bounds of the face irregularity strength of type (α, β, γ) for 2-connected planar graphs, where α, β, γ ∈{0, 1}, and determined the precise values of these parameters for ladders and fan graphs and proved the sharpness of the lower bounds.
The complete star of a graph G is the graph obtained from p+1 copies of the graph G by joining each vertex of G(0) with all corresponding vertices of all the copies G1), . . . , G(p).
Susanti, Khotimah, Hidayati, and Wahyujati determined the total edge irregularity strength of snowflake graphs, water bears graphs, the complete star of Cn, and two other families of ladder related graphs.
In , , , and Jeyanthi and Sudha determine the total edge irreg-ularity strength of fans, helms, closed helms, webs, flowers, gears, sun flowers, tadpoles, armed crowns, split graphs of cycles, split graph of paths, disjoint unions of isomorphic double wheels, and disjoint unions of consecutive non-isomorphic double wheels. Bokhary, Ali, and Maqbool determined the exact values for the total vertex and edge irreg-ularity strength of three wheel related families of graphs. Bokhary, Imran, and Ali determined the exact value of convex polytopes generated by prisms and antiprisms and determined their total vertex irregularity strength and total edge irregularity strength.
Ibrahim, Asif, Ahmad, and Siddiqui investigated the total irregularity strength of fans, helms, closed helms, webs, flower graphs, gears, and sunflowers. In Rat-nasari1, Wahyuni1, Susanti1, and Palupi1 determined the total edge irregularity strength of book graphs, double and triple book graphs, and gave the exact value of the total edge irregularity strength of quadruplet book graphs and quintuplet book graphs. Nurdini, Rosyida, and Mulyono determined the total edge irregularity strength of the chain graph that consists of tadpole graph T(6, n) on each block and constructed an algorithm to find it.
A generalized helm Hm n is a graph obtained by inserting m vertices in every pendent edge of a helm Hn. Indriati, Widodo, and Sugeng proved that for n ≥3, tes(H1 n) = ⌈(4n + 2)/3⌉, tes(H2 n) = ⌈(5n + 2)/3⌉, and tes(Hm n ) = ⌈((m + 3)n + 2))/3⌉for m ≡0 mod 3. They conjecture that tes(Hm n ) = ⌈((m + 3)n + 2))/3⌉, for all n ≥3 and m ≥10.
A cactus graph G is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is Cn with same n is called an n-uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then G is called n-uniform cactus chain graph. Rosyida and Indriati determined the tes of n-uniform cactus chain graphs of length r for some n ≡0 mod 3. They also investigated the tes of tadpole chain graphs. Rosyida and Indriati determined the total edge irregularity strength of the triangular cactus chain with length r and r + 1 pendant vertices (TCr+1 r ) is ⌈(4r + 3)/3⌉.
A para-squares cactus is a graph each of whose blocks is a 4-cycle and two or more squares share a cut vertex. The para-squares cactus chain graph is a cactus graph such that each the electronic journal of combinatorics (2023), #DS6 373 of the two squares has one common cut vertex. Rosyida and Indriati determined that total edge irregularity strength of para square cactus chain graph with length r and r pendant vertices (Qr r) is ⌈(5r + 2)/3⌉. Rosyida, Ningrum, Mulyono, and Indriati determined the total edge irregularity strength of general uniform cactus chain graphs having (n −2)r pendant vertices and length r.
Nurdin, Baskoro, Salman, and Gaos determine the total vertex ir-regularity strength of trees with no vertices of degree 2 or 3; improve some of the bounds given in ; and show that tvs(Pn) = ⌈(n + 1)/3)⌉.
They also made the following two conjectures.
(A) For every tree T, tvs(T) = max ⌈(n1 + 1)/2⌉, ⌈(n1 + n2 + 1)/3⌉, ⌈(n1 + n2 + n3 + 1)/4⌉.
(B) For ev-ery graph G with minimum degree δ and maximum degree ∆, tvs(G) = max ⌈(δ + Pi j=1 nj)/(i + 1)⌉: i ∈[δ, ∆], where nj denotes the number of vertices of degree j. In a 2023 Susanto, Simanjuntak, and Baskoro disproved these conjectures by new giving infinite families of counterexamples.
In Nurdin, Salman, Gaos, and Baskoro prove that for t ≥2, tvs(tP1)= t; tvs(tP2)= t + 1; tvs(tP3)= t + 1; and for n ≥4, tvs(tPn)= ⌈(nt + 1)/3)⌉. Ahmad, Bača, and Bashir proved that for n ≥3 and t ≥1, tvs((n, t)-kite) = ⌈(n + t)/3⌉, where the (n, t)-kite is a cycle of length n with a t-edge path (the tail) attached to one vertex. In Guo, Chen, Wang, and Yao give the total vertex irregularity strength of certain complete m-partite graphs. In Susilawati, Baskaro, and Simanjuntak determined the total vertex irregularity strength of a particular subdivision of any tree.
In , , and Susilawati, Baskoro and Simanjuntak determined the total vertex irregularity strength for trees with maximum degree four or five, for subdivision of several classes of caterpillars, for the subdivision of fire crackers, and for the subdivision of an amalgamation of stars. In they studied the vertex irregularity strength for trees having many vertices of degree 2. Nurdin, Baskoro, Salman and Gaos determined the total vertex irregularity strength for firecrackers and banana trees.
Anholcer, Kalkowski, and Przybylo prove that for every graph with δ(G) > 0, tvs(G)≤⌈3n/δ⌉+1. Majerski and Przybylo prove that the total vertex irregularity strength of graphs with n vertices and minimum degree δ ≥n0.5ln n is bounded from above by (2 + o(1))n/δ + 4.
Their proof employs a random ordering of the vertices generated by order statistics.
Anholcer, Karonński, and Pfender prove that for every forest F with no vertices of degree 2 and no isolated vertices tvs(F)= ⌈(n1 + 1)/2⌉, where n1 is the number of vertices in F of degree 1. They also prove that for every forest with no isolated vertices and at most one vertex of degree 2, tvs(F) = ⌈(n1 + 1)/2⌉.
Anholcer and Palmer determined the total vertex irregularity strength Ck n, which is a generalization of the circulant graphs Cn(1, 2, . . . , k). They prove that for k ≥2 and n ≥2k + 1, tvs(Ck n = ⌈(n + 2k)/(2k + 1)⌉. Przybylo obtained a variety of upper bounds for the total irregularity strength of graphs as a function of the order and minimum degree of the graph.
In Tong, Lin, Yang, and Wang give the exact values of the total edge irreg-ularity strength and total vertex irregularity strength of the toroidal grid Cm × Cn.
In Siddiqui, Miller, and Ryan determine the exact values of the total edge ir-the electronic journal of combinatorics (2023), #DS6 374 regularity strength of octagonal grid graph. In Ahmad, Bača, and Siddiqui gave the exact value of the total edge and total vertex irregularity strength for disjoint union of prisms and for disjoint union of cycles. In Ahmad, Bača, and Numan showed that tes(Sm j=1 Fnj) = 1 + Pm j=1 nj and tvs(Sm j=1 Fnj) = ⌈(2 + 2 Pm j=1 nj)/3⌉, where Sm j=1 Fnj denotes the disjoint union of friendship graphs. Chunling, Xiaohui, Yuansheng, and Lip-ing, showed tvs(Kp) = 2 (p ≥2) and for the generalized Petersen graph P(n, k) they proved tvs(P(n, k)) = ⌈n/2⌉+ 1 if k ≤n/2 and tvs(P(n, n/2))= n/2 + 1. They also obtained the exact values for the total vertex strengths for ladders, Möbius ladders, and Knödel graphs. For graphs with no isolated vertices, Przybylo gave bounds for tvs(G) in terms of the order and minimum and maximum degrees of G. For d-regular (d > 0) graphs, Przybylo gave bounds for tvs(G) in terms d and the order of G.
Ahmad, Ahtsham, Imran, and Gaig determined the exact values of the total vertex irregularity strength for five families of cubic plane graphs. In Ahmad and Bača determine that the total edge-irregular strength of the categorical product of Cn and Pm where m ≥2, n ≥4 and n and m are even is ⌈(2n(m−1)+2)/3⌉. They leave the case where at least one of n and m is odd as an open problem. In and Ahmad, Bača, and Siddiqui determine the exact values of the total edge irregularity strength of the categorical product of two cycles, the total edge (vertex) irregularity strength for the disjoint union of prisms, and the total edge (vertex) irregularity strength for the disjoint union of cycles. In Ahmad, Awan, Javaid, and Slamin study the total vertex irregularity strength of flowers, helms, generalized friendship graphs, and web graphs.
Indriati, Widodo, Wijayanti, Sugeng, and Bača determine the exact value of the total edge irregularity strength of the generalized web graph W(n, m) and two families of related graphs. Ahmad, Bača, and Numan determined the exact values of the total vertex irregularity strength and the total edge irregularity strength of a disjoint union of friendship graphs. Bokhary, Ahmad, and Imran determined the exact value of the total vertex irregularity strength of Cartesian and categorical product of two paths. Koam and Ahmad determined the total vertex irregularity strength for all theta graphs and certain values of the total vertex irregularity strength of the centralized uniform theta graphs. They provide a conjecture for the lower bound of total vertex irregularity strength of the centralized uniform theta graphs. In Bokhary and Faheem proved the conjecture of Bokhary, Ahmam, and Imran that the tvs(Pm□Pn) = ⌊mn+2 5 ⌋for m, n ≥2 for 5 ≤m ≤10 and n ≥1. the state graph for Tower of Hanoi problems with three towers. Farida and Indriati determined the total edge irregularity strength of the state graph for Tower of Hanoi problems with three towers.
In Nurdin, Salman, and Baskoro determine the total edge-irregular strength of the following graphs: for any integers m ≥2, n ≥2, tes(Pm ⊙Pn)= ⌈(2mn + 1)/3⌉; for any integers m ≥2, n ≥3, tes(Pm ⊙Cn)= ⌈((2n + 1)m + 1)/3⌉; for any integers m ≥2, n ≥2, tes(Pm ⊙K1,n)= ⌈(2m(n + 1) + 1)/3⌉; for any integers m ≥2 and n ≥3, tes(Pm ⊙Gn)= ⌈(m(5n + 2) + 1)/3⌉where Gn is the gear graph obtained from the wheel Wn by subdividing every edge on the n-cycle of the wheel; for any integers m ≥2, n ≥2, tes(Pm ⊙Fn)= ⌈m(5n + 2) + 1⌉, where Fn is the friendship graph obtained from W2m by subdividing every other rim edge; for any integers m ≥2 and n ≥3; and the electronic journal of combinatorics (2023), #DS6 375 tes(Pm ⊙Wn)= ⌈((3n + 2)m + 1)/3⌉.
In , , and Rajasingh, Rajan, Teresa Arockiamary, and Quadras provide the total edge irregularity strengths of honeycomb mesh networks, hexagonal networks, butterfly networks, benes networks, and series compositions of uniform theta graphs. Ratnasari and Susanti determined the exact value of the total edge irreg-ularity strength of triangular ladders, diagonal ladders, and circular triangular ladders.
In Nurdin, Baskoro, Salman, and Gaos proved: the total vertex-irregular strength of the complete k-ary tree (k ≥2) with depth d ≥1 is ⌈(kd + 1)/2⌉and the total vertex-irregular strength of the subdivision of K1,n for n ≥3 is ⌈(n+1)/3⌉. They also determined that if G is isomorphic to the caterpillar obtained by starting with Pm and m copies of Pn denoted by Pn,1, Pn,2, . . . , Pn,m, where m ≥2, n ≥2, then joining the i-th vertex of Pm to an end vertex of the path Pn,i, tvs(G)= ⌈(mn + 3)/3⌉. They conjectured that the total vertex irregularity strength of any tree T is determined only by the number of vertices of degrees 1, 2 and 3 in T. This conjecture was confirmed by Susilawati, Baskoro, and Simanjuntak by considering all trees with maximum degree five. They also characterized all such trees having the total vertex irregularity strength either t1, t2, or t3, where ti = ⌈(1 + Pi j=1 nj)/(i + 1)⌉and ni is the number of vertices of degree i.
Ahmad and Bača proved tvs(Jn,2)= ⌈(n+1)/2)⌉(n ≥4) and conjectured that for n ≥3 and m ≥3, tvs(Jn,m)= max{⌈(n(m −1) + 2)/3⌉, ⌈(nm + 2)/4⌉}. They also proved that for the circulant graph (see §5.1 for the definition) Cn(1, 2), n ≥5, tvs(Cn(1, 2))= ⌈(n+4)/5⌉. They conjecture that for the circulant graph Cn(a1, a2, . . . , am) with degree r at least 5 and n ≥5, 1 ≤ai ≤⌊n/2⌋, tvs(Cn(a1, a2, . . . , am)= ⌈(n + r)/(1 + r)⌉. Ahmad, Arshadb, and Iz̆aríková determine tes(G) where G is the generalized helm and tvs(G) where G is the generalized sun graph.
Slamin, Dafik, and Winnona consider the total vertex irregularity strengths of the disjoint union of isomorphic sun graphs, the disjoint union of consecutive nonisomor-phic sun graphs, tvs(∪t i=1Si+2), and disjoint union of any two nonisomorphic sun graphs.
(Recall Sn = Cn⊙K1.) Rajasingh and Annamma determine the total vertex irregu-larity strength of 1-fault tolerant Hamiltonian graphs CH(n), H(n), and W(m). Indriati, Widodo, Wijayanti, Sugeng, Bača, and Semaničová-Feňovčíková determine the ex-act value of the total vertex irregularity strength for generalized helm graphs and for prisms with outer pendent edges. In Asim and Hasni provided an upper bound for es(Kn) that is far better than the previously known upper bound.
In Ahmad shows that the total vertex irregularity strength of the antiprism graph An (n ≥3) is ⌈(2n + 4)/5⌉(see §5.7 for the definition) and gives the vertex irregularity strength of three other families convex polytope graphs. Al-Mushayt, Arshad, and Sid-diqui determined an exact value of the total vertex irregularity strength of some convex polytope graphs. Ahmad, Baskoro, and Imran determined the exact value of the total vertex irregularity strength of disjoint union of helm graphs.
For n ≥3, m ≥2 Jeyanthi and Sudha determine the total vertex irregularity strength of Pn ⊙K1, Pn ⊙K2, Cn ⊙K2, Ln ⊙K1, P2 ⊙Cn, Pn ⊙Km, (Cn × P2) ⊙K1, and Cn ⊙Km. In they determine the total vertex irregularity strength for the graph obtained from a cycle by identifying the endpoint of a path and the vertex of the electronic journal of combinatorics (2023), #DS6 376 a cycle, Cn ⊙Pm, the split graph of a cycle, and split graph of a path. In they determine the total vertex irregularity strength for quadrilateral snakes, sunflowers, double wheels, triangular books, quadrilateral books, and graphs obtained from the wheel Wn and attaching n pendent edges to the center. In Jeyanthi and Sudha determined the total irregularity strength of the n-crossed prism, m copies of crossed prism, necklace and m copies of necklace graph and that these graphs admit totally irregular total k-labeling.
Tilukay, Tomasouw, Rumlawang, and Salman in proved that Kn and Kn,n are both totally irregular total graphs with their ts equal to their tes. Tilukay, Taihuttu, Salman, Rumlawang, and Leleury proved that Km,n is a totally irregular total graph for any positive integer m and n.
A total edge Fibonacci irregular labeling f : V (G) ∪E(G) →{1, 2, . . . , k} of a graph G is a labeling of vertices and edges of G in such a way that for distinct edges xy and x′y′ their weights f(x) + f(xy) + f(y) and f(x′) + f(x′y′) + f(y′) and are distinct Fibonacci numbers. A graph that has a total edge Fibonacci irregular labeling is called a total edge Fibonacci irregular graph. Karthikeyan, Navanaeethakrishnan, and Sridevi proved that stars, bistars Bn,n (n ≥2), and two particular families of star related and bistar related graphs are total edge Fibonacci irregular graphs. The total edge Fibonacci irregularity strength, tefs(G) is the minimum k for which G has total edge Fibonacci irregular labeling. Amutha and Uma Devi determined the exact values of the total edge Fibonacci irregularity strength of fans, double fans, umbrella, and wheels.
In Agustin, Dafik, Marsidi, and Albirri introduced a natural extension of the notation of the total H-irregularity strength of graphs by considering the evaluation of the weight that is not only for each edge but also the weight on each subgraph H of G. They say a total α-labeling is a total H-irregular α-labeling of the graph G if for a subgroup H of G, the total H-weights W(H) = P v∈V (H) f(v) + P e∈E(H) f(e) are distinct. The minimum α for which the graph G has a total H-irregular α-labeling is called the total H-irregularity strength of G, denoted by tHs(G). They study the tHs of shackles and amalgamations of any graphs and their bounds. In Ashraf, Bača, Semaničová-Feňovčíková, and Siddiqui determined the exact value of the total (vertex, edge) H-irregularity strengths for the ladders and fan graphs.
The notion of an irregular labeling of an Abelian group Γ was introduced Anholcer, Cichacz and Milanič in . They defined a Γ-irregular labeling of a graph G with no isolated vertices as an assignment of elements of an Abelian group Γ to the edges of G in such a way that the sums of the weights of the edges at each vertex are distinct. The group irregularity strength of G, denoted sg(G), is the smallest integer s such that for every Abelian group Γ of order s there exists Γ-irregular labeling of G. They proved that if G is connected, then sg(G) = n + 2 when ∼ = K1,32q+1−2 for some integer q ≥1; sg(G) = n + 1 when n ≡2 (mod 4) and G ̸∼ = K1,32q+1−2 for any integer q ≥1; and sg(G) = n otherwise.
Moreover, Anholcer and Cichacz showed that if G is a graph of order n with no component of order less than 3 and with all the bipartite components having both color classes of even order. Then sg(G) = n if n ≡1 (mod 2); sg(G) = n + 1 if n ≡2 (mod 4); and sg(G) ≤n + 1 if n ≡0 (mod 4). In Anholcer, Cichacz, Jura, Marczyk presented some upper bound on group the irregularity strength for all graphs. Moreover, the electronic journal of combinatorics (2023), #DS6 377 they gave the exact values and bounds on sg(G) for disconnected graphs without a star as a component. Anholcer, Cichacz, and Przybyło proved that that sg(G) ≤2n.
They also considered locally irregular labelings where only sums of adjacent vertices are required to be distinct. For the corresponding graph invariant they proved the general upper bound: ∆(G) + col(G) −1 (where col(G) is the coloring number of G) in the case when the identity element is not used as an edge label, and a slightly worse one if additionally the identity is forbidden as the sum of labels around a vertex. In the both cases they provided a sharp upper bound for trees and a constant upper bound for planar graphs. In Cichacz and Krupińska gave a bound and exact values of the group irregular strength for graphs without small stars as components.
Marzuki, Salman, and Miller introduced a new irregular total k-labeling of a graph G called total irregular total k-labeling, denoted by ts(G), which is required to be at the same time both vertex and edge irregular. They gave an upper bound and a lower bound of ts(G); determined the total irregularity strength of cycles and paths; and proved ts(G) ≥ max{tes(G), tvs(G)}.
For n ≥3, Ramdani and Salman proved ts(Sn × P2) = n + 1; ts((Pn + P1) × P2) = ⌈(5n + 1)/3⌉, ts(Pn × P2) = n; and ts(Cn × P2) = n. In Ramdani, Salman, and Assiyatun prove that for a regular graph G ts(mG) ≤m(ts(G)) −⌊(m −1)/2⌊, ts(mCn) = ⌈(mn + 2)/3⌋for n ≡3 mod 3, and ts(m(Cn × P2) = mn + 1. In Ramdani, Salman, Assiyatun, Semaničová-Feňovčíková, and Bača estimate the upper bound of the total irregularity strength of graphs and determine the exact value of the total irregularity strength for three families of graphs. In Jeyanthi and Sudha determined the total irregularity strength of double fans DFn (n ≥3), double triangular snakes DTp (p ≥3), joint-wheel graphs WHn (n ≥3), and Pm + Km (m ≥3). In addition, they show that these graphs admit totally irregular total k-labeling and they determined the exact ts value for each. In Indriati, Widodo, Wijayanti, and Sugeng provided the total irregularity strength of some caterpillars.
In Tilukay, Salman, and Persulessy proved that fans, wheels, triangular books, friendship graphs, double fans DFn, (n ≥3), double triangular snakes DTp (p ≥3), joint-wheel graphs, Pm + Km (m ≥3), stars, double-stars, and caterpillars (see also ) are totally irregular total graphs. Ramdani, Salman, Assiyatun, Semaničová-Feňovčíková, and Bača proved that for any positive integer n ≥2, ts(Kn,n) = ⌈(n2 + 2)/3⌉. Tilukay, Taihuttu, Salman, Rumlawang, and Leleury proved that Km,n is a totally irregular total graph for any positive integer m and n. In Tilukay, Salman, and Persulessy proved that fans, wheels, triangular books, friendship graphs are totally irregular total graphs, double fans DFn, (n ≥3), double triangular snakes DTp, (p ≥3), joint-wheel graphs, Pm + Km (m ≥3), stars, double-stars, and caterpillars (see also ).
In Muthgu Guru Packiam defines a face irregular total k-labeling f from V ∪E∪F to {1, 2, . . . , k} of a 2-connected plane graph G(V, E, F) as a labeling of vertices and edges such that different faces have different weights. The minimum k for which a plane graph G has a face irregular total k-labeling is called total face irregularity strength of G and is denoted by tfs(G). He provides a bound on this parameter and the exact values for shell graphs and a family of planar graphs consisting of an even number of 5-sided faces and the electronic journal of combinatorics (2023), #DS6 378 one external infinite face. In Bača, Lascsḱová, Naseem, and Semaničová-Feňovčíková estimate the lower and upper bounds of the entire face irregularity strength for the disjoint union of multiple copies of a plane graph and prove the sharpness of the lower bound in two cases. Tilukay, Salman, Ilwaru, and Rumlawang estimated the bounds of tfs(G) and prove that the lower bound is sharp for cycles, books with m polygonal pages, and wheels.
For a graph G, Tanna, Ryan, and Semaničová-Feňovčíková define a k-labeling ρ as a labeling such that the edges of G are labeled with {1, 2, . . . , ke} and the vertices of G are labeled with {0, 2, . . . , 2kv}, where k = max{ke, 2kv}. The labeling ρ is called an edge irregular reflexive k-labeling if distinct edges have distinct weights, where the edge weight is defined as the sum of the label of that edge and the labels of its endpoints.
The smallest k for which such a labeling exists is called the reflexive edge strength of G.
The authors give exact values for the reflexive edge strength for prisms, wheels, baskets (graphs obtained by removing a spoke from a wheel), and fans. Guirao, Ahmad, Siddiqui, and Ibrahim investigated the exact value of the reflexive edge strength for disjoint union of s isomorphic copies of generalized Peterson graphs. Zhang, Ibrahim, ul Haq Bokhary, and Siddiqui provided exact value of the reflexive edge strength for disjoint union of gears and prisms. Bača, Irfan, Ryan, Semaničová-Feňovčíková, and Tanna determined the exact value of the reflexive edge strength for cycles, the Cartesian product of two cycles, and for join graphs of the path and cycle with 2K2. Wang, Khan, Ibrahim, Bonyah, Siddiqui, and Khalid determined the reflexive edge irregularity strength for the Cartesian product of a path and a cycle. In Yoong, Hasni, Lau, Asim, and Ahmad obtained the exact reflexive edge strength for antiprisms, two kinds of convex polytopes, and Cn ⊙Pm (n ≥3, m ≥2). Yoong, Hasni, Lau, and Irfan provided the exact value of the reflexive edge strength of three classes of plane graphs. In Basher examined two types of eight-sided and four-faced or six-sided and four-faced planar maps that have an edge irregular reflexive k-labeling. He gave the precise value of the reflexive edge strength for these two classes. In Bača, Irfan, Ryan, Semaničová-Feňovčíková, and Tanna provided the precise values of the reflexive edge strength for the generalized friendship graphs fn,m for n = 3, 4, 5, m ≥1 and made a conjecture of the value for the remaining cases. In they determined the exact value of the reflexive edge strength for cycles, the Cartesian product of two cycles, and for join graphs of the path and cycle with 2K2. Bača, Kovář, and Semaničová-Feňovčíková provided the precise value of the reflexive edge strength of Cn × Pm where n ≥3 and m ≥2. Basher determined the exact value of the reflexive edge strength of toroidal polyhexes. Mato and Wichianpaisarn provided the exact value of the reflexive edge strength of cycles plus one edge that new contains a triangle.
A vertex irregular reflexive k-labeling of a graph G is total k-labeling such that for every two distinct vertices the sums of labels of edges that are incident to each vertex and the vertex label itself are distinct. The reflexive vertex strength of a graph G is a minimum k such that G has a vertex irregular reflexive k-labeling. Agustin, Iman Utoyo, Dafik, and Venkatachalam determined the exact value of reflexive vertex strength of ladders and K2,n.
the electronic journal of combinatorics (2023), #DS6 379 The reflexive vertex strength of a graph G is the minimum k for which G has a ver-tex irregular reflexive k-labeling. In Agustin, Susilowati, Dafik, Cangul, and Mo- new hanapriya determine the exact value of the reflexive vertex strength of regular graphs and regular-like graphs.
A total k-labeling of a graph is a function fe from the edge set to {1, 2, . . . , ke} and a function fv from the vertex set to {0, 2, 4, . . . , 2kv}, where k = max ke, 2kv. A distance irregular reflexive k-labeling of graph G is a total k-labeling if for every two distinct vertices u and u′ of G, wt(u) ̸= wt(u′), where wt(u) = fv(u) + the sum of fe(uv) over all edges uv of G. The minimum k for graph G that has a distance irregular reflexive k-labeling is called the distance reflexive strength of the graph G. In Agustin, Dafik, Mohanapriya, Marsidi, and Cangul determine the lower bound of distance reflexive strength of any graph and the exact value of distance reflexive strength of paths, stars, and friendship graphs.
Recall that an edge-covering of G is a family of subgraphs H1, H2, . . . , Ht such that each edge of E(G) belongs to at least one of the subgraphs Hi, i = 1, 2, . . . , t. In this case we say that G admits an (H1, H2, . . . , Ht)-(edge) covering. If every subgraph Hi is isomorphic to a given graph H, we say that G admits an H-covering.
Motivated by the irregularity strength and the edge irregularity strength of a graph G, Ashraf, Bača, Kimáková, and Semaničová-Feňovčíková introduced two new parameters, edge (vertex) H-irregularity strengths, as the natural extensions of the parameters s(G) and es(G) as follows. Let G be a graph admitting an H-covering. For the subgraph H of G under the edge k-labeling β from E(G) to {1, 2, . . . , k}, the associated H-weight is defined as wtβ(H) = P β(e) over all edges e. An edge k-labeling β is called an H-irregular edge k-labeling of the graph G if for every two different subgraphs H′ and H′′ isomorphic to H we have wtβ(H′) ̸= wtβ(H′′). The edge H-irregularity strength of a graph G, denoted by ehs(G, H), is the smallest integer k such that G has an H-irregular edge k-labeling.
Ibrahim, Gulzar, Fazil, and Azhar compute the exact value of edge H-irregularity strength of hexagonal and octagonal grid graphs.
Ashraf et al. define the vertex H-irregularity strength of a graph G, vhs(G, H), analogously. They estimate the bounds of the parameters ehs(G, H) and vhs(G, H) and determine the exact values of the edge (vertex) H-irregularity strength for paths, ladders, and fans in order to prove the sharpness of lower bounds of these parameters. Nisviasari, Dafik, and Agustin determined the total H-irregularity strength of triangular ladders when H is a windmill or triangular ladder. For a graph G the minimum k for which G has an H-irregular reflexive k-labeling is called the reflexive H strength of graph G and is denoted by rHs(G) . In Marsidil, new Agustin, Dafik, Rahman, and Sullystiawati initiated the study the lower bound of the reflexive H strength of graphs and the reflexive H strength of flowers, Shack(Ct, v, n), and books, where H is isomorphic to C3, Ct, and C4, respectively. In Sullystiawati, new Marsidi, Putra, and Agustin determined lower bounds of the reflexive H-strength of paths, wheels, double fans, triangular ladders, and ladders.
In Ashraf, Bača, Semaničová-Feňovčíková, and Saputro determined the ex-act value of the cycle-irregularity strength of ladders and fan graphs.
Ashraf, Bača, Lascsáková, and Semaničová-Feňovčíková estimated the bounds for the total H-irregularity strength of a graph and determined the exact values of the total H-irregularity the electronic journal of combinatorics (2023), #DS6 380 strength for paths ladders and fans. Ashrafa, Bača, Semaničová-Feňovčíková, and Shab-birc investigated the total (respectively, edge and vertex) G-irregularity strengths of the graphs that contains exactly n subgraphs isomorphic to G. Ahmad, Bača, and Semaničová-Feňovčíková determined the exact values of ehs(G, C4) for grids and gen-eralized prisms.
In Ahmad, Al-Mushayt, and Bača define a vertex k-labeling φ of a graph G from V (G) to {1, 2, . . . , k} to be edge irregular k-labeling if for every two distinct edges e and f, there is wφ(e) ̸= wφ(f), where the weight of an edge e = xy is wφ(xy) = φ(x) + φ(y).
The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). They estimated the bounds of the edge irregularity and determined its exact values for paths, cycles, stars, double stars and Pm × Pn. Tarawneh, Hasni, and Ahmad determined the exact value of the edge irregularity strength of the corona product of graphs with paths. Tarawneh, Hasni, and Ahmad determine the exact value of edge irregularity strength of corona graphs Cn ⊙mK1 (m ≥2). Ahmad determined the exact value of es(Cn ⊙K1). In Ahmad, Bača, and Nadeen determine the exact value of the edge irregularity strength for several classes of Toeplitz graphs. Tarawneh, Hasni, Siddiqui, and Asim determined the exact value of edge irregularity strength of disjoint union of zigzag graphs, grids, and generalized sun graphs. Ahmed, Omar, and Bača obtained estimations of the edge irregularity strength of graphs and determined the precise values for paths, stars, double stars, and the Cartesian product of two paths. Suparta and Suharta determined the edge irregularity strength of the chain graph mK3-path for m ≥3 (mod 4) and the chain graph C[C(m) n ] for n = 0 (mod 4) and provided bounds for the edge irregularity strength of Pm + Kn for m, n ≥3. In Zhang, Mehmood, Rehman, Hussain, and Zhang provided the exact values of the edge irregularity strengths of C4-snakes, complete m-partite graphs, and the middle graphs of paths and cycles.
A chain graph C[B1, B2, . . . , Bn] is a graph with blocks B1, B2, . . . , Bn such that Bi and Bi+1 have a common vertex in such a way that the block-cut vertex graph is a path.
Ahmad, Gupta, and Simanjuntak prove the following: es(C[C(n) 4 ]) = 2n + 1; if Hm is an mK3-path, then es(Hm) has lower bound ⌈3m+3 2 ⌉and upper bound 2m + 1; es(mK4-path) = 3m + 2; and es(K1,n + K1) = n + 2 for n ≥3. They obtained bounds for es(Pm + Kn) and determined that the edge irregularity strength of a graph obtained by joining the vertex of degree m in K1,m to each vertex in K1,n, and the vertex of degree n in K1,n to each vertex in K1,m is m+n+2. They posed the open problems of determining es(mK3-path), es(mKn-path) (m ≥2) and n ≥5, and es(Pm + Kn) for n ≥1 and m ≥7. Tarawneh, Hasni, and Asim determined the exact value of edge irregularity strength for disjoint union of a star graph and the subdivision of a star graph.
The strong product of graphs G1 and G2 has as vertices the pairs (x, y) where x ∈V (G1) and y ∈V (G2). The vertices (x1, y1) and (x2, y2) are adjacent if either x1x2 is an edge of G1 and y1 = y2 or if x1 = x2 and y1y2 is an edge of G2. For m, n ≥2 Ahmad, Bača, Bashir, Siddiqui proved that the total edge irregular strength of the strong product of Pm and Pn is ⌈4(mn+1)/3⌉−(m+n). Al-Mushayt determined the edge irregularity strength of Cartesian product of a star and P2 and a cycle and P2, and the strong product the electronic journal of combinatorics (2023), #DS6 381 of path Pn with P2. Conjectures for the exact value of K1,n × Pm and Cn × Pm are stated.
Bača and Siddiqui determine the exact value of the total edge irregularity strength of the strong product of any two cycles.
An edge e ∈G is called a total positive edge or total negative edge or total stable edge of G if tvs(G + e) > tvs(G) or tvs(G + e) < tvs(G) or tvs(G + e) = tvs(G), respectively.
If all edges of G are total stable (total negative) edges of G, then G is called a total stable (total negative) graph. Otherwise G is called a total mixed graph.
Muthu Guru Packiam and Kathiresan showed that K1,n n ≥4, and the disjoint union of t ≥2 copies of K3 are total negative graphs and that the disjoint union of t ≥2 copies of P3 is a total mixed graph.
For a simple graph G with no isolated edges and at most one isolated vertex Anholcer calls a labeling w : E(G) →{1, 2, . . . , m} product-irregular, if all product degrees pdG(v) = Q e∋v w(e) are distinct. Analogous to the notion of irregularity strength the goal is to find a product-irregular labeling that minimizes the maximum label. This minimum value is called the product irregularity strength of G and is denoted by ps(G). He provides bounds for the product irregularity strength of paths, cycles, Cartesian products of paths, and Cartesian products of cycles. In Anholcer gives the exact values of ps(G) for Km,n where 2 ≤m ≤n ≤(m + 2)(m + 1)/2, some families of forests including complete d-ary trees, and other graphs with d(G) = 1. Darda and Hujdurović proved that ps(X) ≤|V (X)| −1 for any graph X with more than 3 vertices and gave a connection between the product irregularity strength and the multidimensional multiplication table problem. Skowronek-Kaziów proves that for the complete graphs ps(Kn) = 3. In 2023 Bensmail, Hocquard, Lajou, and Sopena proved that the product version of the 1-2-3 conjecture, raised by Skowronek-Kazió in 2012, that for every connected graph with order at least 3, there is an assignment of the labels 1, 2, 3 to the edges in such a way that no two adjacent vertices are incident to the same product of labels.
In Abdo and Dimitrov introduced the total irregularity of a graph. For a graph G, they define irrt(G) = (1/2) P u,v∈V |dG(u)−dG(v)|, where dG(w) denotes the vertex degree of the vertex w. For G with n vertices they proved irrt(G) ≤(1/12)(2n3 −3n2 −2n + 3).
For a tree G with n vertices they prove irrt(G) ≤(n −1)(n −2) and equality holds if and only if G ≈Sn. You, Yang, and You determined the graph with the maximal total irregularity among all unicyclic graphs.
Inspired by the concept of distant chromatic numbers Przybylo calls a labeling f from the edges of a graph G to {1, 2, 3, . . . , k} r-distant irregular, if for every vertex v, the weights of the set of all vertices that are at distance less than or equal to r from v are pairwise distinct, where the weight of the vertex is the sum of the labels of the edges that are incident with that vertex. The minimum k for which there exists an r-distant irregular labeling of G is called r-distant irregularity strength of G and is denoted by sr(G). Muthu Guru Packiam, Manimaran, and Thuraiswamy proved the following: s1(Pn) = 2 for n = 3, 4, 5; s1(Pn) = 3 if n > 5; s1(Cn) = 3; s1(Km,n) = s(Km,n); s1(Fn) = s(Fn) = ⌈(n + 1)/3⌉for n > 2; s1(Km,n) = 3 when 1 < n/2 ≤m < n; s1(Pn × K2) = 3; s1(Cn × K2) = 3; s1(Km,n) = 3 when 1 < n/2 ≤m < n; and provide the exact value for s1(Pm ⊙Kn) for all m and n. They also prove that if G is d-regular with n vertices, then the electronic journal of combinatorics (2023), #DS6 382 s1(G) = s(G) ≤⌈n/2⌉+ 1 for d ≥n/2. Susanto, Wijaya, Purnama, and Slamin derived a new lower bound of distance irregularity strength for graphs with pendant vertices.
They also determined the distance irregularity strength of some families of disjoint unions of paths, suns, helms and friendships graphs. In Bensmail proved that determining the distant irregularity strengths are NP-hard problems.
In a preprint Wijayanti, Noor, Indriati, Alghofari, and Slamin defined a distance vertex irregular total k-labeling of a simple undirected graph G(V, E), as a function f : V (G) ∪E(G) →{1, 2, . . . , k} such that for every pair of distinct vertices u, v ∈V (G) the weights of u and v are distinct. The weight of vertex v ∈V (G) is the sum of the labels of vertices in neighborhood of v and the labels of all incident edges to v. The total distance vertex irregularity strength of G (denoted by tdis(G)) is the minimum k for which G has a distance vertex irregular total k-labeling. In Wijayanti et al.
generalized to distance D where D is any subset of {1, 2, . . . , diam(G)}. Wijayanti, Noor, Indriati, Alghofari determined the exact total distance vertex irregularity strength of fan and wheel graphs for D = {1}. In Wijayanti, Noor, Indriati, Alghofari, and Slamin determined the exact value of the total distance vertex irregularity strength of Km ⊙Kn, Cn ⊙K1, and Pn ⊙K1. Wijayanti, Wijayanti, and Surono provided new a lower bound and determined the value of total distance vertex irregularity strength of the hairy cycle Cn m and proved that for m = 2, 3, 4 and n ≥5 odd, the hairy cycle Cn m graphs admits an D-distance vertex irregular total k-labelings with total distance vertex irregularity strength tdis(C n m ) ≥⌈(mn + 1)/2⌉.
An inclusive distance vertex irregular labeling of a graph G is an assignment of elements of the set {1, 2, . . . , k} to the vertices of G such that the sums of numbers assigned to the closed neighborhoods of all vertices are distinct. The minimum number k for which there exists an inclusive distance vertex irregular labeling of G is denoted by c dis(G).
Bača, Semaničová-Feňovčíková, Slamin, and Sugeng establish a lower bound for the inclusive distance vertex irregularity strength for any graph and determine the exact value of this parameter for several families of graphs. Cichacz, Gömrlich, and Semaničová-Feňovčíková prove that for a simple graph G on n vertices in which no two vertices have the same closed neighborhood is c dis(G) ≤n2.
A mapping φ : V (G) →{1, 2, . . . , k} of a simple graph G is said to be an non-inclusive distance vertex irregular k-labeling of G if the sums of labels of vertices in the open neighborhood of every vertex are distinct. The minimum k for which G has a non-inclusive distance vertex irregular k-labeling is called a non-inclusivedistance irregularity strength and is denoted by dis(G)). Susanto, Wijaya, Slamin, and Semanic̆ová-Fen̆ovc̆íková developed new lower bounds for non-inclusive distance irregularity strength of graphs and proved that such bounds are sharp for some classes of graphs with pendant vertices. They also discussed some properties for non-inclusive distance irregularity strength of trees.
Susanto, Wijaya, Sudarsana, and Slamin studied non-inclusive and inclusive distance irregularity strength for the join product of two given graphs. They gave bounds on non-inclusive and inclusive distance irregularity strength for G+H and showed that in some conditions, the bounds are achievable. They determined the exact value of dis(G + K1) for any given graph G and developed an algorithm for calculating the upper bound the electronic journal of combinatorics (2023), #DS6 383 on inclusive distance irregularity strength of complete multipartite graphs.
Susanto, Betistiyan, Halikin, and Wijaya derived new lower bounds on inclusive distance irregularity strength for arbitrary graph and showed that the bounds are sharp for two stars 2K1,n and three stars 3K1,n. For a integer m ≥2 and non-trivial connected graph G of order p and size q, the m-shadow of a graph G, denoted by Dm(G), is the graph obtained by taking m copies of G, say G1, G2,…, Gm, and joining each vertex u in Gi, i = 1, 2, . . . , m −1, to the neighbors of the corresponding vertex u′ in Gi+1. In new Wijaya, Aulia, Halikin, and Kusbudiono showed that dis(Tn,2) = (n2 + 2)/2, where Tn,2 is a complete n-ary tree to level two.
For a graph G and an integer m ≥3, the closed m-shadow of a graph G, denoted by CDm(G), is a graph obtained from the m-shadow of G by joining an edge from every vertex u in G1 to the neighbors of the corresponding vertex u′ in Gm.
Inayah, Susanto and Semaničová-Feňovčíková proved the following: for every non-trivial connected graph G of order p and size q, and integers m, t with 2 ≤t ≤m−1, Dm(G) is super (a, d)-Dt(G)-antimagic for d ∈{0, 2, . . . , p + 3q} when p + q is even and for d ∈{1, 3, . . . , p + 3q} when p + q is odd; for every non-trivial connected graph G of order p and size q, and integers m, t with 2 ≤t ≤m −1 and gcd(m, t) = 1, CDm(G) is super (a, d)-Dt(G)-antimagic for d ∈{0, 2, . . . , p + q} when p + q is even and for d ∈{1, 3, . . . , p + q} when p + q is odd; for every non-trivial connected graph G of order p and size q, and for any integer m ≥3, CDm(G) is super (a, d)-Dm(G)-antimagic for d ∈{0, 2, . . . , 2q}.
A natural modification of an irregular labeling is a modular irregular labeling intro-duced by Muthugurupackiam and Ramya in 2018. They call an edge k-labeling φ : E(G) →{1, 2, . . . , k} of positive integers to the edges of a graph G of order n a modular irregular labeling of G if the weight function θ : V (G) →Zn defined by θ(v) = wtφ(v) = P φ(uv) over all u ∈N(v) is bijective.
The modular irregularity strength is defined as the minimum k for which G has a modular irregular labeling. If there is no such labeling for a graph then its modular irregularity strength is defined as ∞. They proved the modular irregularity of the tadpole graph and double-cycle graph.
Bača, Muthugurupackiam, Kathiresan, and Ramya in gave a lower bound of the modular irregularity strength, and the exact values for paths, cycles, stars, triangular graphs, and gear graphs are determined.
In Bača, Kimáková, Lascsáková, and Semaničová-Feňovčikovǎ determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs. Bačca, Imran, and Semaničovǎ-Feňovčcíkovǎ showed the existence of an irregular labeling scheme that proves the exact value of the irregularity strength of wheels. By modifying this irregular mapping in six cases, they obtained labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength. Ali, Bača, Lascsaková, Semaničová-Feňovčıková, ALoqaily, and Mlaiki obtained estimations on the modular total vertex irregularity strength, and gave the precise values of this invariant for certain graphs. In Sugeng, Barack, Hinding, and Simanjuntak constructed a modular ir-regular labeling and determined its modular irregularity strength of regular double-star graphs and friendship graphs. Haryeni, Awanis, Bača, and Semaničová-Feňovčikovǎ the electronic journal of combinatorics (2023), #DS6 384 estimated the bounds of the (modular) edge irregularity strength for Pn+Km and Cn+Km and determined the corresponding exact value of the (modular) edge irregularity strength for some fans and wheels in order to prove the sharpness of the presented bounds. In Sugeng, John, Lawrence, Anwar, Bača, and and Semaničová-Feňovčíková deter- new mined the exact values of the modular irregularity strength of rose graphs, daisy graphs and sunfower graphs.
7.13 Geometric Labelings If a and r are positive integers at least 2, we say a (p, q)-graph G is (a, r)-geometric if its vertices can be assigned distinct positive integers such that the value of the edges obtained as the product of the endpoints of each edge is {a, ar, ar2, . . . , arq−1}. Hegde has shown the following: no connected bipartite graph, except the star, is (a, a)-geometric where a is a prime number or square of a prime number; any connected (a, a)-geometric graph where a is a prime number or square of a prime number, is either a star or has a triangle; Ka,b, 2 ≤a ≤b is (k, k)-geometric if and only if k is neither a prime number nor the square of a prime number; a caterpillar is (k, k)-geometric if and only if k is neither a prime number nor the square of a prime number; Ka,b,1 is (k, k)-geometric for all integers k ≥2; C4t is (a, a)-geometric if and only if a is neither a prime number nor the square of a prime number; for any positive integers t and r ≥2, C4t+1 is (r2t, r)-geometric; for any positive integer t, C4t+2 is not geometric for any values of a and r; and for any positive integers t and r ≥2, C4t+3 is (r2t+1, r)-geometric. Hegde has also shown that every Tp-tree and the subdivision graph of every Tp-tree are (a, r)-geometric for some values of a and r (see Section 3.2 for the definition of a Tp-tree). He conjectures that all trees are (a, r)-geometric for some values of a and r.
Hegde and Shankaran prove: a graph with an α-labeling (see §3.1 for the definition) where m is the fixed integer that is between the endpoints of each edge has an (am+1, a)-geometric for any a > 1; for any integers m and n both greater than 1 and m odd, mPn is (ar, a)-geometric where r = (mn + 3)/2 if n is odd and (ar, a)-geometric where r = (m(n + 1) + 3)/2 if n is even; for positive integers k > 1, d ≥1, and odd n, the generalized closed helm (see §5.3 for the definition) CH(t, n) is (kr, kd)-geometric where r = (n −1)d/2; for positive integers k > 1, d ≥1, and odd n, the generalized web graph (see §5.3 for the definition) W(t, n) is (kr, a)-geometric where a = kd and r = (n −1)d/2; for positive integers k > 1, d ≥1, the generalized n-crown (Pm × K3) ⊙K1,n is (a, a)-geometric where a = kd; and n = 2r + 1, Cn ⊙P3 is (kr, k)-geometric.
If a and r are positive integers and r is at least 2 Arumugan, Germina, and Anadavally say a (p, q)-graph G is additively (a, r)-geometric if its vertices can be assigned distinct integers such that the value of the edges obtained as the sum of the endpoints of each edge is {a, ar, ar2, . . . , arq−1}. In the case that the vertex labels are nonnegative integers the labeling is called additively (a, r)∗-geometric. They prove: for all a and r every tree is additively (a, r)∗-geometric; a connected additively (a, r)-geometric graph is either a tree or unicyclic graph with the cycle having odd size; if G is a connected unicyclic graph and not a cycle, then G is additively (a, r)-geometric if and only if either a is even the electronic journal of combinatorics (2023), #DS6 385 or a is odd and r is even; connected unicyclic graphs are not additively (a, r)∗-geometric; if a disconnected graph is additively (a, r)-geometric, then each component is a tree or a unicyclic graph with an odd cycle; and for all even a at least 4, every disconnected graph for which every component is a tree or unicyclic with an odd cycle has an additively (a, r)-geometric labeling.
Vijayakumar calls a graph G (not necessarily finite) arithmetic if its vertices can be assigned distinct natural numbers such that the value of the edges obtained as the sum of the endpoints of each edge is an arithmetic progression. He proves and that a graph is arithmetic if and only if it is (a, r)-geometric for some a and r.
7.14 Strongly Multiplicative Graphs Beineke and Hegde call a graph with p vertices strongly multiplicative if the vertices of G can be labeled with distinct integers 1, 2, . . . , p such that the labels induced on the edges by the product of the end vertices are distinct. They prove the following graphs are strongly multiplicative: trees; cycles; wheels; Kn if and only if n ≤5; Kr,r if and only if r ≤4; and Pm × Pn. They then consider the maximum number of edges a strongly multiplicative graph on n vertices can have. Denoting this number by λ(n), they show: λ(4r) ≤6r2; λ(4r + 1) ≤6r2 + 4r; λ(4r + 2) ≤6r2 + 6r + 1; and λ(4r + 3) ≤6r2 + 10r + 3.
Adiga, Ramaswamy, and Somashekara give the bound λ(n) ≤n(n + 1)/2 + n −2 − ⌊(n + 2)/4⌋−Pn i=2 i/p(i) where p(i) is the smallest prime dividing i. For large values of n this is a better upper bound for λ(n) than the one given by Beineke and Hegde. It remains an open problem to find a nontrivial lower bound for λ(n).
Seoud and Zid prove the following graphs are strongly multiplicative: wheels; rKn for all r and n at most 5; rKn for r ≥2 and n = 6 or 7; rKn for r ≥3 and n = 8 or 9; K4,r for all r; and the corona of Pn and Km for all n and 2 ≤m ≤8. In Seoud and Mahran give some necessary conditions for a graph to be strongly multiplicative.
In Kanani and Chhaya and prove the following graphs are strongly multi-plicative: the total graph, splitting graph, and shadow graph of paths; triangular snakes; splitting graphs of stars and bistars, the degree splitting graph of the bistars Bn,n, and restricted square graph B2 m,n. In and Kanani and Chhaya prove the following graphs are strongly multiplicative: helms, flowers, fans, friendship graphs, bistars, gears, double triangular snakes, double fans, double wheels, snakes, double alternate quadrilat-eral snakes, double quadrilateral snakes, braid graphs, and triangular ladders.
Germina and Ajitha (see also ) prove that K2 + Kt, quadrilateral snakes, Petersen graphs, ladders, and unicyclic graphs are strongly multiplicative. Acharya, Ger-mina, and Ajitha have shown that C(n) k (see §2.2 for the definition) is strongly mul-tiplicative and that every graph can be embedded as an induced subgraph of a strongly multiplicative graph. Germina and Ajitha define a graph with q edges and a strongly multiplicative labeling to be hyper strongly multiplicative if the induced edge labels are {2, 3, . . . , q + 1}. They show that every hyper strongly multiplicative graph has exactly one nontrivial component that is either a star or has a triangle and every graph can be embedded as an induced subgraph of a hyper strongly multiplicative graph.
the electronic journal of combinatorics (2023), #DS6 386 Vaidya, Dani, Vihol, and Kanani prove that the arbitrary supersubdivisions of tree, Kmn, Pn × Pm, Cn ⊙Pm, and Cm n are strongly multiplicative. Vaidya and Kanani prove that the following graphs are strongly multiplicative: a cycle with one chord; a cycle with twin chords (that is, two chords that share an endpoint and with opposite endpoints that join two consecutive vertices of the cycle; the cycle Cn with three chords that form a triangle and whose edges are the edges of two 3-cycles and a n −3-cycle.
duplication of an vertex in cycle (see §2.7 for the definition); and the graphs obtained from Cn by identifying of two vertices vi and vj where d(vi, vj) ≥3. In the same authors prove that the graph obtained by an arbitrary supersubdivision of path, a star, a cycle, and a tadpole (that is, a cycle with a path appended to a vertex of the cycle.
Krawec calls a graph G on n edges modular multiplicative if the vertices of G can be labeled with distinct integers 0, 1, . . . , n −1 (with one exception if G is a tree) such that the labels induced on the edges by the product of the end vertices modulo n are distinct. He proves that every graph can be embedded as an induced subgraph of a modular multiplicative graph on prime number of edges. He also shows that if G is a modular multiplicative graph on prime number of edges p then for every integer k ≥2 there exist modular multiplicative graphs on pk and kp edges that contain G as a subgraph.
In the same paper, Krawec also calls a graph G on n edges k-modular multiplicative if the vertices of G can be labeled with distinct integers 0, 1, . . . , n + k −1 such that the labels induced on the edges by the product of the end vertices modulo n + k are distinct.
He proves that every graph is k-modular multiplicative for some k and also shows that if p = 2n + 1 is prime then the path on n edges is (n + 1)-modular multiplicative. He also shows that if p = 2n + 1 is prime then the cycle on n edges is (n + 1)-modular multiplicative if there does not exist α ∈{2, 3, . . . , n} such that α2 + α −1 ≡0 mod p.
He concludes with four open problems. In Krawec shows that every graph is a subgraph of a modular multiplicative graph. He also defines k-modular multiplicative graphs and proves that certain families of paths and cycles admit such a labeling.
In Nasir, Idrees, Sadiq, Farooq, Kanwal, and Imran show that the join of K1 and a triangular ladder TLn (n ≥3), umbrella graphs, and generalized Petersen graphs GP(n, k) for (n ≤3) and 1 ≤k ≤n/2, double combs, and sunflower planar graphs (obtained by appending one edge to each vertex of the rim of a wheel) admit strongly multiplicative labelings.
7.15 Line-graceful Labelings Gnanajothi has defined a concept similar to edge-graceful. She calls a graph with n vertices line-graceful if it is possible to label its edges with 0, 1, 2, . . . , n such that when each vertex is assigned the sum modulo n of all the edge labels incident with that vertex the resulting vertex labels are 0, 1, . . . , n −1. A necessary condition for the line-gracefulness of a graph is that its order is not congruent to 2 (mod 4). Among line-graceful graphs are (see [pp. 132–181] ) Pn if and only if n ̸≡2 (mod 4); Cn if and only if n ̸≡2 (mod 4); K1,n if and only if n ̸≡1 (mod 4); Pn ⊙K1 (combs) if and only if n is even; (Pn ⊙K1)⊙K1 if and only if n ̸≡2 (mod 4); (in general, if G has order n, G ⊙H is the graph obtained the electronic journal of combinatorics (2023), #DS6 387 by taking one copy of G and n copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H); mCn when mn is odd; Cn ⊙K1 (crowns) if and only if n is even; mC4 for all m; complete n-ary trees when n is even; K1,n ∪K1,n if and only if n is odd; odd cycles with a chord; even cycles with a tail; even cycles with a tail of length 1 and a chord; graphs consisting of two triangles having a common vertex and tails of equal length attached to a vertex other than the common one; the complete n-ary tree when n is even; trees for which exactly one vertex has even degree. She conjectures that all trees with p ̸≡2 (mod 4) vertices are line-graceful and proved this conjecture for p ≤9.
Gnanajothi has investigated the line-gracefulness of several graphs obtained from stars. In particular, the graph obtained from K1,4 by subdividing one spoke to form a path of even order (counting the center of the star) is line-graceful; the graph obtained from a star by inserting one vertex in a single spoke is line-graceful if and only if the star has p ̸≡2 (mod 4) vertices; the graph obtained from K1,n by replacing each spoke with a path of length m (counting the center vertex) is line-graceful in the following cases: n = 2; n = 3 and m ̸≡3 (mod 4); and m is even and mn + 1 ≡0 (mod 4).
Gnanajothi studied graphs obtained by joining disjoint graphs G and H with an edge. She proved such graphs are line-graceful in the following circumstances: G = H; G = Pn, H = Pm and m+n ̸≡0 (mod 4); and G = Pn ⊙K1, H = Pm ⊙K1 and m+n ̸≡0 (mod 4).
In and Vaidya and Kothari proved following graphs are line graceful: fans Fn for n ̸≡1 (mod 4); Wn for n ̸≡1 (mod 4); bistars Bn,n if and only if for n ≡1, 3 (mod 4); helms Hn for all n; S′(Pn) for n ≡0, 2 (mod 4); D2(Pn) for n ≡0, 2 (mod 4); T(Pn), M(Pn), alternate triangular snakes, and graphs obtained by duplication of each edge of Pn by a vertex are line graceful graphs.
7.16 k-sequential Labelings In 1981 Bange, Barkauskas, and Slater defined a k-sequential labeling f of a graph G(V, E) as one for which f is a bijection from V ∪E to {k, k +1, . . . , |V ∪E|+k −1} such that for each edge xy in E, f(xy) = |f(x) −f(y)|. This generalized the notion of simply sequential where k = 1 introduced by Slater.
Bange, Barkauskas, and Slater showed that cycles are 1-sequential and if G is 1-sequential, then G + K1 is graceful. Hegde and Shetty have shown that every Tp-tree (see §4.4 for the definition) is 1-sequential.
In , Slater proved: Kn is 1-sequential if and only if n ≤3; for n ≥2, Kn is not k-sequential for any k ≥2; and K1,n is k-sequential if and only if k divides n. Acharya and Hegde proved: if G is k-sequential, then k is at most the independence number of G; P2n is n-sequential for all n and P2n+1 is both n-sequential and (n + 1)-sequential for all n; Km,n is k-sequential for k = 1, m, and n; Km,n,1 is 1-sequential; and the join of any caterpillar and Kt is 1-sequential. Acharya showed that if G(E, V ) is an odd graph with |E| + |V | ≡1 or 2 (mod 4) when k is odd or |E| + |V | ≡2 or 3 (mod 4) when k is even, then G is not k-sequential. Acharya also observed that as a consequence of results of Bermond, Kotzig, and Turgeon we have: mK4 is not k-sequential for any the electronic journal of combinatorics (2023), #DS6 388 k when m is odd and mK2 is not k-sequential for any odd k when m ≡2 or 3 (mod 4) or for any even k when m ≡1 or 2 (mod 4). He further noted that Km,n is not k-sequential when k is even and m and n are odd, whereas Km,k is k-sequential for all k. Acharya points out that the following result of Slater’s for k = 1 linking k-graceful graphs and k-sequential graphs holds in general: A graph is k-sequential if and only if G + v has a k-graceful labeling f with f(v) = 0. Slater also proved that a k-sequential graph with p vertices and q > 0 edges must satisfy k ≤p −1. Hegde proved that every graph can be embedded as an induced subgraph of a simply sequential graph.
In Acharya conjectured that if G is a connected k-sequential graph of order p with k > ⌊p/2⌋, then k = p−1 and G = K1,p−1 and that, except for K1,p−1, every tree in which all vertices are odd is k-sequential for all odd positive integers k ≤p/2. In Hegde gave counterexamples for both of these conjectures.
In Hegde and Miller prove the following: for n > 1, Kn is k-sequentially additive if and only if (n, k) = (2, 1), (3, 1) or (3,2); K1,n is k-sequentially additive if and only if k divides n; caterpillars with bipartition sets of sizes m and n are k-sequentially additive for k = m and k = n; and if an odd-degree (p, q)-graph is k-sequentially additive, then (p+q)(2k+p+q−1) ≡0 (mod 4). As corollaries of the last result they observe that when m and n are odd and k is even Km,n is not k-sequentially additive and if an odd-degree tree is k-sequentially additive then k is odd.
In Seoud and Jaber proved the following graphs are 1-sequentially additive: graphs obtained by joining the centers of two identical stars with an edge; Sn ∪Sm if and only if nm is even; Cn ⊙Km; Pn ⊙Km; kP3; graphs obtained by joining the centers of k copies of P3 to each vertex in Km; and trees obtained from K by replacing each edge by a path of length 2 when n ≡0, 1 (mod 4). They also determined all 1-sequentially additive graphs of order 6.
7.17 IC-colorings For a subgraph H of a graph G with vertex set V and a coloring f from V to the natural numbers define fs(H) = Σf(v) over all v ∈H. The coloring f is called an IC-coloring if for any integer k between 1 and fs(G) there is a connected subgraph H of G such that fs(H) = k. The IC-index of a graph G, M(G), is max{fs| fs is an IC-coloring of G}.
Salehi, Lee, and Khatirinejad obtained the following: M(Kn) = 2n −1; for n ≥ 2, M(K1,n) = 2n + 2; if ∆is the maximum degree of a connected graph G, then M(G) ≥ 2∆+ 2; if ST(n; 3n) is the graph obtained by identifying the end points of n paths of length 3, then ST(n; 3n) is at least 3n + 3 (they conjecture that equality holds for n ≥4); for n ≥2, M(K2,n) = 3 · 2n + 1; M(Pn) ≥(2 + ⌊n/2⌋)(n −⌊n/2⌋) + ⌊n/2⌋−1; for m, n ≥2, the IC-index of the double star DS(m, n) is at least (2m−1 + 1)(2n−1 + 1) (they conjecture that equality holds); for n ≥3, n(n + 1)/2 ≤M(Cn) ≤n(n −1) + 1; and for n ≥3, 2n +2 ≤M(Wn) ≤2n +n(n−1)+1. They pose the following open problems: find the IC-index of the graph obtained by identifying the endpoints of n paths of length b; find the IC-index of the graph obtained by identifying the endpoints of n paths; and find the IC-index of Km,n. Shiue and Fu completed the partial results by Penrice the electronic journal of combinatorics (2023), #DS6 389 Salehi, Lee, and Khatirinejad by proving M(Km,n) = 3 · 2m+n−2 −2m−2 + 2 for any 2 ≤m ≤n.
7.18 Minimal k-rankings A k-ranking of a graph is a labeling of the vertices with the integers 1 to k inclusively such that any path between vertices of the same label contains a vertex of greater label.
The rank number of a graph G, χr(G), is the smallest possible number of labels in a ranking.
A k-ranking is minimal if no label can be replaced by a smaller label and still be a k-ranking. The concept of the rank number arose in the study of the design of very large scale integration (VLSI) layouts and parallel processing (see , and ). Ghoshal, Laskar, and Pillone were the first to investigate minimal k-rankings from a mathematical perspective. Laskar and Pillone proved that the decision problem corresponding to minimal k-rankings is NP-complete. It is HP-hard even for bipartite graphs . Bodlaender, Deogun, Jansen, Kloks, Kratsch, Müller, Tuza proved that the rank number of Pn is χr(Pn) = ⌊log2(n)⌋+ 1 and satisfies the recursion χr(Pn) = 1 + χr(P⌈(n−1)/2⌉) for n > 1. The following results are given in : χr(Sn) = 2; χr(Cn) = ⌊log2(n−1)⌋+2; χr(Wn) = ⌊log2(n−3)⌋+3(n > 3); χr(Kn) = n; the complete t-partite graph with n vertices has ranking number n+1 - the cardinality of the largest partite set; and a split graph with n vertices has ranking number n + 1 - the cardinality of the largest independent set (a split graph is a graph in which the vertices can be partitioned into a clique and an independent set.) Wang proved that for any graphs G and H χr(G + H) = min{|V (G)| + χr(H), |V (H) + χr(G)}.
In 2009 Novotny, Ortiz, and Narayan determined the rank number of P 2 n from the recursion χr(P 2 n) = 2+χr(P(⌈(n−2)/2⌉) for n > 2. They posed the problem of determin-ing χr(Pm × Pn) and χ(P k n). In 2009 and Alpert determined the rank numbers of P k n, Ck n, P2 × Cn, Km × Pn, P3 × Pn, Möbius ladders and found bounds for rank num-bers of general grid graphs Pm × Pn. About the same time as Alpert and independently, Chang, Kuo, and Lin determined the rank numbers of P k n, Ck n, P2 × Pn, P2 × Cn.
Chang et al. also determined the rank numbers of caterpillars and proved that for any graphs G and H χr(G[H]) = χr(H) + |V (H)|(χr(G) −1).
In 2010 Jacob, Narayan, Sergel, Richter, and Tran investigated k-rankings of paths and cycles with pendent paths of length 1 or 2. Among their results are: for any caterpillar G χr(Pn) ≤χr(G) ≤χr(Pn) + 1 and both cases occur; if 2m ≤n ≤2m+1 then for any graph G obtained by appending edges to an n-cycle we have m+2 ≤χr(G) ≤m+3 and both cases occur; if G is a lobster with spine Pn then χr(Pn) ≤χr(G) ≤χr(Pn) + 2 and all three cases occur; if G a graph obtained from the cycle Cn by appending paths of length 1 or 2 to any number of the vertices of the cycle then χr(Pn) ≤χ(G) ≤χ(Pn) + 2 and all three cases occur; and if G the graph obtained from the comb obtained from Pn by appending one path of length m to each vertex of Pn then χr(G) = χr(Pn)+χr(Pm+1)−1.
Sergel, Richter, Tran, Curran, Jacob, and Narayan investigated the rank number of a cycle Cn with pendent edges, which they denote by CCn, and call a caterpillar cycle.
They proved that χ(CCn) = χr(Cn)) or χ(CCn) = χr(Cn))+1 and showed that both cases the electronic journal of combinatorics (2023), #DS6 390 occur. A comb tree, denoted by C(n, m), is a tree that has a path Pn such that every vertex of Pn is adjacent to an end vertex of a path Pm. In the comb tree C(n, m) (n ≥3) there are 2 pendent paths Pm+2 and n −2 paths Pm+1. They proved χr(C(n, m)) = χr(Pm+1) −1.
They define a circular lobster as a graph where each vertex is either on a cycle Cn or at most distance two from a vertex on Cn.
They proved that if G is a lobster with longest path Pn, then χr(Pn) ≤χr(G) ≤χr(Pn) + 2 and determined the conditions under which each true case occurs. If G is circular lobster with cycle Cn, they showed that χr(Cn) ≤χr(G) ≤χr(Cn) + 2 and determined the conditions under which each true case occurs. An icicle graph In (n ≥3) has three pendent paths P2 and is comprised of a path Pn with vertices v1, v2, . . . , vn where a path Pi−1 is appended to vertex vi. They determine the rank number for icicle graphs.
Richter, Leven, Tran, Ek, Jacob, and Narayan define a reduction of a graph G as a graph G∗ S such that V (G∗ S) = V (G) \ S and, for vertices u and v, uv is an edge of G∗ S if and only if there exists a uv path in G with all internal vertices belonging to S. A vertex separating set of a connected graph G is a set of vertices whose removal disconnects G. They define a bent ladder BLn(a, b) as the union of ladders La and Lb (where Ln = Pn×P2) that are joined at a right angle with a single L2 so that n = a+b+2.
A staircase ladder SLn is a graph with n −1 subgraphs G1, G2, . . . , Gn−1 each of which is isomorphic to C4. (They are ladders with a maximum number of bends.) Richter et al. prove: χr(BLn(a, b)) = χr(Ln) −1 if n = 2k −1 and a ≡2 or 3 (mod 4) and is equal to χr(Ln) otherwise; χr(SLn) = χr(Ln+1) if n = 2k +2k−1 −2 for some k ≥3 and is equal to χr(Ln) otherwise; and for any ladder Ln with multiple bends, the rank number is either χr(Ln) or χr(Ln) + 1).
The arank number of a graph G is the maximum value of k such that G has a minimal k-ranking. Eyabi, Jacob, Laskar, Narayan, and Pillone determine the arank number of Kn × Kn, and investigated the arank number of Km × Kn.
7.19 Set Graceful and Set Sequential Graphs The notions of set graceful and set sequential graphs were introduced by Acharaya in 1983 . A graph is called set graceful if there is an assignment of nonempty subsets of a finite set to the vertices and edges of the graph such that the value given to each edge is the symmetric difference of the sets assigned to the endpoints of the edge, the assignment of sets to the vertices is injective, and the assignment to the edges is bijective. A graph is called set sequential if there is an assignment of nonempty subsets of a finite set to the vertices and edges of the graph such that the value given to each edge is the symmetric difference of the sets assigned to the endpoints of the edge and the the assignment of sets to the vertices and the edges is bijective. The following has been shown: Pn (n > 3) is not set graceful ; Cn is not set sequential ; Cn is set graceful if and only if n = 2m −1 and ; Kn is set graceful if and only if n = 2, 3 or 6 ; Kn (n ≥2) is set sequential if and only if n = 2 or 5 ; Ka,b is set sequential if and only if (a+1)(b+1) is a positive power of 2 ; a necessary condition for Ka,b,c to be set sequential is that a, b, and c cannot have the same parity ; K1,b,c is not set sequential the electronic journal of combinatorics (2023), #DS6 391 when b and c even ; K2,b,c is not set sequential when b and c are odd ; no theta graph is set graceful ; the complete nontrivial n-ary tree is set sequential if and only if n + 1 is a power of 2 and the number of levels is 1 ; a tree is set sequential if and only if it is set graceful ; the nontrivial plane triangular grid graph Gn is set graceful if and only if n = 2 ; every graph can be embedded as an induced subgraph of a connected set sequential graph ; every graph can be embedded as an induced subgraph of a connected set graceful graph , every planar graph can be embedded as an induced subgraph of a set sequential planar graph ; every tree can be embedded as an induced subgraph of a set sequential tree ; and every tree can be embedded as an induced subgraph of a set graceful tree . Hegde conjectures that no path is set sequential. Hegde’s conjecture that every complete bipartite graph that has a set graceful labeling is a star was proved by Vijayakumar .
Shahida and Sunitha prove that the concept of set-gracefulness is equivalent to topologically set-gracefulness in trees and almost all finite trees are not set-graceful. Using this they characterize topologically set-graceful stars and topologically set-graceful paths. In Acharya and Germina survey results on set-valuations of graphs and give open problems and conjectures.
Germina, Kumar, and Princy prove: if a (p, q)-graph is set-sequential with respect to a set with n elements, then the maximum degree of any vertex is 2n−1 −1; if G is set-sequential with respect to a set with n elements other than K5, then for every edge uv with d(u) = d(v) one has d(u) + d(v) < 2n−1 −1; K1,p is set-sequential if and only if p has the form 2n−1 −1 for some n ≥2; binary trees are not set-sequential; hypercubes Qn are not set-sequential for n > 1; wheels are not set-sequential; and uniform binary trees with an extra edge appended at the root are set-graceful and set graceful.
Vijayakumar and Gyri, Balister, and Schelp proved that if a complete bipartite graph G has a set-graceful labeling, then it is a star. Abhishek described a method for constructing a set-graceful bipartite graph of arbitrarily large order and size beginning with a set-graceful bipartite graph. Acharya, Germina, Princy, and Rao proved that K1,m,n is set-graceful if and only if m = 2s −1 and n = 2t −1 and almost all graphs are not set-graceful. In Abhishek surveys results on set-valued graphs. Many open problems and conjectures are included.
Acharya has shown: a connected set graceful graph with q edges and q+1 vertices is a tree of order p = 2m and for every positive integer m such a tree exists; if G is a connected set sequential graph, then G + K1 is set graceful; and if a graph with p vertices and q edges is set sequential, then p+q = 2m−1. Acharya, Germina, Princy, and Rao proved: if G is set graceful, then G ∪Kt is set sequential for some t; if G is a set graceful graph with n edges and n + 1 vertices, then G + Kt is set graceful if and only if m has the form 2t −1; Pn + Km is set graceful if n = 1 or 2 and m has the form 2t −1; K1,m,n is set graceful if and only if m has the form 2t −1 and n has the form 2s −1; P4 + Km is not set graceful when m has the form 2t −1 (t ≥1); K3,5 is not set graceful; if G is set graceful, then graph obtained from G by adding for each vertex v in G a new vertex v′ that is adjacent to every vertex adjacent to v is not set graceful; and K3,5 is not set graceful.
the electronic journal of combinatorics (2023), #DS6 392 Acharya, Germina, Abhishek, and Slater prove Cm is set-graceful if and only if m = (4n −1)/3; mK2 is set-sequential if and only if m = (4n −1)/3; and, for r + s = 2n−1 the bistar B(r, s) is set-sequential if and only if r and s are odd. They also prove that connected planar graphs with an even number of faces, regular polyhedrons, and cacti containing an odd number of cycles are not set-sequential.
Abhishek proved that if G is a set-sequential bipartite graph and H is 2k-set-sequential, then 4kG ∪H is set-sequential. As a corollary, he gets mP3 is set-sequential if and only if m = (16n −1)/5. Abhishek and Agustine characterized the set-sequential caterpillars of diameter four and give a necessary condition for a graph to be set-sequential.
Abhishek characterized the set-sequential caterpillars of diameter five. Golowich and Kim showed that all caterpillars T of diameter k such that k ≤18 or |V (T)| ≥2k−1 are set-sequential, where T has only odd-degree vertices and |V (T)| = 2n−1 for some n.
They also provided a new method of recursively constructing set-sequential trees.
In Mehra and Puneet introduce a topological integer additive set-labeling of signed graphs as follows. Let S = (V, E, s) be a signed graph with corresponding graph G = (V, E) and the signature function s. Here, G is an integer additive set-labeled graph having an injective function f : V (G) →P(X) −{∅} that produces another injective function gf : E(G) →P(X)−{∅, {0}} defined by gf(uv) = f(u)+f(v) for every edge uv, where X is the subset of non-negative integers, P(X) is its power set, and the signature function defined as s : E(G) →{+, −} is such that s(uv) = −1|f(u)+f(v)| for all edges uv.
If f(V (G)) ∪{∅} forms a topology on X then the signed graph S is called a topological integer additive set-labeled signed graph (T-IASL). They proved the following graphs have T-IASL labelings: paths, stars, double stars, tadpoles, and graphs obtained by identifying an end of a path with the center of a star.
7.20 Vertex Equitable Graphs Given a graph G with q edges and a labeling f from the vertices of G to the set {0, 1, 2, . . . , ⌈q/2⌉} define a labeling f ∗on the edges by f ∗(uv) = f(u) + f(v). If for all i and j and each vertex the number of vertices labeled with i and the number of ver-tices labeled with j differ by at most one and the edge labels induced by f ∗are 1, 2, . . . , q, Lourdusamy and Seenivasan call a f a vertex equitable labeling of G. They proved the following graphs are vertex equitable: paths, bistars, combs, n-cycles for n ≡0 or 3 (mod 4), K2,n, C t 3 for t ≥2, quadrilateral snakes, K2 + mK1, K1,n ∪K1,n+k if and only if 1 ≤k ≤3, ladders, arbitrary super divisions of paths, and n-cycles with n ≡0 or 3 (mod 4). They further proved that K1,n for n ≥4, Eulerian graphs with n edges where n ≡1 or 2 (mod 4), wheels, Kn for n > 3, triangular cacti with q ≡0 or 6 or 9 (mod 12), and graphs with p vertices and q edges, where q is even and p < ⌈q/2⌉+ 2 are not vertex equitable. Lourdusamy and Patrick prove that triangular ladders TLn, Ln ⊙mK1, Qn ⊙K1, TLn ⊙K1, and alternate triangular snakes A(Tn) are vertex equitable graphs. In Acharya, Jain, and Kansal introduced vertex equitable labelings of signed graphs and studied vertex equitable behavior of signed paths, signed stars, and signed complete bipartite graphs K2,n.
the electronic journal of combinatorics (2023), #DS6 393 Jeyanthi and Maheswari proved that the following graphs have vertex equitable labelings: the square of the bistar Bn,n; the splitting graph of the bistar Bn,n; C4-snakes; connected graphs for in which each block is a cycle of order divisible by 4 (they need not be the same order) and whose block-cut point graph is a path; Cm ⊙Pn; tadpoles; the one-point union of two cycles; and the graph obtained by starting friendship graphs, C(2) n1 , C(2) n2 , . . . , C(2) nk where each ni ≡0 (mod 4) and joining the center of C(2) ni to the center of C(2) i+1 with an edge for i = 1, 2, . . . , k −1. In Jeyanthi and Maheswari prove that Tp trees, bistars B(n, n + 1), Cn ⊙Km, P 2 n , tadpoles, certain classes of caterpillars, and T ⊙Kn where T is a Tp tree with an even number of vertices are vertex equitable. Jeyanthi and Maheswari gave vertex equitable labelings for graphs constructed from Tp trees by appending paths or cycles.
In Jeyanthi and Maheswari show a number of families of graphs have vertex eq-uitable labelings. Their results include: armed crowns Cm ⊙Pn, shadow graphs D2(K1,n); the graph Cm ∗Cn obtained by identifying a single vertex of a cycle graph Cm with a single vertex of a cycle graph Cn if and only if m + n ≡0, 3 (mod 4); for n ≡0 (mod 4) the graph obtained from m copies of Cn ∗Cn and Pm by joining each vertex of Pm with the cut vertex in one copy of Cn ∗Cn; and graphs obtained by duplicating an arbitrary vertex and an arbitrary edge of a cycle; the total graph of Pn; the splitting graph of Pn; and the fusion of two edges of Cn.
For a graph H with vertices v1, v2, . . . , vn and n copies of a graph G, H b o G is a graph obtained by identifying a vertex ui of the ith copy of G with a vertex vi of H for 1 ≤i ≤n. The graph H e o G is a graph obtained by joining a vertex ui of the ith copy of G with a vertex vi of H by an edge for 1 ≤i ≤n. Jeyanthi, Maheswari, and Laksmi prove that the graphs Lm ˆ o nC4, Lm ˜ o nC4, Cm ˜ o nC4, and Pm ˜ o nC4 are vertex equitable graphs. The graph S∗(G) is obtained from a graph G by replacing every edge e of G with K2,m (m ≥2) with the endpoints of e merged with the two vertices of the 2-vertices part of K2,m after removing the edge e from G. Jeyanthi, Maheswari, and Vijaya Laksmi prove the graphs S∗(Pn ·K1), S∗(B(n, n)), S∗(Pn ×P2), and S∗(Qn) of the quadrilateral snake are vertex equitable.
In Jeyanthi and Maheswari proved the double alternate triangular snake DA(Tn) obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to two new vertices vi and wi is vertex equitable; the double alternate quadrilateral snake DA(Qn) obtained from a path u1, u2, . . . , un by joining ui and ui+1 (alternatively) to two new vertices vi, xi and wi, yi respectively and then joining vi, wi and xi, yi is vertex equitable; and NQ(m) the nth quadrilateral snake obtained from the path u1, u2, . . . , um by joining ui, ui+1 with 2n new vertices vi j and wi j, 1 ≤i ≤m −1, 1 ≤j ≤n is vertex equitable.
Jeyanthi and Maheswari prove DA(Tn)⊙K1, DA(Tn)⊙2K1, DA(Tn), DA(Qn)⊙ K1, DA(Qn) ⊙2K1, and DA(Qn) are vertex equitable.
Jeyanthi, Maheswari, and Vijayalaksmi proved the following graphs are ver-tex equitable: jewel graphs Jn with vertex set {u, v, x, y, ui : 1 ≤i ≤n} and edge set {ux, uy, xy, xv, yv, uui, vui : 1 ≤i ≤n}; jelly fish graphs (JF)n with vertex set {u, v, ui, vj : 1 ≤i ≤n, 1 ≤j ≤n −2} and edge set {uui : 1 ≤i ≤n} ∪{vvj : 1 ≤ j ≤n −2} ∪{un−1un, vun, vun−1}; lobsters constructed from the path a1, a2, . . . , an with the electronic journal of combinatorics (2023), #DS6 394 vertices ai1 and ai2 adjacent to ai for 1 ≤i ≤n and pendent vertices a1 ij, a2 ij, . . . , ak ij joining aij for 1 ≤i ≤n and j = 1, 2; Ln ⊙Km; and the graph obtained from ladder a Ln and 2n copies of K1,m by identifying a non-central vertex of ith copy of K1,m with ith vertex of Ln.
Jeyanthi, Mahewari, and Vijaya Laksmi prove the following graphs are vertex equitable: graphs obtained by joining a vertex of a cycle to a degree 2 vertex of a comb (Pn ⊙K1) with an edge; path unions of quadrilateral snakes; cycle unions of n copies of mC4-snakes where n ≡0, 3 mod 4; the graphs obtained from a path u1, u2, . . . , um by joining the end points of each edge uiui+1 to 2n isolated vertices vi j, wi j for 1 ≤m−1, 1 ≤ j ≤n, where n is even (the nth quadrilateral snake).
Jeyanthi, Maheswari, and Vijaya Laksmi prove that subdivisions of double tri-angular snakes S(D(Tn)), subdivisions of double quadrilateral snakes S(D(Qn)), subdi-visions of double alternate triangular snakes S(DA(Tn)), subdivisions of double alternate quadrilateral snakes S(DA(Qn)), DA(Qm) ⊙nK1, and DA(Tm) ⊙nK1 admit vertex eq-uitable labelings.
The super subdivision graph S∗(G) of a graph G is the graph obtained from G by replacing every edge uv of G by K2,m (m may vary for each edge) and identifying u and v with the two vertices in K2,m that form the partite set with exactly two members.
Jeyanthi, Maheswari, and Vijayalaksmi prove that super subdivision graphs of Pn ⊙K1, bistars B(n, n), Pn × P2, and quadrilateral snakes are vertex equitable.
For a graph H with vertices v1, v2, . . . , vn and n copies of a graph G, H b o G is a graph obtained by identifying a vertex ui of the ith copy of G with a vertex vi of H for 1 ≤i ≤n. The graph H e o G is a graph obtained by joining a vertex ui of the ith copy of G with a vertex vi of H by an edge for 1 ≤i ≤n. Jeyanthi, Maheswari, and Laksmi prove that the graphs Lm ˆ o nC4, Lm ˜ o nC4, Cm ˜ o nC4 and Pm ˜ o nC4 are vertex equitable graphs.
For a graph G with p vertices and q edges and A = {1, 3, . . . , q} if q is odd or A = {1, 3, . . . , q + 1} if q is even, Jeyanthi, Maheswari and Vijaya Laksmi say a vertex labeling f from V (G) to A is an odd vertex equitable even labeling if the induced edge labeling f ∗defined by f ∗(uv) = f(u)+f(v) for all edges uv has the property that for all u and v in A the number of vertices labeled with u and the number of vertices labeled with v differ by at most 1 and the induced edge labels are 2, 4, . . . , 2q. A graph that admits odd vertex equitable even labeling is called an odd vertex equitable even graph. They show that the following graphs have odd vertex equitable even lableings: paths, graphs obtained by identifying an endpoint of Pm with each vertex of Pn, K1,n if and only if n = 1 or 2, K1,n ∪K1,n−2 (n ≥3), K2,n, Tp-trees, Cn when n ≡0 or 1 mod 4, quadrilateral snakes, ladders Ln, Ln ⊙K1, and arbitrary super subdivision of paths. They prove that if every edge of a graph G is an edge of a triangle, then G is not an odd vertex equitable even graph. As a corollary of this they get that the following are not odd vertex equitable even graphs: Kn (n ≥3), wheels, triangular snakes, double triangular snakes, triangular ladders, flower graphs, fans Pn ⊙K1 (n ≥2), double fans Pn ⊙K2, (n ≥2), friendship graphs C3 n, windmills Kn m (m > 3), K2 +mK1, B2 n,n, total graphs T(Pn), and composition graphs Pn[P2]. They also show that if G is a (p, q) graph with p ≤⌈q/2⌉+ 1, then G is the electronic journal of combinatorics (2023), #DS6 395 not an odd vertex equitable even graph. Jeyanthi and Maheswari proved that the subdivision of double triangular snakes and the subdivision of double quadrilateral snakes are odd vertex equitable even graphs.
Lourdusamy and Patrick proved that Pn ⊙mK1, the quadrilateral snake at-tached to each vertex of path Pn, the super splitting graph S∗(Pn ⊙K1), the super splitting graphs of ladders and the bistars Bn,n, B2 n,n, and the splitting S′(Bn,n) ad-mit even vertex equitable even labelings. Maheswari and Jeyanthi have shown the following graphs admit odd vertex equitable even labelings: duplicate graphs of lad-ders; duplicate graphs of the subdivision of ladders; duplicate graphs of a quadrilater-als; and P3 × Pn for odd n > 2. Lourdusamy, Wency, and Patrick prove that S(D(Qn)), S(D(Tn)), DA(Qm)⊙nK1, DA(Tm)⊙nK1, S(DA(Qn)) and S(DA(Tn)) are an even vertex equitable even graphs.
For graphs G1 and G2 that graph G1 ˆ OG2 is obtained from G1 and |V G1)| copies of G2 by identifying one vertex of ith copy of G2 with ith vertex of G1. Jeyanthi, Maheswari, and Vijayalakshmi proved the following graphs have odd vertex equitable even labelings: subdivision graphs of ladders, Lm ˆ OPn, Ln ⊙Km (m > 1), Cn if and only if n ≡0 or 1 (mod 4), K1,n+k ∪K1,n i if and only if k = 1, 2, and ⟨Ln ˆ OK1,m⟩.
Motivated by the concept of vertex equitable labeling first defined by Lourdusamy and Seenivasan in , Lourdusamy, Mary, and Patrick introduced the concept of even vertex equitable even labeling as follows. Let G be a graph with p vertices and q edges and A = {0, 2, 4, . . . , q+1} if q is odd or A = {0, 2, 4, . . . , q} if q is even. A graph G is said to be an even vertex equitable even labeling if there exists a vertex labeling f from V (G) to A that induces an edge labeling f defined by f ∗(uv) = f(u)+f(v) for all edges uv such that for all a and b in A, |vf(a) −vf(b)| ≤1 and the induced edge labels are 2, 4, . . . , 2q, where vf(a) is the number of vertices v with f(v) = a for a ∈A. A graph that admits even vertex equitable even labeling is called an even vertex equitable even graph. They proved that paths, combs, complete bipartite graphs, cycles, K2 + mK1, bistars, ladders, (Pn × P2) ⊙K1, and the subdivision graphs of ladders and bistars Bn,n admit an even vertex equitable even labeling. In Lourdusamy and Patrick proved that Cm ⊙Pn, C4n and C4n+3 with a quadrilateral snake attached to each vertex of the cycle, the graphs obtained by indentifying an edge of Cm and Cn, and the graphs obtained by duplicating an arbitrary vertex and edge of a cycle admit an even vertex equitable even labeling.
Lourdusamy, Shobana Mary, and Patrick proved P 2 n, S(Pn ⊙K1), S′(Pn), T(Pn), graphs obtained by duplication of an edge of a path, quadrilateral snakes, D(Qn), A(Tn), and DA(Tn) have even vertex equitable even labelings. Lourdusamy and Patrick proved that Pn ⊙mK1, the quadrilateral snake attached to each vertex of path Pn, the super splitting graph S∗(Pn ⊙K1), the super splitting graphs of ladders and the bistars Bn,n, B2 n,n, and the splitting S′(Bn,n) admit even vertex equitable even labelings.
7.21 Representations of Graphs modulo n In 1989 Erdős and Evans defined a representation modulo n of a graph G with vertices v1, v2, . . . , vr as a set {a1, . . . , ar} of distinct, nonnegative integers each less than the electronic journal of combinatorics (2023), #DS6 396 n satisfying gcd(ai −aj, n) = 1 if and only if vi is adjacent to vj. They proved that every finite graph can be represented modulo some positive integer. The representation number, Rep(G), is smallest such integer.
Obviously the representation number of a graph is prime if and only if a graph is complete. Evans, Fricke, Maneri, McKee, and Perkel have shown that a graph is representable modulo a product of a pair of distinct primes if and only if the graph does not contain an induced subgraph isomorphic to K2 ∪2K1, K3 ∪K1, or the complement of a chordless cycle of length at least five.
Nešetřil and Pultr showed that every graph can be represented modulo a product of some set of distinct primes. Evans et al. proved that if G is representable modulo n and p is a prime divisor of n, then p ≥χ(G). Evans, Isaak, and Narayan determined representation numbers for specific families as follows (here we use qi to denote the ith prime and for any prime pi we use pi+1, pi+2, . . . , pi+k to denote the next k primes larger than pi): Rep(Pn) = 2 · 3 · · · · · q⌈log2(n−1)⌉; Rep(C4) = 4 and for n ≥3, Rep(C2n) = 2 · 3 · · · · · q⌈log2(n−1)⌉+1; Rep(C5) = 3 · 5 · 7 = 105 and for n ≥4 and not a power of 2, Rep(C2n+1) = 3 · 5 · · · · · q⌈log2n⌉+1; if m ≥n ≥3, then Rep(Km −Pn) = pipi+1 where pi is the smallest prime greater than or equal to m −n + ⌈n/2⌉; if m ≥n ≥4, and pi is the smallest prime greater than or equal to m −n + ⌈n/2⌉, then Rep(Km −Cn) = qiqi+1 if n is even and Rep(Km −Cn) = qiqi+1qi+2 if n is odd; if n ≤m −1, then Rep(Km −K1,n) = psps+1 · · · ps+n−1 where ps is the smallest prime greater than or equal to m −1; Rep(Km) is the smallest prime greater than or equal to m; Rep(nK2) = 2 · 3 · · · · · q⌈log2n⌉+1; if n, m ≥2, then Rep(nKm) = pipi+1 · · · pi+m−1, where pi is the smallest prime satisfying pi ≥m, if and only if there exists a set of n −1 mutually orthogonal Latin squares of order m; Rep(mK1) = 2m; and if t ≤(m −1)!, then Rep(Km + tK1) = psps+1 · · · ps+m−1 where ps is the smallest prime greater than or equal to m. Narayan proved that for r ≥3 the maximum value for Rep(G) over all graphs of order r is psps+1 · · · ps+r−2, where ps is the smallest prime that is greater than or equal to r −1. Agarwal and Lopez determined the representation numbers for complete graphs minus a set of stars.
Evans used matrices over the additive group of a finite field to obtain various bounds for the representation number of graphs of the form nKm.
Among them are Rep(4K3) = 3 · 5 · 7 · 11; Rep(7K5) = 5 · 7 · 11 · 13 · 17 · 19 · 23; and Rep((3q −1)/2)Kq) ≤ pqpq+1 · · · p(3q−1)/2) where q is a prime power with q ≡3 (mod 4), pq is the smallest prime greater than or equal to q, and the remaining terms are the next consecutive (3q −3)/2 primes; Rep(2q −2)Kq) ≤pqpq+1 · · · p(3q−3)/2) where q is a prime power with q ≡3 mod 4, and pq is the smallest prime greater than or equal to q; Rep((2q−2)Kq) ≤pqpq+1 · · · p2q−3.
In Narayan asked for the values of Rep(C2k+1) when k ≥3 and Rep(G) when G is a complete multipartite graph or a disjoint union of complete graphs.
He also asked about the behavior of the representation number for random graphs. Yahyaei and Katre gave upper and lower bounds for the representation number of a caterpillar and exact values in some cases.
Akhtar, Evans, and Pritikin characterized the representation number of K1,n using Euler’s phi function, and conjectured that this representation number is always of the form 2a or 2ap, where a ≥1 and p is a prime. They proved this conjecture for “small” the electronic journal of combinatorics (2023), #DS6 397 n and proved that for sufficiently large n, the representation number of K1,n is of the form 2a, 2ap, or 2apq, where a ≥1 and p and q are primes. In they showed that for sufficiently large n ≥m, rep(Km,n) = 2a, 3a, 2apb, or 2apq, where a, b ≥1 and p and q are primes; and for sufficiently large order, rep(Kn1,n2,...,nt = pa, paqb, or paqbu, where p, q, u are primes with p, q < u. Akhtar determined the representation number of graphs of the form K2 ∪nK1 (he uses the notation K2 + nK1) and studies their prime decom-positions. Using relations between representation modulo r and product representations, he determined representation number of binary trees and gave an improved lower bound for hypercubes.
7.22 Sequentially Additive Graphs Bange, Barkauskas, and Slater defined a k-sequentially additive labeling f of a graph G(V, E) to be a bijection from V ∪E to {k, . . . , k + |V ∪E| −1} such that for each edge xy, f(xy) = f(x) + f(y).
They proved: Kn is 1-sequentially additive if and only if n ≤3; C3n+1 is not k-sequentially additive for k ≡0 or 2 (mod 3); C3n+2 is not k-sequentially additive for k ≡1 or 2 (mod 3); Cn is 1-sequentially additive if and only if n ≡0 or 1 (mod 3); and Pn is 1-sequentially additive. They conjecture that all trees are 1-sequentially additive. Hegde proved that K1,n is k-sequentially additive if and only if k divides n.
Hajnal and Nagy investigated 1-sequentially additive labelings of 2-regular graphs. They prove: kC3 is 1-sequentially additive for all k; kC4 is 1-sequentially ad-ditive if and only if k ≡0 or 1 (mod 3); C6n ∪C6n and C6n ∪C6n ∪C3 are 1-sequentially additive for all n; C12n and C12n∪C3 are 1-sequentially additive for all n. They conjecture that every 2-regular simple graph on n vertices is 1-sequentially additive where n ≡0 or 1 (mod 3).
Acharya and Hegde have generalized k-sequentially additive labelings by allowing the image of the bijection to be {k, k+d, . . . , (k+|V ∪E|−1)d}. They call such a labeling additively (k, d)-sequential.
7.23 Difference Graphs Analogous to a sum graph, Harary calls a graph a difference graph if there is an bijection f from V to a set of positive integers S such that xy ∈E if and only if |f(x)−f(y)| ∈S. Bloom, Hell, and Taylor have shown that the following graphs are difference graphs: trees, Cn, Kn, Kn,n, Kn,n−1, pyramids, and n-prisms. Gervacio proved that wheels Wn are difference graphs if and only if n = 3, 4, or 6. Sonntag proved that cacti (that is, graphs in which every edge is contained in at most one cycle) with girth at least 6 are difference graphs and he conjectures that all cacti are difference graphs. Sugeng and Ryan provided difference labelings for cycles; fans; cycles with chords; graphs obtained by the one-point union of Kn and Pm; and graphs made from any number of copies of a given graph G that has a difference labeling by identifying one vertex the first with a vertex of the second, a different vertex of the second with the third the electronic journal of combinatorics (2023), #DS6 398 and so on. Vaithilingam proved that gears, ladders, fans, friendship graphs, helms, and wheels admit difference labelings.
Hegde and Vasudeva call a simple digraph a mod difference digraph if there is a positive integer m and a labeling L from the vertices to {1, 2, . . . , m} such that for any vertices u and v, (u, v) is an edge if and only if there is a vertex w such that L(v) −L(u) ≡L(w) (mod m). They prove that the complete symmetric digraph and unidirectional cycles and paths are mod difference digraphs.
In Seoud and Helmi provided a survey of all graphs of order at most 5 and showed the following graphs are difference graphs: Kn, (n ≥4) with two deleted edges having no vertex in common; Kn, (n ≥6) with three deleted edges having no vertex in common; gear graphs Gn for n ≥3; Pm × Pn (m, n ≥2); triangular snakes; C4-snakes; dragons (that is, graphs formed by identifying the end vertex of a path and any vertex in a cycle); graphs consisting of two cycles of the same order joined by an edge; and graphs obtained by identifying the center of a star with a vertex of a cycle.
7.24 Square Sum Labelings and Square Difference Labelings A bijective mapping f : V (G) to {0, 1, 2, . . . , |V (G)| −1} is said to be a square sum labeling if the induced function f ∗from E(G) to the positive integers defined by f ∗(xy) = (f(x)2+(f(y))2 is injective. A graph that has a square sum labeling is called a square sum graph. A square difference graph is defined the same way with f ∗(xy) = (f(x)2 + (f(y))2 replaced with f ∗(xy) = |(f(x)2−(f(y))2|. Maheswari and Srividya proved that every cycle Cn (n ≥6) with parallel P3 chords admit a vertex odd mean labeling, a vertex even mean labeling, and a square sum labeling. Maheswari, Azhagarasi, and Samuvel proved cycles with at least 6 vertices with parallel P4 chords are vertex odd mean graphs and vertex even mean graphs and they admit square sum labelings, and square difference labelings.
In they proved that the following graphs are square sum graphs: cycles with parallel chords, graphs obtained by attaching an arbitrary number of pendant edges at a vertex of degree 2 of a cycle with parallel chords, duplication of a vertex of degree 2 of a cycle with parallel chords, crowns with parallel chords, chains of even cycles with parallel chords, and graphs obtained from copies of Cn by joining a vertex from each copy of Cn to a common vertex. Patel and Ghodasara proved that the graph obtained by joining two copies of a specific graph by a path of arbitrary length admits a square sum labeling. They gave some results about the square sum graphs of arbitrary super subdivisions. Zhang, Naeem, Tariq, and Zhao investigated the square sum labeling of generalized Petersen graphs and double generalized Petersen graphs. Shiama proved that the total graph of paths and cycles, and the middle graphs of paths and cycles, admit square sum labelings. Parameswari and Margret proved that the octopus graph and Vanessa graph admit square sum labelings. Krithika and Keerthana proved that shell bow graphs admit square difference labelings and square difference new prime labelings.
Kulli defined a semitotal-block graph Tb(G) of a graph G as the graph whose the electronic journal of combinatorics (2023), #DS6 399 set of vertices is the union of the set of vertices and blocks of G and in which two points are adjacent if and only the corresponding vertices of G are adjacent or the corresponding blocks are incident. Mirajkar and Sthavarmath proved the following graphs admit square sum and square difference labelings: Tb(Pn ⊙K1) (n ≥3); Tb(Cn ⊙K1) (n ≥3) except for n = 0 (mod 8); and Tb(Fn ⊙K1) (n ≥4). In Cynthia and Poorani proved new circulant networks and a torus are square difference graphs. In Cynthia, Poorani, new and Mercy provide an algorithm to find the square difference labelings of bloom graphs.
Ajitha, Arumugam, and Germina prove the following graphs have square sum labelings: trees; cycles; K2 + mK1; Kn if and only if n ≤5; C(t) n (the one-point union of t copies of Cn); grids Pm × Pn; and Km,n if m ≤4. They also prove that every strongly square sum graph except K1, K2, and K3 contains a triangle.
In Shiama proved that the total graphs of paths and cycles and the middle graphs of paths and cycles are square sum graphs. A lilly graph is obtained by identifying one endpoint of each of two copies of Pn with the center of K1,2n (n > 1). Samuvel and Kalaivani proved the following graphs are square sum graphs: the graph obtained by the duplication of the center vertex and another vertex of a lilly graph; the graph obtained by identifying any two vertices of a lilly graph, and the graph obtained by the switching of the center vertex and another vertex of a lilly graph. Ghodasara and Patel proved the following graphs are square sum graphs: restricted square graphs; splitting graphs and shadow graphs of the bistar Bn,n; the restricted total, the restricted middle, and the degree splitting graph of Bn,n; and the duplication of a vertex and arbitrary super subdivision of Bn,n. Subhashini, Ramanathan, and Manimekalai proved that pyramids with at least 3 rows, hanging pyramids, graphs obtained from starting with r copies of a hanging pyramid and joining each copy with the next one with an edge, and the one point union of r copies of a pyramid and hanging pyramids admit square sum labelings. Maheswari, Azhagarasi, and Samuvel showed that the following graphs are square sum graphs: the corona product of Pm (m ≥2) and C2n (n ≥3) with P3 parallel chords; the corona product Pm (m ≥2) and C2n+1 (n ≥3) with P3 parallel chords; C2n ⊙K1 (n ≥3) with P3 parallel chords; C2n+1 ⊙K1 with P3 parallel chords; the chain of cycle C2n,m (n ≥3) with P3 parallel chords (a chain of m copies of C2n with P3 parallel chords where each copy shares exactly one vertex of the next copy); and edge connected cycle C2n (n ≥3) with P3 chords.
In Agasthi and Narvathi provided ways to construct square sum, square differ-ence, root mean square, strongly multiplicative, even mean and odd mean labelings for triangular snakes, and the central graph of triangular snake graphs.
In Ghodasara and Patel gave a counterexample to the conjecture by Germina and Sebastian that if G1 and G2 are square sum graphs then G1 ∪G2 is a square sum graph. They proved that the duplication graphs of any vertex of the following graphs are square sum graphs: Kn if and only if n ≤7, the Petersen graph P(5, 2), K1,n, and Cn They also proved that cycle Cn with [ n 2] concurrent chords is a square sum graph.
In Ghodasara and Patel proved that the following constructions based on the bistar Bn,n are square sum graphs: the restricted square, the splitting graph, the shadow graph, the degree splitting graph, the arbitrary super subdivision graph, and the du-the electronic journal of combinatorics (2023), #DS6 400 plication of any vertex of Bn,n. They defined restricted total graph of Bn,n as a graph with vertex set = V (Bn,n) ∪E(Bn,n) = {u, v, w, ui, vi, u′ i, v′ i/1 ≤i ≤n}, where u and v are apex vertices, ui and vi are pendent vertices, w, u′ i and v′ i are vertices corresponding to the edges of Bn,n and edge set = E(Bn,n) ∪{uw, vw, wu′ i, wv′ i, uu′ i, vv′ i, uiu′ i, viv′ i, /1 ≤ i ≤n}. They also defined restricted middle graph of Bn,n as a graph with vertex set = V (Bn,n) ∪E(Bn,n) = {u, v, w, ui, vi, u′ i, v′ i/1 ≤i ≤n}, where u and v are apex vertices, ui and vi are pendent vertices, w, u′ i and v′ i are vertices corresponding to the edges of Bn,n and edge set = {uw, vw, wu′ i, wv′ i, uu′ i, vv′ i, uiu′ i, viv′ i, /1 ≤i ≤n}. They proved that restricted total graph and restricted middle graph of Bn,n are square sum graphs.
Germina and Sebastian proved that the following graphs are square sum graphs: trees; unicyclic graphs; mCn; cycles with a chord; the graphs obtained by joining two copies of cycle Cn by a path Pk; and graphs that are a path union of k copies of Cn and the path is P2. In Seoud and Al-Harere give several necessary conditions for a graph to be a square sum graph and show that 2Cn, P2n, and C2n are square sum graphs.
Huilgol and Sriram prove that if G1 and G2 are square sum, then G1 ∪G2 ∪G3 is also square sum, where G3 is a set of isolated vertices.
In Somashekara and Veena used the term “square sum labeling” to mean “strongly square sum labeling.” They proved that the following graphs have strongly square sum labelings: paths, K1,n1 ∪K1,n2 ∪· · ·∪K1,nk, complete n-ary trees, and lobsters obtained by joining centers of any number of copies of a star to a new vertex. They observed that that if every edge of a graph is an edge of a triangle then the graph does not have strongly square sum labeling. As a consequence, the following graphs do not have a strongly square sum labelings: Kn, n ≥3; wheels; fans Pn + K1 (n ≥2); double fans Pn + K2 (n ≥2); friendship graphs C(n) 3 ; windmills K(n) m (m > 3); triangular ladders; triangular snakes; double triangular snakes; and flowers. They also proved that helms are not strongly square sum graphs and the graphs obtained by joining the centers of two wheels to a new vertex are not strongly square sum graphs.
Govindan, Pinelas, and Dhivya introduced the notion a cube sum graph as a graph G(V, E) with p vertices and q edges such that if there exists a bijection f : V →{0, 1, 2, . . . , p −1}, then the induced function f ∗: E →N, defined by f(∗uv) = (f(u))3 + (f(v))3, is injective. They proved that paths, cycle, stars, wheels, and fans are cube sum graphs. Patel and Ghodasara proved that the following graphs are cube sum graphs: trees, gears, shell graphs, helms, Kn if and only if n is at most 11, and K2,n for all n. Krithika and Keerthana proved that shell bow graphs, the path union new of two subdivided shell graphs, and the disjoint union of two copies of shell bow graphs admit cube difference labelings.
Let G1, G2, . . . , Gn be subdivided shell graphs of any order. The graph SSG(n) is obtained by adding an edge to apexes of Gi and Gi+1 for i = 1, 2, . . . , n −1. The graph SSG(n) is called a path union of n subdivided shell graphs.
In Krithika and new Keerthana proved that SSG(2) admits square difference labeling and square difference prime labeling.
In Sonchhatra and Ghodasara call a (p, q)-graph G = (V, E) sum perfect square if there exists a bijection f from V to {0, 1, 2, . . . , p−1} such that the function f ∗from E the electronic journal of combinatorics (2023), #DS6 401 defined by f ∗(uv) = (f(u)) + (f(v))2 for all edges uv is an injection. Such an f is called a sum perfect square labeling of G. In a series of four papers the following graphs are proved to be sum perfect square graphs: cycles, cycles with one chord, cycles with twin chords, trees ; several snake related graphs ; K1,n + K1, K2 + mK1, Cn ⊙K1, graphs obtained from K1,n with endpoint vertices v1, v2, . . . , vn by joining vi and vi+1 with an edge for i = 1, 2, . . . , v⌊n/2⌋(“half wheel”), the middle graphs of paths, the total graphs of paths ; P 2 (n > 1), mK1,n, mCn,, and the splitting graph and the shadow graph of a star . In they prove that the union of two stars and that for any sum perfect square graph G, G ∪Pn is sum perfect square. They conjecture that the union of any two sum perfect square graphs is sum perfect square.
Ajitha, Princy, Lokesha, and Ranjini defined a graph G(p, q) to be a square differ-ence graph if there exist a bijection f from V (G) to {0, 1, 2, . . . , p−1} such that the induced function f ∗from E(G) to the natural numbers given by f ∗(uv) = |(f(u))2 −(f(v))2| for every edge uv of G is a bijection. Such a the function is called a square difference label-ing of the graph G. They proved that following graphs have square difference labelings: paths, stars, cycles, Kn if and only if n ≤5, Km,n if m ≤4, friendship graphs C(n) 3 , triangular snakes, and K2 + mK1. They also prove that every graph can be embedded as a subgraph of a connected square difference graph and conjecture that trees, complete bipartite graphs and C(n) k are square difference graphs.
Kulli defined a semitotal-block graph Tb(G) of a graph G as the graph whose set of vertices is the union of the set of vertices and blocks of G and in which two points are adjacent if and only the corresponding vertices of G are adjacent or the corresponding blocks are incident. Mirajkar and Sthavarmath proved the following graphs admit square sum and square difference labelings: Tb(Pn ⊙K1) (n ≥3); Tb(Cn ⊙K1) (n ≥3) except for n = 0 (mod 8); and Tb(Fn ⊙K1) (n ≥4). In Cynthia, Poorani, and new Mercy provide an algorithm to find the square difference labelings of bloom graphs.
Tharmaraj and Sarasija proved that following graphs have square difference labelings: fans Fn (n ≥2); Pn + K2; the middle graphs of paths and cycles; the total graph of a path; the graphs obtained from m copies of an odd cycle and the path Pm with consecutive vertices v1, v2, . . . , vm by joining the vertex vi to a vertex of the ith copy of the odd cycle; and the graphs obtained from m copies of the star Sn and the path Pm by joining the vertex vi of Pm to the center of the ith copy of Sn. Sebastian and Germina proved that certain planar graphs and higher order level joined planar grid admit square sum labeling. They also study square sum properties of several classes of graphs with many odd cycles.
In Geetha and Kalamani showed that the following graphs have square difference labelings: two copies of the same star whose centers are joined by a path; and two copies of the same cycles whose centers are joined by a path, the restricted square of bistar Bn,n; the restricted total graph of bistar Bn,n; the restricted middle graph of Bn,n Shiama showed that cycles, complete graphs, cycle cacti, ladders, lattice grids, quadrilateral snakes, K2 + mK1 admit square difference labelings.
In Sunoj and Mathew Varkey defined the square difference prime labeling of new a graph G(V, E) as a bijection f from V to {0, 1, 2, . . . , |V |1} such that the induced the electronic journal of combinatorics (2023), #DS6 402 function f ∗ spd from E to the natural numbers given by f ∗ spd(uv) = |(f(u)2 −(f(v))2| has the property that the greatest common divisor of the labels of the edges incident with a vertex of degree greater than one is 1. In , , , and they proved new new new new the following graphs admit square dofference prime labelings: certian triangular snakes, double triangular snakes and alternate triangular snakes, centipedes, twigs, coconut trees, combs, stars, subdivision of stars, bistars, cycles, middle graphs of cycles, total graphs of cycles, lillies graph, and some star related graphs.
Vaghela and Parmar say a graph G admits a difference perfect square cordial labeling if there is a bijection f : V (G) →{1, 2, . . . , |V (G|} such that for each edge uv the induced map f ∗: E(G) →{0, 1} defined by f ∗(uv) = 1 if u2 −2uv + v2 = 1, and 0 otherwise, has the property that the number of edges labeled with 0 and the number of edges labeled with 1 difference by at most 1. A graph that admits a difference perfect square cordial labeling is said to be a difference perfect square cordial graph. They obtained difference perfect square cordial labelings for paths, cycles, wheels, fans, combs, crowns, (Cm⊙K1)∪(Pn⊙K1), D2(Pn), P 2 n, K2⊙Cn, graphs consisting of two copies of Cn that share a common edge, the vertex switching of Cn, and the graph obtained by starting with Pn (n ≥6) and two new vertices u and v of Pn and joining v to first two vertices and last two vertices of Pn and joining u to the remaining vertices of Pn (called the shipping graph) In Vaghela and Parmar provided difference perfect square cordial labelings of the H-graphs of paths, some corona graphs, total graphs of paths, and graphs obtained from Pn×P2 where Pn has consecutive vertices v1, v2, . . . , vn by joining vi in left Pn to vi+1 in the right copy of Pn with an edge for i = 1, 2, . . . , n −1. In Vaghela and Parmar obtain difference perfect square cordial labeling of triangular snake graphs, quadrilateral snake graphs, alternate triangular snake graphs, alternate quadrilateral snake graphs, irregular triangular snake graphs, irregular quadrilateral snake graphs, double triangular snake graphs, double quadrilateral snake graphs, double alternate triangular snake graphs, and double alternate quadrilateral snake graphs.
A graph G is said to admit a square difference labeling if there exists a injection f : V (G) →{0, 1, 2, . . . , n −1} for some n such that when each edge of G is assigned the absolute square difference of its end-vertices, the resulting labels are distinct. Cube difference labeling are defined analogously.
Shiana proved that paths, cycles, stars, fans, wheels, crowns, helms, dragons, coconut trees, and shell graphs admit cube difference labelings. Sharon Philomena and Thirusangu proved the cycle cactus graph C(3) n , the tree of diameter 4 obtained from the bistar Bn,n by subdividing the middle edge with a new vertex, and the graph obtained by joining one vertex of a cycle and one vertex of degree 2 of a comb by an edge have square and cube difference labelings (that is, the absolute cube difference of end-vertices of the edges are distinct). Sherman proved the path union of nC3 and the disjoint union of m stars K1,n1, K1,n2, . . . , K1,nm are square difference graphs Subashini, Bhuvaneswari, and Manimekalai proved the following graphs have square difference labelings: theta graphs, the duplication of any vertex of degree 3 in the cycle of a theta graph, the one point union of any number of theta graphs, the path union of any number of copies of a theta graph, the fusion of any two vertices in the cycle of a the electronic journal of combinatorics (2023), #DS6 403 theta graph, and the switching of a central vertex of a theta graph.
7.25 Permutation and Combination Graphs Hegde and Shetty define a graph G with p vertices to be a permutation graph if there exists a injection f from the vertices of G to {1, 2, 3, . . . , p} such that the induced edge function gf defined by gf(uv) = f(u)!/|f(u) −f(v)|! is injective. They say a graph G with p vertices is a combination graph if there exists a injection f from the vertices of G to {1, 2, 3, . . . , p} such that the induced edge function gf defined as gf(uv) = f(u)!/|f(u) −f(v)|!f(v)! is injective. They prove: Kn is a permutation graph if and only if n ≤5; Kn is a combination graph if and only if n ≤5; Cn is a combination graph for n > 3; Kn,n is a combination graph if and only if n ≤2; Wn is a not a combination graph for n ≤6; and a necessary condition for a (p, q)-graph to be a combination graph is that 4q ≤p2 if p is even and 4q ≤p2 −1 if p is odd. They strongly believe that Wn is a combination graph for n ≥7 and all trees are combinations graphs. Baskar Babujee and Vishnupriya prove the following graphs are permutation graphs: Pn; Cn; stars; graphs obtained adding a pendent edge to each edge of a star; graphs obtained by joining the centers of two identical stars with an edge or a path of length 2); and complete binary trees with at least three vertices. Seoud and Salim determine all permutation graphs of order at most 9 and prove that every bipartite graph of order at most 50 is a permutation graph. Seoud and Mahran give an upper bound on the number of edges of a permutation graph and introduce some necessary conditions for a graph to be a permutation graph. They show that these conditions are not sufficient for a graph to be a permutation graph.
Ghodasara and Patel proved that the following graphs are permutation graphs: the Petersen graph P(5, 2), trees, K3,n (n ≥1) for n+3 prime, Wn (n ≥3) for n+1 prime, shell graph Sn (n ≥3) for prime n, dumbbell graph Dn,k,2 (n, k ≥3), Cn ⊙K1 (n ≥3), and the one point union C(k) n (k ≥2, n ≥3) of k copies of cycle Cn. A t-ply Pt(u, v) is a graph with t paths, each of length at least two and such that no two paths have a vertex in common except for the end vertices u and v. Ghodasara and Patel defined t∗-ply Pt∗(u, v) as a special case of t-ply Pt(u, v) graph with every t path have same length and proved that t∗-ply Pt∗(u, v) is a permutation graph.
The graph obtained from two copies of an (m, n) kite graph by connecting the degree 1 vertex of one copy to the vertex of degree 3 and the second copy is called the 1-join (m,n) kite. The graph obtained by repeating this construction with t copies of an (m, n) kite is called a is called the t-join (m,n) kite. Sriramr and Govindarajan proved t-join (m, n) kites are permutation graphs.
Ghodasara and Patel proved that the following graphs are combination graphs: Cn × P2 for n ≥6, umbrella graph U(m, n) for m, n > 2, armed crown Cn ⊕Pm for n ≥4 and m ≥1, the graphs obtained by joining C2m (m ≥2) to each pendent vertex of K1,n(n ≥2), the duplication of any rim vertex of Wn for n ≥7, Cn with [ n−4 2 ] concurrent chords for n ≥6, and the duplication of vertex in Cn for n ≥5.
Hegde and Shetty say a graph G with p vertices and q edges is a strong k-the electronic journal of combinatorics (2023), #DS6 404 combination graph if there exists a bijection f from the vertices of G to {1, 2, 3, . . . , p} such that the induced edge function gf from the edges to {k, k + 1, . . . , k + q −1} defined by gf(uv) = f(u)!/|f(u) −f(v)|!f(v)! is a bijection. They say a graph G with p vertices and q edges is a strong k-permutation graph if there exists a bijection f from the vertices of G to {1, 2, 3, . . . , p} such that the induced edge function gf from the edges to {k, k + 1, . . . , k + q −1} defined by gf(uv) = f(u)!/|f(u) −f(v)|! is a bijection.
Seoud and Anwar provided necessary conditions for combination graphs, permutation graphs, strong k-combination graphs, and strong k-permutation graphs.
Seoud and Al-Harere showed that the following families are combination graphs: graphs that are two copies of Cn sharing a common edge; graphs consisting of two cycles of the same order joined by a path; graphs that are the union of three cycles of the same order; wheels Wn (n ≥7); coronas Tn ⊙K1, where Tn is the triangular snake; and the graphs obtained from the gear Gm by attaching n pendent vertices to each vertex which is not joined to the center of the gear. They proved that a graph G(n, q) having at least 6 vertices such that 3 vertices are of degree 1, n −1, n −2 is not a combination graph, and a graph G(n, q) having at least 6 vertices such that there exist 2 vertices of degree n −3, two vertices of degree 1 and one vertex of degree n −1 is not a combination graph.
Seoud and Al-Harere proved that the following families are combination graphs: unions of four cycles of the same order; double triangular snakes; fans Fn if and only if n ≥6; caterpillars; complete binary trees; ternary trees with at least 4 vertices; and graphs obtained by identifying the pendent vertices of stars Sm with the paths Pni, for 1 ≤ni ≤m. They include a survey of trees of order at most 10 that are combination graphs and proved the following graphs are not combination graphs: bipartite graphs with two partite sets with n ≥6 elements such that n/2 elements of each set have degree n; the splitting graph of Kn,n (n ≥3); and certain chains of two and three complete graphs. Seoud and Anwar proved the following graphs are combination graphs: dragon graphs (the graphs obtained from by joining the endpoint of a path to a vertex of a cycle); triangular snakes Tn (n ≥3); wheels; and the graphs obtained by adding k pendent edges to every vertex of Cn for certain values of k.
In and Seoud and Al-Harere proved the following graphs are non-combination graphs: G1 + G2 if |V (G1)|, |V (G2)| ≥ 2 and at least one of |V (G1)| and |V (G2)| is greater than 2; the double fan K2 + Pn; Kl,m,n; Kk,l,m,n; P2[G]; P3[G]; C3[G]; C4[G]; Km[G]; Wm[G]; the splitting graph of Kn (n ≥3); Kn (n ≥4) with an edge deleted; Kn (n ≥5) with three edges deleted; and Kn,n (n ≥3) with an edge deleted. They also proved that a graph G(n, q) (n ≥3) is not a combination graph if it has more than one vertex of degree n −1.
In and Tharmaraj and Sarasija defined a graph G(V, E) with p vertices to be a beta combination graph if there exist a bijection f from V (G) to {1, 2, . . . , p} such that the induced function Bf from E(G) to the natural numbers given by Bf(uv) = (f(u) + f(v))!/f(u)!f(v)! for every edge uv of G is injective. Such a function is called a beta combination labeling. They prove the following graphs have beta combination label-ings: Kn if and only if n ≤8; ladders Ln (n ≥2); fans Fn (n ≥2); wheels; paths; cycles; friendship graphs; Kn,n (n ≥2); trees; bistars; K1,n (n > 1); triangular snakes; quadrilat-the electronic journal of combinatorics (2023), #DS6 405 eral snakes; double triangular snakes; alternate triangular snakes (graphs obtained from a path v1, v2, . . . , vn, where for each odd i ≤n −1, vi and vi+1 are joined to a new vertex ui,i+1; alternate quadrilateral snakes (graphs obtained from a path v1, v2, . . . , vn, where for each odd i ≤n −1, vi and vi+1 are joined to two new vertices ui,i+1,1 and ui,i+1,2); helms; gears; combs Pn ⊙K1; and coronas Cn ⊙K1.
7.26 Strongly -graphs A variation of strong multiplicity of graphs is a strongly -graph. A graph of order n is said to be a strongly -graph if its vertices can be assigned the values 1, 2, . . . , n in such a way that, when an edge whose vertices are labeled i and j is labeled with the value i + j + ij, all edges have different labels. Adiga and Somashekara have shown that all trees, cycles, and grids are strongly -graphs. They further consider the problem of determining the maximum number of edges in any strongly -graph of given order and relate it to the corresponding problem for strongly multiplicative graphs. In and Seoud and Mahan give some technical necessary conditions for a graph to be strongly -graph, Baskar Babujee and Vishnupriya have proved the following are strongly -graphs: Cn × P2, (P2 ∪Km) + K2, windmills K(n) 3 , and jelly fish graphs J(m, n) obtained from a 4-cycle v1, v2, v3, v4 by joining v1 and v3 with an edge and appending m pendent edges to v2 and n pendent edges to v4.
Baskar Babujee and Beaula prove that cycles and complete bipartite graphs are vertex strongly -graphs. Baskar Babujee, Rajesh Kannan, and Vishnupriya prove that wheels, paths, fans, crowns, (P2 ∪mK1) + K2, and umbrellas (graphs obtained by appending a path to the central vertex of a fan) are vertex strongly -graphs.
In Seoud, Roshdy, and AboShady gave an upper bound for the number of edges of any graph in terms of the number of vertices to be a strongly ∗-graph and some new families to be strongly∗- graphs. They also provided an algorithm for checking if a graph is a strongly ∗-graph or not.
7.27 Triangular Sum Graphs Hegde and Shankaran call a labeling of graph with q edges a triangular sum labeling if the vertices can be assigned distinct non-negative integers in such a way that, when an edge whose vertices are labeled i and j is labeled with the value i + j, the edges labels are {k(k + 1)/2| k = 1, 2, . . . , q}. They prove the following graphs have triangular sum labelings: paths, stars, complete n-ary trees, and trees obtained from a star by replacing each edge of the star by a path. They also prove that Kn has a triangular sum labeling if and only if n is 1 or 2 and the friendship graphs C(t) 3 do not have a triangular sum labeling.
They conjecture that Kn (n ≥5) are forbidden subgraphs of graph with triangular sum labelings.
They conjectured that every tree admits a triangular sum labeling.
They show that some families of graphs can be embedded as induced subgraphs of triangular sum graphs. They conclude saying “as every graph cannot be embedded as an induced the electronic journal of combinatorics (2023), #DS6 406 subgraph of a triangular sum graph, it is interesting to embed families of graphs as an induced subgraph of a triangular sum graph”. In response, Seoud and Salim showed the following graphs can be embedded as an induced subgraph of a triangular sum graph: trees, cycles, nC4, and the one-point union of any number of copies of C4 (friendship graphs).
Vaidya, Prajapati, and Vihol showed that cycles, cycles with exactly one chord, and cycles with exactly two chords that form a triangle with an edge of the cycle can be embedded as an induced subgraph of a graph with a triangular sum labeling. They proved that several classes of graphs do not have triangular sum labelings. Among them are: helms, graphs obtained by joining the centers of two wheels to a new vertex, and graphs in which every edge is an edge of a triangle. As a corollary of the latter result they have that Pm + Kn, Wm + Kn, wheels, friendship graphs, flowers, triangular ladders, triangular snakes, double triangular snakes, and flowers.
do not have triangular sum labelings.
Seoud and Salim proved the following are triangular sum graphs: Pm∪Pn, m ≥ 4; the union of any number of copies of Pn, n ≥5; Pn ⊙Km; symmetrical trees; the graph obtained from a path by attaching an arbitrary number of edges to each vertex of the path; the graph obtained by identifying the centers of any number of stars; and all trees of order at most 9.
For a positive integer i the ith pentagonal number is i(3i −1)/2. Somashekara and Veena define a pentagonal sum labeling of a graph G(V, E) as one for which there is a one-to-one function f from V (G) to the set of nonnegative integers that induces a bijection f + from E(G) to the set of the first |E| pentagonal numbers. A graph that admits such a labeling is called a pentagonal sum graph. Somashekara and Veena proved that the following graphs have pentagonal sum labelings: paths, K1,n1 ∪K1,n2 ∪· · · ∪K1,nk, complete n-ary trees, and lobsters obtained by joining centers of any number of copies of a star to a new vertex. They conjecture that every tree has a pentagonal sum labeling and as an open problem they ask for a proof or disprove that cycles have pentagonal labelings. They observed that if every edge of a graph is an edge of a triangle then the graph does not have pentagonal sum labeling. As was the case for triangular sum labelings the following graphs do not have a pentagonal sum labeling: Pm + Kn, and Wm + Kn wheels, friendship graphs, flowers, triangular ladders, triangular snakes, double triangular snakes, and flowers. Somashekara and Veena also proved that helms and the graphs obtained by joining the centers of two wheels to a new vertex are not pentagonal sum graphs.
7.28 Divisor Graphs Santhosh and Singh call a graph G(V, E) a divisor graph if V is a set of integers and uv ∈E if and only if u divides v or vice versa. They prove the following are divisor graphs: trees; mKn; induced subgraphs of divisor graphs; cocktail party graphs Hm,n (see Section 7.1) for the definition); the one-point union of complete graphs of different orders; complete bipartite graphs; Wn for n even and n > 2; and Pn + Kt. They also prove that the electronic journal of combinatorics (2023), #DS6 407 Cn (n ≥4) is a divisor graph if and only if n is even and if G is a divisor graph then for all n so is G + Kn.
Chartrand, Muntean, Saenpholphat, and Zhang proved complete graphs, bipar-tite graphs, complete multipartite graphs, and joins of divisor graphs are divisor graphs.
They also proved if G is a divisor graph, then G × K2 is a divisor graph if and only if G is a bipartite graph; a triangle-free graph is a divisor graph if and only if it is bipartite; no divisor graph contains an induced odd cycle of length 5 or more; and that a graph G is divisor graph if and only if there is an orientation D of G such that if (x, y) and (y, z) are edges of D then so is (x, z).
In and Al-Addasi, AbuGhneim, and Al-Ezeh determined precisely the values of n for which P k n (k ≥2) are divisor graphs and proved that for any integer k ≥2, Ck n is a divisor graph if and only if n ≤2k + 2. In they gave a characterization of the graphs G and H for which G×H is a divisor graph and a characterization of which block graphs are divisor graphs. (Recall a graph is a block graph if every one of its blocks is complete.) They showed that divisor graphs form a proper subclass of perfect graphs and showed that cycle permutation graphs of order at least 8 are divisor graphs if and only if they are perfect. (Recall a graph is perfect if every subgraph has chromatic number equal to the order of its maximal clique.) In Al-Addasi, AbuGhneim, and Al-Ezeh proved that the contraction of a divisor graph along a bridge is a divisor graph; if e is an edge of a divisor graph that lies on an induced even cycle of length at least 6, then the contraction along e is not a divisor graph; and they introduced a special type of vertex splitting that yields a divisor graph when applied to a cut vertex of a given divisor graph.
AbuHijleh, AbuGhneim, and Al-Ezeh prove that for any tree T, T 2 is a divisor graph if and only if T is a caterpillar and the diameter of T is less than six. For any caterpillar T and a positive integer k with diam(T) < 2k, they show that T k is a divisor graph. Moreover, for a caterpillar T and k ≥3 with diam(T) = 2k or diam(T) = 2k + 1, they show that T k is a divisor graph if and only if the centers of T have degree two.
In AbuHijleh, AbuGhneim, and Al-Ezeh prove that the k-th power Qk n of Qn is a divisor graph if and only if n = 2, 3 or n ≥4 and k ≥n −1 hold. In the case of the n-dimensional folded-hypercube FQn (that is, the graph obtained from Qn by adding to it a perfect matching that connects opposite pairs of the vertices of Qn) they show that FQn is a divisor graph for odd n, but not for even n ≥4. They also prove (FQn)k is not a divisor graph if and only if 2 ≤k ≤⌈n/2⌉, where n ≥5.
Ganesan and Uthayakumar proved that G ⊙H is a divisor graph if and only if G is a bipartite graph and H is a divisor graph. Frayer proved Kn × G is a divisor graph for each n if and only if G contains no edges and Kn × K2 (n ≥3) is a divisor graph.
Vinh proved that for any n > 1 and 0 ≤m ≤n(n −1)/2 there exists a divisor graph of order n and size m. She also gave a simple characterization of divisor graphs due to Chartrand, Muntean, Saenpholphat, and Zhang . Gera, Saenpholphat, and Zhang established forbidden subgraph characterizations for all divisor graphs that contain at most three triangles. Tsao investigated the vertex-chromatic number, the clique number, the clique cover number, and the independence number of divisor graphs and their complements. In Seoud, El Sonbaty, and Mahran discuss here the electronic journal of combinatorics (2023), #DS6 408 some necessary and sufficient conditions for a graph to be a divisor graph.
In Kasthuri, Karuppasamy, and Nagarajan introduced the notions of SD-divisor labelings and SD-divisor cordial labelings as follows. Let G be finite, simple graph G with n vertices and f be a bijection from V (G) to {1, 2, . . . , n}. Define a function f ′ induced by f by assigning f ′(uv) = 1 if S = f(u)+f(v) divides D = |f(u)−f(v)| and f ′(uv) = 0, otherwise. They say that f ′ is a SD-divisor labeling if f ′(uv) = 1 for all edges uv. In Revathi and Mary Jeya Jothi provided SD-divisor labelings for paths and cycles and certain path and cycle related graphs. They further find SD-divisor labelings of some subdivision graphs, coronas, and splitting graphs, and prove that stars, complete graphs, and wheels do not admit SD-divisor labelings.
A graph G with n vertices has a modular multiplicative divisor (MMD) labeling if there exist a bijection f from vertices of G to the set of all natural numbers from 1 to n such that when the edge uv is labeled f(u)f(v)(mod n), then n divides the sum of all edge labels of G. They prove that r-dimensional butterfly networks admit MMD labelings. Revathi, Dharmakkan, and Jeya Jothi proved that Kl,m,n admits a modular multiplicative divisor labeling. Revathi and Rajeswari proved Pn, Pa,b (the graph that connects two vertices by means of b internally disjoint paths of length a each), the shadow graph of a path, and Pn × P1, (where n is not a multiple of 6) admit modular multiplicative divisor labelings. They also discuss the upper bound for the number of edges in a modular multiplicative divisor graphs. In Revathi and Rajeswari proved that the split graphs of cycles, helms, and flower graphs admit modular multiplicative divisor labelings.
A divisor 3-equitable labeling of a graph G is a bijection d : V (G) →{1, 2, . . . , |V (G)|} such that induced map d∗defined on the edges of G by, for any edge uv with d(u) ≤ d(v), d∗(uv)) = 1 if d(v)/d(u) = 1; d∗(uv) = 2 if d(v)/d(u) = 2; and d∗(uv) = 0, otherwise, has the property that the number of edges labeled with i and the number of edges labeled with j differ by at most 1 for all i and j. A graph that admits a divisor 3-equitable labeling is called a divisor 3-equitable graph. The existence and non-existence of divisor 3-equitable labelings have been determined for wheels , complete graphs and stars , ladders, triangular snakes, and lollipop graphs , and the degree splitting graph of ladders .
7.29 Other Kinds of Labelings Zumkeller labelings A positive integer n is said to be a Zumkeller number if all the positive factors of n can be partitioned into two disjoint parts so that the sum of the two parts are equal. (For example, 6 partitions as {1, 2, 3} and {6} and 20 partitions as {2, 12} and {1, 3, 4, 6}.
An injective function f from the vertices of a graph G to the natural numbers N is said to be a Zumkeller labeling of G, if the induced function f ∗: E(G) →N defined by f ∗(xy) = f(x)f(y) is a Zumkeller number for all edges xy. A graph that admits a Zumkeller labeling is called a Zumkeller graph. Balamurugan, Thirusangu, Thomas, and Murali investigated the existence of Zumkeller labelings of paths, cycles, and ladders.
Wilson and Bebincy proved that the splitting graph of paths, the total graphs of the electronic journal of combinatorics (2023), #DS6 409 paths, the shadow graphs of paths, and the middle graphs of paths admit Zumkeller labelings. Wilson and Bebincy investigated the existence of Zumkeller labelings for closed helms, double wheels, sunflowers, flower graphs, and the prisms of wheels. Patodia and Saikia provided algorithms to label complete bipartite graphs, wheels, cycles, and paths with m-Zumkeller numbers.
A simple graph G(V, E) is said to be a k-Zumkeller graph if there is an injection f from the vertices of G to the natural numbers such that when each edge xy is assigned the label f(x)f(y), the resulting edge labels are k distinct Zumkeller numbers. Balamurugan, Thirusangu, and Thomas proved that twig graphs admit a 4-Zumkeller labelings.
Basher showed that the super subdivision of paths, cycles, combs, ladders, crowns, circular ladders, grids, and prism are k-Zumkeller graphs. An injection is called a t-m-Zumkeller labeling of a graph G if, for every edge uv, the induced function defined by f ∗(uv) = f(u)f(v) is an m-Zumkeller number and |f ∗(E)| = t, where t denotes the number of distinct m-Zumkeller numbers on the edges of G. A graph that admits a t−m-Zumkeller labeling is called a t-m-Zumkeller graph.. In Patodia and Saikia gave t-m-Zumkeller labelings for paths, cycles, combs graphs, ladders graphs, and twigs.
A positive integer is said to be a half-Zumkeller number if its proper positive divisors can be partitioned into two disjoint non-empty subsets of equal sum. A half-Zumkeller labeling of a graph is an injective mapping f from the vertex set into the set of natural numbers N such that the induced mapping f ∗from the edges to N given by f ∗(uv) = f(u)f(v) is a half-Zumkeller number. A graph that admits a half-Zumkeller labeling is called a half-Zumkeller graph. Zeen El Deen, Elmahdy, Elkholy, and El Sherbiny gave half-Zumkeller labelings for stacked books (Km,1 × Pn), Pm × Pn, prisms of ladders, grids, gears, and flowers. They also showed that if G is a half-Zumkeller graph and H is a non-totally disconnected subgraph of G, then H is a half-Zumkeller graph.
In 2015 Murali, Thirusangu, and Meenakshi introduced Zumkeller cordial labeling of graphs as an injective function f from the vertices to the natural numbers such that the induced function f ∗from the edges to {0, 1} defined by f ∗(xy) = f(x)f(y) is 1 if f(x)f(y) is a Zumkeller number and 0 otherwise with the condition that the number of edges labeled with 1 and the number labeled with 0 differ by at most 1. They proved the existence of Zumkeller cordial labeling for paths, cycles, and stars. Murali, Thirusangu, and Balamurugan showed the existence of Zumkeller cordial labelings for helms, wheels, flower graphs, and crowns.
Lucky labelings A labeling of the vertices of a graph with natural numbers is said to be lucky if the sum of labels over all neighbors of vertex v and the sum of labels over all neighbors of vertex u are distinct whenever u and v are adjacent. (An isolated vertex is defined to have sum 0). The smallest nonzero label of a lucky labeling of a graph G is called the lucky number of the graph and is denoted by η(G). A lucky labeling of a graph that is injective is called a proper lucky graph. The proper lucky number of graph G is denoted by ηp(G). Sateesh Kumar and Meenakshi determined the proper lucky numbers for Km,n, friendship graphs, certain triangular books, and certain rectangular books.
the electronic journal of combinatorics (2023), #DS6 410 Chiranjilal Kujur, Xavier, and Raja gave proper lucky labelings for K-identified triangular meshs and K-identified Sierpinski gasket graphs. Ahadi, Dehghan, Kazemi, and Mollaahmadi proved that for a given planar 3-colorable graph G, determining whether η(G) = 2 is NP-complete, and that for every k ≥2, it is NP-complete to decide whether η(G) = k for a given graph G. Using algebraic methods Czerwiński, Grytczuk, and Zelazny proved that η(T) ≤2 for every tree T, and η(G) ≤3 for every bipartite planar graph G. They obtained a bound for the lucky number in terms of the acyclic chromatic number and conjecture that for every graph G, η(G) ≤χ(G). In Kujur gave the lucky number and proper lucky number for the bloom graph (a particular planar, tripartite, 4-regular graph).
Elrokh, Sateesh Kumar, and Meenaksh computed the lucky numbers for jelly fish graphs, cocktail party graphs, and crowns.
Sateesh Kumar and Meenakshi determined the lucky numbers and proper lucky numbers for quadrilateral snakes, double quadrilateral snakes, alternate quadrilateral snakes, and double alternate quadrilateral snakes.
Xavier and Rathi determined the proper lucky numbers of hexagonal meshes and honeycomb networks. Indira, Selvam, and Thirusangu gave proper lucky labelings for extended duplicate graphs of quadrilateral snakes.
A labeling of the vertices of a graph with natural numbers is said to be a lucky edge labeling if for any two adjacent edges the sums of the vertex labels incident to the two edges are distinct. Thesmallest positive integer k for which a graph G has a lucky edge labeling from the set {1, 2, . . . , k} is called the lucky number of G and denoted by η(G).
A graph that admits a lucky edge labeling is called a lucky edge graph. Esakkiammal, Thirusangu, and Seethalakshmi showed that the super subdivisions of stars and wheels are lucky edge labeled graphs and determined their lucky numbers.
In Murugan and Chitra provided lucky edge labelings for Pn, Cn and Pn ⊙Cn.
In they proved that triangular snakes, book with triangular pages, and Pn × C3 (triangular prisms) are lucky edge graphs. Indira, Selvam, and Thirusangu gave lucky edge labelings for extended duplicate graphs of quadrilateral snakes. Chitra and Murugan proved that graphs obtained from two copies Pn in which the ith vertex of one path is joined with the (i + 1)th vertex of the second copy, fish graphs subindexgraphsfish (the one-point union of a vertex of Cn and a vertex of K3), butterfly graphs, double triangular snakes, flower graphs, and P 2 n are lucky edge graphs. Nagarajan and Priyadharsini provided lucky edge labelings and lucky numbers for lotus graphs, prisms, and C2m@Pt.
Mohana priya and S Santhiya gave lucky edge labelings for graphs obtained by joining Cm and Cn with an edge, bistars, wheels, fans, and butterfly graphs. Murugan and Chitra gave lucky edge labeling of fans, spider, and twig graphs. Aishwarya provided lucky edge labelings and luck numbers for ladders, shell graphs, and books with triangular pages.
S3 cordial remainder labelings In Lourdusamy, Jenifer Wency, and Patrick introduced the concept of the group S3 cordial remainder labeling as follows. Let G(V, E) be a graph and let g : V →S3 be a function. For each edge xy assign the label r where r is the remainder when o(g(x)) the electronic journal of combinatorics (2023), #DS6 411 is divided by o(g(y)) or o(g(y)) is divided by o(g(x)) according as |g(x)| ≥|g(y)| or |g(y)| ≥|g(x)|. The function g is called a group S3 cordial remainder labeling of G if |vg(x) −vg(y)| ≤1 and |eg(1) −eg(0)| ≤1, where vg(x) denotes the number of vertices labeled with x and eg(i) denotes the number of edges labeled with i (i = 0, 1). A graph G that admits a group S3 cordial remainder labeling is called a group S3 cordial remainder graph . They prove the following graphs admit a group S3 cordial remainder labeling: paths, cycles, stars, bistars, complete bipartite graphs, wheels, fans, combs, crowns, the lotus inside a circle, double fans, ladders, slanting ladders, and triangular ladders.
In they proved that shadow graph of cycle and path, splitting graph of cycle, armed crown, umbrella graphs, and dumbbell graphs admit a group S3 cordial remainder label-ing. Also they proved that certain snake related graphs are a group S3 cordial remainder graphs. In [2109–2111], they investigated the behavior of group S3 cordial remainder labelings of subdivision of stars, subdivision of bistars, subdivision of wheels, subdivision of combs, subdivision of crowns, subdivision of fans, subdivision of ladders, helms, flower graphs, closed helms, gears, sunflowers, triangular snakes, quadrilateral snakes, square of paths, the duplication of a vertex by a new edge in path and cycle graphs, the dupli-cation of an edge by a new vertex in paths and cycles, and total graphs of cycles and paths.
Pair difference cordial labelings For a (p, q)-graph G(V, E), let ρ = p 2 if p is even and p−1 2 , if p is odd, and L = {±1, ±2, ±3, . . . , ±ρ}.
Ponraj, Gayathri, and Somasundaram say G is new a pair difference cordial if the mapping f : V − →L obtained by assigning different labels in L to the different elements of V when p is even, and different labels in L to p −1 elements of V and repeating a label for the remaining one vertex when p is odd, has the property that for each edge uv of G labeled with |f(u) −f(v)| it holds that ∆f1 −∆fc 1 ≤1, where ∆f1 and ∆fc 1 respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. They investigated the pair difference cordial labeling behavior of paths, cycles, stars, bistarsa, snakes, butterfly graphs, some wheels and path related graphs, splitting graphs, shadow graphs, laddar related graphs, subdivision of graphs the corona of some graphs, mangolian tents, planar grids, star related graphs, m−copies of some graphs, Franklin graphs, Heawood graphs, Tietze graphs, and Durer graphs. They further investigated the pair difference cordial labeling behavior of certain trees, certain graphs derived from cube graphs, Petersen graphs p(n, k), and broken wheels in [2529–2538], [2539–2542]. In [2527, 2528] Ponraj and [2529–2538] new [2539–2542] new [2527, 2528] new Gayathri investigated the pair difference cordial labeling behavior of double alternate snake graphs in . Ponraj, Gayathri, and Sivakumar investigated the pair difference new cordial labelings of m-copies of paths, cycles, stars, ladders, and the subdivision of wheels and combs.
In [2552–2556] Ponraj and Prabu investigated the pair mean cordial labeling [2552–2556] new behavior of paths, stars, cycles, bistars, triangular snakes, alternate triangular snakes, quadrilateral snakes, alternate quadrilateral snakes, middle graphs, splitting graphs, friendshiip graphs, crowns, double fans, and some corona graphs.
the electronic journal of combinatorics (2023), #DS6 412 Even sum labelings Andharia and Kaneria introduced the concept of even sum labeling as follows.
A graph G = (V, E) is said to admit even sum labeling if there exist an injective function f : V (G) →{0, ±2, ±4, . . . , ±2|V (G)|} such that the induced mapping f ∗: E(G) →{2, 4, . . . , 2|E(G)|} defined by f ∗(uv) = f(u) + f(v), ∀uv ∈E(G) is bijective. The function f is called an even sum labeling of G. The graph that admits even sum labeling is called an even sum graph. In , , , and Andharia and Kaneria proved the following graphs are even sum graphs: slanting ladders SLn (1 < n < 9), Pn, C4n, the complete bipartite graphs, Pm × Pn, mirror graphs M(Pn) (2 ≤n ≤6), jelly fish graphs, the splitting graph of stars, the degree splitting graph of stars, the splitting graph of K2,n, the splitting graph of K1,n,n, jewel graphs, triangular books, the triangular book graph with a bookmark, Pm (+) Kn, and (Kn ∪P3) + 2K1.
In Krishna, Raj, and Mary obtained some results on even sum graphs.
new Odd-even sum labelings Monika and Murugan proved the graphs obtained by the following duplications admit odd-even sum labelings: the apex vertex of stars, the pendent vertices of stars, the apex vertex of stars by an edge, a vertex of Pn (n ≥2), an edge by a vertex of Pn (n ≥3), a pendant vertex by an edge of Pn (n ≥2), and a pendent edge of Pn (n ≥2) by edge, They also proved that C3 is not an odd-even sum graph.
Prime distance labelings Eggleton, Erdős, and Skilton, call a graph a prime distance graph if its vertices can be labeled with distinct integers in such a way that for any two adjacent vertices, the absolute difference of their labels is a prime number. Laison, Starr, and Walker prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Twin Prime Conjecture and the Polignac’s Conjecture that for every positive even integer n there are infinitely many cases of two consecutive prime numbers with difference n are true, respectively. Parthiban and David proved the following graphs admit prime distance labelings: the graph obtained by the duplication of every vertex by an edge in any prime distance graph H if the Twin prime conjecture is true; the graph obtained by fusing any two vertices vi and vj, where d(vi, vj) ≥3, of Cn; the graph obtained by duplicating any vertex of Cn (n ≥6); and the graph obtained from m copies of Cn by joining a vertex in the ith copy of Cn to a vertex in the (i + 1)th copy of Cn where 1 ≤i ≤m −1.
2-odd labelings Laison, Starr, and Walker say a graph G is a 2-odd graph if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the absolute difference of their labels is either an odd integer or 2. They give a characterization of 2-odd graphs in terms of edge colorings and use this characterization to determine which circulant graphs of a particular form are 2-odd and to prove results on circulant prime distance graphs.
the electronic journal of combinatorics (2023), #DS6 413 Pir and Parathiban proved that diamonds, double fans, double alternate trian-gular snakes, king graphs, which consist of the m × n chessboard with edges being the moves that a king chess piece can make, and antiprisms admit 2-odd labelings.
In Abirami, Parthiban, and Srinivasan prove the following graphs admit 2-odd labelings: Wn (n ≥4), fans, graphs with 2n vertices obtained from an n-cycle and joining every two consecutive edges to a new vertex, and shell graphs C(n, n −3) (n ≥4).
Abirami, Parthiban, and Srinivasan proved the following graphs admit 2-odd labelings: helms Hn (n ≥4), double wheels, umbrellas, graphs obtained by performing the duplication of a vertex by an edge at all the vertices of 2-odd graph if the Twin Prime conjecture is true, Ka + mK1, butterfly graphs Bn,m (n ≥3) if Goldbach’s conjecture is true, and graphs obtained from a cycle by performing duplication of a vertex by a vertex at all the vertices of the cycle if Goldbach’s conjecture is true.
Absolute difference labelings In Shalini and Dhayaran introduced the notion of absolute differences of cubic and square difference labelings of a graph as labelings for which every edge label is the absolute difference of the cubes of the vertices and the difference of the squares of the vertices. They proved that paths, stars, cycles, fans, windmills, and wheels admit differences of cubic and squared labelings. They also observed that the labels of the edges must be even. Shalini, Gowri, and Dhayabaran proved that barbells, cycle cac-tuses, coconut trees, shells, and dragons admit absolute differences of cubic and squared labelings. In , , , , , , , , , , and Mathew Varkey and Sunoj proved that planar grids, web graphs, kayak paddle graphs, snake graphs, armed crowns, fans, friendship graphs, windmill graphs, cycles, wheels, gears, helms, 2-tuple graphs, middle graphs, total graphs, shadow graphs barbells, K(m) 2,n , and K1,n⊙2Pm admit absolute difference of cubic and square sum labelings.
Quotient labelings In Sumathi and Rathi introduced the notion of quotient labeling as follows. Let G(V, E) be a finite, non-trivial, simple,undirected graph.
For a one-one assignment f : V →{1, 2, . . . , |V |} let f ∗(uv) = ⌊f(u)/f(v)⌋. Then f ∗is a said to be a quotient labeling of G if f ∗: E →{1, 2, . . . , |V |} and f(u) > f(v). (The edge labels need not be distinct.) The maximum value of f ∗(E(G)), denoted ql(f ∗), is called the q-labeling number of G. The quotient labeling number of G, QL(G), is the minimum value among all ql(f ∗). They found the quotient labeling number of the following graphs: the graph obtained by duplicating an arbitrary vertex of a cycle by an edge, the graph obtained by duplicating an arbitrary edge of cycle by a vertex, two copies Cn sharing a common edge, the twig graphs, the splitting graph of a path, the composition graph Pm[P2] and joint sum of two copies of Cn. In Sumathi and Rathi determine the quotient number of various families of ladder-related graphs. In Sumathi and Rathi provided the quotient labeling number of quadrilateral snakes, double quadrilateral snakes, alternate triangular snakes, alternate double triangular snakes, and the subdivision of triangular and quadrilateral snakes along the main path.
the electronic journal of combinatorics (2023), #DS6 414 Edge-sum distingishing game The edge-sum distinguishing game (ESD game) is a graph labeling game proposed by Tuza in 2017. In such a game, the players, traditionally called Alice and Bob, alternately new assign an unused label f(v) ∈{1, 2, . . . , s} to an unlabeled vertex v of a graph G, and the induced edge label φ(uv) of an edge uv ∈E(G) is given by φ(uv) = f(u) + f(v). Alice’s goal is to end up with an injective vertex labeling of all vertices of G that induces distinct edge labels, and Bob’s goal is to prevent this. Tuza also posed the following questions about the ESD game: Given a simple graph G, for which values of s can Alice win the ESD game? And if Alice wins the ESD game with the set of labels {1, 2, . . . , s}, can she also win with {1, 2, . . . , s+1}? Oliveira, Artigas, Dantas, and Luiz partially answer new these questions by presenting bounds on the number of consecutive non-negative integer labels necessary for Alice to win the ESD game on general and classical families of graphs.
Oblong labelings Oblong numbers are the numbers of dots that can be placed in rows and columns in a rectangular array, each row containing one more dot than each column. (The first five oblong numbers are 2, 6, 12, 20, and 30.) In Prema and Murugan proved the following graphs have oblong labelings: H-graphs, jelly fish, shrubs, G⊙K1 where G is an H-graph that has an oblong labeling, banana trees, and graphs obtained by identifying an endpoint of K1,3 with an end vertex of Pn. Muthumanickavel and Murugan investigated the existence of oblong sum labelings of the union of graphs involving stars and subdivisions of stars and bistars. They also prove that helms do not admit oblong labelings.
Signed cordial labelings In 2011 two versions of cordial labelings utilizing −1 and 1 as vertex and edge labels were introduced. Devaraj and Delphy defined the notion of signed cordial graphs as follows. A graph G(V, E) is called signed cordial if the edges can be assigned −1 and 1 in such a way that when each vertex v is assigned the product of the labels of the edges incident with v the resulting graph has the property that the number of vertices labeled with i and the number of edges labeled with i differ by at most 1 for i = −1 and 1. A graph is called a signed cordial graph if it admits a signed cordial labeling.
Similarly, Babujee and Loganathan say graph G(V, E) is signed product cordial if the vertices can be assigned −1 and 1 in such a way that when each edge assigned the product of its endpoints, the number of vertices labeled with i and the number of edges labeled with i differ by at most 1 for i = −1 and 1. A graph is called signed product cordial if it admits a signed product cordial labeling. In Nada, Elrokh, Elmshtaye, and El-hay gave necessary and sufficient conditions for which cones (Kn + Cm) and their second powers are signed product cordial. In Babujee and Loganathan proved that paths, cycles, stars, and bistars are signed product cordial. Devaraj and Delphy investigated signed-cordiality of complete graphs, books, Jahangir graphs, flower graphs, and Petersen graphs. Cynthia and Padmavathy proved that Cn × Pn and certain the electronic journal of combinatorics (2023), #DS6 415 banana trees admit signed cordial and signed product cordial labelings.
k-vertex weighting In Shiu, Laub, and Ng introduced the following notion. For a simple, finite and undirected graph G(V, E) of order n, a k-vertex weighting of a graph G is a mapping w : V (G) →{1, . . . , k}. A k-vertex weighting induces an edge labeling fw : E(G) →N such that fw(uv) = w(u) + w(v).
Such a labeling is called an edge-coloring k-vertex weighting if fw(e) ̸= fw(e′) for any two adjacent edges e and e′. They determined the minimum k that admit an edge-coloring k-vertex weighting for the following graphs: wheels, gears, grids, P2 × Cn, lollipops, theta graphs, and two cycles joined by a path (long dumbbells).
Harmonic mean cordial Let f be a function from V (G) to {1, 2}.
For each edge uv of G, assign the label ⌊2(f(uf(v))/(f(u) + f(v))⌋.
Such an f is a harmonic mean cordial labeling of G if |vf(i) −vf(j)| ≤1 for i and j in {1, 2}, where vf(x) and ef(x) denote the number of vertices and edges labeled with x, respectively. A graph with a harmonic mean cordial labeling is called a harmonic mean cordial graph. In Parejiya, Jani, and Hathi, proved that new CHn ⊙K1 (CHn is the closed helm obtained from Wn) and the tensor product of two paths admit harmonic mean cordial labelings. They also proved that Km,n, Kn ∨Cm (join of Km,n and Cn) and Cn ∨Cm are not do not admit harmonic mean cordial labelings.
Tri sum perfect square cordial labeling A graph G(V, E) is said to be tri sum perfect square cordial if there exists a func-tion f from V to {−1, 0, 1} such that the function f ∗from E to {0, 1} defined by f ∗(uv) = f(u)2 + 2f(u)f(v) + f(v)2 has the property that the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. Such an f is called a tri sum perfect square cordial labeling of G. In Vaghela and Parmar gave tri new sum perfect square cordial labelings for paths, combs, stars, fans, the H-graph of Pn, the H ⊙K1 graph of Pn, and the union of two stars.
Radio contra harmonic mean labeling In Ashika and Asha defined the radio contra harmonic mean labeling of a connected new graph G(V, E) as an injective function f : V (G) →Z such that for each pair of distinct vertices u and v of G, d (u, v) + l f(u)+f(v) 2 m ≥1 + diam (G) or d (u, v) + j f(u)+f(v) 2 k ≥1 + diam (G). The radio contra harmomic mean number of f is the maximum number assigned to any vertex of G. The radio contra harmonic mean number of G is the minimum value of the radio contra harmonic mean number taken over all radio contra harmonic mean labelings f of G. Ashika and Asha determined the radio contra harmonic mean numbers and labelings of Gn ⊙K1 (Gn is the gear graph), (Wn ⊙mK1) ⊙K1, the subdivision of jellyfish, and the path union for two copies of the bistar graph Bm,n)).
the electronic journal of combinatorics (2023), #DS6 416 SD-harmonious labelings For a graph G with q edges, an injection f : V (G) →{0, 1, 2, . . . , q} is said to be SD-harmonious labeling if the induced function f ∗: E(G) →{0, 2, . . . , 2q −2} defined by f(uv) = S + D (mod 2q) is bijective, where S = f(u) + f(v) and D = |f(u)f(v)|, for every edge uv in E(G). In Lourdusamy, Wency, and Patrick investigated SD-harmonious labelings of paths, trees, star related graphs, and the disjoint union of graphs.
Dividing graceful labelings Zahraa, Nabeel, and Fawzi say a graph G(V, E) has a dividing graceful labeling if there is a one-to-one mapping φ from V to {1, 2, 3, . . . , |E|} such that for every edge uv the induced function φ∗defined by φ∗(uv) = ⌈(φ(u) + φ(v))/|E|⌉has the property that φ∗(E) = {1, 2, 3, . . . , |E|}. They showed the following graph have dividing graceful labelings: stars, graphs obtained by joining the centers of two copies of K1,t, and spiders with diameter 4.
Square product labelings In Mirajkar and Sthavarmath introduced the notion a square product graph as one for which there exists a bijection f from V (G) to {1, 2, 3, . . . , p} that induces an injective labeling f ∗from E(G) to N defined by f ∗(uv) = f(u)2f(v)2. In they proved that Cn ⊙K1 and certain kinds of cactus graphs are square product graphs.
Numbering of graphs In Ichishima, Oshima, and Takahashi define a numbering f of a graph G of order n as a labeling that assigns distinct elements of the set {1, 2, . . . , n} to the vertices of G.
The strength strf(G) of a numbering f : V (G) →{1, 2, . . . , n} of G is defined by strf(G) = max{f(u) + f(v) | uv ∈ E(G)}, that is, strf(G) is the maxi-mum edge label of G and the strength str(G) of a graph G itself is str(G) = min{strf(G) | f is a numbering of (G)}. They proved necessary and sufficient conditions for the strength of a graph G of order n to meet str(G) = 2n −2β(G) + 1 and str(G) = n + δ(G) = 2n −2β(G) + 1, where β(G) and δ(G) denote the independence number and the minimum degree of G, respectively. They answer open problems posed by Gao, Lau and Shiu, and the earlier result leads us to determine a formula for the strength of graphs containing a particular class of graphs as a subgraph. They also extend what is known in the literature about k-stable properties. In Ichishima, López, Muntaner-Batle, Takahashi present a sharp lower bound for the strength of a graph in terms of its domination number as well as its (edge) covering and (edge) independence number.
They also provide a necessary and sufficient condition for the strength of a graph to attain the earlier bound in terms of their subgraph structure and establish a sharp lower bound for the domination number of a graph under certain conditions. In Ichishima, Muntaner-Batle, and Takahashi study general conditions for graphs new that allow one to determine which graphs have the property that lower and upper bounds for the strength coincide. In Ichishima1, Muntaner-Batle, and Takahashi use the new the electronic journal of combinatorics (2023), #DS6 417 concept of independence number of a graph to determine formulas for the strength of powers of paths and cycles.
In Ichishima, Oshima, and Takahashi define an edge numbering f of a graph G of size m as a labeling that assigns distinct elements of the set {1, 2, . . . , m} to the edges of G. The edge-strength, estr(G), of G is defined as the minimum estrf(G) where estrf(G) = max{f(e1) + f(e2) | e1, e2 are adjacent edges of G} over all edge numberings of G. They present formulas for estr(G) when G is the forest whose components are stars of order at least three and the complete bipartite graph whose partite sets consist of at least two vertices. The edge-strength of a graph G is the strength of the line graph of G, and thus this work extends what was known about the edge-strength and strength.
Fibonacci range labelings A simple finite graph G with p vertices and q edges is said to be a Fibonacci range graph if there is a bijective f : V (G) →{F2, F3, F4, . . . , Fp+1}, where Fi is the ith Fibonacci num-ber, such that the induced edge labels given by f ∗(uv) = ⌈(f(u)2 + f(v)2)/(f(a) + f(b))⌉ or f ∗(uv) = ⌊(f(u)2 + f(v)2)/(f(a) + f(b))⌋can be assigned subscripts 1, 2, . . . , n in such a way that the ratios e −1/e2, e2/e3, . . . , en−1/en approach the golden ratio φ = 1.618 . . .
. In Odyuo and Mercy classified k-copies of shell graph C[n, (n3)]k with a union of K2 for n = 4, 2K2 for n = 6, and 3K2 for n = 8 having a common end vertex joined to the apex of the shell are Fibonacci range graphs.
Super arithmetic graceful labelings A connected graph is said to be highly irregular if each of its vertices is adja-cent only to vertices with distinct degrees.
Let Hi(m, n) denote the bipartite graph of order n = 2m (m ≥2) having partite sets, V1 = {u1, u2, . . . , um} and V2 = {v1, v2, . . . , vm} and edge set E(H) = {uivj : 1 ≤i ≤m, 1 ≤j ≤m + 1 −i} with degH(ui) = degH(vi) = m + 1 −i for i = 1, 2, . . . , m.
The highly irregular graph H(1) i (m, m) of order 2m + 1 ≥9 is obtained by subdividing the edge u2vm−1 of Hi(m, m) for m ≥4.
An E-super arithmetic graceful labeling of a graph G is a bijection f from the union of the vertex set and edge set to {1, 2, 3, . . . , |V (G) ∪E(G)|} such that the edges have the labels from the set {1, 2, 3, . . . , |E(G)|} and the in-duced mapping given by f(uv) = f(u) + f(v)f(uv) for uv ∈E(G) has the range {|V (G) ∪E(G)| + 1, |V (G)E(G)| + 2, . . . , |V (G)| + 2|E(G)|}.
In Anubala and new Ramachandran proved that Hi(m, m), H(1) i (m, m), and chain of even cycles C4,n, C6,n are E-super arithmetic graceful.
E-super arithmetic graceful labelings In Sekar and Varatharajaprumal define a (p, q) graph G to be E-super arithmetic graceful if there exists a bijection f from V (G) ∪E(G) to {1, 2, . . . , p + q} such that f(E(G)) = {1, 2, . . . , q}, f(V (G) = {q + 1, q + 2, . . . , q + p}, and the induced mapping f ∗given by f ∗(uv) = f(u) + f(v)f(uv) for uv ∈E(G) has the range {p + q + 1, p + q + 2, . . . , p + 2q}. They provided E-super arithmetic graceful labelings for four families of cycle related graphs.
the electronic journal of combinatorics (2023), #DS6 418 Gaussian tribonacci r-graceful labelings For natural number r, Sunitha and Sheriba say an injective function φ from V (G) new to {0, ki, 1+ki, 2+ki, . . . , GTq+r−1} for all k, where GTq+r−1 is the (q +r −1)th Gaussian tribonacci number in the Gaussian tribonacci sequence is a Gaussian tribonacci r-graceful labelings if the induced edge labeling φ∗: E(G) →{GT1, GT2, . . . , GTq+r−1} is bijective.
A graph that admits Gaussian tribonacci r-graceful labeling is called Gaussian tribonacci r-graceful A graph is said to be Gaussian tribonacci arbitrarily graceful if it is Gaussian tribonacci r-graceful for all r.
They prove that paths, combs, coconut trees, regular caterpillars, bistars, and the subdivision of bistars are Gaussian tribonacci arbitrarily graceful for all nontrivial cases.
Radio heronian mean k-graceful labelings For a graph G and a positive integer k a mapping g from V (G) to {k, k+1, . . . , k+N −1} is a radio heronian mean k-labeling if for any two distinct vertices s and t of G, d(s, t) + ⌈(g(s) + g(t) + p g(s)g(t) )/3⌉≥1 + D, where D is the diameter of G.
The radio heronian mean k-number of g, rrhmnk(g), is the maximum number as-signed to any vertex of G.
The radio heronian mean number of G, rhmnk(g), is the minimum value of rhmnk(g) taken overall radio heronian mean labelings g of G.
If rhmnk(g) = |V (G)| + k −1, G is said to be a radio heronian mean k-graceful.. In new Sunitha and Rani investigated the radio heronian mean number of Tn ⊙K1 (obtained by joining a single pendent edge to each vertex of a triangular snake), ITn ⊙K1 (ITn is the irregular triangular snake), and Qn ⊙K1 (Qn is the quadrilateral snake). In new Sunitha and Rani proved that the degree splitting of graphs of combs, rooted trees, graphs obtained from a path by attaching a pendant edge to every internal vertex of the path (hurdle graphs) and twigs admit radio heronian mean k-graceful labelings.
Heronian Dd-distance mean labelings In Bosco and Dinesh introduced the concept of radio heronian Dd-distance new mean labeling graphs as follows.
A radio Heronian Mean Dd-distance label-ing of a connected graph G is an injective map f from the vertex set to the positive integers such that for two distinct vertices u and v of G we have DDd(u, v) + ⌈(f(u) + p f(u)f(v) + f(v))/3⌉≥1 + diamDd(G), where where DDd(u, v) denotes the Dd-distance between u and v and diamDd(G) denotes the Dd-diameter of G. The radio Heronian Dd-distance number of f (rhmnDd(f)) is the maximum label assigned to any vertex of G. The radio heronian Dd-distance number of G (rhmnDd(G) is the minimum value of rhmnDd(f) taken over all radio heronian Dd distance labeling of G. They determined radio heronian mean Dd-distance number of complete graphs, paths, stars, subdivisions of stars, and fans. In Bosco and Dinesh determined the new radio heronian mean Dd-distance number of degree splitting stars, flower graphs, shadow graphs, switching wheels, and switching helms.
Super fibonacci graceful antimagic graph the electronic journal of combinatorics (2023), #DS6 419 In Shanmuga and Amuthavalli introduced the concept of a super fibonacci new graceful antimagic graph G with q edges as one that has an injective function φ : V (G) →{0, F2, F3, . . . , Fq+1} (Fi is the ith Fibonacci number) such that the induced edge labeling φ∗(uv) = |φ(u)φ(v)| is a bijection onto the set {F2, F3, . . . , Fq+1}, the vertex sums are pairwise distinct, and all the edges labels are distinct. They proved that following graphs have such labelings: graphs obtained by joining n copies of C4 +e and m copies K2 with a common vertex (clematis flower); graphs obtained by joining n copies of C3 and m copies K2 with a common vertex (cherry blossom), and graphs obtained by joining n copies of C6 with a common vertex (rose).
Radio even mean graceful graphs A radio even mean graceful labeling of a connected graph G is a bijec-tion π from the vertex set V (G) to {2, 4, 6, . . . , 2|V |} satisfying the condition (d(s, t) + ⌈φ(s) + φ(t)⌉)/2 ≥1 + diam(G) for every s and t in V (G).
A graph that admits radio even mean graceful labeling is called a radio even mean graceful graph.radio even mean. In Brindha Mary, Raj, and Jayasekaran investigate the new radio even mean graceful labeling on degree splitting of some special graphs.
Sudoku number of graphs In Jeyaseeli, Lau, Shiu, and Arumugam introduced a new concept of Sudoku new number as follows. Let G be a graph with chromatic number k. The minimum number of vertices in G have been pre-colored such that this pre-coloring can be extended uniquely to a k-coloring of G is called the Sudoku number of the graph G. They gave the Sudoku numbers of the amalgamations of complete graphs, tadpole graphs, lollipop graphs, and graphs obtained by identifying every alternate edge of C2n with an edge of a distinct complete graph Km and its related graph.
Prime graceful In 2018 Selvarajan and Subramoniam introduced the notion of prime grace- new ful graphs as follows. A graph G with m vertices and n edges is said to be a prime graceful graph if there is an injection φ from the vertices of G to {1, 2, . . . , k} where k = min(2m, 2n) such that gcd(φ(u), φ(v)) = 1 and the induced injective function φ∗ from the edges of G to {1, 2, . . . , k1} defined by φ(uv) = |φ(u) −φ(v)|, the resulting edge labels are distinct. They proved paths, cycles, stars, friendship graphs, bistars Bn,n, C4 ∪Pn, Km,2 and Km,2 ∪Pn admit prime graceful Labelings.
In 2021 Panma and Rochanakul introduced a different version for prime- new graceful graphs (Selvarajan and Subramoniam did not use a hypen).
A graph G with n vertices and m edges is said to be prime-graceful if there is an injection ψ : V (G) →{1, 2, . . . , m + 1}, where gcd(ψ(u), ψ(v)) = 1 for all e = {u, v} ∈E(G) and the induced function ψ∗: E(G) →{1, 2, . . . , m} defined as ψ∗(e) = |ψ(u) −ψ(v)| is injective. They showed stars, bistars Bn,n, bistars Bn,p−2, where p is an odd prime, K2,n, the electronic journal of combinatorics (2023), #DS6 420 tristars SL(3, n), triangular books B(3) n , and some spiders are prime-graceful. They also study k-prime-graceful labelings where the range of ψ is extended to min{kn, km} for k > 1.
LH graphs In Farisa and Parvathy introduced the notion of LH graphs as follows. A graph new G with n vertices is said to have an LH labeling if there exists a bijective function f : V (G) to {1, 2, 3, . . . , n} such that the induced map f ∗from E(G) to set of natural numbers defined by f ∗(uv) = LCM(f(u), f(v))/HCF(f(u), f(v)) is injective (LCM and HCF denote the least common multiple and highest common factor, respectively). A graph that admits an LH labeling is called an LH graph. They proved that theta graphs, sparklers, H graphs, and the splitting graphs of paths and combs are LH graphs. In they proved that combs, helms, triangular snakes, quadrilateral snakes, twigs, and new bistars are LH graphs. Obaed and Arif proved that cycles, cactus graphs, coconut new trees, combs, triangular snakes, and K2 ⊙Cn admit LH-labelings.
k-distant edge total labelings; k-distant vertex total labelings; k-distant edge chromatic number In Rana introduced the following graph labelings.
Given an undirected new connected graph G(V, E), a labeling f from V ∪E to {1, 2, . . . , r} is called a k-distant vertex total labeling if |w(x)w(y)| ≥k for every edge xy, where w(x) = f(x) + P(x, y) over all edges xy is the weight of the vertex x. The minimum r for which a graph G has a k-distant vertex total labeling is called the total k-distant chromatic number of G and is denoted by γkd(G). A labeling f from V ∪E to {1, 2, . . . , r} is called a k-distant edge total labeling if for any two adjacent edges e1 and e2 we have |w(e1) −w(e2)| ≥k, where w(e) = f(e) + f(x) + f(y) and x, y ∈V are the two end vertices of the edge e.
The minimum r for which a graph G has a k-distant edge total labeling is called the total k-distant edge chromatic number of G and is denoted by γ′ kd(G). Rana determines the k-distant vertex total labeling for paths, cycles, complete graphs, stars, bistars, friendship graphs, and the value of the parameter γkd for these graph classes. Moreover, k-distant edge total labelings for paths, cycles and stars are inveatigated and an upper bound of γkd and a lower bound of γ′ kd are presented.
Super totient labelings A positive integer n is called super totient if the residues of n that are prime to n can be partitioned into two disjoint subsets of equal sums. An injective function f defined on the vertices of a graph G(V, E) to a subset of integers is called a super totient labeling if the function f ∗from E to the natual numbers defined by f ∗(xy) = f(x)f(y) assigns a super totient number for all edges xy of G.
A graph that admits such a labeling is called a super totient graph. In Mahmood and Ali proved that every graph new with at least one edge admits a super totient labeling. If the range of a super totient labeling f is {1, 2, . . . , |V |}, then f is said to be a restricted super totient labeling. They the electronic journal of combinatorics (2023), #DS6 421 provided super totient labelings for friendship graphs, wheels, complete graphs, and complete bipartite graphs. In Harrington and Wong classified all restricted super new totient labelings of complete bipartite graphs, trees, wheel graphs, and friendship graphs.
Furthermore, they introduced the notation of the sum index of a graph G, which is the minimum positive integer k such that there exists an injective vertex labeling f of G with k being the cardinality of the range of f. They showed that the sum index is related to super totient labelings, and determined the sum index of complete graphs, complete bipartite graphs, and certain families of trees including caterpillar graphs.
Fuzzy vertex-antimagic labelings A fuzzy vertex anti-magic (FVAM) labeling of a graph G is a 1-1 correspondence f : E(G) to {1, 2, . . . , |E(G)} such that for any two different vertices v and w, the total of the labels on edges incident to v are distinct from the total of labels on edges incident to w.
In Selvarasu and Murugan proved that a graph that is formed by joining new two terminal edges to every internal vertex of the path (twing) and combs admit FVAM labeling by providing an algorithm.
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J. Jeba Jesintha and K. Subashini, On gracefulness of vertex switching of even cycles, International J. Pure Appl. Math.,(IJPAM), 109(10) (2016) 11-18. http: //acadpubl.eu/jsi/2016-109-si/10/2.pdf J. Jeba Jesintha and K. Subashini, On cordial labeling of vertex switching of cycles, Middle East J. Sci. Res.,(MEJSR), 25(04) (2017) 791-794.
org/mejsr/mejsr25(4)17/13.pdf J. Jeba Jesintha and K. Subashini, Gracefulness in the path union of vertex switch-ing of odd cycle graphs, Math. Sci. Internat. Res. J(IMRFJ), 07(Special Issue 2) (2018) 71-75.
J. Jeba Jesintha and K. Subashini, Cordial labeling for new class of graphs, IAETSD J. for Adv. Res. in Appl. Sci.,(IAETSDJARAS), 05(04) (2018), 98-101.
J. Jeba Jesintha and K. Subashini, Gracefulness in the path union of vertex switching of even cycles in increasing order, South East Asian J. Math. and Mathl. Sci., (SEAJMAMS), 14(3) (2018)15-20. articleinfo.php?articleid=317&tag=seajmams J. Jeba Jesintha and K. Subashini, Cordiality in the path union of vertex switch-ing of cycles in increasing order, South East Asian J. Math.Mathl. Sci., (SEA-JMAMS), 14(3) (2018) 111-118.
php?articleid=327&tag=seajmams J. Jeba Jesintha and K. Subashini, Gracefulness of the join of cycle with different graphs, Internat. J of Pure and Appl. Math., (IJPAM), 120 (Special Issue)(08) (2018) 49-57. J. Jeba Jesintha and K. Subashini, Gracefulness in the path union of vertex switch-ing of cycles, Internat. J Res. Eng., IT Social Sci., (IJREISS), 09(Spl. issue) (2019) 06-10. J. Jeba Jesintha and K. Subashini, On gracefulness of quadrilateral snakes In-ternat. J. Res. Eng. IT Social Sci., (IJREISS), 09(Spl. issue) (2019) 15-23.
J. Jeba Jesintha and K. Subashini, On gracefulness in the path union and cycle of PmθSn, Bull. Pure Appl. Sci. Sec. E. Math. Stat.,(BPAS) 38E(Special Issue)(2S) (2019) 85-91. upload/dynamic2/85-91-bpas-e-dec.2019-special%20issue.pdf the electronic journal of combinatorics (2023), #DS6 525 J. Jeba Jesintha and K. Subashini, Cordiality in the path union of vertex switching of wheel graphs in increasing order,Bull. Pure Appl. Sci. Sec. E. Math. Stat., (BPAS) 38E(Special Issue)(2S) (2019) 79-84. admin/upload/dynamic2/79-84-bpas-e-dec.2019-special%20issue.pdf J. Jeba Jesintha and K. Subashini, Cordiality in the path union and cycle of jelly fish graphs J. Inf. Comp. Sci.,(JOICS) 10(7) (2020) 740-749.
org/gallery/ics-3920.pdf J. Jeba Jesintha, K. Subashini, and Allu Merin Sabu, Graceful labeling on a new family of graphs, Bull. Pure Appl. Sci. Sec. E. Math. Stat.,(BPAS) 38E(Special Issue)(2S) (2019) 1-6. dynamic2/1-6-bpas-e-dec2019-special-issue.pdf J. Jeba Jesintha, K. Subashini, and Allu Merin Sabu, Graceful labeling on twig dia-mond graph with pendant edges, Bull. Pure Appl. Sci. Sec. E. Math. Stat., (BPAS) 39E(2) (2020) 188-192. 188-192-bpas-e-dec.2020.pdf J. Jeba Jesintha, K. Subashini, and H. Aysha Siddiqa, Gracefulness in the path union of vertex switching of odd and even cycle graphs J. Maharaja Sayajirao Univ. Baroda, 54(2)(XI) (2020) 140-144.
J. Jeba Jesintha, K. Subashini, and J. R. Rashmi Beula, Graceful labeling of some new graphs, Bull. Pure Appl. Sci. Sec. E. Math. Stat.,(BPAS) 38E(Special Issue)(2S) (2019) 60-64. 60-64-bpas-e-dec.2019-special%20issue.pdf J. Jeba Jesintha and G. Sethuraman, A new class of graceful rooted trees, A new class of graceful rooted trees, J. Discrete Math. Sci. Cryptogr., 11 (2008) 421-435.
J. Jeba Jesintha and G. Sethuraman, Generation of graceful trees, Internat. J.
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J. Jeba Jesintha and G. Sethuraman, All arbitrarily fixed generalized banana trees are graceful, Math. Comput. Sci., 5(1) (2011) 51-62.
J. Jeba Jesintha and K. Subashini, Gracefulness in path union of vertex switching of even cycles in increasing order, South East Asian J. Math. Math. Sci., 14(3) (2018) 15-20.
J. Jeba Jesintha, N. K. Vinodhini, and S. Divya Lakshmi, Antimagic labeling of triangular book graph and double fan graph, Bull. Pure Appl. Sci., Sect. E Math.
Stat., 42E(1) (2023) 27-30.
the electronic journal of combinatorics (2023), #DS6 526 K. Jeya Daisy, R. Santrin Sabibha, P. Jeyanthi, and M. Z. Youssef, k-Product cordial behaviour of union of graphs, J.
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Soc., 28(1) (2022) 1-7. k-Product_Cordial_Behaviour_of_Union_of_Graphs K. Jeya Daisy, R. Santrin Sabibha, P. Jeyanthi and M. Z. Youssef, k-Product cordial labeling of Napier bridge graphs, Nepal J. Math. Sci., 3(2) (2022) 59-70.
doi:10.3126/njmathsci.v3i2.49201 K. Jeya Daisy, R. Santrin Sabibha, P. Jeyanthi, and M. Z. Youssef, k-Product cordial labeling of cone graphs, Internat. J. Math. Combin., 2 (2022) 72-80. http: //fs.unm.edu/IJMC/k-Product_Cordial_Labeling_of_Cone_Graphs.pdf K. Jeya Daisy, R. S. Sabibha, P. Jeyanthi, and M. Z. Youssef, k-product cordial labeling of fan graphs, TWMS J. App. Eng. Math., 13(1) (2023) 11-20.
CORDIAL_LABELING_OF_FAN_GRAPHS K. Jeya Daisy, R. Santrin Sabibha, P. Jeyanthi, and M. Z. Youssef, k-Product cordial labeling of cone graphs, Internat. J. Math. Combin., 2 (2022) 72-80 https: //fs.unm.edu/IJMC/k-Product_Cordial_Labeling_of_Cone_Graphs.pdf K. Jeya Daisy, R. Santrin Sabibha, P. Jeyanthi, and M. Z. Youssef, k-Product cordial labeling of product of graphs, Discrete Math., Algorithms and Appl., 16(1) (2024) 2250187 (8 pages).
K. Jeya Daisy, R. S. Sabibhab, P. Jeyanthi, and M. Z. Youssef, k-Product cordial la-beling of splitting graph of star graphs, Commun. Combin., Cryptogr. & Computer Sci., 1 (2024) 68-80. k-Product_Cordial_Labeling_of_Splitting_Graph_of_Star_Graphs P. Jeyanthi, k-equitable mean labeling, J. Algor. Computation, 44(1) (2013) 21-30.
P. Jeyanthi, M.Selvi and D.Ramya, Skolem difference mean labeling of union of graphs, Util. Math., 111 (2019) 139-150.
P. Jeyanthi, N. Angel Benseera and M. Immaculate Mary, On totally magic cordial labeling, SUT J. Math., 49(1) (2013) 13-18.
P. Jeyanthi and N. Angel Benseera, Totally vertex-magic cordial labeling, SUT J.
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P. Jeyanthi and N. Angel Benseera, Totally magic cordial labeling of one-point union of n copies of a graph, Opuscula Math., 34(1) (2014) 115-133.
P. Jeyanthi and N. Angel Benseera, Totally magic cordial labeling of some graphs, J. Algorithms Comput., 46 (2015) 1-8.
the electronic journal of combinatorics (2023), #DS6 527 P. Jeyanthi, N. Angel Benseera, and I. Cahit, Totally magic cordial labeling of mPn and mKn, Proyecciones J. Math., 35(4) (2016) 371-381.
P. Jeyanthi, N. Angel Benseera and G.-C, Lau, On k-totally magic cordial labeling of graphs, Disc. Math., Algor. Appl., 7(3)(2015) 1550024, 7 pp.
P. Jeyanthi and N. Angel Benseera, Totally magic cordial deficiency of some graphs, Utilitas Math., 105 (2017) 87-94.
P. Jeyanthi, N. Angel Benseera, and G.-C. Lau, Hypo-k-totally magic cordial la-beling of graphs, Proyecciones J. Math., 34(4) (2015) 347-355.
P. Jeyanthi and R. Gomathi, Odd mean labeling of chain of graphs, Electron. Notes Disc. Math., 48 (2015) 157-160.
022 P. Jeyanthi and R. Gomathi, Analytic odd mean labeling of super subdivision and H-super subdivision of graphs, J. Emerging Tech. Innov. Res. (JETIR), 6(6) (2019) 541-551. P. Jeyanthi and R. Gomathi, Analytic odd mean labeling of corona graphs, Malaya J. Mat., 8(3)(2020) 898-902.
P. Jeyanthi, R. Gomathi, and G.-C. Lau, Analytic odd mean labelings of some graphs, Palestine J. Math., 8(2) (2019) 392-399.
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Results_on_Analytic_Odd_Mean_Labeling_of_Graphs P. Jeyanthi, R. Gomathi, G. C. Lau, and W. C. Shiu, Analytic odd mean labeling of union and identification of some graphs, Proyecciones J. Mathematics, 42(5) (2023) 1141-1155.
P. Jeyanthi and K. Jeya Daisy, Zk-magic labeling of subdivision graphs, Discrete Math., Algor. Appl., 8(3) (2016) 1650046 (19 pages).
P. Jeyanthi and K. Jeya Daisy, Certain classes of magic graphs, J. Graph Labeling, 4(1) (2018) 38-47.
P. Jeyanthi and K. Jeya Daisy, Zk-magic labeling of some families of graphs, J.
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the electronic journal of combinatorics (2023), #DS6 528 P. Jeyanthi and K. Jeya Daisy, Zk-Magic labeling of star of graphs, Proyecciones J. Math., 39(1) (2020) 31-51.
P. Jeyanthi and K. Jeya Daisy, Zk-Magic labeling of open star of graphs, Bull. Int.
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P. Jeyanthi, K. Jeya Daisy, and A. Semaničová-Feňovčíková, Zk-Magic labeling of path union of graphs, CUBO J. Math., 21(2) (2019) 15-35.
P. Jeyanthi and K. Jeya Daisy, Some results on Zk-magic labeling, Palestine J.
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P. Jeyanthi and K. Jeya Daisy, Zk-magic labeling of cycle of graphs, Internat. J.
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P. Jeyanthi, R. Gomathi and G.-C. Lau, Analytic odd mean labeling of some standard graphs, Palestine J. Math., 8(1) (2019) 444-450. sites/default/files/papers/PJM_Dec_2019_444to450.pdf P. Jeyanthi, R. Gomathi, and G.-C. Lau, Analytic odd mean labeling of square and H graphs, Internat. J. Math. Combin., Special Issue 1 (2018) 61-67.
P. Jeyanthi, R. Kalaiyarasi and D. Ramya, Centered triangular mean graphs, Inter. J. Math. Comb., 1 (2019) 126-133. publication/332230070_Centered_Triangular_Mean_Graphs P. Jeyanthi, R. Kalaiyarasi, D. Ramya, and T. Saratha Devi, Some results on Skolem odd difference mean labeling, Proyecciones J. Math., 35(4) (2016) 407-417.
P. Jeyanthi and A. Maheswari, Vertex equitable labeling of cycle and path related graphs, Util. Math., 98 (2015) 215-226.
P. Jeyanthi and A. Maheswari, Some results on vertex equitable labeling, Open J.
Discrete Math., 2 (2012) 51-57. NEW Nov. 2016 P. Jeyanthi and A. Maheswari, 3-product cordial labeling, SUT J. Math., 48 (2) (2012) 231-240.
P. Jeyanthi and A. Maheswari, 3-product cordial labeling of some graphs, Internat.
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P. Jeyanthi and A. Maheswari, Vertex equitable labeling of transformed trees, J.
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P. Jeyanthi and A. Maheswari, One modulo three mean labeling of graphs, Amer.
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P. Jeyanthi and A. Maheswari, Vertex equitable labeling of graphs, Graph Theory Notes New York, LXVII (2014) 34-42.
P. Jeyanthi and A. Maheswari, Further results on vertex equitable graphs, Elec-tron. Notes Disc. Math., 48 (2015) 161-164.
endm.2015.05.023 P. Jeyanthi and A. Maheswari, 3-product cordial labeling of star graphs, Southeast Asian Bull. Math., 39(3) (2015) 429-437.
P. Jeyanthi and A. Maheswari, Vertex equitable labeling of cyclic snakes and bistar graphs, J. Scientific Res., 6(1) (2014) 79-85.
P. Jeyanthi and A. Maheswari, Odd vertex equitable even labeling of cycle re-lated graphs, CUBO A Math. J., 20 (02) (2018) 13-21. doi:10.4067/S0719-06462018000200013 P. Jeyanthi, A. Maheswari, and P. Pandiaraj, One modulo three mean labeling of cycle related graphs, Internat. J. Pure and Appl. Math., 103(4)(2015) 625-633.
P. Jeyanthi, A. Maheswari, and P. Pandiaraj, On one modulo three mean labeling of graphs, J. Discrete Math. Science & Cryptography, 19:2 (2016) 375-384.
P. Jeyanthi, A. Maheswari, and P. Pandiaraj, One modulo three geometric mean graphs, J. Algor. Comput., 50(1) (2018) 101-108.
P. Jeyanthi, A. Maheswari, and P. Pandiaraj, One modulo three mean labeling of transformed trees, Proyecciones J. Math., 35(3) (2016) 277-289.
P. Jeyanthi, A. Maheswari and M. VijayaLaksmi, Odd vertex equitable even la-beling of graphs, Proyecciones J. Math., 36(1) (2017) 1-11.
conicyt.cl/pdf/proy/v36n1/art01.pdf P. Jeyanthi, A. Maheswari, and M. VijayaLaksmi, New results on vertex equitable labeling, J. Algebra Combin. Discrete Structures Appl., 3(2)(2016) 97-104. https: //dergipark.org.tr/en/download/article-file/267792 P. Jeyanthi, A. Maheswari and M. VijayaLaksmi, Vertex equitable labeling of union of cyclic snake related graphs, Proyecciones J. Mathematics, 35(1) (2016) 177-186.
the electronic journal of combinatorics (2023), #DS6 530 P. Jeyanthi, A. Maheswari, and M. VijayaLaksmi, Vertex equitable labeling of double alternate snake graph, J. Algor. Comput., 46 (2015) 27-34.
ut.ac.ir/article_7923_76d1a6298f68c081c967627653edc287.pdf P. Jeyanthi, A. Maheswari and M. VijayaLaksmi, Vertex equitable labeling of cycle and star related graphs, J. Scientific Res., 7(3)(2015) 33-42.
banglajol.info/index.php/JSR/article/view/22810 P. Jeyanthi, A. Maheswari, and M. VijayaLaksmi, Vertex equitable label-ings of super subdivision graphs, Scientific Internat., 27(5) (2015) 3881-3883.
284182322_VERTEX_EQUITABLE_LABELING_OF_SUPER_SUBDIVISION_GRAPHS/ links/564ee37a08aefe619b103a26.pdf P. Jeyanthi, A. Maheswari, and M. VijayaLakshmi, 3-Product cordial labeling of some snake graphs, Proyecciones J. Math. 38(1) (2019) 13-30.
P. Jeyanthi, A. Maheswari, and M. VijayaLaksmi, Further results on 3-product cordial labeling, Proyecciones J. Math., 38(2) (2019) 191-202.
P. Jeyanthi, A. Maheswari, and M. VijayaLakshmi, Vertex switching in 3-product cordial graphs, J. Algor. Computation, 50(1) (2018) 185-193.
P. Jeyanthi, A. Maheswari, and M. VijayaLaksmi, Vertex equitable labeling of cyclic snake related graphs, Proyecciones J. Math., 37(4) (2018) 613-625.
equitable_even_labeling_of_ladder_graphs P. Jeyanthi, A. Maheswari, and M. Vijayalaksmi, 3-product cordial labeling of some snake graphs, Proyecciones J. Math., 38 P. Jeyanthi, A. Maheswari, and M. Vijayalakshmi, Odd vertex equitable even labeling of ladder graphs, Jordan J.
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P. Jeyanthi, N. T. Muthuraja, A. Semaničová-Feňovčíková and S. J. Dharshikha, More classes of cycle super antimagic graphs, Australasian J. Combin., 67(1) (2017) 46-64.
P. Jeyanthi, S. Philo, and Kiki A.Sugeng, Odd harmonious labeling of some new families of graphs, SUT J. Math., 51(2)(2015), 53-65.
the electronic journal of combinatorics (2023), #DS6 531 P. Jeyanthi and S. Philo, Odd harmonious labeling of some cycle related graphs, Proyecciones J. Math., 35(1) 2016) 85-98.
cl/index.php/proyecciones/article/view/1236/949 P. Jeyanthi and S. Philo, Odd harmonious labeling of plus graphs, Bull. Int. Math.
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P. Jeyanthi and S. Philo, Odd harmonious labeling of some new graphs, Southeast Asian Bull. Math., 43 (2019) 509-523.
net/publication/329626405_Cordial_Labeling_of_Generalized_Jahangir_ Graph P. Jeyanthi and S. Philo, Odd harmonious labeling of some super subdivision graphs, Proyecciones J. Math., 38(1) (2019) 1-11. pdf/proy/v38n1/0716-0917-proy-38-01-00001.pdf P. Jeyanthi and S. Philo, Odd harmonious labeling of complete bipartite graphs, Southeast Asian Bull. Math., 46 (2022) 23-34.
P. Jeyanthi and S. Philo, New results on odd harmonious labeling of graphs, TWMS J. App. Eng. Math., 12(4) (2022) 1301-1310. images/articles/vol.12.no.4/14.pdf P. Jeyanthi and S. Philo, Odd harmonious labeling of line and disjoint union of graphs, Chinese J. Math. Sci., (1)(1) (2021) 61-68.
P. Jeyanthi and S. Philo, Odd harmonious labeling of subdivided shell graphs, Internat. J. Comput. Sci. Eng., 7(5) (2019) 77-80.
P. Jeyanthi and S. Philo, Odd harmonious labeling of step ladder graphs, Util.
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P. Jeyanthi and S. Philo, Odd harmonious labelings of certain graphs, JASC: J. Appl. Sci. Comput., VI(IV) (2019) 1224-1232.
net/profile/P_Jeyanthi2/publication/332320709_ODD_HARMONIOUS_ LABELING_OF_CERTAIN_GRAPHS/links/5cadc56492851ccd4ac087b5/ ODD-HARMONIOUS-LABELING-OF-CERTAIN-GRAPHS.pdf P. Jeyanthi and S. Philo, Some results on odd harmonious labeling of graphs, Bull.
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P. Jeyanthi and S. Philo, Odd harmonious labeling of some classes of graph, CUBO, A Math. J., 22(3) (2020) 299-314.
P. Jeyanthi, S. Philo and M. Z. Youssef, Odd harmonious labeling of grid graphs, Proyec. J. Math., 38(3) (2019) 411-428. v38n3/0716-0917-proy-38-03-00411.pdf the electronic journal of combinatorics (2023), #DS6 532 P. Jeyanthi and D. Ramya, Some results on super mean graphs, Util. Math., 92 (2013) 149-160.
P. Jeyanthi and D. Ramya, On construction of mean graphs, J. Scientific Research 5(2) (2013) 265-273.
P. Jeyanthi and D. Ramya, Super mean graphs, Util. Math., 96 (2015) 101-109.
P. Jeyanthi, D. Ramya, and A. Maheswari, On mean labeling of some trees, Util.
Math., 93 (2014) 161-191.
P. Jeyanthi and D. Ramya Super mean labeling of some classes of graphs, Inter-national J. Math. Combin., 1 (2012) 83-91.
P. Jeyanthi, D. Ramya, and M. Selvi, Even vertex odd mean labeling of transformed trees, TWMS J. App. Eng. Math., 10(2) 2020 338-345.
P. Jeyanthi, D. Ramya, and P. Thangavelu, On super mean graphs, AKCE J.
Graphs Combin., 6(1) (2009) 103-112.
P. Jeyanthi, D. Ramya, and P. Thangavelu, Some constructions of k-super mean graphs, Inter. J. Pure Applied Math., 56 (2009) 77-86.
P. Jeyanthi, D. Ramya, and P. Thangavelu, On super mean labeling of some graphs, SUT J. Math., 46 (2010) 53-66.
P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling, J. Scientific Research, 5 (3) (2013) 457-467.
P. Jeyanthi and T. Sarada Devi, On edge pair sum labeling of graphs, Internat. J.
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P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling of spider graph, J. Algor.
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P. Jeyanthi and T. Sarada Devi, Some edge pair sum graphs, J. Discr. Math.
Science & Crypt., 18(5) (2015) 481-493.
P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling of some Cartesian product graphs, Discrete Math., Algor. Appl., 8 (23) (2016) 16500245 (10 pages).
P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling of butterfly graph with shell order, Malaya J. Matematik, 4(2) (2016) 205-210.
P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling of cycle related graphs, J. Algor. Comput., 48 (2016) 57-68.
P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling of some subdivision of graphs, Bull. Internat. Math. Virtual Inst., 8 (2018) 67-80.
the electronic journal of combinatorics (2023), #DS6 533 P. Jeyanthi and T. Saraha Devi, Some new families of edge pair sum graphs, Palettine J. Math., 8(2) (2019) 61-69.
P. Jeyanthi and T. Sarada Devi, Edge pair sum labeling of some classes of graphs, J. Discr. Math. Sci. Crypt.,24(4) (2021) 1075-1084..
P. Jeyanthi, T. Sarada Devi, and G.-C Lau, Some results on edge pair sum labeling, Electronic Notes Disc. Math., 48 (2015) 169-173. j.endm.2015.05.025 P. Jeyanthi, T. Sarada Devi, and G.-C. Lau, New results on edge pair sum graphs, Internat. J. Math. Appl., 4(1-B) (2016) 57-65.
P. Jeyanthi, T. Saratha Devi, and G.-C. Lau, Edge pair sum labeling of WT(n : k) tree, Global J. Pure and Appl. Math., 11(3) (2015) 1523-1539.
P. Jeyanthi and P. Selvagopal, Magic covering of chain of an arbitrary 2-connected simple graph, SUT J. Math., 44 (2008) 23-31.
P. Jeyanthi and P. Selvagopal, More classes of H-supermagic graphs, Internat. J.
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P. Jeyanthi and P. Selvagopal, H-supermagic strength of some graphs, Tokyo J.
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P. Jeyanthi and P. Selvagopal, Some C4-supermagic graphs, Ars Combin., 111 (2013) 129-136.
P. Jeyanthi and P. Selvagopal, Supermagic coverings of some simple graphs, Inter-nat. J. Math. Combin., 1 (2011) 33-48.
P. Jeyanthi, P. Selvagopal, and S. Soma Sundaram, Some C3-supermagic graphs, Util. Math., 89 (2012) 357-366.
P. Jeyanthi, M. Selvi, and D. Ramya, Skolem difference mean labeling of discon-nected graphs, Proyecciones J. Math., 36(2) (2017) 347-361.
P. Jeyanthi, M. Selvi, and D. Ramya, Skolem difference mean labeling of union of graphs, Util. Math., 111 (2019) 139-150.
P. Jeyanthi, M. Selvi, and D. Ramya, Restricted triangular difference mean graphs, Proyecciones J. Math., 39(2) (2020) 275-286.
P. Jeyanthi, M. Selvi, and D. Ramya, Odd mean labeling of T ˆ oCn and T ˜ oCn, Util.
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P. Jeyanthi, M. Selvi, and D. Ramya, Centered triangular difference mean graphs, Util. Math., 112 (2019) 125-138.
the electronic journal of combinatorics (2023), #DS6 534 P. Jeyanthi, M. Selvi and D. Ramya, Triangular difference mean graphs, Internat.
J. Math. Combin., 2 (2020) 109-117.
P. Jeyanthi and A. Sudha, Total edge irregularity strength of disjoint union of wheel graphs, Elect. Notes Disc. Math., 48 (2015) 175-182. 10.1016/j.endm.2015.05.026 P. Jeyanthi, M. Selvi and D. Ramya, One modulo N-difference mean graphs, Pales-tine J. Math., 12(3) (2023) 163-169.
P. Jeyanthi, M. Selvi, and D. Ramya, New results on one modulo N-difference mean graphs, Open J. Discrete Math.,13 (2023) 100-112.
P. Jeyanthi and A. Sudha, Total edge irregularity strength of wheel related graphs, J. Graph Labeling, 2(1) (2015) 45-57.
P. Jeyanthi and A. Sudha, On the total irregularity strength of wheel related graphs, Utilitas Math., 110 (2019) 131-134.
P. Jeyanthi and A. Sudha, Some results on edge irregular total labeling, J. Internat.
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the electronic journal of combinatorics (2023), #DS6 690 Index Symbols (α1, α2, . . . , αk)-cordial, 319 (α1, α2, . . . , αk)-cordial graph, 319 (α1, α2, . . . , αk)-cordial labeling, 319 (a, d)-F-antimagic, 235 (a, d)-1-vertex-antimagic vertex, 238 (a, d)-distance antimagic, 231 (a, d) −D antimagic, 237 (a, r)-geometric, 385 (k, d)-Heronian mean, 364 (k, d)-Skolem graceful, 83 (k, d)-graceful labeling, 80 (k, d)-hooked Skolem graceful, 39 (m, n)-gon star, 278 < Km1,n1, . . . , Kmt,nt >, 35 A-antimagic, 224 A-cordial graph, 100 A-magic, 209 B(n, r, m), 25 B∗ n,n, 314 Bm, 23 Bn,n, 314 C(G1, G2, . . . , Gn), 35 C(n · G), 35 CDm(G), 384 Cn, 121 Pn, 104 Cm ∗Cn, 394 C+ n,k, 17 C(t) n , 19 Circ(n, s), 232 D-distance, 231 D-distance antimagic, 231 D-distance magic, 237 D-weight, 237 DStn), 36 D2(G), 85, 114 Dm(G), 87, 135, 384 E-super vertex magic, 186 Ek-cordial, 99 Ek-regular, 187 F-geometric mean, 356 Fn, 49 Fln, 294, 322 G ⊙H, 21 G ⊗H, 81 G ▽H, 53, 56 G∗, 95, 104 G′, 35 G1 ⊕G2, 54 G1 ˆ o G2, 284 G1[G2], 24 H-E-super magic, 197 H-E-super magic decomposable, 197 H-cordial, 98 H-covering, 160, 245, 380 H-decomposable, 197 H-graph of Pn, 342 H-graph of a path, 351 H-magic, 193 H-supermagic strength, 197 H-union, 115 H −V -super magic decomposable, 197 Hn, 342 Hn-cordial, 98 Hn-graph, 121 JFn, 394 Jn, 394 KP(r, s, l), 69 K(m) n , 25 M(G), 38, 95 Mm(G), 135 Mn, 23 P(G, f), 38 P(n, k), 32 P(n · G), 35 P t n, 36 P t n(tn · H), 36 K1,n, 123 P k n, 31 the electronic journal of combinatorics (2023), #DS6 691 Pt(G), 96 Pt(u, v), 96 Pa,b, 33, 339 Pln, 37 R-ring-magic, 212 Rm(G), 38 S(G1, G2, . . . , Gn), 35 S(n · G), 35 Sm, 23 Sn, 243 Sm,n, 116 Splm(G), 135 St(n), 29 St(n1, n2, . . . , nk), 83 T(G), 109 T(Pn), 38 Tp-tree, 80 W(t, n), 15 Γ-distance magic, 202 α-labeling eventually, 58 free, 62 near, 63 strong, 61 weakly, 61, 72 α-deficit, 58 α-labeling, 19, 50, 72, 84 α-mean labeling, 339 α-size, 60 α-valuation, 50 β-valuation, 5 χslat(G), 241 δ-optimal, 275 δ-optimal summable, 275 γ-labeling, 70 ˆ ρ-labelings, 67 ρ-labeling, 68 ρ-valuation, 68 ρ+-labeling, 69 ρ⋆, 68 θ-labeling, 69 ˜ ρ-labelings, 72 a-vertex consecutive bimagic labeling, 217 a-vertex consecutive magic labeling, 216 a-vertex multiple magic, 147 b-edge consecutive magic labeling, 216 b-edge multiple magic, 147 d-antimagic, 229 d-graceful, 60 f-permutation graph, 38 grac(G), 41 k-cordial labeling, 100 k-difference cordial, 332 k-even mean graph, 350 k-even mean labeling, 350 k-even sequential harmonious, 123 k-fold, 176 k-graceful, 78 k-graceful digraph, 82 k-magic, 149 k-mean graph, 342 k-multilevel corona, 142 k-prime, 284 k-prime cordial, 337 k-product cordial, 316 k-ranking, 390 minimal, 390 k-remainder cordial, 103 k-super mean, 342 k-total product cordial, 316 k-totally magic cordial, 214 k-ubiquitously graceful, 10 k-vertex amalgamation, 56 kCn-snake, 20, 67 linear, 20 m index, 108 m-gracefulness, 73 m-mirror graph, 135 m-shadow graph, 135 m-splitting graph, 135 mG, 27 n-cone, 15 n-cube, 24, 50 n-point suspension, 15 nth quadrilateral snake, 395 n · ⃗ Cm, 41 the electronic journal of combinatorics (2023), #DS6 692 q-labeling number, 414 r-distant irregular, 382 r-distant irregularity strength, 382 rHs(G), 380 rn ⋆(f), 305 s(G), 369 sg(G), 377 t-fold, 56 t-join (m, n) kite, 404 t-ply graph, 96 tdis(G), 383 ts(G), 378 w-graph, 158 w-tree, 158 y-tree, 12 0-magic, 214 1-vertex bimagic, 207 2-link fence, 55 3-equitable prime cordial, 337 3-product cordial, 314 3-total super sum cordial graph, 318 3-total super sum cordial labeling, 318 A abbreviated double tree of T, 150 absolutely harmonious graph, 122 additively (a, r)-geometric, 385 adjacency matrix, 68 almost graceful labeling, 67 almost-bipartite graph, 70 alpha-number, 172 alternate hexagonal snake, 20 alternate quadrilateral snake, 284, 312, 329 alternate shell, 95 alternate triangular snake, 284, 312, 329 amalgamation, 196 analytic mean graph, 363 antimagic edge chromatic, 229 antimagic orientation, 227 antipodal balanced, 206 antiprism, 197, 236, 260, 376 apex, 17, 113 arank number, 391 arbitrarily distance antimagic, 231 arbitrarily graceful, 78 arbitrary supersubdivision, 33, 95 arithmetic, 125 armed crown, 327 B balance index set, 110 balanced cordial, 104 balanced distance graphs, 205 bamboo tree, 9, 85 banana tree, 12, 75, 85 barycentric subdivision, 35 bent ladder, 391 beta combination graph, 405 beta-number, 71 bi-odd sequential, 120 bicomposition, 69 bigraceful graph, 39 bipartite labeling, 60 bistar, 162, 168 block, 19, 171 block graph, 408 block-cut-vertex graph, 171 block-cutpoint, 54 block-cutpoint graph, 19 book, 6, 18, 23, 157 generalized, 281 stacked, 24 boundary value, 57 bow graph, 17 broom, 144 C Cn, 209 cactus k-angular, 89 triangular, 19 Cartesian product, 22, 294 cartoon flower graph, 144 caterpillar, 9, 50, 63, 75, 118, 163 caterpillar cycle, 390 cells, 54 the electronic journal of combinatorics (2023), #DS6 693 chain graph, 54, 172 chain of cycles, 18 chain tree, 55 chord, 16 chordal ring, 186, 237 circulant graph, 145 circular lobster, 391 closed m-shadow of a graph, 384 closed helm, 15 coalescence, 54 cocktail party graph, 135, 186, 273 coconut tree, 289 color number, 240 comb tree, 391 combination graph, 404 combs, 34 complete n-partite graph, 91, 266 bipartite graph, 19, 24 graph, 24 tripartite graph, 25 complete mixed generalized sausage graph, 221 complete star, 373 component, 285 composition, 24, 89, 294 conjunction, 311 consecutive radio labeling, 306 consecutively super edge-magic, 167 consecutively super edge-magic deficiency, 167 contra harmonic mean, 356 converse skew product, 56 convex polytope, 198, 259 cordial graph, 91 cordial labeling, 89 corona, 21, 157 covering, 246 critical number, 57 crown, 21, 116, 118, 270, 388 cube, 23, 39 cube divisor cordial, 322, 323 cubic graph, 174 cycle, 5, 273 cycle of a graph, 212 cycle of graphs, 35, 340 cycle with a Pk-chord, 16 cycle with parallel Ck−chord, 17 cycle with parallel Pk chords, 16 cyclic G-decomposition, 63 cyclic decomposition, 68 cylinders, 198 D Dd-length, 309 decomposition, 5, 50, 63, 67, 70 deficiency edge-magic, 170 super edge-magic, 170 degree splitting graph, 331 degree-magic, 142 difference cordial labeling, 329 difference graph, 398 direct product, 203 directed edge-graceful, 302 directed graceful graph, 42 dis(G), 383 disjoint union, 27 distance k-antimagic, 230 distance antimagic, 230 labeling, 222 distance magic labeling, 201 distance reflexive strength, 380 divisor cordial, 319 divisor graph, 407 dodecahedron, 39 double alternate hexagonal snake, 20 double alternate quadrilateral snake, 284 double alternate quadrilateral snake, 312, 329 double alternate triangular snake, 284, 312, 329 double coconut trees, 289 double cone, 15 double fans, 34 double graph of G, 143 the electronic journal of combinatorics (2023), #DS6 694 double hexagonal snake, 20 double path union, 82 double quadrilateral snake, 284, 312, 329 double star, 150 double step grid graph, 36 double tree, 150 double triangular snake, 284, 312, 329, 341 dragon, 18 duplication of a vertex, 35, 311 duplication of an edge, 35, 96, 311 Dutch t-windmill, 19 Dutch windmill, 148 E EBI(G), 112 edge H-irregularity strength, 380 edge amalgamation, 281 edge bimagic total , 207 edge comb product, 137 edge even graceful labeling, 73 edge irregular total labeling, 370 edge irregularity strength, 381 edge linked cyclic snake, 339 edge magic graceful, 160 edge magic strength, 148 edge pair sum, 367 edge parity, 61 edge product cordial labeling, 327 edge reduced integral sum number, 272 sum number, 272 edge trimagic total labeling, 175 edge-antimagic graceful, 238 edge-antimagic total, 226 edge-balance index, 112 edge-coloring k-vertex weighting, 416 edge-covering, 380 edge-decomposition, 63 edge-friendly index, 110 edge-graceful deficiency, 295 edge-graceful spectrum, 296 edge-magic index, 176 edge-magic injection, 162 edge-magic total labeling, 149 edge-odd graceful, 86, 88 edge-prime graph, 288 ehs(G, H), 380 elegant, 127 elegant labeling, 127 elem. parallel transformation, 80 elementary transformation, 52, 159 envelope graph, 112 EP-cordial graph, 316 EP-cordial labeling, 316 Eulerian graph, 113 even 2a-sequential, 139 even 1-vertex bimagic, 208 even graceful, 56 even mean labeling, 350 even vertex equitable even, 396 even vertex magic total, 189 even vertex odd mean, 351 even-even, 88 even-odd harmonious, 140 exclusive sum labeling, 274 exclusive sum number, 274 extended w-tree, 158 extended edge vertex cordial labeling, 107 extended jewel graph, 361 F face, 198, 259 face irregular total k-labeling, 378 fan, 49, 127, 141, 154, 157, 169, 186, 198, 277 fence, 55 FI(G), 107 Fibonacci graceful, 76 firecracker, 12 flag, 92, 123, 302 flower, 15, 184, 277 folded hypercube graphs, 205 forest, 170 free α-labeling, 62 the electronic journal of combinatorics (2023), #DS6 695 friendly index set, 107 friendship graph, 19, 89, 169, 184, 186, 198, 274 full r-ary tree, 12 full edge-friendly index, 110 full friendly index set, 111 full hexagonal caterpillars, 56 full product-cordial index, 314 fully magic, 211 fully product-cordial, 313 functional extension, 150 fuzzy, 422 G gamma-number, 40 gear graph, 15 generalized book, 281 bundle, 97 fan, 97 wheel, 97 generalized kCn-snake, 339 generalized antiprism, 252 generalized caterpillar, 34 generalized edge linked cyclic snake, 339 generalized helm, 184, 373 generalized Jahangir graph, 184 generalized prisms, 305 generalized sausage graph, 220 generalized shackle, 245 generalized spider, 34 generalized web, 15, 184 geometric mean 3-equitable, 369 geometric mean cordial, 369 glutting number, 325 Golomb ruler, 26 graceful almost super Fibonacci, 77 graceful center, 62 graceful game, 74 graceful graph, 5 gracesize, 60 gracious k-labeling, 62 gracious labeling, 62 graph, 154, 331, 348, 420 (α1, α2, . . . , αk)-cordial, 319 (ω, k)-antimagic, 228 (a, d)-F-antimagic, 235 (a, d)-antimagic, 233 (a, d)-distance antimagic, 231 (a, r)-geometric, 385 (k + 1)-equitable mean, 368 (k, λ)-magically total labeling, 208 (k, d)-Heronian mean, 364 (k, d)-Skolem graceful, 83 (k, d)-arithmetic, 124 (k, d)-balanced, 82 A-antimagic, 225 A-cordial, 100 A-vertex magic, 146 C-geometric, 357 D-distance, 231 D-distance antimagic, 231 E-cordial, 301 E-super arithmetic graceful, 418 E-super vertex magic, 186 Ek-cordial, 99 G-distance magic, 202 G-snake, 20 H-cordial, 98 H-elegant, 129 H-group magic, 146 H-harmonious, 129 Hk-cordial, 99 Hn-cordial, 98 S-magic, 146 V -cordial, 325 V -super vertex magic, 147 Vk super vertex in-magic, 218 Vk-super vertex magic, 188 ∆-optimum summable, 274 Γ irregular, 377 θ-Petersen, 297 a-vertex multiple magic, 147 b-edge multiple magic, 147 d-graceful, 60 the electronic journal of combinatorics (2023), #DS6 696 f-permutation, 38 g-graph, 273 k-antimagic, 228 k-balanced, 106, 115 k-difference cordial, 332 k-distance magic, 202 k-edge-magic, 149 k-enriched fan, 18 k-even edge-graceful, 296 k-magic, 149 k-modular multiplicative, 387 k-multilevel corona, 142 k-prime cordial, 337 k-prime total, 285 k-product cordial, 316 k-shifted antimagic, 223 k-super cube root cube mean, 347 k-super graceful, 73 k-super root square mean, 346 k-ubiquitously, 10 kth Fibonacci prime, 290 k(G) snake, 133 m-level wheel, 298 m-mirror, 135 m-shadow, 135 m-splitting, 135 n-uniform, 373 n-uniform cactus chain, 373 p-distance magic, 203 t-m-Zumkeller, 410 t-uniform homeomorph, 96 w-graph, 158 w-tree, 158 (1,0,0)-F-face magic mean, 358 2-odd, 413 3-equitable prime cordial, 337 3-product cordial, 314 3-total super sum cordial labeling, 318 F-root square mean, 345 k-odd edge mean, 349 aboreale star, 331 absolutely antimagic, 223 absolutely harmonious, 122 additively (a, r)-geometric, 385 additively (a, r)∗-geometric, 385 almost-bipartite, 70 alternate hexagonal snake, 20 alternate quadrilateral snake, 284, 312, 329, 406 alternate shell, 95 alternate triangular snake, 284, 312, 329, 406 analytic mean, 363 analytic odd mean, 363 anti skolem, 344 antimagic, 219 arbitrarily graceful, 78 arbitrary calendula, 352 arithmetic, 125, 386 armed crown, 327 armed helms, 92 balanced distance, 205 balloon, 121 barbell, 371 basket, 379 bent ladder, 391 beta combination, 405 bi-odd sequential, 120 bicomposition, 69 biconditional cordial, 107 bicyclic, 297, 310 bigraceful, 39 block, 408 bow, 17, 320 braid, 175 braided star, 285, 330 branched-prism, 252 broken wheel, 109 broom, 144, 187 butterfly, 123, 297, 302 cactus, 373 calendula, 193 caterpillar cycle, 390 centered triangular difference mean, 354 centered triangular mean, 354 centroidal mean, 343 the electronic journal of combinatorics (2023), #DS6 697 chain, 197, 381 cherry blossom, 420 chordal ring, 186, 237 circle, 134 circulant, 145 circular ladders, 98 circular lobster, 391 circulent, 232 clematis flower, 420 closed distance magic, 204 closed helm, 15 cocktail party, 135, 186, 273 coconut tree, 289 comb tree, 391 combs, 34 complete, 24 complete mixed generalized sausage graph, 221 composition, 24 cone, 315 conservative, 142 contra harmonic mean, 356 cordial, 169 countable infinite, 156 cube root cube mean, 347 cube sum, 401 cycle butterfly, 335 cycle with parallel chords, 28 cyclic snake, 349 dandelion, 305 decomposable, 166 degree-magic, 142 diamond, 79 difference, 398 difference cordial, 329 difference perfect square cordial, 403 directed, 7 directed Γ-distance magic, 205 directed edge-graceful, 302 disconnected, 27 distance, 309 distance k-antimagic, 230 distance antimagic, 230 distance irregular k, 380 divisor, 407 divisor 3-equitable, 409 double alternate hexagonal snake, 20 double alternate quadrilateral snake, 284, 312, 329 double alternate triangular snake, 284, 312, 329 double arrow, 212 double coconut tree, 289 double divisor cordial, 324 double fans, 34 double graph of G, 143 double hexagonal snake, 20 double quadrilateral snake, 284, 312, 329 double squid, 137 double step grid, 36 double triangular, 20 double triangular snake, 284, 312, 329 double wheels, 24 double-sided arrow, 366 dragon, 366 dumbbell, 123, 297 edge bimagic harmonnious, 175 edge corona path, 195, 236 edge linked cyclic snake, 339 edge magic graceful, 160 edge pair sum, 367 edge product cordial, 327, 329 edge vertex prime, 289 edge-friendly, 106 edge-magic, 174 edge-prime, 288 EP-cordial, 316 even 2a-sequential, 139 even edge-graceful, 299 even sum, 413 even vertex odd mean, 351 even-multiple subdivision, 93 extended w-tree, 158 extended jewel, 361 extended vertex edge additive cordial, the electronic journal of combinatorics (2023), #DS6 698 107 extra Skolem difference mean, 353 face integer edge cordial, 326 face-magic toridal, 206 fan, 49 festoon, 123, 302 Fibonacci, 318 Fibonacci divisor cordial, 319 Fibonacci graceful, 76 Fibonacci mean antimagic, 227 Fibonacci range, 418 firecracker, 136 flower, 294, 322 flower snark, 335 friendship, 19 fully product-cordial, 313 Gaussian tribonacci r-graceful, 419 Gaussian tribonacci arbitrarily graceful, 419 generalize shacke, 245 generalized caterpillar, 34 generalized cycle star, 289 generalized edge corona, 221 generalized edge linked cyclic snake, 339 generalized helm, 184, 371, 373 generalized Jahangir, 171, 184 generalized sausage, 220 generalized spider, 34 generalized web, 15, 184 generalized wheel, 313 globe, 108, 326 graceful, 5 graceful antimagic, 238 gracefulness, 73 graph-block chain, 34 grid-like, 52 group S3, 336 group S3 cordial remainder, 412 group S3 mean cordial, 359 hair-kC4 graph, 133 half-Zumkeller, 410 Halin, 149 Hamming-graceful, 117 handicap distance d-antimagic, 222 Harary, 251 harmonic mean, 355 harmonious, 6 hefty V4-magic, 147 Heronian mean, 364 hetro-cordial, 326 hexagonal snake, 20 highly irregular, 418 highly vertex prime, 286 holiday star, 330 homo-cordial, 326 hybrid quadrilateral snake, 132 hyper strongly multiplicative, 386 ideal magic, 163 indexable, 126 integer cordial, 105 integer edge cordial, 326 integral sum, 267 irregular quadrilateral snake, 330 irregular triangular snake, 329 Jahangir, 91 jelly fish, 160, 406 jewel, 340, 394 join, 30 join sum, 35, 134 kayak paddle, 19 king, 414 kite, 18, 168 Knödel, 186, 279 komodo dragon with many tails, 38 komodo dragons, 37 Kusadama, 330 ladder, 22 Lehmer-4 mean, 356 LH, 421 lict, 86 lilly, 105, 400 line-graceful, 387 linear cactus, 130 litact, 86 local edge antimagic, 229 lollipop, 370 the electronic journal of combinatorics (2023), #DS6 699 long brush, 202 lotus, 79, 318 Lucas divisor cordial, 323 lucky edge, 411 middle, 95 millipede, 81 minimally k-equitable, 116 mirror, 38 mixed generalized sausage, 220 modified arrow, 137 modified double arrow, 137 modular multiplicative, 387 multiple shell, 114 Mycielskian, 202 Mycieski, 114 Napier bridge, 315 nested triangle, 98 node-graceful, 83 north star, 318 odd (a, d)-antimagic, 238 odd antimagic, 238 odd Fibonacci mean, 349 odd prime, 289 odd sum, 121 odd vertex equitable even, 395 odd vertex magic, 144 one modulo N graceful, 75 one modulo N difference mean, 362 one modulo three square mean, 362 one-sided arrow, 366 ordered, 226 orientable Gamma-distance magic, 205 pagoda, 358 pair difference cordial, 412 pair mean, 368 pair sum, 366 pair sum modulo, 368 para-squares cactus, 373 para-squares cactus chain, 373 parity combination cordial, 325 path-block chain, 34 pentagonal sum, 407 perfect, 408 perfect super edge-magic, 161 Perrin graceful, 77 plus, 37, 136 polar grid, 56 prime, 278, 284 prime distance, 413 prime graceful, 41, 420 prime odd mean, 349 prime-graceful, 420 pronic graceful, 64 pseudo-magic, 147 pumpkin, 194 pyramid, 79, 136 radio mean, 359 reduction, 391 relaxed mean, 343 remainder cordial, 103, 324 replicated, 38 restricted k-mean, 342 restricted triangular difference mean, 363 rigid ladders, 340 root square mean, 347 rose, 420 SD-divisor, 409 SD-harmonious, 417 SD-prime cordial, 283 semi Jahangir, 175, 212 semi-edge-prime, 288 semi-magic, 141 semismooth graceful, 81 semitotal-block, 399, 402 set graceful, 391 set sequential, 391 set-cordial, 325 shacke, 244 shackle, 196 shadow, 85, 114 sharp, 226 shell-butterfly, 17 shell-type, 17 shellflower, 17 shipping, 403 the electronic journal of combinatorics (2023), #DS6 700 signed cordial, 415 signed product cordial, 334, 415 simply sequential, 389 Skolem difference Lucas mean, 352 Skolem difference mean, 352 Skolem even difference mean, 354 Skolem labeled, 83 Skolem-graceful, 82 slanting ladder, 121, 212 smooth graceful, 37 sparkler, 296 sparklers, 123 splitting, 34 square difference, 399, 402 square harmonious, 123 square product, 417 square sum, 399 squid, 137 SSG(n), 86, 135 star, 23 star extension, 128 star of, 95, 104 stem-lotus, 131 step grid, 36, 340 step ladder, 135 strong edge-graceful, 295 strong face plane, 222 strong magic, 163 strong sum, 267 strong super edge-magic, 161 strongly c-elegant, 131 strongly k-indexable, 169 strongly 1-harmonious, 169 strongly felicitous, 131 strongly harmonious, 118 strongly indexable, 125 strongly multiplicative, 386 subdivided shell, 75, 86, 135 sum divisor cordial, 321 sun, 243 sunflower, 92 super (a, d)-F-antimagic, 236 super edge magic graceful, 161 super edge-graceful, 58 super fibonacci graceful antimagic, 420 super graceful, 73 super Lehmer-3 mean, 356 super pair sum, 367 super root square mean, 346 super subdivision, 395 super totient, 421 super vertex mean, 342 supermagic, 141 supersubdivision, 32 swastik, 37 tadpoles, 18 theta, 128 theta graph, 34 Toeplitz, 254 torch, 34, 279 tortoise, 344 total, 38, 310 total edge Fibonacci irregular, 377 total mean cordial, 359, 361 total mean labeling, 352 total mixed, 382 total prime, 285 totally antimagic total, 226 totally magic, 190 tri sum perfect square cordial, 416 triangular belt, 175 triangular difference mean, 354 triangular ladders, 314 triangular snake, 19 triangular tree, 50 trimagic harmonious, 175 twing, 422 twisted cylinder, 112, 314 T–IASL signed graph, 393 umbrella, 108, 160 unicyclic, 16 uniform bow, 17 uniformly balanced, 106 uniformly cordial, 104 universal alpha-graceful, 62 universal graceful, 62 the electronic journal of combinatorics (2023), #DS6 701 vanessa, 105 VECN prime, 287, 288 vertex k-prime, 309 vertex even mean, 348 vertex magic, 146 vertex odd divisor cordial, 323 vertex odd graceful, 89 vertex product, 317 vertex switching, 35, 77, 95, 220 vertex-edge neighborhood-prime, 287 weak antimagic, 224 weak magic, 163 weak sum, 275 web graph without a center, 280 weighted-k-antimagic, 230 zero divisor, 322 zero-sum A-magic, 209 zig-zag triangle, 156 Zumkeller, 409, 410 graph labeling, 5 super edge bimagic harmonious, 175 graph-block chain, 34 graphs double-sided step, 366 dragon, 343 exponential, 365 hurdle, 419 lilly, 213 long dumbell, 416 middle, 346 multi-bridge, 223 octopus, 213 one-sided step, 365 path union of n subdivided shell graphs, 401 radio heronian mean k-graceful, 419 radio mean graceful, 360 super-rooted, 52 superstar, 52 vanessa, 213 weighted k-list-antimagic, 230 grid, 22, 79 grid-like graph, 52 group irregularity strength, 377 H half-Zumkeller, 410 Halin graph, 149 Hamming-graceful graph, 117 handicap distance antimagic graphs, 222 handicap incomplete tournament, 222 harmonic labeling mean cordial, 416 harmonic mean, 355 harmonic mean cordial, 416 harmonious graph, 6 harmonious number, 123 harmonious order, 40 Heawood graph, 39, 62 helm, 15, 277 closed, 92 generalized, 92 Heronian mean, 364 Herschel graph, 39, 233 hexagonal lattice, 198 hexagonal snake, 20 holey α-labeling, 67 homeomorph, 111 honeycomb graph, 261 hooked Skolem sequence, 84 host graph, 58 hybrid quadrilateral snake, 132 hypercycle, 272 strong, 272 hypergraph, 145, 176, 228, 272 hyperwheel, 272 I IC-coloring, 389 IC-index, 389 icicle graph, 391 icosahedron, 39 index of cordiality, 96 index of product cordiality, 318 integer-antimagic spectrum, 225 integer-magic spectrum, 149, 211 integral radius, 272 the electronic journal of combinatorics (2023), #DS6 702 integral sum number, 270 tree, 270 integral sum-diameter, 268 irregular crown, 161 irregular fence, 55 irregular labeling, 369 irregular quadrilateral snake, 330 irregular triangle snake, 329 irregularity strength, 369 J jewel graph, 340 join product, 171 join sum, 35 K kayak paddle, 19, 69 kite, 18, 58, 168, 188 Kotzig’s Conjecture, 68 L L-cordial, 104 labeling (α1, α2, . . . , αk)-cordial, 319 (ω, k)-antimagic, 228 (a, b)-consecutive, 296 (a, b; n)-graceful, 40 (a, d)- vertex-antimagic edge, 233 (a, d)-H-antimagic total labeling, 245 (a, d)-1-vertex-antimagic vertex, 238 (a, d)-distance antimagic, 231 (a, d)-edge-antimagic total, 246 (a, d)-edge-antimagic vertex, 246 (a, d)-face antimagic, 259 (a, d)-indexable, 246 (a, d)-vertex-antimagic total, 243 (a, r)-geometric, 385 (k, λ)-magically total labeling, 208 (k, d)-Heronian mean, 364 (k, d)-Skolem, 83 (k, d)-arithmetic, 124 (k, d)-even mean, 350, 351 (k, d)-graceful, 80 (k, d)-hooked Skolem graceful, 39 (k, d)-odd mean, 349 (k, d)-super mean, 355 A-magic, 209 A-vertex magic, 146 C-geometric mean, 357 D-magic, 206 E-cordial, 301 E-sum cordial, 317 F-centroidal mean, 343 F-geometric, 356 G-distance magic, 202 H-E-super magic, 197 H-group magic, 146 H-irregular k, 380 H-magic, 193 P(a)Q(1)-super vertex-graceful, 300 Q(a)P(b)-super edge-graceful, 300 R-ring-magic, 212 V -cordial, 325 V -super vertex in-antimagic total grace-ful labeling, 227 V -super vertex magic, 147 VK-super vertex magic, 187 Vk-super vertex in-magic labeling, 218 ∆-exclusive sum labeling, 274 Γ irregular, 377 Θ-graceful, 77 α-, 50 α-mean, 339 ρ⋆, 68 σ-, 69 a-vertex consec. edge bimagic, 217 a-vertex-consecutive magic, 189 d-antimagic, 229 d-antimagic of type (1, 1, 1), 259 d-graceful, 60 k-DML, 202 k-antimagic, 228 k-balanced, 106 k-cordial, 100 k-distance magic, 202 k-distant edge total labeling, 421 the electronic journal of combinatorics (2023), #DS6 703 k-edge graceful, 296 k-edge-magic, 149 k-equitable, 113, 116 k-even edge-graceful, 296 k-even mean, 350 k-even sequential harmonious, 123 k-graceful, 82 k-indexable, 125 k-labeling, 309 k-mean, 342 k-odd edge mean, 349 k-odd mean, 350 k-prime, 284 k-prime cordial, 337 k-prime total, 285 k-product, 315 k-product cordial, 316 k-remainder cordial, 103 k-sequential, 388 k-sequentially additive, 398 k-super cube root mean labeling, 347 k-super harmonic, 364 k-super labeling, 73 k-super mean, 342, 355 k-total product cordial, 316 k-totally magic cordial, 214 kth Fibonacci prime, 290 m-bonacci graceful, 63 t-m-Zumkeller, 410 t-harmonious, 40 w-sum, 275 (1,0,0)-F-face magic, 358 1-vertex bimagic, 207 1-vertex magic, 201 1-vertex magic vertex, 215 3-equitable prime cordial, 337 3-product cordial, 314 3-total super sum cordial labeling, 318 (k, d)-indexable, 125 absolute differences of cubic and square difference, 414 absolute graceful, 366 absolutely harmonious, 122 additively (a, r)-geometric, 385 additively (a, r)∗-geometric, 385 additively (k, d)-sequential, 398 additively graceful, 124 almost graceful, 67 almost magic, 207 analytic mean, 363 analytic odd mean, 363 antimagic, 219, 227 arbitrarily graceful, 78 arithmetic sequential graceful, 63 balanced, 50, 105 balanced cordial, 104 balanced mean cordial, 361 beta combination, 405 bi-odd sequential, 120 bigraceful, 63 binary magic total, 214 bipartite, 60 C-exponential mean, 365 centered triangular difference mean, 354 centered triangular mean, 354 closed distance magic, 204 complete k-equitable, 116 consecutive, 131 consecutive radio, 306 coprime, 289 cordial, 89 cordial edge deficiency, 104 cordial vertex deficiency, 104 cube difference, 403 cube divisor cordial, 322, 323 cube root mean labeling, 347 cubic roots cordial, 101 difference cordial, 329 difference perfect square cordial, 403 directed Γ-distance magic, 205 directed edge-graceful, 302 distance k-antimagic, 230 distance magic, 201, 215 distance vertex irregular total k-labeling, 383 dividing graceful, 417 the electronic journal of combinatorics (2023), #DS6 704 divisor 3-equitable, 409 divisor cordial, 319 double divisor cordial, 324 edge bimagic, 207 edge bimagic total, 217 edge even graceful, 73 edge irregular k-labeling, 381 edge irregular reflexive k-labeling, 379 edge irregular total, 370 edge numbering, 418 edge pair sum, 367 edge product cordial, 327 edge trimagic total, 175 edge vertex prime, 288 edge-antimagic graceful, 238 edge-friendly, 106 edge-graceful, 293 edge-magic, 153, 174 edge-magic total, 153 edge-odd graceful, 86, 88 edge-prime, 288 elegant, 127 EP-cordial, 316 equitable, 207 even 2a-sequential, 139 even 1-vertex bimagic, 208 even falicitous, 131 even mean labeling, 350 even sequential harmonious, 122 even sum, 413 even vertex equitable even, 396 even vertex magic total, 189 even vertex odd mean, 351 even-even, 88 even-odd harmonious, 140 exponential, 365 extended edge vertex cordial labeling, 107 F-face mean, 365 face integer cordial, 105 face integer edge cordial, 326 face irregular total k-labeling, 378 felicitous, 129 Fibonacci, 318 Fibonacci divisor, 318 Fibonacci graceful, 76 Fibonacci mean antimagic, 226 friendly, 104 fuzzy quotient-3 cordial, 326, 338 Gaussian tribonacci r-graceful, 419 geometric mean, 362 geometric mean 3-equitable, 369 geometric mean cordial, 369 graceful antimagic, 238 graceful data structure, 67 graceful difference, 42 graceful-harmonious, 41 gracefully consistent, 57 gracious, 62 group S3, 336 group S3 cordial, 412 group S3 mean cordial, 359 half-Zumkeller, 410 handicap distance d-antimagic, 222 harmonious numbering, 123 Heronian mean, 364 Heronian mean Dd-distance, 419 hetro-cordial, 326 highly vertex prime, 286 homo-cordial, 325 in-magic total, 165 inclusive distance vertex irregular, 383 indexable, 126 integer cordial, 104 integer cordial edge, 326 interlaced, 50 irregular, 369 L-cordial, 104 Lehmer-4, 356 LH, 421 line-graceful, 387 local antimagic, 240 local edge antimagic, 229 Lucas divisor cordial, 323 lucky, 410 lucky edge, 411 the electronic journal of combinatorics (2023), #DS6 705 magic, 141, 147 consecutive, 198 of type (0,1,1), 198 of type (1,0,0), 199 of type (1,1,0), 198 of type (1,1,1), 198 magic valuation, 153 mean, 338 mean cordial, 358 mean square cordial, 345 minimum coprime, 289 MMD, 409 modular irregular, 384 modular multiplicative divisor, 409 modulo antimagic, 229 near mean, 366 near-elegant, 127 nearly distance magic, 206 nearly graceful, 67 neighborhood-prime, 286 nice (1, 1) edge-magic, 163 non-inclusive distance vertex irregular k-labeling, 383 numbering, 173, 417 odd 1-vertex bimagic, 208 odd Fibonacci mean, 349 odd harmonious, 132, 137 odd mean, 348 odd prime, 289 odd sum, 121 odd vertex equitable, 395 odd-elegant, 129 odd-even, 81, 88 odd-graceful, 67, 84 one modulo N graceful, 75 one modulo N-difference, 363 one modulo three graceful, 75 one modulo three mean, 361 one modulo three root square mean, 362 optimal k-equitable, 116 optimal sum graph, 266 ordered, 226 orientable Γ-distance magic, 205 pair mean, 368 pair sum, 366 parity combination cordial, 325 partial vertex, 106 partitional, 119 Pell graceful, 77 pentagonal sum, 407 perfect super edge-magic, 161 Perrin graceful, 77 polychrome, 129 prime, 277 prime cordial, 333 prime-magic, 145 product antimagic, 264 product cordial labeling, 310 product edge-antimagic, 265 product edge-magic, 265 product integer cordial, 105 product magic, 264 product-irregular, 382 pronic graceful, 64 pronic Heron mean, 365 proper lucky, 410 properly even harmonious, 138 pseudo α, 71 pseudograceful, 70 quotient, 414 radial radio mean, 310 radio ⋆, 305 radio antipodal, 307 radio antipodal geometric mean, 357 radio geometric mean, 357 radio heronian k-labeling, 419 radio mean, 359 radio mean D-distance, 360 radio mean Dd-distance, 309 radio mean graceful, 420 radio mean square, 346 rainbow antimagic vertex, 223 range-relaxed graceful, 74 real-graceful, 40 relaxed mean, 343 remainder cordial, 103, 324 the electronic journal of combinatorics (2023), #DS6 706 restricted k-mean, 342 restricted super totient, 421 restricted triangular difference mean, 363 reverse edge magic, 150 reverse edge-trimagic, 176 reverse super edge magic, 150 reverse super edge-trimagic, 176 rosy, 68 SD-divisor, 409 SD-prime cordial, 283 semi (k, d)-arithmetic, 124 semi harmonious, 124 semi-elegant, 127 sequential, 118 set-ordered odd-graceful, 87 set-ordered strongly k-elegant, 128 sharp ordered, 226 sigma, 201 signed cordial, 415 signed product cordial, 334, 415 simply sequential, 388 Skolem difference Lucas mean, 352 Skolem difference mean, 352 Skolem even difference mean, 354 Skolem even vertex odd difference mean, 353 Skolem odd difference mean, 353 Skolem-graceful, 82 square difference, 402, 403 square difference prime, 402 square divisor cordial, 322 square harmonious, 124 square sum, 399 strength sum, 168 strong edge-graceful, 295 strong rainbow antimagic, 223 strong super edge-magic, 161 strongly (k, d)-indexable, 125 strongly c-harmonious, 118 strongly k-elegant, 127 strongly balanced, 105 strongly edge-magic, 163 strongly even harmonious, 138 strongly graceful, 50, 61 strongly harmonious, 32, 118, 121 strongly indexable, 125 strongly odd harmonious, 132 strongly super edge-graceful, 301 strongly vertex-magic total, 188 sum divisor cordial, 321 sum graph, 266 sum perfect square, 402 super (a, d)-F-antimagic, 236 super (a, d)-edge-antimagic graceful, 238 super (a, d)-vertex-antimagic total, 243 super arithemic graceful, 418 super edge bimagic cordial, 209 super edge-antimagic total, 248 super edge-graceful, 297 super edge-magic, 163 super edge-magic total, 154 super Fibonacci graceful, 76 super geometric mean, 357 super graceful, 73 super Lehmer-3 mean, 356 super mean, 341 super pair sum, 367 super root mean, 346 super vertex in-out-antimagic total, 228 super vertex local antimagic total, 241 super vertex mean, 342 super vertex-graceful, 300 super vertex-magic total, 185 supermagic, 141, 170 total, 225 total H-irregular α, 377 total cordial, 107 total edge product cordial, 328 total irregular total k, 378 total magic cordial, 213 total mean, 352 total mean cordial, 359, 361 total neighborhood prime, 286 total prime, 285 the electronic journal of combinatorics (2023), #DS6 707 total product cordial labeling, 315 totally antimagic total, 226 totally magic, 190 totally magic cordial, 216 totally vertex-magic cordial, 215 tri sum perfect square cordial, 416 triangular difference mean, 354 triangular graceful, 74 triangular sum, 406 universal antimagic, 232 VECN prime, 287, 288 vertex k-prime, 309 vertex balanced cordial, 104 vertex equitable, 393 vertex even mean, 348 vertex in-out-antimagic total, 228 vertex irregular reflexive k-labeling, 379 vertex irregular total, 370 vertex magic total, 147 vertex odd divisor cordial, 323 vertex odd graceful, 89 vertex odd mean, 348 vertex prime, 285 vertex product cordial, 317 vertex-bimagic, 207 vertex-edge neighborhood-prime, 287 vertex-friendly, 111 vertex-graceful, 299 vertex-magic total, 182 vertex-relaxed graceful, 74 weak antimagic, 224 zero-sum A-magic, 209 Zumkeller, 409 Zumkeller cordial, 410 labeling number, 58 lableing radio contra harmonic mean, 416 ladder, 22, 118, 198, 199 Langford sequence, 160 level joined planar grid, 127 lexicographic product, 148 linear cyclic snake, 20 lobster, 11, 68 local antimagic chromatic number, 238 local antimagic vertex coloring, 238 lotus inside a circle, 199 Lucas divisor cordial, 323 lucky number, 411 M Möbius grid, 253 Möbius ladder, 23, 119, 141, 147, 198, 278, 295 magic b-edge consecutive, 189 magic constant, 153, 206 magic square, 141 magic strength, 148, 162 magic sum index, 146 mean cordial, 358 mean graph, 338 mean number, 358 middle graph, 95 minimum coprime number, 289 mirror graph, 38 mixed generalized sausage graph, 220 mod difference digraph, 399 mod integral sum graph, 273 mod integral sum number, 273 mod sum graph, 273 mod sum number, 273 mod sum graph, 275 mod sum number, 275 modular irregularity strength, 384 modulo local antimagic chromatic number, 229 Mongolian tent, 22, 79 Mongolian village, 22, 79 MSG, 273 multigraph, 170, 176 multiple shell, 17 multiply divisor cordial, 327 mutation, 188 mutual duplication, 340 N near α-labeling, 63 nearly distance magic, 206 the electronic journal of combinatorics (2023), #DS6 708 nearly graceful labeling, 67 neighborhood-prime, 286 non-inclusive distance irregularity strength, 383 nullset, 146 numbering, 173 O Oberwolfash Problem, 40 oblong numbers, 415 odd 1-vertex bimagic, 208 odd harmonious, 132, 137 odd mean graph, 348 odd mean labeling, 348 odd prime graph, 289 odd-elegant, 129 odd-even, 81, 88 odd-graceful labeling, 67, 84 olive tree, 9 one modulo N graceful, 75 one modulo three graceful labeling, 75 one-point union, 19, 25, 51, 84, 89, 130 open star of G, 212 optimal sum graph, 266 P pair mean, 368 pair mean graph, 368 pair sum, 366 pair sum graph, 366 parachutes, 233 parallel chord, 109 path, 17, 127 path union, 35, 97 path-block chain, 34 pendent edge, 59 pentagonal number, 407 pentagonal sum labeling, 407 perfect Golomb ruler, 27 perfect system of difference sets, 80 permutation graph, 404 Perrin sequence, 77 Petersen graph, 39 generalized, 31, 89, 157, 169, 183, 234, 243, 246 planar bipyramid, 198 planar graph, 198, 259 Platonic family, 198 plus graph, 37, 136 polar grid, 56 polyminoes, 79 polyominoes, 56 power graph, 39 prime cordial strongly, 336 prime cordial labeling, 333 prime graceful, 41 prime graph, 278, 284 prime labeling, 277 prism, 23, 198, 243, 259 product cordial, 310 product cordial labeling, 310 product graph, 266 product irregularity strength, 382 product-cordial index, 313 product-cordial set, 313 pronic number, 64, 365 properly even harmonious, 138 pseudo α-labeling, 71 pseudo-magic graph, 147 pseudograceful labeling, 70 Q quadrilateral snakes, 20 quotient labeling number, 414 R radial radio mean number, 310 radio k-chromatic number, 308 radio k-coloring, 308 radio k-number, 309 radio star-number, 305 radio antipodal geometric number, 357 radio antipodal labeling, 307 radio antipodal mean number, 307 radio antipodal number, 307 radio contra harmonic mean number, 416 the electronic journal of combinatorics (2023), #DS6 709 radio geometric mean number, 357 radio graceful, 306 radio Heronian Dd-distance number, 419 radio heronian mean number, 419 radio labeling, 304 radio mean D-distance number, 360 radio mean labeling, 359 radio mean number, 359 radio mean square number, 346 radio number, 304 rainbow antimagic connection number, 223 range-relaxed graceful game, 74 range-relaxed graceful labeling, 74 rank number, 390 real sum graph, 266 reflexive H strength, 380 reflexive edge strength, 379 reflexive vertex strenght, 380 reflexive vertex strength, 379 regular graph, 141, 145, 157, 184, 215 regular tree, 55 relaxed mean graph, 343 remainder cordial, 103 replicated graph, 38 representation, 396 representation number, 397 restricted triangular difference mean, 363 rigid ladders, 340 Ringel-Kotzig, 9 root, 92 root-union, 109 S saturated vertex, 267 SD-divisor, 409 SD-prime cordial, 283 semi-edge-prime graph, 288 semismooth graceful, 81 separating set, 391 sequential join, 59 sequential number, 172 set-ordered odd-graceful, 87 shackle, 244 shadow graph, 85, 114 shadow of a graph G, 384 shell, 17, 92, 94, 113 multiple, 17 shell graph, 102 Skolem labeled graph, 83 Skolem sequence, 12, 27 Skolem-graceful labelings, 82 SLAT, 241 smooth graceful, 37 snake, 19, 54 n-polygonal, 75 double triangular, 20 edge linked cyclic, 339 generalized edge linked cyclic, 339 quadrilateral, 53 triangular, 19, 67 snake polyomono, 54 sparse semi-magic square, 190 special super edge-magic, 165 spider, 9 split graph, 221 splitting graph, 34, 85, 331 spum, 266 square difference graph, 402 square divisor cordial, 322 SSG(n), 86 SSG(n), 135 stable set, 33, 38, 54 star, 29, 30, 184, 388 star of G, 316 star of a G, 95, 104 star of graphs, 35 star super edge-magic deficiency, 156 step grid graph, 36, 340 step ladder, 135 straight simple polymonial caterpillars, 56 strength edge magic, 148 magic, 148, 162 maximum magic, 163 strength sum, 168 strong product of graphs, 204 the electronic journal of combinatorics (2023), #DS6 710 strong A-magic, 146 strong k-combination graph, 405 strong k-permutation graph, 405 strong beta-number, 71 strong edge antimagic, 229 strong edge-graceful, 295 strong gamma-number, 40 strong harmonious number, 123 strong product, 381 strong rainbow antimagic connection num-ber, 223 strong sequential number, 172 strong sum graph, 267 strong supersubdivision, 33 arbitrary, 33 strong vertex-graceful, 300 strongly c-harmonious, 118 strongly -graph, 406 strongly antimagic, 228 strongly even harmonious, 138 strongly graceful labeling, 61 strongly harmonious, 32, 118 strongly odd harmonious, 132 strongly prime cordial, 336 strsG, 168 stunted tree, 68 subdivided shell graph, 75, 86, 135 subdivision, 11, 22, 85, 199 Sudoku number, 420 sum graph, 266 mod, 273 mod integral, 273 real, 266 sum index, 422 sum number, 266 sum perfect square, 401 sum graph, 275 sum number, 275 sum-diameter, 268 sunflower, 92, 294 super (a, d)-F-antimagic, 236 super (a, d)-H-antimagic total labeling, 245 super (a, d)-edge-antimagic graceful, 238 super d-antimagic, 229 super edge magic graceful, 161 super edge-magic deficiency, 156 super edge-magic total labeling, 149 super Fibonacci graceful, 76 super geometric mean, 357 super graceful, 73 super labeling, 225 super magic frame, 144 super magic strength, 148, 168 super mean, 341 super mean number, 347 super subdivision, 395 super totient, 421 super vertex local antimagic total chromatic number, 241 super vertex mean, 342 super vertex-magic total, 185 super weak sumgraph, 275 supersubdivision, 32, 97 arbitrary, 33 swastik graph, 37 switching invariant, 282 symmetric product, 24, 54 T tadpoles, 18 tensor product, 81, 87, 106, 319 tes(G), 370 theta graph, 128 theta graphs, 34 toroidal polyhex, 252 torus grid, 23 total H-irregularity strength, 377 total k-distant chromatic number, 421 total edge (vertex) irregular strength, 370 total edge irregularity strength, 370 total edge product cordial labeling, 328 total graph, 38, 109, 310 total labeling, 225 total mean cordial, 359 total mixed, 382 the electronic journal of combinatorics (2023), #DS6 711 total negative, 382 total negative edge, 382 total positive edge, 382 total product cordial, 315 labeling, 317 total product cordial labeling, 315 total stable, 382 total stable edge, 382 totally magic cordial, 216 totally magic cordial deficiency, 214 totally vertex-magic cordial labeling, 215 tr(G), 306 tree, 5, 221, 273 binary, 157 path-like, 159 symmetrical, 9 triameter, 306 triangular graceful labeling, 74 triangular snake, 19 tvs(G), 370 U umbrella, 160 unicyclic graph, 18 uniform cycle snake, 278 uniform-distant tree, 12 union, 26, 153, 168, 170, 184, 270, 278, 285 universal antimagic, 232 unlabeled vertices, 106 V vertex H-irregularity strength, 380 vertex balance index set, 111 vertex balanced cordial, 104 vertex equitable, 393 vertex irregular total labeling, 370 vertex parity, 61 vertex prime labeling, 285 vertex switching, 35, 77, 95, 220, 282 vertex weight, 241 vertex-antimagic total, 226 vertex-graceful, 299 vertex-relaxed graceful labeling, 74 vhs(G, H), 380 W weak sum graph, 275 weak tensor product, 59, 63 weakly α-labeling, 61 web, 15 generalized, 163 weight, 259 weight of vertex, 383 weight universal antimagic, 232 weighted-k-antimagic, 230 wheel, 15, 118, 141, 153, 186, 198, 219, 246 windmill, 25, 92 working vertex, 274 wreath product, 130 Y Young tableau, 22, 79 Z zero-sum A-magic, 209 zero-sum h-magic, 146 the electronic journal of combinatorics (2023), #DS6 712 |
3492 | https://pmc.ncbi.nlm.nih.gov/articles/PMC2681281/ | Association between Vitamin D Deficiency and Primary Cesarean Section - PMC
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J Clin Endocrinol Metab
. 2008 Dec 23;94(3):940–945. doi: 10.1210/jc.2008-1217
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Association between Vitamin D Deficiency and Primary Cesarean Section
Anne Merewood
Anne Merewood
1 Department of Pediatrics (A.M.), Department of Medicine (T.C.C.), Division of Endocrinology, Diabetes, and Nutrition, Boston University School of Medicine, and Division of General Pediatrics (A.M., H.B., M.F.H.), Boston Medical Center, Boston, Massachusetts 02118; Department of Medicine, Physiology, and Biophysics, Boston University School of Medicine (M.F.H.), Boston University Medical Center, Boston, Massachusetts 02118; and Department of Epidemiology and Biostatistics (S.D.M.), University of Illinois Chicago School of Public Health, Chicago, Illinois 60612
Find articles by Anne Merewood
1, Supriya D Mehta
Supriya D Mehta
1 Department of Pediatrics (A.M.), Department of Medicine (T.C.C.), Division of Endocrinology, Diabetes, and Nutrition, Boston University School of Medicine, and Division of General Pediatrics (A.M., H.B., M.F.H.), Boston Medical Center, Boston, Massachusetts 02118; Department of Medicine, Physiology, and Biophysics, Boston University School of Medicine (M.F.H.), Boston University Medical Center, Boston, Massachusetts 02118; and Department of Epidemiology and Biostatistics (S.D.M.), University of Illinois Chicago School of Public Health, Chicago, Illinois 60612
Find articles by Supriya D Mehta
1, Tai C Chen
Tai C Chen
1 Department of Pediatrics (A.M.), Department of Medicine (T.C.C.), Division of Endocrinology, Diabetes, and Nutrition, Boston University School of Medicine, and Division of General Pediatrics (A.M., H.B., M.F.H.), Boston Medical Center, Boston, Massachusetts 02118; Department of Medicine, Physiology, and Biophysics, Boston University School of Medicine (M.F.H.), Boston University Medical Center, Boston, Massachusetts 02118; and Department of Epidemiology and Biostatistics (S.D.M.), University of Illinois Chicago School of Public Health, Chicago, Illinois 60612
Find articles by Tai C Chen
1, Howard Bauchner
Howard Bauchner
1 Department of Pediatrics (A.M.), Department of Medicine (T.C.C.), Division of Endocrinology, Diabetes, and Nutrition, Boston University School of Medicine, and Division of General Pediatrics (A.M., H.B., M.F.H.), Boston Medical Center, Boston, Massachusetts 02118; Department of Medicine, Physiology, and Biophysics, Boston University School of Medicine (M.F.H.), Boston University Medical Center, Boston, Massachusetts 02118; and Department of Epidemiology and Biostatistics (S.D.M.), University of Illinois Chicago School of Public Health, Chicago, Illinois 60612
Find articles by Howard Bauchner
1, Michael F Holick
Michael F Holick
1 Department of Pediatrics (A.M.), Department of Medicine (T.C.C.), Division of Endocrinology, Diabetes, and Nutrition, Boston University School of Medicine, and Division of General Pediatrics (A.M., H.B., M.F.H.), Boston Medical Center, Boston, Massachusetts 02118; Department of Medicine, Physiology, and Biophysics, Boston University School of Medicine (M.F.H.), Boston University Medical Center, Boston, Massachusetts 02118; and Department of Epidemiology and Biostatistics (S.D.M.), University of Illinois Chicago School of Public Health, Chicago, Illinois 60612
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1
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1 Department of Pediatrics (A.M.), Department of Medicine (T.C.C.), Division of Endocrinology, Diabetes, and Nutrition, Boston University School of Medicine, and Division of General Pediatrics (A.M., H.B., M.F.H.), Boston Medical Center, Boston, Massachusetts 02118; Department of Medicine, Physiology, and Biophysics, Boston University School of Medicine (M.F.H.), Boston University Medical Center, Boston, Massachusetts 02118; and Department of Epidemiology and Biostatistics (S.D.M.), University of Illinois Chicago School of Public Health, Chicago, Illinois 60612
Address all correspondence and requests for reprints to: Michael Holick, Professor of Medicine, Division of Endocrinology, Diabetes, and Nutrition, Department of Medicine, Boston University School of Medicine, Room M-1022, 715 Albany Street, Boston Massachusetts 02118. E-mail: mfholick@bu.edu.
Received 2008 Jun 5; Accepted 2008 Dec 12; Issue date 2009 Mar.
Copyright © 2009 by The Endocrine Society
PMC Copyright notice
PMCID: PMC2681281 PMID: 19106272
Abstract
Background: At the turn of the 20th century, women commonly died in childbirth due to rachitic pelvis. Although rickets virtually disappeared with the discovery of the hormone vitamin D, recent reports suggest vitamin D deficiency is widespread in industrialized nations. Poor muscular performance is an established symptom of vitamin D deficiency. The current U.S. cesarean birth rate is at an all-time high of 30.2%. We analyzed the relationship between maternal serum 25-hydroxyvitamin D [25(OH)D] status, and prevalence of primary cesarean section.
Methods: Between 2005 and 2007, we measured maternal and infant serum 25(OH)D at birth and abstracted demographic and medical data from the maternal medical record at an urban teaching hospital (Boston, MA) with 2500 births per year. We enrolled 253 women, of whom 43 (17%) had a primary cesarean.
Results: There was an inverse association with having a cesarean section and serum 25(OH)D levels. We found that 28% of women with serum 25(OH)D less than 37.5 nmol/liter had a cesarean section, compared with only 14% of women with 25(OH)D 37.5nmol/liter or greater (P = 0.012). In multivariable logistic regression analysis controlling for race, age, education level, insurance status, and alcohol use, women with 25(OH)D less than 37.5 nmol/liter were almost 4 times as likely to have a cesarean than women with 25(OH)D 37.5 nmol/liter or greater (adjusted odds ratio 3.84; 95% confidence interval 1.71 to 8.62).
Conclusion: Vitamin D deficiency was associated with increased odds of primary cesarean section.
Vitamin D deficiency is common in pregnant women. A multivariate logistic regression analysis revealed women who were vitamin D deficient were more likely to have a caesarian section than women who were vitamin D sufficient.
At the turn of the 20th century, rickets ran rampant in the newly industrialized cities of Europe and North America, and rachitic pelvis was a common cause of death in childbirth (1). Cesarean sections became established, in part, to manage this condition: “… malformed pelvises often prohibited normal delivery. As a result the rate of cesarean section went up markedly.” (2) Although rickets virtually disappeared with the discovery of the hormone vitamin D and its subsequent addition to milk, recent reports suggest its reemergence, (3,4) and that vitamin D deficiency is widespread in industrialized nations (5,6,7). Meanwhile, research into vitamin D deficiency and awareness of its range of acute and chronic consequences have proliferated (8,9,10,11,12). Poor muscular performance (5,11,12,13,14,15,16) is an established symptom of vitamin D deficiency. The term rachitic pelvis has fallen into disuse, but an association has been noted between cesarean birth and a narrow pelvis (17).
The current U.S. cesarean birth rate is 30.2% (18), a record high for the nation (18), up from 5% in 1970, and characterized by steep increases in primary as well as repeat cesareans (19). Common reasons for cesareans in industrialized nations include dystocia (20) and failure to progress (21). Recent research suggests that maternal calcium status plays a role both in preterm labor (22) and in the initiation of labor (23). This analysis assessed the relationship between maternal vitamin D status (serum 25-hydroxyvitamin D [25(OH)D]) at birth and primary cesarean section.
Subjects and Methods
Recruitment
Women were enrolled between March 21, 2005, and March 20, 2007, on the postpartum unit at Boston Medical Center, an urban teaching hospital in Boston, MA, within 72 h of giving birth.
Enrollment was evenly distributed over time to ensure data were representative of season because seasonal sunlight exposure affects vitamin D status (3). Women were ineligible if they had spent more than 2 months away from Boston during pregnancy; if they were not of black, white, or Hispanic race/ethnicity (due to low numbers of women available in other racial/ethnic groups); if they did not speak English, French, or Spanish; if the infant was admitted to intensive care, premature; or if the mother had a history of parathyroid, renal, or liver disease or was using illegal drugs. Women having a repeat cesarean were also excluded from this analysis because of the strong causal relationship between primary and repeat cesareans. Enrolled women answered a questionnaire on the postpartum unit within 72 h of birth, and venipuncture was performed on the mother before discharge. None of the patients were receiving an iv infusion at the time of the venipuncture.
Diagnostic test
Serum 25(OH)D, accepted as the indicator of vitamin D status in children and adults, was measured by competitive protein binding as described by Chen et al. (24) Because the serum half-life for 25(OH)D is about 21 d, it was minimally influenced by fasting or changes in dietary intake during a short fasting. This method measures both 25(OH)D 2 and 25(OH)D 3 equally well and was compared with liquid chromatography tandem mass spectroscopy with excellent correlation (25). The lower limit of detection was 12.5 nmol/liter (5 ng/ml), and the intra- and interassay coefficients of variation 5.0–10 and 10%–15%, respectively. The reference range was 50–250 nmol/liter (20–100 ng/ml).
Definitions
Certain definitions pertinent to U.S., or more specific regional descriptions, are summarized here. GED indicates graduate equivalency degree, a high school diploma alternative for individuals who do not graduate high school. Healthy Start is perinatal, state-funded insurance for women who are ineligible for other insurance programs; it is frequently used by women of illegal immigration status. The Special Supplemental Nutrition Program for Women, Infants, and Children provides food and nutrition counseling for low income families with children younger than 5 yr. In Massachusetts, applicant income must fall at or below 185% of the federal poverty guidelines.
Outcome measures and data analysis
The dependent variable was maternal vitamin D deficiency, defined as serum 25(OH)D less than 37.5 nmol/liter (15 ng/ml), per the Centers for Disease Control and Prevention definition for adults (26). Recent research has found 25(OH)D less than 37.5 to be an unacceptably low state of deficiency and that less than 50 nmol/liter is considered to be vitamin D deficiency and 51–74 nmol/liter as vitamin D insufficiency (3,7,8,11,12). Race/ethnicity and skin color were analyzed as two variables because skin color by ethnicity alone can vary from white to dark brown, and skin pigmentation affects vitamin D synthesis (3). Skin color (black, brown, or white) was based on the skin type matrix of Fitzpatrick et al. (27). Prenatal vitamin use was analyzed by any reported use in each trimester (data in Table 1) as well as by frequency of use in each trimester; neither measure was associated with type of delivery. Body mass index (BMI) information before pregnancy was obtained from the medical record. Continuous values of BMI were categorized for analysis (17 to <25, 25 to <30, 30 to <35, 35+ kg/m 2) according to World Health Organization ranges ( Reasons for cesarean were obtained from the mother’s medical record. Response frequencies for categorical variables are presented in Table 1. Continuous variables (maternal age, BMI, and 25(OH)D level) are presented as medians with binomially obtained 95% confidence intervals (CI) in Table 1.
Table 1.
Characteristics of study sample by delivery mode
| | Vaginal, n = 210, n (%) | Primary cesarean section, n = 43, n (%) | P value |
:---: :---: |
| Median age, yr (95% CI) | 26.2 (24.8–27.1) | 24.8 (22.8–28.1) | 0.579a |
| Race/ethnicity | | | 0.006 |
| Black non-Hispanic | 81 (86) | 13 (14) | |
| Caucasian non-Hispanic | 18 (62) | 11 (38) | |
| Hispanic | 111 (85) | 19 (15) | |
| Skin color | | | 0.734 |
| Black | 72 (83) | 15 (17) | |
| Brown | 79 (85) | 14 (15) | |
| White | 57 (80) | 14 (20) | |
| Season of birth | | | 0.648 |
| Spring | 53 (80) | 13 (20) | |
| Summer | 43 (83) | 9 (17) | |
| Fall | 47 (89) | 6 (11) | |
| Winter | 67 (82) | 15 (18) | |
| Infant gender | | | 0.255 |
| Male | 107 (80) | 26 (20) | |
| Female | 103 (86) | 17 (14) | |
| Maternal birthplace | | | 0.023 |
| United States | 141(87) | 21 (13) | |
| Not United States | 69 (76) | 22 (24) | |
| Median maternal BMI, kg/m 2 | 25.0 | 24.0 | 0.634a |
| 95% CI | (24.0–26.1) | (23.1–25.8) | |
| Maternal BMI, kg/m 2 | | | 0.281 |
| 17 to <25 | 101 (80) | 26 (20) | |
| 25 to <30 | 56 (86) | 9 (14) | |
| 30-<35 | 29 (91) | 3 (12) | |
| 35+ | 14 (74) | 5 (26) | |
| Education | | | 0.484 |
| Less than high school | 91 (82) | 20 (18) | |
| High school/GED | 67 (87) | 10 (13) | |
| More than high school | 51 (80) | 13 (20) | |
| Insurance status | | | 0.389 |
| Private | 20 (71) | 8 (29) | |
| Public | 167 (84) | 31 (16) | |
| Healthy Start | 15 (83) | 3 (17) | |
| Other | 7 (88) | 1 (13) | |
| WIC status | | | 0.236 |
| No | 33 (77) | 10 (23) | |
| Yes | 176 (84) | 33 (16) | |
| Marital status | | | 0.421 |
| Single/divorced/widowed | 147 (82) | 33 (18) | |
| Married | 61 (86) | 10 (14) | |
| Any prenatal vitamin use | | | |
| First trimester | | | 0.499 |
| No | 48 (80) | 12 (20) | |
| Yes | 160 (84) | 31 (16) | |
| Second trimester | | | 0.978 |
| No | 24 (83) | 5 (17) | |
| Yes | 185 (83) | 38 (17) | |
| Third trimester | | | 0.978 |
| No | 24 (83) | 5 (17) | |
| Yes | 185 (83) | 38 (17) | |
| | | | (Continued) |
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Table 1A.
Continuous
| | Vaginal, n = 210, n (%) | Primary cesarean section, n = 43, n (%) | P value |
:---: :---: |
| Calcium supplements | | | 0.857 |
| Never | 145 (83) | 29 (17) | |
| Occasionally | 23 (79) | 6 (21) | |
| Almost daily | 41 (84) | 8 (16) | |
| Drank milk in pregnancy | | | 0.083b |
| Yes | 177 (81) | 41 (19) | |
| No | 31 (94) | 2 (6) | |
| Ever used sunscreen | | | 0.073 |
| Yes | 67 (77) | 20 (23) | |
| No | 141 (86) | 23 (14) | |
| Drank alcohol in pregnancy | | | 0.039 |
| Yes | 11 (65) | 6 (35) | |
| No | 198 (84) | 37 (16) | |
| Median maternal 25(OH)D, nmol/liter | 62.5 | 45.0 | 0.007a |
| 95% CI | (57.4–68.2) | (36.5–62.0) | |
| Maternal 25(OH)D, nmol/liter | | | 0.012 |
| 25(OH)D, ≥37.5 | 168 (86) | 27 (14) | |
| 25(OH)D, <37.5 | 41 (72) | 16 (28) | |
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Not all cells total 253 because of missing data. WIC, Special Supplemental Nutrition Program for Women, Infants, and Children.
a
P value determined by Wilcoxon rank sum test. 95% CI for mothers’ median age, BMI; 25(OH)D is binomially obtained.
b
Fisher’s exact test used where expected cell value was less than 5.
The statistical significance of differences in maternal factors associated with cesarean delivery was assessed using Pearson’s χ 2 test. Multivariate logistic regression analysis tested all variables with P< 0.25 in univariate analysis. Backward selection techniques were used to derive the final model, which maintained variables with P ≤ 0.05. Statistical analyses were conducted using Stata/SE 9.2 for Windows (Stata Corp., College Station, TX). The study obtained approval from the Boston University Medical Center Institutional Review Board, and women signed informed consent before participating.
Results
Between March 21, 2005, and March 20, 2007, we enrolled 253 women, of whom 43 (17%) had a primary cesarean section. In this same time period, 370 women refused enrollment, primarily because they did not want blood drawn on themselves or their infant. Based on a random sample of 95 refusers, 51% of enrolled women were Hispanic, whereas only 30% of women who refused had Hispanic race/ethnicity recorded in their medical record (P< 0.001). This difference probably arose from the practice of asking women who are enrolled in the study their race and whether they are Hispanic, which is not done for the record. No other significant differences existed between consented and refusing women with regard to age, infant birth weight, or gestational age. The method of delivery did not differ between mothers who enrolled in the study and those who refused (76% vaginal delivery, and 24% cesarean section delivery, both groups).
Of the 277 women enrolled, 210 had vaginal deliveries and 67 had cesarean deliveries, of which 43 were primary cesareans. This analysis was limited to women with vaginal deliveries or primary cesareans. Thus, of the 253 eligible women enrolled, 43 (17%) had a primary cesarean. Reasons for cesarean included failure to progress (17 of 43); nonreassuring fetal tracing (11 of 43); malpresentation (such as breech) (six of 43); and three each of cephalopelvic disproportion, variable fetal heart rate, and other.
We found that 28% of women with serum 25(OH)D less than 37.5 nmol/liter had a primary cesarean section, compared with only 14% of women with 25(OH)D 37.5 nmol/liter or greater [P = 0.012; unadjusted odds ratio (OR) = 2.43; 95% CI 1.20–4.92]. In addition, women who had cesareans had a lower median 25(OH)D level than women who delivered vaginally (45.0 vs. 62.5 nmol/liter, P = 0.007). Compared with women who had vaginal births, women who had cesareans were also significantly more likely to be Caucasian/non-Hispanic than black and/or Hispanic (P = 0.006); OR 3.81; 95% CI 1.47–9.86), to be U.S. born (P = 0.023; OR 2.14; 95% CI 1.10–4.16), and to have used alcohol in pregnancy (P = 0.039; OR 2.92; 95% CI 1.02–8.38) (Table 1).
In multivariate logistic regression analysis controlling for race, age, education level, insurance status, maternal birthplace, and alcohol use (Table 2), women with vitamin D deficiency (<37.5 nmol/liter) were almost 4 times as likely to have a primary cesarean section as women without deficiency (OR 3.84 95% CI 1.71–8.62). Caucasian women and those who reported any alcohol use during pregnancy also had increased likelihood of primary cesarean. Maternal birthplace was no longer statistically significant. Figure 1 shows the association between mother’s increasing 25(OH)D level in nanomoles per liter and decreasing predicted probability (from multivariate analysis) of having a Cesarean section vs. vaginal delivery. Thus, women with the highest serum 25(OH)D had the lowest probability of requiring a cesarean section.
Table 2.
Multivariate logistic regression analysis: factors associated with having primary cesarean section delivery, n = 250a
| | Adjusted OR | 95% CI | P value |
:---: :---: |
| Mother’s race/ethnicity | | | |
| Black, non-Hispanic | Ref | | |
| Caucasian, non-Hispanic | 4.79 | 1.60–14.4 | 0.005 |
| Hispanic | 1.16 | 0.48–2.88 | 0.733 |
| Mother’s 25(OH)D, nmol/liter | | | |
| ≥37.5 | Ref | | |
| <37.5 | 3.84 | 1.71–8.62 | 0.001 |
| Used alcohol during pregnancy | | | |
| No | Ref | | |
| Yes | 3.21 | 1.02–10.11 | 0.046 |
| Mother’s educational status | | | |
| Less than high school or GED | Ref | | |
| High school or GED | 0.63 | 0.25–1.56 | 0.317 |
| More than high school | 0.75 | 0.27–2.11 | 0.588 |
| Mother’s insurance status | | | |
| Private | Ref | | |
| Public | 0.49 | 0.16–1.56 | 0.230 |
| Healthy start | 0.87 | 0.16–4.83 | 0.874 |
| Other | 0.29 | 0.03–3.13 | 0.307 |
| Mother’s age, yr | 0.97 | 0.91–1.04 | 0.364 |
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The model is adjusted for all variables presented. Ref, Reference category.
a
Women with missing data were excluded from multivariate analysis.
Figure 1.
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Mother’s 25(OH)D level (nanomoles per liter) and predicted probability of primary cesarean section delivery.
Discussion
In our analysis, women who were severely vitamin D deficient [25(OH)D <37.5 nmol/liter] at the time of delivery had almost 4 times the odds of cesarean birth than women who were not deficient. One explanation for our findings is the fact that skeletal muscle contains the vitamin D receptor (8,11). Vitamin D deficiency has been associated with proximal muscle weakness (11) as well as suboptimal muscle performance and strength (5,11,12,13,14,15,16). Moreover, vitamin D deficiency is a possible risk factor for preeclampsia (9,28).
Serum calcium status, which is regulated by vitamin D, plays a role in smooth muscle function in early labor (23,29). Papandreou et al. (23) reported significantly higher serum calcium levels in pregnant women at the time of vaginal delivery compared with term women not in labor or women who did not labor but delivered by scheduled cesarean. It was speculated that the higher serum calcium levels played a role in the mechanism of initiation of labor. Because vitamin D is critically important for the maintenance of calcium homeostasis, it is possible that vitamin D deficiency, which causes a slight lowering of the serum calcium, is related to both skeletal muscle and smooth muscle strength and may play a role in initiation of early labor. It is also possible that vitamin D deficiency might be related to specific types of cesareans (such as cephalopelvic disproportion or failure to progress) than to others (such as breech), although we did not have a large enough sample to be able to analyze this. This would be a critical area for future research.
There are other potential explanations for the association of vitamin D deficiency and cesarean section. The simplest explanation is that the iv hydration would have diluted the blood and give an artificially lower level of 25(OH)D. However, the amount of iv fluids relative to the blood loss is essentially the same. We did a subset analysis of five women in whom we obtained serum 25(OH)D before and after the cesarean section, and they were statistically the same, i.e. the 25(OH)D before surgery was 80 ± 25 nmol/liter and after surgery 97 ± 33 nmol/liter P> 0.1. Vitamin D status is linked to immune status (30,31). Certain infections have been associated with preeclampsia (32), and preeclampsia in turn increases the odds of cesarean (33). Vitamin D deficiency may thus be a marker for a compromised immune system and an associated, higher risk of cesarean.
A study performed in 1994–1995 by Brunvand et al. (34) found no association between vitamin D deficiency at the time of delivery and obstructed labor in a case control study of Indian women giving birth in Karachi. Their findings bear little relevance for the present study however; outcomes were measured only for cesareans due to cephalopelvic disproportion; the sample consisted of largely undernourished impoverished women, and 71% of study participants were severely vitamin D deficient [25(OH)D <30 nmol/liter]. In addition, the paper does not satisfactorily clarify the use of the term cephalopelvic disproportion (as opposed, for example, to alternative, yet closely related reasons such as fetal distress). The specific validity of the term cephalopelvic disproportion has been questioned for more than 50 yr (35,36).
We also found some interesting trends regarding other interactions in our baseline data (Table 1). Whereas skin color, independent of race/ethnicity and based on the skin type matrix of Fitzpatrick et al. (27) was not related to risk of cesarean delivery, women who self-identified as being non-Hispanic whites had a significantly elevated risk of cesarean delivery; 38% of non-Hispanic whites underwent primary cesareans, compared with 14% of non-Hispanic blacks, and 15% of Hispanics (P = 0.006). This is not reflective of national data, in which the primary cesarean rate among black, non-Hispanic primiparas is approximately one percentage point higher than the rate in Hispanic and white, non-Hispanic women (37).
Drinking milk in pregnancy and reported use of sunscreen were both marginally related with an increased risk of cesarean, but both these risks disappeared in the multivariate model and were probably a result of small sample size or confounding variables. On the other hand, maternal report of alcohol use during pregnancy was strongly associated with an increased risk of cesarean even after multivariate analysis, an unanticipated finding that clearly requires further investigation.
Maternal vitamin D deficiency is a widespread public health problem. Lee et al. (38) reported that 50% of mothers and 65% of newborns infants were severely vitamin D deficient [25(OH)D <30 nmol/liter] at the time of birth despite the fact that the mother was taking a prenatal vitamin containing 400 IU of vitamin D and drinking two glasses of vitamin D fortified milk (100 IU per 8 oz glass). This mirrors the observation of Hollis and Wagner (39) and Bodnar et al. (40), who found a high incidence of vitamin D deficiency in pregnant and lactating women. Whereas other factors, such as increased liability for obstetricians (41), undoubtedly play a role in the staggering rise in primary cesareans, the rate of primary cesareans with no medical or obstetrical indication is also rising; the current U.S. rate of primary cesareans with no reported medical or obstetrical indication is between 3 and 7% (19). As Declercq et al. (37) noted in a study to examine reasons for cesarean increases, apparent risk factors remained stable: shifts in primary cesarean rates during the study period were not related to shifts in maternal risk profiles. A randomized clinical trial is now needed to determine whether adequate vitamin D supplementation during pregnancy to raise blood levels of 25(OH)D above at least 37.5 nmol/liter can reduce the cesarean section rate and whether increasing it above 75 nmol/liter provides any additional reduction as suggested by our data.
Footnotes
This work was supported by the Department of Health and Human Services, Bureau of Maternal Child Health (Grant R40MC03620-02-00) and the Department of Agriculture Cooperative State Research, Education, and Extension Service Award 2005-35200-15620.
Disclosure Statement: A.M., S.D.M., T.C.C., and H.B. have nothing to declare. M.F.H. consults for Merck, Procter & Gamble, and Quest Diagnostics and received lecture fees from Merck and Procter & Gamble.
First Published Online December 23, 2008
Abbreviations: BMI, Body mass index; CI, confidence interval; GED, graduate equivalency degree; 25(OH)D, 25-hydroxyvitamin D; OR, odds ratio.
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3493 | https://scholars.duke.edu/publication/1283676 | Scholars@Duke publication: Are Periventricular Lesions Specific for Multiple Sclerosis?
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Are Periventricular Lesions Specific for Multiple Sclerosis?
Publication,Journal Article
Casini, G; Yurashevich, M; Vanga, R; Dash, S; Dhib-Jalbut, S; Gerhardstein, B; Inglese, M; Toe, W; Balashov, KE
Published in: J Neurol Neurophysiol
May 3, 2013
Published version (DOI)Link to item
BACKGROUND: The presence of periventricular lesions (PVL) on MRI scans is part of the revised McDonald multiple sclerosis (MS) diagnostic criteria. However, PVL can be found in other neurological diseases including stroke and migraine. Migraine is highly prevalent in patients with MS. OBJECTIVE: To determine if PVL are specific for patients with MS compared to stroke and migraine. METHODS: We studied patients diagnosed with clinically isolated syndrome (CIS), relapsing-remitting MS (RRMS), migraine, and ischemic stroke. The number, location and the volume of PVL were identified on brain MRI scans and analyzed. RESULTS: The number and volume of PVL adjacent to the body and the posterior horn of the lateral ventricles were significantly increased on fluid-attenuated inversion recovery MRI in RRMS compared to migraine. There were no significant differences in the total number and volume of PVL in ischemic stroke patients compared to the age-matched RRMS patients nor in the number and volume of PVL adjacent to the anterior and temporal horns of the lateral ventricles on FLAIR images in migraine compared to CIS or RRMS. CONCLUSION: In contrast to PVL adjacent to the body and the posterior horn of the lateral ventricles, PVL adjacent to the anterior and temporal horns of the lateral ventricles may not be specific for CIS/RRMS when compared to migraine, the disease highly prevalent among patients with MS. PVL are not specific for MS when compared to ischemic stroke.
Duke Scholars
Author Mary Yurashevich Anesthesiology, Women's
Published In
J Neurol Neurophysiol
DOI
10.4172/2155-9562.1000150
ISSN
2155-9562
Publication Date
May 3, 2013
Volume
4
Issue
2
Start / End Page
150
Location
United States
Citation
APA
Chicago
ICMJE
MLA
NLM
Casini, G., Yurashevich, M., Vanga, R., Dash, S., Dhib-Jalbut, S., Gerhardstein, B., … Balashov, K. E. (2013). Are Periventricular Lesions Specific for Multiple Sclerosis?J Neurol Neurophysiol, 4(2), 150.
Casini, Gianna, Mary Yurashevich, Rohini Vanga, Subasini Dash, Suhayl Dhib-Jalbut, Brian Gerhardstein, Matilde Inglese, Win Toe, and Konstantin E. Balashov. “Are Periventricular Lesions Specific for Multiple Sclerosis?” J Neurol Neurophysiol 4, no. 2 (May 3, 2013): 150.
Casini G, Yurashevich M, Vanga R, Dash S, Dhib-Jalbut S, Gerhardstein B, et al. Are Periventricular Lesions Specific for Multiple Sclerosis? J Neurol Neurophysiol. 2013 May 3;4(2):150.
Casini, Gianna, et al. “Are Periventricular Lesions Specific for Multiple Sclerosis?” J Neurol Neurophysiol, vol. 4, no. 2, May 2013, p. 150. Pubmed, doi:10.4172/2155-9562.1000150.
Casini G, Yurashevich M, Vanga R, Dash S, Dhib-Jalbut S, Gerhardstein B, Inglese M, Toe W, Balashov KE. Are Periventricular Lesions Specific for Multiple Sclerosis? J Neurol Neurophysiol. 2013 May 3;4(2):150.
Published In
J Neurol Neurophysiol
DOI
10.4172/2155-9562.1000150
ISSN
2155-9562
Publication Date
May 3, 2013
Volume
4
Issue
2
Start / End Page
150
Location
United States
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3494 | https://ems.press/content/serial-article-files/34664 | Zeitschrift fur Analysis und ihre Anwendunqor' Vol. 11(1992)2,277283
General Sufficient Conditions for the Convexity of a Function
D. ZAGRODNY
Sufficient conditions for a given function to be convex on a given segment, in terms of upper subderivative, are proved.
Key words: Convex functions, upper subderivati yes, monotonicity of subdifferentials AMS subject classification: 49A52
Introduction
Recently, the problem of characterizing an interesting class of convex functions has arisen (see [3,91). In the case of R 2 , this class consists of, roughly speaking, functions for which the limit of directional derivatives lim,,..,,,,, f'(x +),,Y +t,);(p, q)) exists, where (x,y),(p, q) € 1R2 are given and {(x, l')} c R2 is an arbitrary sequence of points of a given set such that (Xv , ii) - 0. The importance of that one can see, for example, in [91, where one find an algorithm for calculating a subgradient for a function from this class. We must agree upon that taking out "well" behaving convex functions leads to investigations on "bad" one, which may prove diffi-cult (see [11 -13]). Another motivation for seeking new conditions for the convexity can be found in [8: Theorem 3.2], where, loosely speaking, we should ensure that a given function on a product set, say on R 2 , is upper semicontinuous and convex with respect to the second va-riable.
Herein, we provide general sufficient conditions for a function f to be convex on a given segment [a, b] of a Banach cpace X (see Theorems 3.1 and 3.2). The basis virtue of them is that to obtain the convexity it is enough that, in case of upper semicontinuity, the inequality
urn s.ip :5 urn irif x-i'as(b-aJ y-s.at(b-a)
xcdomfn do Of, XE c'f(x) y€domfndom af,ycaf(y)
f(x)— 'f( as( b-a)) f(y)—i'f(a+t( b-a))
holds for any 0 :5 s < t :^ 1 (see Theorems 3.1 and 3.2 and Lemma 3.3). The conditions encom-pass the lower and upper semicontinuity case on a Banach space. When f is Lipschitzian it yields the monotonicity of the subdifferential, see [1,2,4 - 71.
Recently, there has been obtained a new result characterizing the convexity of lower se-micontinuous functions on the whole - space (see [4,51), i.e., a lower semicontinuous function is convex if and only if the subdifferential is monotone. The result has been obtained for fini-te-dimensional spaces (see [41) or reflexive Banach spaces (see [51). In the case when X is one-dimensional we compare it with the above condition. 278 D. ZAGRODNY
Basic facts on upper subderivativea
The apparatus of the paper is taken from Nonsmooth Analysis (see [1,2, 7 ]). Below we shall summarize those basic facts about generalized derivatives which are used in the sequel.
Let 1: X- Ru(+oD) be a lower semicontinuous function on a Banach space X. The upper subderivative of fat X X, 1(x) € R, with respect to veXis defined by
f 1 (x;v) = skip limsti i
f(y+tu)-f(y)
c>o (y,f(y))_(x,Jx)) IIu-vII<c t4o
and the subdifferential àf(x) by
ëf(x) = {€ XI :5 f(x;v) for all ye
where denotes the value of the linear continuous functional fat v. Let us recall the mean value theorem for these two notions.
Theorem 2.1 [ 10 ] : Let X be a real Banach space, a,b e X, a b andf: X- Ru {+oD) be lo-
wer semicontinuous and finite at a and b. Then, for every x €[a, b] such
that x I' b and the in-
equality
f(b) - f(a) 1(x) + llx - bll ^ 1(y) + 1(b) - 1(a) - bll (2.1) lIb - all lb - all
holds for all y € [a, b], there exist sequences { Xk} c X and {x} c X such that
lim xk = x and lim sup f(xk) I 1(a) + f(b) — f(a)il - all
k-0 llb - all
X€aI(X k ) V
and liminf<%,b -a> z- 1(b) - 1(a).
k-
Throughout the paper for any a, b E X we denote by [a, b] the set (a t(b - a)l 0 :5 r :5 1). We write x <[a,b]y if there exist 0 ^ s < t :5 I such that x = a +s(b - a)and y = a + t(b -a).
Further we denote by domaf the set {x E XI af(x) }.
Convexity on a segment
In this section we provide sufficient conditions for lower and upper semicontinuous functions to be convex on a given segment [a, b]. Before we do it, let us refer to known facts on convex functions.
From the classical differential calculus we know that the convexity of a real function is related to the monotonicity of its derivative. When we use a more sophisticated tool, for example, the subdifferential calculus, we still have to do with monotonicity (see, e.g., [7: Proposition 7A]). This strongly suggest that the monotonicity of a derivative is essential for the convexity. However, it is worth mentioning that the subdifferential can be empty on a given segment (see [10: Example 4. 1]), so we can not follow directly the methods of subdiffe-rential calculus and some refinements are needed. Let us also notice that the funcion f: R2
R, where f(x,y) = - lyl 112 - x 2 is not convex on the line L = {(x,0)l x € R} but fis equal to -co on it. So we have the monotonicity, but f1L is not convex (examples where 1' does not General Sufficient Conditions 279
exist can be obtained from this one replacing -x 2 by a proper non-convex function (P: IR - R which does not possess the right derivative).
Theorem 3.1: Let X be a Banaxh space and a, b € Xwith a b. Assume that f: X-+ Ru {+co}
is lower semicontinuous. If the implication
X <[a,bjY for all x,y€ [a,b]n domf
urn sp (u,b -a> urn if (3.1)
(u,
f(u))— . (x, f(x)) (v, f(v))—(y, f(y))
ucdomcIf ucc)f(u) vc domcf, vc,f(v)
holds, then "j [a, b] is convex.
Proof: Let us introduce an auxiliary function g on [0,1] by g(s) 1(a +s(b - a)). If f l[a bJ is not convex, then there exist 0 :5 s, < 2 <s 3 :^ I such that
g(s2) > s'-
S
2 S2 - S1
53 s g(s1 ) + (3.2)
This implies that 9(s1 ), g(s3 ) € R and (9(s 2 ) - g(SO)/(S 2 - s) > (9(s3 ) - 9(s2 ))/(s3 - s2 ). Let x1
a + s1(b - a) for I = 1, 2,3. We get
f(x2 ) - f(x1 ) 1(x3) - f(x2)
11 X2
. X j 11 11X3X211 (3.3)
Now let us consider the case when f(x2 ) € R, choose x € [x 1 , x 2 ) and y€ (X 21 X11 such that
f(x) + f(x2 ) - f(x1) lix11 )
ilx2 - X1 11 11 X2- x1Il :5 1(z) + f(x2) - f(x1 liz -x2 ll for every z € [x 1 ,x 2 )- x2
f(x2 ) - f(x3 ) f(x2) - f(x3)
1(y) + ilx2 - x31I il y - x 2 li :5 1(z)
+
1x2 - X 3 11
liz -x 2Ii for every z [x3,x1.
In particular, we have
+
f(x2 ) - f(x1) :5
+
f(x2 ) - 1(x3)
lix - x 2 li f(x2 ) and 1(y)
"X2 - xli 11X2 - X.,11Ily - X 2 ll ^ f(X2)1
thus the auxiliary functions p and q, where
f(x2) -f(x 1)
Ip(Z) llx2-x1li ilx-x 2 11,z=x2 _If(Y)+f2)3)llyX2ii, zx2
and q(z) - iiX2 x3
1( 2) ,zx2 11(z)
are lower semicontinuous and Op(z) Of(z) = 1q(z) for z x2 .Theorem 2.1 applied for the func-tions p and q ensures the existence of sequences {Uk} c domp, {Vk} c domq, {u} and such that
lim ( Uk, p(uk)) = (x, 1(x)) (3.4)
k-
lim (v,q(v)) = (y, 1(y)) (3.5)
k-+co
u ap(uk) and v € aq(v) for all k (3.6) 280 D.ZAGRODNY
limsup(u;, \ p(x2 )-p(x) = f(x2)-f(x1)
k--^-,
\ IIx-xlI/ IXXII 11X2 - x111
1imsup<"v, x2_ Y 2:
q(x2)-q(y) f(x2)-f(x3)
k-co
\ Ix, -yll
>
IIx2-yII 11X2 - X311
The last two inequalities, by (3.2), imply urn sup k > urn inf,., which, by (3.4) -(3.6), contradicts (3.1). When 1(x 2 ) = +co we can run a proof as before, replacing! by ?given as ?(z) = 1(z) for z x 2 and ?z) = ot for z x2 ,where a is such that (3.2) still holds I
In the view of the above proof, it might seem that instead of f we can consider the func-tion g,g(t) = f(ta +0 - t)b) on [0,11. In this case, we may admit in (3.1) only those pairs (t,g(t)) for which t E (0,1). However, in some particular situation, this restriction would lead to a false assertion. For example, let us consider the function 1: R - R, where 1(t) = I for t < 0 and 1(t)
=_tL'2
for t ^t 0. This function is lower semicontinuous but is not convex on the segment
[-1,0], condition (3.1) is violated for x = -1/2 and 0. The function g is equal to 0 for t = 0
and to 1 for 0 < t -1 I. Since cg(t) {0} for t € (0,1), so (3.1) is fulfilled. It is worth noticing that if we reduce our considerations to the one-dimensional case, then (3.1) is equivalent to the monotonicity of the multifunction t— 6f(t) (if wee assume that (3.1) holds for every a,b
R). In fact, if the multifunction is monotone (i.e., for all t,, t 2 R, the inclusions t: E c)f(t1)
and t2 E 6f(t 2 ) imply the inequality (t - t)(t 1 - t 2 ) a 0), then
urn stip :5 urn siLip
(t,f(t))-'(x,f(x))
tcdomc'(f), tca(f)(t) tc(fXt)
s
lirri irif :r liryi irif .
t-+y (t. f( t))-( , , f(y))
tc a(f)(t) tcdom(f), t c 06 'X t)
So the monotonicity forces that (3.1) holds, and vice verse, if (3.1) holds, then
sup :5 inf .
t1 cf(t 1 ) t2càf(t2)
Thus, in the one-dimensional case, Theorem 3.1 is equivalent to Poliquin's result (; for an extension see [41) which states that a lower semicontinuous function Ion IR' 1 is convex if and only if the multifunction of(') is monotone.
The next theorem deals with upper semicontinuous functions. Unexpectedly, the monoto-nicity alone is not sufficient for its convexity. We give a proper example after that theorem.
Theorem 3.2: Let X be a Banach space and a,b E .X'with a $ b. Assume that 1: A'- R u( - co)
is an upper semicontinuous function such that the following implication holds:
For every x,<[.b] < X2 <[.b] x3 there is an e [x1,x3]{x2} such that
IC 1 ) !( 3 ) - f(X2)
f(x 2) -f(x 1) f(x3)-f(x2) II2-x1II > "X3 _X 21 1 and x1 <[a,bj X2 <[a,b]X2
11X
2 - X , 11II X X21I = or (37)
f(x2) - f(x 1 ) f( X3 ) -
f2) and X2 <La,bJ2<[a,bjX3.
11x2 X , 11 11X3-211
If the implication General Sufficient Conditions 281
X <[8,bJY ' x,yE[a,b]ndom(-f)
urn t.1D <-u,b -a> s liri-i irif <-v,b -a> (3.8)
(u, f(u))-(x,ftx)) (vf(v))-.(,f(y))
ucdomc(-f), uc.(-f)(u) vedoma(- f),v ca(-f)(v)
holds, then i[a, b] IS convex.
Proof: Let us introduce an auxiliary function g by g(s) =f(a + s(b - a)) for all s c [0,1]. If
"I[a,bJ is not convex, then there exist 0 s s < s2 < s3 :5 I such that
S
2 ) > -: - g(s) + 2
g(
This implies that 8(s2 ) € IR and ( s 2 ) g(s j ) 9(s3) - 8(s2) Let x = a +s .(b -a) for i = 1,2,3. s2 -s1 s3-s2
We get
(-f)(x) - (-f)(x2 ) (-f)(x) - (-f)(x3)
11X
2 - x i ii IiXa - X 211
Now, by (3.7), there are 2' A 2 E [x 5 , x3 ] n dom(-f) such that
X
1 <[a,b]Xa < L ab ] 2 < [ a,bJ X3 and Xx1) - (-f)( 2 ) > (-)(2) - (-f)(x3)
11
X2 - .' 1 iI iL'3 - Xzii
Repeating he method of the proof of Theorem 3.1 we get sequences { U k}, {u}, { vk},{ V } such that { U k} and {vk} converge to x E [xi , 21 and y E [, x3 1, respectively, and u a(-r)(uk ), v
ô(-f)(v k ) for every k e N, and
• x \ (-f)(x 1 ) - (-f)(2)
l lmsupu k , - ) a -
k-co \ 11x 5 -x2 11 / IIx - x 2 ii
lim sup /;, X3 A2 ) (-f)(x 3) - (-f)()
k-->— \ Iix 3 -x 2 ii IiX3 2ii
The last two inequalities are a contradiction to (3.8) I
Let us notice that if a function I is continuous, then (3.7) holds automatically. However, this assumption can not be dropped when the function is upper semicontinuous. Let us consi-der the non-convex function 1(x) = 0 for x 0 and f(x) = 1 for x = 1, which satisfies only (3.8). For the time being, we focus our attention on locally Lipschitzian functions. We know that in this case -àf(x) = c(-f)(x) (see [2: Proposition 2.3.1]). Thus Theorems 3.1 and 3.2 yield the same condition. Moreover, (3.1) is equivalent to the monotonicity of the multifunction f on [a, b] (see [2: Proposition 2.1.5]). Now, we may say that Theorems 3.1 and 3.2 in the case are neither more nor less than the "classical" sufficient condition for the convexity (see [7: Pro-position 7A]).
Finally observe that the relation a(-f)(x) = -c)f(x) may fail, even when the function is continuous. However, (3.1) and (3.8) are equivalent in this case, as it will be shown in the next lemma: But for some classes of functions conditions (3.1) and (3.8) are not equivalent. Indeed, let us consider the function f: R - R, where 1(x) = -x for x € [0,1) and 1(x) = 0 for x E [1, 2]. Conditions (3.1) and (3.7) are fulfilled, with a = 0 and b = 2, but the function f is not convex. Thus (3.8) is violated since
19 Analysis. Bd. II. Heft 2(1992) 282 D. ZAGRODNY
=(-,O]
and urn irf <- t, 2> = 0.
(t,(-fX:))-3(3/2,(-fX3/2))
rc)(-fXt)
Lemma3.3: Let X be a Banach space and!: X- R a Continuous function. Then
urn SLAP = urn sp <-v,p>
U^x vucdomaf, ucf(u) vcdoma(-f),vcc(-fXv)
and
urn irif = urn if <-v',p> v--.),. xucdomàf, ucaf(u) vEdom)(-f), vc,(-fXv)
for every p E X.
Proof:
Let us consider the non-trivial case when p 0. Let a be equal to the left-hand side of the first equality and 13 to the right-hand one. For any E > 0 we can find sequences {x}, I t,,)
C Xsuch that
(-f)(x) - (-f)(x+t p)
X" - x, t1 'l t and a a - E for all n.
tn
By Theorem 2.1 we can find sequences {vn} c domà(-f) and {v} C X such that
v - x and v, € a(-fXv), lirristip a a -
Thus 13 a a, similarly a The second equality can be obtained from the first one when we take -p instead of 61
Below we present a result which can be helpful to prove the convexity of a continuous function (see, for example, [iii).
Corollary
3.4: Let a, b e R, a 4' b, and!: [a, b] - R a continuous function such that the Ii-
mit
!(x; 1) = lim f(x + t) - f(x)
t4'O
exists for every x €[a, b]. If the function f'( . ;l) is non-decreasing, then (3.1) holds on every segment [c, d] C [a, b] and fis convex.
Proof:
Assume that for some c <d (3.1) is violated. By Lemma 3.3 for some x,y E [c, d]
with x <[C.dJyWe get
a urn stip > 1irr irif <-vd-c> (3.
v—y
ucdomc'f, uc.f(u) v€doma(-f), v€a(-fXv)
We infer the existence of sequences {x}, { y,,}, { t,}, {s, 3} such that x, 1 - x, y,, - y, t, 3 l0, s,, 4' 0 and numbers Me (a,(3), e > 0 such that, for all n,
f(x+t(d - c)) - f(x) ^: M+E and (-!Xy +s(d-c))-(-f)(y)
tn Sn
The continuity off forces that there exist k. [x,x +t(d -c)) and y,1 E [y,y, +s(J - c)) General Sufficient Conditions 283
such that
f'(n;d-c)?-M+t and (-f)'(Yn;dc)2: -Mt
Thus
M+c M- E af'(Yn ;l) for all n. dc
This is a contradiction I
REFERENCES
CLARKE, F. H.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205 (1975), 247 - 266.
CL.ARKE, F. H.: Optimization and Nonsmooth Analysis. New York: Wiley-Jntersci. 1983.
GIANNESSI, F.: A problem on convex functions. J. Opt. Theory App!. 59 (1989), 525. [4J CORREA, R., JOFRE, A. and L. THIBAULT: Characterization of lower sezuicontinuous convex functions. Submitted.
POLIQUIN, R.: Subgradient rnonotonicity and convex functions. Nonlin. Anal. Theory, Methods, App!. 14 (1990), 305 - 317.
ROCKAFELL.AR, R. T.: Convex Analysis. Princeton: University Press 1972.
ROCKAFELL.AR, R.T.: The Theory of Subgradients and its Applications: Convex and
Nonconvex Functions. Berlin: Heldermann Verlag 1981.
STUDNIARSKI, M.: Mean value theorems and sufficient optimality conditions for non-
smooth functions. J. Math.Anal. AppI. 111 (1985), 313 - 326.
STUDNIARSKI,
M.: An algorithm for calculating one subgradient of a convex function of
two variables. Num. Math. 55 (1989), 685 - 693.
ZAGRODNY, D.: Approximate mean value theorem for upper subderivatives. Nonlin. Anal. Theory, Methods,Appl. 12 (1988), 1413 - 1428.
ZAGRODN y, D.: An example of bad convex function. J. Opt. Theory Appl. 70 (1991), 631 - 637.
Received 03.05.1991
Dr. Dariusz Zagrodny
Technical University of Lcdi
Institute of Mathematics
Al. Politechniki 11
P -
90-924 Lddi
Added in proof:
PONTINI, C.: Solving in the affirmative a conjecture about a limit of gradients. J. Opt. Theory AppI. 70 (1991), 623 - 629.
ROCKAFELLAR, R.T.: On a special class of convex functions. J. Opt. Theory App!. 70 (1991), 619 - 621.
19 |
3495 | https://goldbook.iupac.org/terms/view/MT06972 | IUPAC - molecular shape (MT06972)
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Determining the sum of an infinite series and assessing its convergence or divergence is not always straightforward. Several methods are available to study convergence, one of which involves comparing the series to an improper integral. This test applies to series with positive terms and relies on the principle that the convergence of the series can be determined by comparing it to the behavior of an associated improper integral.
Let f be a positive, decreasing function defined on [1,+∞), such as a rational or polynomial function. Then the series
∑n=1∞f(n)
converges or diverges if and only if the improper integral
∫1∞f(x)d x
does the same, assuming that f is continuous on [1,+∞).
The graph illustrates the connection between a series and an improper integral as stated by the Integral Test.
The curve f(x) represents the continuous function.
The gray area shows a portion of the improper integral (the area under the curve from x=1 to some x=n).
The vertical rectangles represent the terms of the series f(n), each with base 1 and height f(n).
This visual helps compare the discrete sum (the series) and the continuous accumulation (the integral). Since the rectangles overestimate or underestimate the area depending on the function’s behavior, the integral can be used to determine the convergence of the series.
Proof
Let us consider the partial sum of the series:
s k=∑n=1 k f(n)k∈N
This represents the sum of the first k terms of the series ∑f(n). Since the series has positive terms, the sequence of partial sums s k is increasing and admits a limit as k→∞:
lim k→+∞s k=s∈[0,+∞]
Just as the series is defined by the limit of its partial sums, the improper integral is defined as the limit of the definite integral as the upper bound tends to infinity:
lim k→+∞∫1 k f(x)d x=∫1+∞f(x)d x
By the linearity of the integral, and in particular its additivity over adjacent intervals, we can write:
∫1 k f(x)d x=∑n=1 k−1∫n n+1 f(x)d x
This holds because the definite integral over [1,k] can be decomposed into a sum of integrals over the unit-length subintervals [n,n+1], which are disjoint and consecutive. Since f is assumed to be decreasing, we obtain the following inequality for all x∈[n,n+1]:
f(n+1)≤f(x)≤f(n)
By applying the inequality within the integral, we obtain:
∫n n+1 f(n+1)d x≤∫n n+1 f(x)d x≤∫n n+1 f(n)d x
By the properties of definite integrals, the first and third terms represent integrals of constant functions. Therefore, the constants can be factored out of the integrals, giving:
f(n+1)≤∫n n+1 f(x),d x≤f(n)
Now summing these inequalities from n=1 to k−1:
∑n=1 k−1 f(n+1)≤∑n=1 k−1∫n n+1 f(x)d x≤∑n=1 k−1 f(n)
By taking the limit as k→∞, we obtain:
∑n=2∞f(n)≤∫1∞f(x)d x≤∑n=1∞f(n)
which shows that the improper integral is bounded between two versions of the series differing only by the first term f(1). Because the integral lies between two versions of the series that differ only by the first term, if the integral converges, so does the series, and if the integral diverges, the series diverges as well.
Example
Determine whether the following series converges or diverges using the integral test:
∑n=2∞1 n logn
First, consider the associated function:
f(x)=1 x logx
defined on the interval x≥2. This function is positive, continuous, and decreasing on [2,+∞), so the conditions for using the integral test are satisfied.
Now we evaluate the improper integral:
∫2∞1 x logx d x
To compute this, use the substitutionu=logx, which implies d u=1 x d x. The integral becomes:
∫log2∞1 u d u=lim t→∞∫log2 t 1 u d u=lim t→∞[logu]log2 t=∞
Since the integral diverges, the integral test tells us that the series also diverges.
Glossary
Infinite Series: an expression of the form ∑n=1∞a n=a 1+a 2+a 3+…, where a n are the terms of the series.
Convergence of a series: an infinite series converges if its sequence of partial sums approaches a finite limit.
Divergence of a series: an infinite series diverges if its sequence of partial sums does not approach a finite limit (either it goes to infinity or oscillates).
Improper integral: a definite integral where at least one of the limits of integration is infinite, or the integrand has a discontinuity within the interval of integration.
Decreasing function: a function f(x) is decreasing on an interval if for any x 1<x 2 in that interval, f(x 1)≥f(x 2).
Continuous function: a function whose graph can be drawn without lifting the pen, meaning there are no abrupt jumps or breaks.
Partial sum s k: the sum of the first k terms of an infinite series, denoted as s k=∑n=1 k a n.
Limit of a sequence: the value that the terms of a sequence approach as the index tends to infinity.
Additivity of integrals: the property that the definite integral over a compound interval is the sum of the definite integrals over the disjoint subintervals that make up the compound interval.
Continuing on this topic
349Series
327Cauchy’s Convergence Criterion for Series
329Series with Positive Terms
353Harmonic Series
466Geometric Series
250Root Test for Series Convergence
344Leibniz’s Criterion
80Function Series
80Power Series
104Taylor Series
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Why am I seeing a negative R^2 value?
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For nonlinear regression models where the distinction between dependent and independent variables is unambiguous, the calculator will display the coefficient of determination, R2.
In most cases this value lies between 0 and 1 (inclusive), but it is technically possible for R2 to lie outside of that range.
This might initially appear strange. First, a common way to interpret R2 is as the fraction of variability in the dependent variable that the model accounts for, and this interpretation only makes sense for values between 0 (accounts for none of the variability) and 1 (accounts for all of it). Second, the name itself gives the impression that some quantity R is being squared to produce a result. Either way, it seems that R2 should probably lie in [0,1], or at the very least it should be nonnegative.
The computational definition of R2, however, is divorced from both the notation and this common interpretation. Apart from the special case of a linear regression model with an intercept term, R2 is not actually equal to the square of any particular quantity. It is calculated by taking the mean of the squared errors, dividing by the variance of the dependent variable, and subtracting this ratio from 1. Since there is no limit to how bad a model’s predictions can be—and thus no limit to how big the errors can get—it’s possible for this ratio to become arbitrarily large, and 1 minus a large value is negative.
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3499 | https://www.freemathhelp.com/forum/threads/trick-for-finding-the-square-of-a-number-with-a-known-square.130923/ | Trick for finding the square of a number with a known square | Free Math Help Forum
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Trick for finding the square of a number with a known square
Thread startersepulorl000
Start dateAug 2, 2021
S
sepulorl000
New member
Joined Aug 2, 2021 Messages 1
Aug 2, 2021
#1
I was looking over the internet and I was unable to find this trick that I randomly stumbled upon when I was just randomly thinking about squares and was wondering if there was already a paper about it that I could look at? Also this is a great trick that should have been taught in high school or even middle school. I also did not know under what category this would fall into so if this post is not under this category, please let me know. Thanks!
The trick is
(a+b)(b-a)+a^2=b^2 , and this holds true as
(a+b)(b-a)+a^2=b^2
ab-a^2+b^2-ab+a^2=b^2
b^2=b^2
so if you did not have access to a calculator you could solve for 43^2 just by knowing 2^2 which would go like
(43+2)(43-2)+2^2=43^2
4541+4=1849
Steven G
Elite Member
Joined Dec 30, 2014 Messages 14,594
Aug 2, 2021
#2
You are correct that this should have been taught in school.
Many things are taught in school but they are never used to do calculations.
Every math student should now that a^2 - b^2 = (a+b)(a-b). This is how you factor the difference of squares.
Just add b^2 to both sides and you'll get a^2 = (a+b)(a-b) + b^2 for ANY choice of b.
In calculating 53^2 you should use 3 for b, because 53-3 = 50 and multiplying a number by 50 is similar to multiplying a number by the single digit 5.
So 53^2 = (53-3)(53+3) + 3^2 = (50)(56) + 9 = 2800 + 9 = 2809.
Personally I would not compute 53^2 that way. I know that (a+b)^2 = a^2 + 2ab + b^2
I would think of (53)^2 as (50+3)^2. So a = 50 and b = 3. Then 50^2 = (50+3)^2 = 50^2 + 2(50)(3) + 3^2 =2500 + 300 + 9 = 2809. I can do this in my head. All teachers show their student's that (a+b)^2 = a^2 + 2ab + b^2 but few show their students that this works for numbers as well.
Teachers spent a whole lecture on additive inverses (3+(-3) =0, -9 + 9 = 0, -2/3 + 2/3 = 0) yet when it comes to solving (x + 3) = 0, most teachers would say to subtract 3 from both sides. Now I do not necessarily have a problem with a teacher saying to subtract 3 from both sides. I just wonder why they spend anytime on teaching additive inverses.
I have a saying that goes like this: If it works for letters, then it works for numbers. If it works for numbers, then it works for letters. If it does not work for letters, then it does not work for numbers. If it does not work for numbers, then it does not work for letters.
Steven G
Elite Member
Joined Dec 30, 2014 Messages 14,594
Aug 2, 2021
#3
sepulorl000 said:
(43+2)(43-2)+2^2=43^2
4541+4=1849
Click to expand...
Although you could, I would not use 2 for b. I would use 3 for b.
(43-3)(43+3) + 3^2 = (40)(46) + 9 = 1840 + 9 = 1849.
I have a better chance of being able to figure out (40)(46) in my head compared to (4541). Multiplying 40 by 46 is extremely close to multiplying 4 by 46.
Here is another thing that is not always taught well in school.
All students have learned the distributed law. Well this works for numbers as well!
To multiply 4 by 46 in your head, just use the distribute law! 4(46) = 4(40 + 6) = 160 +24 = 160 + 20 + 4 = 180 + 4 = 184. So 40(46) = 1840
Otis
Elite Member
Joined Apr 22, 2015 Messages 4,592
Aug 2, 2021
#4
sepulorl000 said:
...if you did not have access to a calculator you could solve for 43^2 ... like ... 4541+4=1849
Click to expand...
If a person knows how to multiply 45x41, then why wouldn't they just multiply 43×43 to begin with?
By the way, this identity is already taught in schools:
(a+b)(a-b) = a^2 - b^2
Maybe you forgot it.
?
Steven G
Elite Member
Joined Dec 30, 2014 Messages 14,594
Aug 2, 2021
#5
Otis said:
If a person knows how to multiply 45x41 then why don't they just multiply 43×43 to begin?
By the way, this identity is already taught in schools:
(a+b)(a-b) = a^2 - b^2
Maybe you forgot it.
?
Click to expand...
Otis,
Yes, this identity is taught in schools. However, it is not generally taught using numbers.
For example, 51^2-50^2 is a trivial problem using the above formula. But seriously, how many students do you know who would say 51^2-50^2 = (51 + 50)(51-50) = 101?
Dr.Peterson
Elite Member
Joined Nov 12, 2017 Messages 16,844
Aug 2, 2021
#6
sepulorl000 said:
I was looking over the internet and I was unable to find this trick that I randomly stumbled upon when I was just randomly thinking about squares and was wondering if there was already a paper about it that I could look at? Also this is a great trick that should have been taught in high school or even middle school. I also did not know under what category this would fall into so if this post is not under this category, please let me know. Thanks!
The trick is
(a+b)(b-a)+a^2=b^2 , and this holds true as
(a+b)(b-a)+a^2=b^2
ab-a^2+b^2-ab+a^2=b^2
b^2=b^2
so if you did not have access to a calculator you could solve for 43^2 just by knowing 2^2 which would go like
(43+2)(43-2)+2^2=43^2
4541+4=1849
Click to expand...
There are, of course, many "tricks" you could use; the hard part is to pick one that will actually save time. Often, we really use tricks mostly because they are impressive.
Since I don't happen to know 4541 offhand, I might do something like what Jomo suggested for 53, and use 7:
43^2 - 7^2 = (43+7)(43-7) = 5036, so 43^2 = 5036 + 49 = 10018 + 49 = 1849............................................ corrected
which is very pretty! (Note that I didn't memorize your idea as a formula, but just used what I have already memorized, that a^2 - b^2 = (a+b)(a-b).)
The good thing you are illustrating is the value of "randomly thinking" about math, rather than just sticking with what you've been taught. That is really what schools should teach -- don't just learn, discover! It's what worked for me.
Last edited by a moderator: Aug 2, 2021
Steven G
Elite Member
Joined Dec 30, 2014 Messages 14,594
Aug 2, 2021
#7
A mathematician saying don't just learn, discover! It's what worked for me is just so classic. I think that the probability of finding a mathematician who disagrees with that statement will be a true 0 (no limit is needed).
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