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Hierarchy theory is a means of studying ecological systems in which the relationship between all of the components is of great complexity. Hierarchy theory focuses on levels of organization and issues of scale, with a specific focus on the role of the observer in the definition of the system. Complexity in this context... | Wikipedia/Hierarchy_theory |
In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems that can be reduced to finding good paths through graphs. Artificial ants represent multi-agent methods inspired by the behavior of real ants.
The pheromone-based comm... | Wikipedia/Ant_colony_optimization_algorithms |
Network motifs are recurrent and statistically significant subgraphs or patterns of a larger graph. All networks, including biological networks, social networks, technological networks (e.g., computer networks and electrical circuits) and more, can be represented as graphs, which include a wide variety of subgraphs.
Ne... | Wikipedia/Network_motif |
Systemic therapy is a type of psychotherapy that seeks to address people in relationships, dealing with the interactions of groups and their interactional patterns and dynamics.
Early forms of systemic therapy were based on cybernetics and systems theory. Systemic therapy practically addresses stagnant behavior pattern... | Wikipedia/Systemic_therapy |
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems via biologic... | Wikipedia/Genetic_algorithm |
In control theory, affect control theory proposes that individuals maintain affective meanings through their actions and interpretations of events. The activity of social institutions occurs through maintenance of culturally based affective meanings.
== Affective meaning ==
Besides a denotative meaning, every concept... | Wikipedia/Affect_control_theory |
Systems thinking is a way of making sense of the complexity of the world by looking at it in terms of wholes and relationships rather than by splitting it down into its parts. It has been used as a way of exploring and developing effective action in complex contexts, enabling systems change. Systems thinking draws on a... | Wikipedia/Systems_thinking |
Systemic design is an interdiscipline that integrates systems thinking and design practices. It is a pluralistic field, with several dialects including systems-oriented design. Influences have included critical systems thinking and second-order cybernetics. In 2021, the Design Council (UK) began advocating for a system... | Wikipedia/Systemic_design |
Systems ecology is an interdisciplinary field of ecology, a subset of Earth system science, that takes a holistic approach to the study of ecological systems, especially ecosystems. Systems ecology can be seen as an application of general systems theory to ecology. Central to the systems ecology approach is the idea ... | Wikipedia/Systems_ecology |
Ecological systems theory is a broad term used to capture the theoretical contributions of developmental psychologist Urie Bronfenbrenner. Bronfenbrenner developed the foundations of the theory throughout his career, published a major statement of the theory in American Psychologist, articulated it in a series of propo... | Wikipedia/Ecological_systems_theory |
In computer science, robustness is the ability of a computer system to cope with errors during execution and cope with erroneous input. Robustness can encompass many areas of computer science, such as robust programming, robust machine learning, and Robust Security Network. Formal techniques, such as fuzz testing, are ... | Wikipedia/Robustness_(computer_science) |
Systems theory in anthropology is an interdisciplinary, non-representative, non-referential, and non-Cartesian approach that brings together natural and social sciences to understand society in its complexity. The basic idea of a system theory in social science is to solve the classical problem of duality; mind-body, s... | Wikipedia/Systems_theory_in_anthropology |
In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that can be solved using set-theoretic methods, for example, Suslin's problem.
== Objects studied in set-theoretic topology ==
=== Dowker spaces ===
In the mathematical field of ge... | Wikipedia/Set-theoretic_topology |
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups. This operation, in turn, allows one to associate vari... | Wikipedia/Homology_theory |
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function.
Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all ... | Wikipedia/Urysohn's_lemma |
In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.
== Definitions ==
A continuum that contains more than o... | Wikipedia/Continuum_(topology) |
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object... | Wikipedia/Algebraic_K-theory |
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be a... | Wikipedia/Continuity_(topology) |
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simpli... | Wikipedia/Combinatorial_topology |
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy t... | Wikipedia/Higher_category_theory |
In algebraic topology, the cellular approximation theorem states that a map between CW-complexes can always be taken to be of a specific type. Concretely, if X and Y are CW-complexes, and f : X → Y is a continuous map, then f is said to be cellular, if f takes the n-skeleton of X to the n-skeleton of Y for all n, i.e. ... | Wikipedia/Cellular_approximation_theorem |
In the mathematical field of geometric topology, the Poincaré conjecture (UK: , US: , French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces t... | Wikipedia/Poincaré_conjecture |
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.
A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations... | Wikipedia/Weak_equivalence_(homotopy_theory) |
In algebraic topology, a k-chain
is a formal linear combination of the k-cells in a cell complex. In simplicial complexes (respectively, cubical complexes), k-chains are combinations of k-simplices (respectively, k-cubes), but not necessarily connected. Chains are used in homology; the elements of a homology group are ... | Wikipedia/Chain_(algebraic_topology) |
Digital topology deals with properties and features of two-dimensional (2D) or three-dimensional (3D) digital images
that correspond to topological properties (e.g., connectedness) or topological features (e.g., boundaries) of objects.
Concepts and results of digital topology are used to specify and justify important (... | Wikipedia/Digital_topology |
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
== Higher-dimensional categories ==
A first step towards defining higher dimensional algebras is the concept... | Wikipedia/Higher-dimensional_algebra |
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant, under ... | Wikipedia/Algebraic_invariant |
In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G.
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful... | Wikipedia/Algebraic_topology_(object) |
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embed... | Wikipedia/Knot_theory |
In mathematics, specifically in topology,
the interior of a subset S of a topological space X is the union of all subsets of S that are open in X.
A point that is in the interior of S is an interior point of S.
The interior of S is the complement of the closure of the complement of S.
In this sense interior and closure... | Wikipedia/Interior_(topology) |
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology... | Wikipedia/Low-dimensional_topology |
Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century.
Further p... | Wikipedia/Approximations_of_π |
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
Often, the term "polynomial ring" refers implicitly to the specia... | Wikipedia/Polynomial_algebra |
A timeline of number theory.
== Before 1000 BCE ==
ca. 20,000 BCE — Nile Valley, Ishango Bone: possibly the earliest reference to prime numbers and Egyptian multiplication although this is disputed.
== About 300 BCE ==
300 BCE — Euclid proves the number of prime numbers is infinite.
== 1st millennium AD ==
250 — ... | Wikipedia/Timeline_of_number_theory |
The following timeline of algorithms outlines the development of algorithms (mainly "mathematical recipes") since their inception.
== Antiquity ==
Before – writing about "recipes" (on cooking, rituals, agriculture and other themes)
c. 1700–2000 BC – Egyptians develop earliest known algorithms for multiplying two numb... | Wikipedia/Timeline_of_algorithms |
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.
For example,
{
3
x
+
... | Wikipedia/Simultaneous_linear_equations |
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebr... | Wikipedia/Boolean_algebra_(logic) |
In the history of calculus, the calculus controversy (German: Prioritätsstreit, lit. 'priority dispute') was an argument between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first discovered calculus. The question was a major intellectual controversy, beginning in 1699 and reaching its peak in... | Wikipedia/Leibniz–Newton_calculus_controversy |
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizin... | Wikipedia/Algebraic_curves |
In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjust... | Wikipedia/False_position_method |
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for exam... | Wikipedia/Theory_of_numbers |
In algebra, a cubic equation in one variable is an equation of the form
a
x
3
+
b
x
2
+
c
x
+
d
=
0
... | Wikipedia/Cubic_equations |
In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers.
Cantor's theorem implies that there are sets having cardinality greater... | Wikipedia/Controversy_over_Cantor's_theory |
A timeline of calculus and mathematical analysis.
== 500BC to 1600 ==
5th century BC - The Zeno's paradoxes,
5th century BC - Antiphon attempts to square the circle,
5th century BC - Democritus finds the volume of cone is 1/3 of volume of cylinder,
4th century BC - Eudoxus of Cnidus develops the method of exhaustion,... | Wikipedia/Timeline_of_calculus_and_mathematical_analysis |
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as:
Categories of abstract algebraic structures including representation theory and universal algebra;
Homological algebra;
Homotopical algebra;
Topology using categories, including algebraic topology, categorical... | Wikipedia/Timeline_of_category_theory_and_related_mathematics |
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse t... | Wikipedia/Morse_theory |
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes und... | Wikipedia/Hodge_theory |
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However... | Wikipedia/Surface_(topology) |
In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.
== Topology ==
=== Orientable surfaces ===
The genus of a connected, orientable surface is an integer representing t... | Wikipedia/Genus_(topology) |
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well... | Wikipedia/Differentiable_function |
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain... | Wikipedia/Symplectic_topology |
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.
A subset of a topological space
... | Wikipedia/Connected_(topology) |
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Ri... | Wikipedia/Geometrization_conjecture |
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor (1961). Milnor called this technique surgery, while Andrew Wallace called it spherical modification. The "surgery... | Wikipedia/Surgery_theory |
In mathematics, the rank of a differentiable map
f
:
M
→
N
{\displaystyle f:M\to N}
between differentiable manifolds at a point
p
∈
M
{\displaystyle p\in M}
is the rank of the derivati... | Wikipedia/Rank_(differential_topology) |
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to s... | Wikipedia/Deformation_theory |
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants.
While TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the th... | Wikipedia/Topological_quantum_field_theory |
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold that is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is
Every homotopy sphere (a closed n-... | Wikipedia/Generalized_Poincaré_conjecture |
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notati... | Wikipedia/Boundary_(topology) |
In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e... | Wikipedia/Dual_graph |
A graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key concept of the system is the graph (or edge or relationship). The graph relates the data items in the store to a collection of nodes and edges, the edges representing... | Wikipedia/Graph_database |
In the mathematical field of graph theory, a path graph (or linear graph) is a graph whose vertices can be listed in the order v1, v2, ..., vn such that the edges are {vi, vi+1} where i = 1, 2, ..., n − 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices of degree 1)... | Wikipedia/Path_graph |
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction).
Graph theory i... | Wikipedia/Complete_graph |
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The conne... | Wikipedia/Connected_graph |
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph.
== Oriented graphs ==
A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, th... | Wikipedia/Orientation_(graph_theory) |
In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union.
C... | Wikipedia/Cograph |
In computational biology, power graph analysis is a method for the analysis and
representation of complex networks. Power graph analysis is the computation, analysis and visual representation of a power graph from a graph (networks).
Power graph analysis can be thought of as a lossless compression algorithm for graphs... | Wikipedia/Power_graph_analysis |
In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops.
Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple ed... | Wikipedia/Loop_(graph_theory) |
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity.
Formally, an automorphism of a graph G = (V, E) is a permutation σ of the vertex set V, such that the pair of vertices (u, v) form an edge i... | Wikipedia/Graph_automorphism |
In graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed.
The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected.
Edge connectivity and the enumeration of k-edge-connected graphs was studied by Camille Jordan in 1869.
== ... | Wikipedia/K-edge-connected_graph |
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central... | Wikipedia/Cayley_graph |
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets
U
{\displaystyle U}
and
V
{\displaystyle V}
, that is, every edge connects a ve... | Wikipedia/Bipartite_graph |
In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that
the vertex set of G ∙ H is the cartesian product V(G) × V(H); and
any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent to x in G or u = x and v is adjacent to y in H.
If... | Wikipedia/Lexicographic_product_of_graphs |
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed lo... | Wikipedia/Directed_acyclic_graph |
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in Cn equals the number of edges, and every verte... | Wikipedia/Cycle_graph |
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and the distance between v and w.
Some authors exclude the complete graphs and disconnected graphs f... | Wikipedia/Distance-regular_graph |
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The conne... | Wikipedia/Connectivity_(graph_theory) |
In graph theory, the Cartesian product G □ H of graphs G and H is a graph such that:
the vertex set of G □ H is the Cartesian product V(G) × V(H); and
two vertices (u,v) and (u' ,v' ) are adjacent in G □ H if and only if either
u = u' and v is adjacent to v' in H, or
v = v' and u is adjacent to u' in G.
The Cartesi... | Wikipedia/Cartesian_product_of_graphs |
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function
W
:
[
0
,
1
]
2
→
[
0
,
1
]
... | Wikipedia/Continuous_graph |
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph.
== Definitions ==
While graph drawing and graph representation are valid topics in graph theory, in order to fo... | Wikipedia/Graph_property |
In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary (one input) and binary (two input) operations.
== Unary operations ==
Unary operations create a new graph from a single initial graph.
=== Elementary operations ===
Elementa... | Wikipedia/Graph_operations |
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertic... | Wikipedia/Regular_graph |
A geographic information system (GIS) consists of integrated computer hardware and software that store, manage, analyze, edit, output, and visualize geographic data. Much of this often happens within a spatial database; however, this is not essential to meet the definition of a GIS. In a broader sense, one may consider... | Wikipedia/Geographic_information_systems |
In computer science, graph transformation, or graph rewriting, concerns the technique of creating a new graph out of an original graph algorithmically. It has numerous applications, ranging from software engineering (software construction and also software verification) to layout algorithms and picture generation.
Grap... | Wikipedia/Graph_rewriting |
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
Formally, a directed hypergraph is a pair
(
X
,
E
)
{\displaysty... | Wikipedia/Hypergraph |
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics.
A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of un... | Wikipedia/Graph_(abstract_data_type) |
In graph theory, a branch of mathematics, the disjoint union of graphs is an operation that combines two or more graphs to form a larger graph.
It is analogous to the disjoint union of sets and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs and by maki... | Wikipedia/Disjoint_union_of_graphs |
In graph theory, a mixed graph G = (V, E, A) is a graph consisting of a set of vertices V, a set of (undirected) edges E, and a set of directed edges (or arcs) A.
== Definitions and notation ==
Consider adjacent vertices
u
,
v
∈
V
{\displaystyle... | Wikipedia/Mixed_graph |
Graph may refer to:
== Mathematics ==
Graph (discrete mathematics), a structure made of vertices and edges
Graph theory, the study of such graphs and their properties
Graph (topology), a topological space resembling a graph in the sense of discrete mathematics
Graph of a function
Graph of a relation
Graph paper
Chart... | Wikipedia/Graph_(disambiguation) |
In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have exactly three vertices. The chordal graphs may als... | Wikipedia/Chordal_graph |
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge be... | Wikipedia/Line_graph |
In the mathematical field of graph theory, an automorphism is a permutation of the vertices such that edges are mapped to edges and non-edges are mapped to non-edges. A graph is a vertex-transitive graph if, given any two vertices v1 and v2 of G, there is an automorphism f such that
f
(... | Wikipedia/Vertex-transitive_graph |
In the mathematical field of graph theory, a graph G is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices
(
u
1
,
v
1
)
... | Wikipedia/Arc-transitive_graph |
In graph theory, series–parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits.
== Definition and terminology ==
In this context, the term graph means multigraph.
There are se... | Wikipedia/Series–parallel_graph |
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arc... | Wikipedia/Vertex_(graph_theory) |
In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").
== Order-zero graph ==
The order-zero graph, K0, is the unique graph having no vertices (hence its order is zero). It ... | Wikipedia/Empty_graph |
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers
λ
,
μ
≥
0
{\displaystyle \lambda ,\mu \geq 0}
every two adjacent vertices have λ common neighbours, an... | Wikipedia/Strongly_regular_graph |
A conceptual graph (CG) is a formalism for knowledge representation. In the first published paper on CGs, John F. Sowa used them to represent the conceptual schemas used in database systems. The first book on CGs applied them to a wide range of topics in artificial intelligence, computer science, and cognitive science.... | Wikipedia/Conceptual_graph |
In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set of G, and a subgroup of G. The Schreier graph encodes the abstract structure of the group modulo an equivalence relation formed by the cosets of the subgroup.
The graph is named ... | Wikipedia/Schreier_coset_graph |
In graph theory, the tensor product G × H of graphs G and H is a graph such that
the vertex set of G × H is the Cartesian product V(G) × V(H); and
vertices (g,h) and (g',h' ) are adjacent in G × H if and only if
g is adjacent to g' in G, and
h is adjacent to h' in H.
The tensor product is also called the direct produ... | Wikipedia/Tensor_product_of_graphs |
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