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In this paper it is shown that for any network there is a uniquely determined network based on a structure tree that provides a convenient way of determining a minimal cut separating a pair $s, t$ where each of $s, t$ is either a vertex or an end in the original network. A Max-Flow Min-Cut Theorem is proved for any ne... |
Given a convex $n$-gon $P$ and a positive integer $m$ such that $3\le m\le n-1$, let $Q$ denote the largest area convex $m$-gon contained in $P$. We are interested in the minimum value of $\Delta(Q)/\Delta(P)$, the ratio of the areas of these two polygons. |
We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial distance ideal. |
In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial numbers, integer numbers that are obtained from the binomial coefficients by replacing each term by its corresponding Fibonacci number. In this paper, we present a combinatorial description for the $q$-analog and elliptic analog of the Fibon... |
Given an $n$-vertex pseudorandom graph $G$ and an $n$-vertex graph $H$ with maximum degree at most two, we wish to find a copy of $H$ in $G$, i.e.\ an embedding $\varphi\colon V(H)\to V(G)$ so that $\varphi (u)\varphi |
The least eigenvalue of a graph $G$ is the least eigenvalue of adjacency matrix of $G$. In this paper we determine the graphs which attain the minimum least eigenvalue among all complements of connected simple graphs with given connectivity. |
We show that the $n$'th diagonal sum of Barry's modified Pascal triangle can be described as the continuant of the run lengths of the binary representation of $n$. We also obtain an explicit description for the row sums. |
In this article, a combinatorial characterization of the family of parabolic hyperplanes of a hyperbolic (respectively, elliptic) quadric of PG(2n + 1, q), using their intersection properties with the points and subspaces of codimension 2, is given. |
In 1935, Pauling proposed an estimate for the number of Eulerian orientations of a graph in the context of the theoretical behaviour of water ice. The logarithm of the number of Eulerian orientations, normalised by the number of vertices, is called the residual entropy. |
We study Poncelet's Theorem in the four non-isomorphic finite projective planes of order 9. Among these planes, only the Desarguesian plane turns out to be a Poncelet plane, while the other three planes which are constructed over the miniquaternion near-field of order 9, are not. |
We explore how the asymptotic structure of a random permutation of $[n]$ with $m$ inversions evolves, as $m$ increases, establishing thresholds for the appearance and disappearance of any classical, consecutive or vincular pattern. The threshold for the appearance of a classical pattern depends on the greatest number ... |
In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$. We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove that the Ehrhart polynomials of $C_d^*$ for $d=1,2,\cdots$ has the interlacing p... |
We investigate eigenvalues of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $\Gamma(R)$. The graph $\Gamma(R)$ is defined as the graph with vertex set consisting of all non... |
In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. |
The geodesic flow on a finite discrete q-manifold with or without boundary is defined as as a permutation of its ordered q-simplices. This allows to define geodesic sheets and a notion of sectional curvature. |
In this paper, we investigate the algebraic structure underlying the acyclic decomposition. This decomposition applies to directed metric ribbon graphs and enables the recursive computation of the volumes of their moduli spaces. |
Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point locus in $\mathbb{C}^n$. The cyclic sieving phenomenon is a notion in enumerative ... |
In this paper we give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. |
A 1d random geometric graph (1d RGG) is built by joining a random sample of $n$ points from an interval of the real line with probability $p$. We count the number of $k$-hop paths between two vertices of the graph in the case where the space is the 1d interval $[0,1]$. |
For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix, and the distance energy is defined as the sum of the absolute values of the eigenvalues of its distance matrix. We establish lower and upper bounds for the distance spectral radius of graphs and bipartite graphs, lowe... |
We prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ${n \choose 2}! $ for both the code weights and the Chevalley weights. |
A digraph is semicomplete if any two vertices are connected by at least one arc and is locally semicomplete if the out-neighbourhood and the in-neighbourhood of any vertex induce a semicomplete digraph. In this paper we study various subclasses of locally semicomplete digraphs for which we give structural decompositio... |
A nondegenerate toric hypersurface of negative Kodaira dimension can be characterized by the empty Fine interior of its Newton polytope according to recent work by Victor Batyrev, where the Fine interior is the rational subpolytope consisting of all points which have an integral distance of at least 1 to all integral s... |
In the infinite regular tree $\mathbb{T}_{q+1}$ with $q \in \mathbb{Z}_{\ge 2}$, we consider families $\{\mu_u^n\}$, indexed by vertices $u$ and nonnegative integers ("discrete time steps") $n$, of probability measures such that $\mu_u^n(v) = \mu_{u'}^n(v')$ if the distances $\operatorname{dist}(u,v)$ a... |
The explicit double sum for the associated Laguerre polynomials is derived combinatorially. The moments are described using certain statistics on permutations and permutation tableaux. |
Let $U_\theta$ be a unital defined in a shift plane of odd order $q^2$, which are constructed recently by the authors. In particular, when the shift plane is desarguesian, $U_\theta$ is a special Buekenhout-Metz unital formed by a union of ovals. |
In this paper we inititate the study of abstract simplicial complexes which are initial segments of qualitative probability orders. This is a natural class that contains the threshold complexes and is contained in the shifted complexes, but is equal to neither. |
Complete colorings have the property that any two color classes has at least an edge between them. Parameters such as the Grundy, achromatic and pseudoachromatic numbers comes from complete colorings, with some additional requirement. |
Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. |
We study lattice walks in a Weyl chamber of type A with fixed or free end points. For lattice walk models with zero drift that may be counted by means of a reflection argument, we determine asymptotics for the number of such walks as their length tends to infinity. |
Let $p_\mathrm{c}$ and $q_\mathrm{c}$ be the threshold and the expectation threshold, respectively, of an increasing family $\mathcal{F}$ of subsets of a finite set $X$, and let $l$ be the size of a largest minimal element of $\mathcal{F}$. Recently, Park and Pham proved the Kahn-Kalai conjecture, which says that $p_\... |
We define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices; we prove that the Riemann-Roch theorem holds in both cases. We extend Baker's Specialization Lemma to weighted graphs. |
We prove that the number of iterations taken by the Weisfeiler-Leman algorithm for configurations coming from Schreier graphs is closely linked to the diameter of the graphs themselves: an upper bound is found for general Schreier graphs, and a lower bound holds for particular cases, such as for Schreier graphs with $G... |
Let the Kneser graph $K$ of a distance-regular graph $\Gamma$ be the graph on the same vertex set as $\Gamma$, where two vertices are adjacent when they have maximal distance in $\Gamma$. We study the situation where the Bose-Mesner algebra of $\Gamma$ is not generated by the adjacency matrix of $K$. |
We define the notion of a painted tropical $A$-complex and describe a poset structure on the set of all such complexes. This poset is equivalent to the face lattice of a secondary polytope $\Sigma (\bar{A}_\alpha )$ where $\bar{A}_\alpha$ is built from $A$ and an additional point $\alpha$. |
It is known that for the Young diagram of any partition of an integer $n$, the sum of squares of the hook lengths of its cells is exactly $n^2$ more than that of the contents of its cells. That is, for any partition $\lambda$ of an integer $n$, \begin{equation*} <br>\sum_{u \in \lambda} h(u)^2 = n^2 + \sum_{u \in \lam... |
Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds o... |
Factorial Schur functions are generalizations of Schur functions that have, in addition to the usual variables, a second family of "shift" parameters. We show that a factorial Schur function times a deformation of the Weyl denominator may be expressed as the partition function of a particular statistical-mecha... |
Novák conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic $(v,k,\lambda)$-designs with $1 \leq \... |
A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exi... |
Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over the previously best known bounds. |
We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees we obtain a simple identity relating hitting times to the Wiener index. |
We investigate the generalized bilinear forms graph $\Gamma_d$ over a residue class ring $\mathbb{Z}_{p^s}$. We show that $\Gamma_d$ is a connected vertex transitive graph, and completely determine its independence number, clique number, chromatic number and maximum cliques. |
A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by Croot, Lev, and Pach in their breakthrough work on arithmetic-progression-free set... |
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to... |
In 1976 Faudree and Schelp conjectured that in a hamiltonian-connected graph on $n$ vertices, any two distinct vertices are connected by a path of length $k$ for every $k \ge n/2$. In 1978 Thomassen constructed a (non-cubic and non-planar) family of counterexamples, showing that there exist hamiltonian-connected $n$-v... |
A seminal technique of theoretical physics called Wick's theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogo... |
We study the reconstruction problem of permutation sequences from their $k$-minors, which are subsequences of length $k$ with entries renumbered by $1,2,\ldots,k$ preserving order. We prove that the minimum number $k$ such that any permutation of length $n$ can be reconstructed from the multiset of its $k$-minors is b... |
We estimate the number of incidences in a configuration of $m$ lines and $n$ points in dimension 3. The main term is $mn^{1/3}$ if we work over the real or complex numbers but $mn^{2/5}$ over finite fields. |
In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra $T$ of a distance-regular graph that has $q$-Racah type; |
In this paper, we disprove EMSO(FO$^2$) convergence law for the binomial random graph $G(n,p)$ for any constant probability $p$. More specifically, we prove that there exists an existential monadic second order sentence with 2 first order variables such that, for every $p\in(0,1)$, the probability that it is true on $... |
Let $v(n)$ be the minimum number of voters with transitive preferences which are needed to generate any strong preference pattern (ties not allowed) on $n$ candidates. Let $k=\lfloor \log_2 n\rfloor$. |
We prove a precise min-max theorem for the following problem. Let $G$ be an Eulerian graph with a specified set of edges $S \subseteq E(G)$, and let $b$ be a vertex of $G$. |
This is a continuation of the early paper concerning matroid base polytope decomposition. Here, we will present sufficient conditions on $M$ so its base matroid polytope $P(M)$ has a {\em sequence} of hyperplane splits. |
We construct an addition and a multiplication on the set of planar binary trees, closely related to addition and multiplication on the integers. This gives rise to a new kind of (noncommutative) arithmetic theory. |
In the Matching Cut problem we ask whether a graph $G$ has a matching cut, that is, a matching which is also an edge cut of $G$. We consider the variants Perfect Matching Cut and Disconnected Perfect Matching where we ask whether there exists a matching cut equal to, respectively contained in, a perfect matching. |
Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\dot x=-{\cal L}x$ for consensus and $\dot p=-p{\cal L}$ for diffusion. |
In 1984, Wilson proved the Erdős-Ko-Rado theorem for $t$-intersecting families of $k$-subsets of an $n$-set: he showed that if $n\ge(t+1)(k-t+1)$ and $\mathcal{F}$ is a family of $k$-subsets of an $n$-set such that any two members of $\mathcal{F}$ have at least $t$ elements in common, then $|\mathcal{F}|\le\binom{n-t}{... |
We give an upper bound on the number of cycles in a simple graph in terms of its degree sequence, and apply this bound to resolve several conjectures of Király and Arman and Tsaturian and to improve upper bounds on the maximum number of cycles in a planar graph. |
This paper deals with two problems about splitting fairly a path with colored vertices, where "fairly" means that each part contains almost the same amount of vertices in each color. Our first result states that it is possible to remove one vertex per color from a path with colored vertices so that the remaini... |
Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if $e_{1}, e_{2}$ and $e_{3}$ are consecutive edges in $G$, then $c(e_{1})\neq c(e_3)$.... |
Following problems posed by Gyárfás, we show that for every $r$-edge-colouring of $K_n$ there is a monochromatic triple star of order at least $n/(r-1)$, improving a previous result by Ruszinkó. <br>An edge colouring of a graph is called a local $r$-colouring if every vertex spans edges of at most $r$ distinct colours... |
The conditional $h$-vertex($h$-edge) connectivity of a connected graph $H$ of minimum degree $ k > h$ is the size of a smallest vertex(edge) set $F$ of $H$ such that $H - F$ is a disconnected graph of minimum degree at least $h. $ Let $G$ be the Cartesian product of $r\geq 1$ cycles, each of length at least four and... |
We study the problem of packing arborescences in the random digraph $\mathcal D(n,p)$, where each possible arc is included uniformly at random with probability $p=p(n)$. Let $\lambda(\mathcal D(n,p))$ denote the largest integer $\lambda\geq 0$ such that, for all $0\leq \ell\leq \lambda$, we have $\sum_{i=0}^{\ell-1} (... |
A {\em restraint} on a (finite undirected) graph $G = (V,E)$ is a function $r$ on $V$ such that $r(v)$ is a finite subset of ${\mathbb N}$; a proper vertex colouring $c$ of $G$ is {\em permitted} by $r$ if $c(v) \not\in r(v)$ for all vertices $v$ of $G$ (we think of $r(v)$ as the set of colours {\em forbidden} at $v$).... |
The inner Tutte group of a matroid is a finitely generated abelian group introduced as an algebraic counterpart of Tutte's homotopy theory of matroids. The aim of this work is to provide a new presentation for this group with a set of generators that is smaller than those previously known. |
The lexicographical ordering of hypergraphs via spectral moments is called the $S$-order of <a href="http://hypergraphs.In" rel="external noopener nofollow" class="link-external link-http">this http URL</a> this paper, the $S$-order of hypergraphs is <a href="http://investigated.We" rel="external noopener nofollow" cla... |
We develop the theory of matroids over one-dimensional algebraic groups, with special emphasis on positive characteristic. In particular, we compute the Lindström valuations and Frobenius flocks of such matroids. |
We study continuous-time quantum walks on normal Cayley graphs of certain non-abelian groups, called extraspecial groups. By applying general results for graphs in association schemes we determine the precise conditions for perfect state transfer and fractional revival, and use partial spreads to construct graphs on e... |
In this article we describe an algorithm that can be applied for the generation of various classes of maps on orientable surfaces. It uses existing generators for abstract graphs and combines them with an efficient embedding and isomorphism rejection routine. |
A function $f: \mathbb{Z} \to \mathbb{Q}^n$ is a $c$-quasihomomorphism if the Hamming distance between $f(x+y)$ and $f (x)+f |
We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree $T$ of order $n$, $cfc(T)\geq cfc(P_n)=\lceil \log_{2} n\rceil$, which completely confirms the conjecture of Li and Wu. |
The sequence counting the number of unique tournaments with n people is the same as the sequence counting non-tracking binary strings corresponding to n-2 radar observations with the tracking rule "3 out of 5 with loss 2." This fact allows us to build a bijection between unique tournaments and non-tracking bina... |
A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric subset with measure D(x). |
Analogous to the case of the binomial random graph $G(d+1,p)$, it is known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, which we denote by $Q^d_p$, undergoes a phase transition around the critical value of $p=\frac{1}{d}$. More p... |
Although the Pólya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of unique colorings of a finite set under the action of a finite group. |
The aim of this thesis is to investigate how the structure of a digraph affects its dichromatic number and to extend various results on undirected colouring to digraphs. <br>In the first part of this thesis, we examine how the dichromatic number interacts with other metrics. |
We show that the sequentially $(S_r)$ condition for simplicial complexes is a topological property. Along the way, we present an elementary proof for the fact that the Serre's condition $(S_r)$ is a topological property. |
For graphs $G$ and $F$, the saturation number $\textit{sat}(G,F)$ is the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$. In 2017, Korándi and Sudakov initiated the study of saturation in random graphs. |
Nous refutons, sous une certaine hypothese combinatoire, la "nonrevisiting path conjecture". <br>Abstract: In this article, we give, under some hypothesis, a couterexample to the nonrevisiting path conjecture. |
Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the free cumulants of the associated Young diagram. We present two positivity conjectures for their coefficients. |
We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner; a new proof of a theorem of Liggett stating that ultra-log-... |
We give a new proof Tverberg's famous theorem: For every set $X \subset \R^d$ with $|X|=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\ldots,X_r$ such that $\bigcap_{p=1}^r \conv X_p\ne \emptyset$. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and... |
In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial representations of double affine Hecke algebras with respect to higher rank Chere... |
Given a geometric hypergraph (or a range-space) $H=(V,\cal E)$, a coloring of its vertices is said to be conflict-free if for every hyperedge $S \in \cal E$ there is at least one vertex in $S$ whose color is distinct from the colors of all other vertices in $S$. The study of this notion is motivated by frequency assig... |
For a positive integer $k$, we say that a graph is $k$-existentially complete if for every $0 \leq a \leq k$, and every tuple of distinct vertices $x_1,\ldots,x_a$, $y_1,\ldots,y_{k-a}$, there exists a vertex $z$ that is joined to all of the vertices $x_1,\ldots,x_a$ and none of the vertices $y_1,\ldots,y_{k-a}$. Whil... |
A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $\Delta(G)\geq 2$ has an equitable coloring with $\Delta(G)$ colors, except when $... |
In this paper we study a classical Maxwell question on the existence of self-stresses for frameworks, which are called tensegrities. We give a complete answer on geometric conditions of at most $(d+1)$-valent tensegrities in $\mathbb{R}^d$ both in terms of discrete multiplicative 1-forms and in terms of "meet"... |
Motivated by the bootstrap percolation process for graphs, we define a new, high-order generalisation to $k$-uniform hypergraphs, in which we infect $j$-sets of vertices for some integer $1\le j \le k-1$. We investigate the smallest possible size of an initially infected set which ultimately percolates and determine t... |
Answering a question of Cameron, Pretzel and Siemons proved that every integer partition of $n\ge 2(k+3)(k+1)$ can be reconstructed from its set of $k$-deletions. We describe a new reconstruction algorithm that lowers this bound to $n\ge k^2+2k$ and present examples showing that this bound is best possible. |
A unique sink orientation (USO) is an orientation of the hypercube graph with the property that every face has a unique sink. A number of well-studied problems reduce in strongly polynomial time to finding the global sink of a USO; most notably, linear programming (LP) and the P-matrix linear complementarity problem (... |
The regular graph of ideals of the commutative ring $R$, denoted by ${\Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this pa... |
Win [\emph{J. Graph Theory} {\bf 6}(1982), 489--492] conjectured that a graph $G$ on $n$ vertices contains $k$ disjoint perfect matchings, if the degree sum of any two nonadjacent vertices is at least $n+k-2$, where $n$ is even and $n\geq k+2$. |
The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an $\alpha $-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*... |
Beautiful formulas are known for the expected cost of random two-dimensional assignment problems, but in higher dimensions even the scaling is not known. In three dimensions and above, the problem has natural "Axial" and "Planar" versions, both of which are NP-hard. |
Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. |
Kostant's partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra $\mathfrak{g}$ as a nonnegative integral linear combination of the positive roots of $\mathfrak{g}$. Multiplex juggling sequences are generalizations of juggling sequences that speci... |
Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual Ramsey theory. |
Nath and Paul (Linear Algebra Appl. ,460(2014),97-110) have shown that the largest distance Laplacian eigenvalue of a path is simple and the corresponding eigenvector has properties similar to the Fiedler vector. |
We consider the problem of determining which matrices are permutable to be supmodular. We show that for small dimensions any matrix is permutable by a universal permutation or by a pair of permutations, while for higher dimensions no universal permutation exists. |
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