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Bidirected graphs are a generalisation of directed graphs that arises in the study of undirected graphs with perfect matchings. Menger's famous theorem - the minimum size of a set separating two vertex sets $X$ and $Y$ is the same as the maximum number of disjoint paths connecting them - is generally not true in bidirected graphs. We introduce a sufficient condition for $X$ and $Y$ which yields a version of Menger's Theorem in bidirected graphs that in particular implies its directed counterpart., Comment: 23 pages, 6 figures

Given a graph $G=(V,E)$, for a vertex set $S\subseteq V$, let $N(S)$ denote the set of vertices in $V$ that have a neighbor in $S$. Extending the concept of binding number of graphs by Woodall~(1973), for a vertex set $X \subseteq V$, we define the binding number of $X$, denoted by $\bind(X)$, as the maximum number $b$ such that for every $S \subseteq X$ where $N(S)\neq V(G)$ it holds that $|N(S)|\ge b {|S|}$. Given this definition, we prove that if a graph $V(G)$ contains a subset $X$ with $\bind(X)= 1/k$ where $k$ is an integer, then $G$ possesses a matching of size at least $|X|/(k+1)$. Using this statement, we derive tight bounds for the estimators of the matching size in planar graphs. These estimators are previously used in designing sublinear space algorithms for approximating the maching size in the data stream model of computation. In particular, we show that the number of locally superior vertices is a $3$ factor approximation of the matching size in planar graphs. The previous analysis by Jowhari (2023) proved a $3.5$ approximation factor. As another application, we show a simple variant of an estimator by Esfandiari \etal (2015) achieves $3$ factor approximation of the matching size in planar graphs. Namely, let $s$ be the number of edges with both endpoints having degree at most $2$ and let $h$ be the number of vertices with degree at least $3$. We prove that when the graph is planar, the size of matching is at least $(s+h)/3$. This result generalizes a known fact that every planar graph on $n$ vertices with minimum degree $3$ has a matching of size at least $n/3$., Comment: 9 pages

Aldous and Fill (2002) conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^{2}}{2\pi ^{2}}$. A conjecture by Guiduli and Mohar (1996) predicts the structure of graphs whose algebraic connectivity $\mu $ is the smallest among all connected graphs whose minimum degree $\delta $ is a given $d$. We prove that this conjecture implies the Aldous--Fill conjecture for odd $d$. We pose another conjecture on the structure of $d$-regular graphs with minimum $\mu $, and show that this also implies the Aldous--Fill conjecture for even $d$. In the literature, it has been noted empirically that graphs with small $\mu $ tend to have a large diameter. In this regard, Guiduli (1996) asked if the cubic graphs with maximum diameter have algebraic connectivity smaller than all others. Motivated by these, we investigate the interplay between the graphs with maximum diameter and those with minimum algebraic connectivity. We show that the answer to Guiduli problem in its general form, that is for $d$-regular graphs for every $d\ge 3$ is negative. We aim to develop an asymptotic formulation of the problem. It is proven that $d$-regular graphs for $d\ge 5$ as well as graphs with $\delta =d$ for $d\ge 4$ with asymptotically maximum diameter, do not necessarily exhibit the asymptotically smallest $\mu$. We conjecture that $d$-regular graphs (or graphs with $\delta =d$) that have asymptotically smallest $\mu $, should have asymptotically maximum diameter. The above results rely heavily on our understanding of the structure as well as optimal estimation of the algebraic connectivity of nearly maximum-diameter graphs, from which the Aldous--Fill conjecture for this family of graphs also follows., Comment: 25 pages, final version, to appear in J. Combin. Theory Ser. B

In this note we present an infinite family of 2-connected graphs such that their matching polynomials have only integer zeros. This answers in negative a question of Akbari et al. [Graphs with integer matching polynomial zeros, Discrete Appl. Math. 224 (2017), 1-8]., Comment: Some typos corrected

Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted $\Delta_n$ and $\Gamma_n$ which are more `symmetric' compared to the other graphs in these two families, respectively. We prove that $(1,\sqrt{5}]$ is a gap interval for $\Delta_n$, and $[(-1+\sqrt{17})/2,3]$ is a gap interval for $\Gamma_n$. We conjecture that these two are indeed maximal gap intervals. As a by-product, we show that the eigenvalues of $\Delta_n$ lying in the interval $[-3,-\sqrt{5}]$ (in particular, its minimum eigenvalue) converge to $(1-\sqrt{33})/2$ and the eigenvalues of $\Gamma_n$ lying in the interval $[-4,-(1+\sqrt{17})/2]$ (and in particular, its minimum eigenvalue) converge to $1-\sqrt{13}$ as $n$ tends to infinity. The proofs of the above results heavily depend on the following property which can be of independent interest: with few exceptions, all the eigenvalues of connected cubic and quartic graphs with minimum spectral gap are simple., Comment: 31 pages, final version, to appear in Discrete Mathematics

Littlewood--Offord Problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of this problem where we estimate the number of subsums that are $(0,1)$-vectors. We then utilize this to find the maximum order of graphs with given rank or corank. The rank of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the corank of $G$ is the rank of $A(G)+I$., Comment: 18 pages, to appear in Electronic Journal of Combinatorics

The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the corrected version of) a conjecture by Koledin and Stani\'c (2017)., Comment: 14 pages, final version, to appear in Discrete Mathematics

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2\pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected $k$-regular graph on $n$ vertices is at least $(1+o(1))\frac{2k\pi^2}{3n^2}$, and the bound is attained for at least one value of $k$. We determine the structure of connected quartic graphs on $n$ vertices with minimum spectral gap which enable us to show that the minimum spectral gap of connected quartic graphs on $n$ vertices is $(1+o(1))\frac{4\pi^2}{n^2}$. From this result, the Aldous--Fill conjecture follows for $k=4$., Comment: 31 pages, final version, to appear in Journal of Graph Theory

The $k$-coprime graph of order $n$ is the graph with vertex set $\{k, k+1, \ldots, k+n-1\}$ in which two vertices are adjacent if and only if they are coprime. We characterize Hamiltonian $k$-coprime graphs. As a particular case, two conjectures by Tout, Dabboucy, Howalla (1982) and by Schroeder (2019) on prime labeling of $2$-regular graphs follow. A prime labeling of a graph with $n$ vertices is a labeling of its vertices with distinct integers from $\{1, 2,\ldots , n\}$ in such a way that the labels of any two adjacent vertices are relatively prime., Comment: Typos corrected

A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed graphs have a symmetric spectrum but not the other way around. We present constructions of signed graphs with symmetric spectra which are not sign-symmetric. This, in particular answers a problem posed by Belardo, Cioab\u{a}, Koolen, and Wang (2018).

This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which $-1$ and $0$ are considered as trivial eigenvalues. We show that threshold graphs have no non-trivial eigenvalues in the interval $\left[(-1-\sqrt{2})/2,\,(-1+\sqrt{2})/2\right]$. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018)., Comment: Comments from referee incorporated; Final version

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph $G$ is called the Dilworth number of $G$. We prove that for any cograph $G$, the multiplicity of any eigenvalue $\lambda\ne0,-1$, does not exceed the Dilworth number of $G$ and show that this bound is tight. G. F. Royle [The rank of a cograph, Electron. J. Combin. 10 (2003), Note 11] proved that if a cograph $G$ has no pair of vertices with the same neighborhood, then $G$ has no 0 eigenvalue, and asked if beside cographs, there are any other natural classes of graphs for which this property holds. We give a partial answer to this question by showing that an $H$-free family of graphs has this property if and only if it is a subclass of the family of cographs. A similar result is also shown to hold for the $-1$ eigenvalue., Comment: 13 pages, Comments from referees incorporated

A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. Alazemi, Andeli\'c and Simi\'c conjectured that no chain graph shares a non-zero (adjacency) eigenvalue with its vertex-deleted subgraphs. We disprove this conjecture. However, we show that the assertion holds for subgraphs obtained by deleting vertices of maximum degrees in either of color classes. We also give a simple proof for the fact that chain graphs have no eigenvalue in the interval $(0,1/2)$.

A prime labeling of a graph with $n$ vertices is a labeling of its vertices with distinct integers from $\{1, 2,\ldots , n\}$ in such a way that the labels of any two adjacent vertices are relatively prime. T. Varkey conjectured that ladder graphs have a prime labeling. We prove this conjecture.

A nuciferous graph is a simple graph with a non-singular $0$-$1$ adjacency matrix $A$ such that all the diagonal entries of $A^{-1}$ are zero and all the off-diagonal entries of $A^{-1}$ are non-zero. Sciriha et al. conjectured that except $K_2$, no nuciferous graph exists. We disprove this conjecture. Moreover, we conjecture that there infinitely many nuciferous Cayley graphs., Comment: Minor error corrected, acknowledgments updated

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an eigenvalue in the interval $(-1,0)$. It is also shown that the multiplicity of any eigenvalue of a cograph $G$ does not exceed the sum of multiplicities of $0$ and $-1$ as eigenvalues of $G$. We introduce a partial order on the vertex set of graphs $G$ in terms of inclusions among the open and closed neighborhoods of vertices, and conjecture that the multiplicity of any eigenvalue of a cograph $G$ except for $0,-1$ does not exceed the maximum size of an antichain with respect to that partial order. In two extreme cases (in particular for threshold graphs), the conjecture is shown to be true. Finally, we give a simple proof for the result that bipartite $P_5$-free graphs have no eigenvalue in the intervals $(-1/2,0)$ and $(0,1/2)$., Comment: 12 pages; A missing reference is added comparing with the published version in Linear Multilinear Algebra. arXiv admin note: text overlap with arXiv:1803.00246

Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both Laplacian and signless Laplacian eigenvalues from which the conjecture follows., Comment: Incorporated refree's commnets

In this paper, we show that the eigenvalues of certain classes of Cayley graphs are integers. The (n,k,r)-arrangement graph A(n,k,r) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they differ in exactly r positions. We establish a relation between the eigenvalues of the arrangement graphs and the eigenvalues of certain Cayley graphs. As a result, the conjecture on integrality of eigenvalues of A(n,k,1) follows., Comment: 12 pages, final version

The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k-1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n,k). This equitable partition can be employed to compute the complete set of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine the eigenvalues of A(n,k) for small values of k. Finally, we show that any eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n,k) and that -k is the smallest eigenvalue of A(n,k) with multiplicity O(n^k) for fixed k., Comment: 18 pages. Revised version according to a referee suggestions

To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A^{-1} have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree., Comment: 9 pages

For a graph $G$, let $S(G)$ be the Seidel matrix of $G$ and $\te_1(G),...,\te_n(G)$ be the eigenvalues of $S(G)$. The Seidel energy of $G$ is defined as $|\te_1(G)|+...+|\te_n(G)|$. Willem Haemers conjectured that the Seidel energy of any graph with $n$ vertices is at least $2n-2$, the Seidel energy of the complete graph with $n$ vertices. Motivated by this conjecture, we prove that for any $\al$ with $0<\al<2$, $|\te_1(G)|^\al+...+|\te_n(G)|^\al\g (n-1)^\al+n-1$ if and only if $|{\rm det} S(G)|\g n-1$. This, in particular, implies the Haemers' conjecture for all graphs $G$ with $|{\rm det} S(G)|\g n-1$.

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\lambda\equiv2\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\lambda\equiv0\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if $\tau(G)=2^ts$ with $s$ odd, then the multiplicity of any even integer eigenvalue of $Q(G)$ is at most $t+1$. Among other things, we prove that if $L(G)$ or $Q(G)$ has an even integer eigenvalue of multiplicity at least $2$, then $\tau(G)$ is divisible by $4$. As a very special case of this result, a conjecture by Zhou et al. [On the nullity of connected graphs with least eigenvalue at least $-2$, Appl. Anal. Discrete Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows., Comment: Final version. To appear in Discrete Math

We determine all graphs whose matching polynomials have at most five distinct zeros. As a consequence, we find new families of graphs which are determined by their matching polynomial., Comment: 14 pages, 7 figures, 1 appendix table. Final version. Some typos are fixed

Let $G$ be a graph on $n$ vertices with $r := \lfloor n/2 \rfloor$ and let $\lambda_1 \geq...\geq \lambda_{n} $ be adjacency eigenvalues of $G$. Then the H\"uckel energy of $G$, HE($G$), is defined as $$\he(G) = {ll} 2\sum_{i=1}^{r} \lambda_i, & \hbox{if $n= 2r$;} 2\sum_{i=1}^{r} \lambda_i + \lambda_{r+1}, & \hbox{if $n= 2r+1$.} $$ The concept of H\"uckel energy was introduced by Coulson as it gives a good approximation for the $\pi$-electron energy of molecular graphs. We obtain two upper bounds and a lower bound for HE$(G)$. When $n$ is even, it is shown that equality holds in both upper bounds if and only if $G$ is a strongly regular graph with parameters $(n, k, \lambda, \mu) = (4t^2 +4t +2, 2t^2 +3t +1, t^2 +2t, t^2 + 2t +1),$ for positive integer $t$. Furthermore, we will give an infinite family of these strongly regular graph whose construction was communicated by Willem Haemers to us. He attributes the construction to J.J. Seidel., Comment: 13 pages; historical points and some references added

A pseudo $(v,\, k,\, \la)$-design is a pair $(X, {\cal B})$ where $X$ is a $v$-set and ${\cal B}=\{B_1,...,B_{v-1}\}$ is a collection of $k$-subsets (blocks) of $X$ such that each two distinct $B_i, B_j$ intersect in $\la$ elements; and $0\le\la

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved that E(G)>= 2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>= 2ch(G) for all graphs G except for those in a few specified families, where \bar{G}, \chi(G), and ch(G) are the complement, the chromatic number, and the choice number of G, respectively., Comment: to appear in Linear Algebra and its Applications