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In this paper, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices $n$ and number of edges $e$, the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this paper, we answer Hong's question positively for various values of $n$ and $e$ and in several cases, we determined the graphs with minimum spectral radius., Comment: 22 pages, 6 figures, revised paper taking into consideration the comments from the referees

In this paper, we devote to investigating the circumstances under which the addition of an edge to a graph will cause the Laplacian matching roots to change only by integer quantities. We prove that the Laplacian matching root integral variation in one place is impossible and the Laplacian matching root integral variation in two places is also impossible under some constraints.

In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph., Comment: 12 pages, 1 figure

A recent result of Cioab\u{a}, Dewar and Gu implies that any $k$-regular Ramanujan graph with $k\geq 8$ is globally rigid in $\mathbb{R}^2$. In this paper, we extend these results and prove that any $k$-regular Ramanujan graph of sufficiently large order is globally rigid in $\mathbb{R}^2$ when $k\in \{6, 7\}$, and when $k\in \{4,5\}$ if it is also vertex-transitive. These results imply that the Ramanujan graphs constructed by Morgenstern in 1994 are globally rigid. We also prove several results on other types of framework rigidity, including body-bar rigidity, body-hinge rigidity, and rigidity on surfaces of revolution. In addition, we use computational methods to determine which Ramanujan graphs of small order are globally rigid in $\mathbb{R}^2$., Comment: 23 pages, 9 figures

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter $d$, the most contracted pairs of vertices are those at distance $d$. In this paper, we confirm Vallentin's conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance $d{-}1$. We suggest three alternative conjectures and prove them for several families of distance-regular graphs., Comment: 19 pages

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $\delta \ge 2m+2 \ge 4$ satisfies $\lambda_2(G) < \delta - \frac{2m+1}{\delta+1}$, then $G$ contains at least $m+1$ edge-disjoint spanning trees, which verified a generalization of a conjecture by Cioab\u{a} and Wong. We show this bound is essentially the best possible by constructing $d$-regular graphs $\mathcal{G}_{m,d}$ for all $d \ge 2m+2 \ge 4$ with at most $m$ edge-disjoint spanning trees and $\lambda_2(\mathcal{G}_{m,d}) < d-\frac{2m+1}{d+3}$. As a corollary, we show that a spectral inequality on graph rigidity by Cioab\u{a}, Dewar, and Gu is essentially tight., Comment: 13 pages, 2 figures

Recently, Brouwer, Cioab\u{a}, Ihringer and McGinnis obtained some new results involving the eigenvalues of various graphs coming from association schemes and posed some conjectures related to the eigenvalues of Grassmann graphs, bilinear forms graphs and Hermitian forms graphs. In this paper, we prove some of their conjectures., Comment: 22 pages

In 2011, Haemers asked the following question: If $S$ is the Seidel matrix of a graph of order $n$ and $S$ is singular, does there exist an eigenvector of $S$ corresponding to $0$ which has only $\pm 1$ elements? In this paper, we construct infinite families of graphs which give a negative answer to this question. One of our constructions implies that for every natural number $N$, there exists a graph whose Seidel matrix $S$ is singular such that for any integer vector in the nullspace of $S$, the absolute value of any entry in this vector is more than $N$. We also derive some characteristics of vectors in the nullspace of Seidel matrices, which lead to some necessary conditions for the singularity of Seidel matrices. Finally, we obtain some properties of the graphs which affirm the above question.

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity and plays a key role in the theory of expander graphs. In this paper, we extend previous work done for graphs and bipartite graphs and present a linear programming method for obtaining an upper bound on the order of a regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore, we obtain a general upper bound on the order of a regular uniform hypergraph whose second eigenvalue is bounded by a given value. Our results improve and extend previous work done by Feng-Li (1996) on Alon-Boppana theorems for regular hypergraphs and by Dinitz-Schapira-Shahaf (2020) on the Moore or degree-diameter problem. We also determine the largest order of an $r$-regular $u$-uniform hypergraph with second eigenvalue at most $\theta$ for several parameters $(r,u,\theta)$. In particular, orthogonal arrays give the structure of the largest hypergraphs with second eigenvalue at most $1$ for every sufficiently large $r$. Moreover, we show that a generalized Moore geometry has the largest spectral gap among all hypergraphs of that order and degree., Comment: references updated, fixed some typos, added explanation describing the differences between the graphs and hypergraphs results, 27 pages, 3 tables, European Journal of Combinatorics, accepted for publication

In this paper, we study the flip graph on the perfect matchings of a complete graph of even order. We investigate its combinatorial and spectral properties including connections to the signed reversal graph and we improve a previous upper bound on its chromatic number., Comment: 15 pages, 6 figures, 2 tables

In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana theorems for regular graphs and in the classical degree-diameter/Moore problem., Comment: 14 pages, 1 table,to appear in RIMS K\^oky\^uroku

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair $(G,p)$ of graph $G$ together with a map $p$ of the vertices of $G$ into the Euclidean plane. We view the edges of $(G, p)$ as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combinatorial characterization of rigid graphs in the Euclidean plane. In 1982, Lov\'asz and Yemini discovered a new characterization and proved that every $6$-connected graph is rigid. Combined with a characterization of global rigidity, their proof actually implies that every 6-connected graph is globally rigid. Consequently, if Fiedler's algebraic connectivity is greater than 5, then $G$ is globally rigid. In this paper, we improve this bound and show that for a graph $G$ with minimum degree $\delta\geq 6$, if its algebraic connectivity is greater than $2+\frac{1}{\delta -1}$, then $G$ is rigid and if its algebraic connectivity is greater than $2+\frac{2}{\delta -1}$, then $G$ is globally rigid. Our results imply that every connected regular Ramanujan graph with degree at least $8$ is globally rigid. We also prove a more general result giving a sufficient spectral condition for the existence of $k$ edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains $k$ edge-disjoint spanning $2$-connected subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees., Comment: 15 pages, 1 figure; this updated version includes a new co-author and new results involving global rigidity

Let $\Gamma$ be a $Q$-polynomial distance-regular graph of diameter $d\geq 3$. Fix a vertex $\gamma$ of $\Gamma$ and consider the subgraph induced on the union of the last two subconstituents of $\Gamma$ with respect to $\gamma$. We prove that this subgraph is connected., Comment: Journal of Combinatorial Theory, Series A, accepted for publication, 6 pages

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds., Comment: 24 pages, Discussiones Mathematicae Graph Theory, Special Issue in honor of Slobodan K. Simi\'c, accepted for publication

Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs. Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs., Comment: 22 pages, 4 figures, submitted to The Art of Discrete and Applied Mathematics, https://adam-journal.eu/index.php/ADAM

We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree $k \leq 18$ and their complements, the Latin square graphs and their complements, and the triangular graphs and their complements. Moreover, using a recent result of Ferber and Jain it is shown that an SRG of even order $n$, which is not the block graph of a Steiner 2-design or its complement, has chromatic index $k$, when $n$ is big enough. Except for the Petersen graph, all investigated connected SRGs of even order have chromatic index equal to their degree, i.e., they are class 1, and we conjecture that this is the case for all connected SRGs of even order., Comment: 10 pages

Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is minimum possible. In this paper, we determine an addressing of length $k(n-k)$ for the Johnson graphs $J(n,k)$ and we show that our addressing is optimal when $k=1$ or when $k=2, n=4,5,6$, but not when $n=6$ and $k=3$. We study the addressing problem as well as a variation of it in which the alphabet used has more than three symbols, for other graphs such as complete multipartite graphs and odd cycles. We also present computations describing the distribution of the minimum length of addressings for connected graphs with up to $10$ vertices. Motivated by these computations we settle a problem of Graham, showing that most graphs on $n$ vertices have an addressing of length at most $n-(2-o(1))\log_2 n$., Comment: 18 page, 24 tables, accepted for publication to Experimental Mathematics

Let $\Gamma$ be a finite group acting transitively on $[n]=\{1,2,\ldots,n\}$, and let $G=\mathrm{Cay}(\Gamma,T)$ be a Cayley graph of $\Gamma$. The graph $G$ is called normal if $T$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $G$ in terms of the second eigenvalues of certain subgraphs of $G$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $S_n$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $S_n$ with $\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$, where $\mathrm{supp}(\tau)$ is the set of points in $[n]$ non-fixed by $\tau$., Comment: 26 pages

Let $b(k,\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\theta$. In this paper, we obtain a general upper bound for $b(k,\theta)$ for any $0\leq \theta< 2\sqrt{k-1}$. Our bound gives the exact value of $b(k,\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\theta$, diameter $d$ and girth $g$ such that $g\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\geq 15$, we prove that there are no bipartite distance-regular graphs with $g\geq 2d-2$., Comment: 24 pages, 7 tables; In the revised version, a new result (Theorem 4.12) is added at the end of Section 4 showing that the bipartite bound from this paper (Theorem 4.7) is always better than the general upper bound from our previous SIDMA paper (Theorem 4.9)

A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a path of length $\ell-1$ which is a subgraph of a complete graph $K_n$. C\'{a}mara and Haemers~\cite{MC} conjectured that $K_n \backslash P_{\ell}$ is determined by its adjacency spectrum for every $2\leq \ell \leq n$. In this paper we show that the conjecture is true for $7\leq \ell \leq9$.

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical $P$- and $Q$-polynomial association schemes., Comment: 30 pages, accepted by JCTB

Let $k\ge 3$ be an integer, $q$ be a prime power, and $\mathbb{F}_q$ denote the field of $q$ elements. Let $f_i, g_i\in\mathbb{F}_q[X]$, $3\le i\le k$, such that $g_i(-X) = -\, g_i(X)$. We define a graph $S(k,q) = S(k,q;f_3,g_3,\cdots,f_k,g_k)$ as a graph with the vertex set $\mathbb{F}_q^k$ and edges defined as follows: vertices $a = (a_1,a_2,\ldots,a_k)$ and $b = (b_1,b_2,\ldots,b_k)$ are adjacent if $a_1\ne b_1$ and the following $k-2$ relations on their components hold: $$ b_i-a_i = g_i(b_1-a_1)f_i\Bigl(\frac{b_2-a_2}{b_1-a_1}\Bigr)\;,\quad 3\le i\le k. $$ We show that graphs $S(k,q)$ generalize several recently studied examples of regular expanders and can provide many new such examples., Comment: 21 pages

Let $n\geq k\geq 2$ be two integers and $S$ a subset of $\{0,1,\dots,k-1\}$. The graph $J_{S}(n,k)$ has as vertices the $k$-subsets of the $n$-set $[n]=\{1,\dots,n\}$ and two $k$-subsets $A$ and $B$ are adjacent if $|A\cap B|\in S$. In this paper, we use Godsil-McKay switching to prove that for $m\geq 0$, $k\geq \max(m+2,3)$ and $S = \{0, 1, ..., m\}$, the graphs $J_S(3k-2m-1,k)$ are not determined by spectrum and for $m\geq 2$, $n\geq 4m+2$ and $S = \{0,1,...,m\}$ the graphs $J_{S}(n,2m+1)$ are not determined by spectrum. We also report some computational searches for Godsil-McKay switching sets in the union of classes in the Johnson scheme for $k\leq 5$., Comment: 9 pages, no figures, 3 tables; 2nd version contains improved results compared to the 1st version

Graham and Pollak showed that the vertices of any connected graph $G$ can be assigned $t$-tuples with entries in $\{0, a, b\}$, called addresses, such that the distance in $G$ between any two vertices equals the number of positions in their addresses where one of the addresses equals $a$ and the other equals $b$. In this paper, we are interested in determining the minimum value of such $t$ for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs., Comment: 10 pages, 2 figures; This version is identical to the first in content, but it includes more explicit attributions to David A. Gregory. In particular, for Lemmas 3.1, 3.2, Remark 3.4 and Theorem 3.5 and includes an added acknowledgement

In this paper, we present some new results describing connections between the spectrum of a regular graph and its generalized connectivity, toughness, and the existence of spanning trees with bounded degree., Comment: 13 pages; accepted to Czechoslovak Mathematical Journal, special issue dedicated to Professor Miroslav Fiedler

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to $-2$, or $0$, and determine which of these graphs are determined by their adjacency spectrum.

Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most $\frac{v}{2}(1-1/g)$. We use this fact to also study the extendability of matchings in distance-regular graphs. A graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t

Many regular graphs admit a natural partition of their edge set into cliques of the same order such that each vertex is contained in the same number of cliques. In this paper, we study the mixing rate of certain random walks on such graphs and we generalize previous results of Alon, Benjamini, Lubetzky and Sodin regarding the mixing rates of non-backtracking random walks on regular graphs., Comment: 31 pages; to appear in the CRM Proceedings Series, published by the American Mathematical Society as part of the Contemporary Mathematics Series

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $\lambda<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we investigate the largest number of vertices of such graphs., Comment: SIAM Journal of Discrete Mathematics 30, 1509-1525, (2016)

The simplicial rook graph ${\rm SR}(m,n)$ is the graph of which the vertices are the sequences of nonnegative integers of length $m$ summing to $n$, where two such sequences are adjacent when they differ in precisely two places. We show that ${\rm SR}(m,n)$ has integral eigenvalues, and smallest eigenvalue $s = \max (-n, -{m \choose 2})$, and that this graph has a large part of its spectrum in common with the Johnson graph $J(m+n-1,n)$. We determine the automorphism group and several other properties.