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In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral versions of Mantel's theorem., 14 pages, 4 figures

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$ for which $Q_q$ is positive semidefinite is fundamental in asymptotic spectral analysis of graphs from the viewpoint of quantum probability theory. Assume that $G$ has at least two vertices. Then $\pi(G)$ is easily seen to be a nonempty closed subset of the interval $[-1,1]$. In this note, we show that $\pi(G)=[-1,1]$ if and only if $G$ is isometrically embeddable into a hypercube (infinite-dimensional if $G$ is infinite) if and only if $G$ is bipartite and does not possess certain five-vertex configurations, an example of which is an induced $K_{2,3}$., Comment: 6 pages

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $$\alpha(\phi(X))= \alpha(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=\phi(J_n-I_n)$ with all off-diagonal entries positive.

We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach. We also establish a criterion for the positivity and log-convexity of such sequences., 13 pages

Let $G$ be a graph of size $m$ and $\rho(G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free non-star graph with $m\geq (4r+2)^2+1$, then $\rho(G)\leq \rho(S_m^1)$, with equality if and only if $G\cong S_m^1$. Recently, Li, Sun and Wei showed that for any $\theta_{1,2,3}$-free graph of size $m\geq 8$, $\rho(G)\leq \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$. However, this bound is not attainable when $m$ is even. We proved that if $G$ is $\theta_{1,2,3}$-free and $G\ncong S_{\frac{m+3}{2},2}$ with $m\geq 22$, then $\rho(G)\leq \rho(F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $\rho(G)\leq \rho(F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$., Comment: 15 pages, 3 figures

Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different connectivity matrices associated with hypergraphs. In fact, we introduce here a unified framework for studying different variations of the connectivity operators associated with hypergraphs at the same time. Eigenvalues and corresponding eigenspaces of the general connectivity operators associated with some classes of hypergraphs are computed. Applications such as random walks on hypergraphs, dynamical networks, and disease transmission on hypergraphs are studied from the perspective of our newly introduced operators. We also derive spectral bounds for the weak connectivity number, degree of vertices, maximum cut, bipartition width, and isoperimetric constant of hypergraphs.

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between $0$ and $1$. Finally, we state several open conjectures that are motivated by empirical data.

We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that $c_{2}\in [1/2, \sqrt{2+\sqrt{5}}]$, where $c_2$ is the minimum real number $c$ for which every real number greater than $c$ is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem., 36 pages, 2 tables

Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, a subset $\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on some element of $\Omega$. The \emph{intersection density} of $G$, denoted by $\rho(G)$, is the maximum of the rational number $|\mathcal{F}|\left(\frac{|G|}{|\Omega|}\right)^{-1}$ when $\mathcal{F}$ runs through all intersecting sets in $G$. In this paper, we prove that if $G$ is the group $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $\rho(G)=1$. Our proof relies on the representation theory of the symmetric group and the ratio bound., Comment: 31 pages

We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant, a matrix function which has generated a lot of interest in recent times. Recently, the Cayley--Hamilton theorem for mixed discriminants was proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a Phillips-type generalization of the Bapat--Roy theorem involving $2nk$ matrices, where $n$ is the size of the matrices, among which $nk$ commute pairwise. Our proofs generalize the univariate proof of Straubing (Disc. Math., 1983) for the original Cayley--Hamilton theorem in a nontrivial way, and involve decorated permutations and decorated paths., Comment: 20 pages, 9 figures, minor corrections, final version

Let $\mathcal{H}$ be an $m$-uniform hypergraph, and let $\mathcal{A}(\mathcal{H})$ be the adjacency tensor of $\mathcal{H}$ which can be viewed as a system of homogeneous polynomials of degree $m-1$. Morozov and Shakirov generalized the traces of linear systems to nonlinear homogeneous polynomial systems and obtained explicit formulas for multidimensional resultants. Sun, Zhou and Bu introduced the Estrada index of uniform hypergraphs which is closely related to the traces of their adjacency tensors. In this paper we give formulas for the traces of $\mathcal{A}(\mathcal{H})$ when $\mathcal{H}$ contains cut vertices, and obtain results on the traces and Estrada index when $\mathcal{H}$ is perturbed under local changes. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index.

Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$ symmetric matrices $M$ over $\mathbb{F}$ with all diagonal entries zero and the $(i, j)$th element of $M$ either $a_i$ or $a_j$ for $i < j$. In this short paper, we show that all matrices in a certain subclass of $\mathcal{M}_n(\mathbf{a})$ -- which can be naturally associated with transitive tournaments -- have rank at least $\lfloor 2n/3 \rfloor - 1$. We also show that if $\operatorname{char}(\mathbb{F}) \neq 2$ and $M$ is a matrix chosen uniformly at random from $\mathcal{M}_n(\mathbf{a})$, then with high probability $\operatorname{rank}(M) \geq \bigl(\frac{1}{2} - o(1)\bigr)n$., 12 pages; addendum added

With $k$ an infinite field and $\tau_1,\tau_2$ endomorphisms of $k^m$, we provide a dimension bound on an open locus of a determinantal scheme, under which, for a general subspace $V \subseteq k^m$ of dimension $n \le m/2$, for $v_1,v_2 \in V$ we have $\tau_1(v_1)=\tau_2(v_2)$ only if $v_1=v_2$. Specializing to permutations composed by coordinate projections, we obtain an abstract proof of the unlabeled sensing theorem., Comment: enhanced technical clarity at some places, 15 pages

An arithmetical structure on a finite, connected graph without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Associated to each arithmetical structure is a finite abelian group known as its critical group. Keyes and Reiter gave an operation that takes in an arithmetical structure on a finite, connected graph without loops and produces an arithmetical structure on a graph with one fewer vertex. We study how this operation transforms critical groups. We bound the order and the invariant factors of the resulting critical group in terms of the original arithmetical structure and critical group. When the original graph is simple, we determine the resulting critical group exactly., 19 pages, 3 figures

We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices, and give various examples. This work resolves two open questions of Chan et.~al. ["Quantum Fractional Revival on graphs". Discrete Applied Math, 269:86-98, 2019.], Comment: 38 pages

Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices $a$ and $b$ in a graph, there cannot be perfect state transfer from $a$ to any other vertex. No examples of unweighted graphs with fractional revival between non-cospectral vertices were known; here we give an infinite family of such graphs. No examples of unweighted graphs where the pairs involved in fractional revival overlapped were known; we give examples of such graphs as well.

We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.

We study the spectrum of the join of several circulant matrices. We apply our results to compute explicitly the spectrum of certain graphs obtained by joining several circulant graphs., To appear in Linear Algebra and Its Applications

We show that there exists no algorithm that decides for any bilinear system $(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui who showed that if the coefficients are positive the growth rate is computable (i.e., there is an algorithm that outputs the sequence of digits of the growth rate of $(B,v)$). Our proof is based on a reduction of the computation of the joint spectral radius of a set of matrices to the computation of the growth rate of a bilinear system. We also use our reduction to deduce that there exists no algorithm that approximates the growth rate of a bilinear system with relative accuracy $\varepsilon$ in time polynomial in the size of the system and of $\varepsilon$. Our two results hold even if all the coefficients are nonnegative rationals.

In this paper, we define and obtain several properties of the (adjacency) energy of a hypergraph. In particular, bounds for this energy are obtained as functions of structural and spectral parameters, such as Zagreb index and spectral radius. We also study how the energy of a hypergraph varies when a vertex/edge is removed or when an edge is divied. In addition, we solved the extremal problem energy for the class of hyperstars, and show that the energy of a hypergraph is never an odd number.

Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several well-known eigenvalue bounds. In this work we aim to further our algebraic and computational understanding of weight-equitable partitions. We do so by showing several spectral properties and algebraic characterizations, and by providing a method to find coarse weight-equitable partitions.

The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erd\H{o}s, F\"uredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100] determined the maximum number of edges in an $n$-vertex graph that does not contain $F_{k,3}$ as a subgraph. Furthermore, Chen, Gould, Pfender and Wei [J. Combin. Theory Ser. B 89 (2003) 159--171] proved the analogous result on $F_{k,r}$ for the general case $r\ge 3$.In this paper, we show that for sufficiently large $n$, the graphs of order $n$ that contain no copy of $F_{k,r}$ and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have $\mathrm{ex}(n, F_{k,r})$ edges., Comment: arXiv admin note: substantial text overlap with arXiv:2106.00587

The signless Laplacian spectral radius of a graph $G$, denoted by $q(G)$, is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles or long cycles. Let $\mathcal{G}(m,g)$ be the family of graphs on $m$ edges with girth $g$ and $\mathcal{H}(m,c)$ be the family of graphs on $m$ edges with circumference $c$. More precisely, we obtain the unique extremal graph with maximal $q(G)$ in $\mathcal{G}(m,g)$ and $\mathcal{H}(m,c)$, respectively., 13 pages, one figure

Arithmetical structures on graphs were first introduced in \cite{Lorenzini89}. Later in \cite{arithmetical} they were further studied in the setting of square non-negative integer matrices. In both cases, necessary and sufficient conditions for the finiteness of the set of arithmetical structures were given. More precisely, an arithmetical structure on a non-negative integer matrix $L$ with zero diagonal is a pair $(\mathbf{d},\mathbf{r})\in \mathbb{N}_+^n\times \mathbb{N}_+^n$ such that \[ (\textrm{Diag}(\mathbf{d})-L)\mathbf{r}^t=\mathbf{0}^t\text{ and }\gcd(r_1,\ldots,r_n)=1. \] Thus, arithmetical structures on $L$ are solutions of the polynomial Diophantine equation \[ f_L(X):=\det(\text{Diag}(X)-L)=0. \] Therefore, it is of interest to ask for an algorithm that compute them. We present an algorithm that computes arithmetical structures on a square integer non-negative matrix $L$ with zero diagonal. In order to do this we introduce a new class of Z-matrices, which we call quasi $M$-matrices., 14 pages. Major changes, sections 4 and 5 was deleted. Section 4 is the base of the article "Arithmetical structures on dominated polynomials"

A $(k,g)$-cage is a $k$-regular simple graph of girth $g$ with minimum possible number of vertices. In this paper, $(k,g)$-cages which are Moore graphs are referred as minimal $(k,g)$-cages. A simple connected graph is called distance regular(DR) if all its vertices have the same intersection array. A bipartite graph is called distance biregular(DBR) if all the vertices of the same partite set admit the same intersection array. It is known that minimal $(k,g)$-cages are DR graphs and their subdivisions are DBR graphs. In this paper, for minimal $(k,g)$-cages we give a formula for distance spectral radius in terms of $k$ and $g$, and also determine polynomials of degree $[\frac{g}{2}]$, which is the diameter of the graph. This polynomial gives all distance eigenvalues when the variable is substituted by adjacency eigenvalues. We show that a minimal $(k,g)$-cage of diameter $d$ has $d+1$ distinct distance eigenvalues, and this partially answers a problem posed in [5]. We prove that every DBR graph is a $2$-partitioned transmission regular graph and then give a formula for its distance spectral radius. By this formula we obtain the distance spectral radius of subdivision of minimal $(k,g)$-cages. Finally we determine the full distance spectrum of subdivision of some minimal $(k,g)$-cages., 22 pages

Let B ( n , q ) be the class of block graphs on n vertices having all their blocks of the same size. We prove that if G ∈ B ( n , q ) has at most three pairwise adjacent cut vertices then the minimum spectral radius ρ ( G ) is attained at a unique graph. In addition, we present a lower bound for ρ ( G ) when G ∈ B ( n , q ) .

A matrix-weighted graph is an undirected graph with a $k\times k$ positive semidefinite matrix assigned to each edge. There are natural generalizations of the Laplacian and adjacency matrices for such graphs. These matrices can be used to define and control expansion for matrix-weighted graphs. In particular, an analogue of the expander mixing lemma and one half of a Cheeger-type inequality hold for matrix-weighted graphs. A new definition of a matrix-weighted expander graph suggests the tantalizing possibility of families of matrix-weighted graphs with better-than-Ramanujan expansion., 21 pages, 2 figures

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint t-cliques. The extremal graphs attaining the bounds are exactly the block graphs of Steiner 2-designs and the regular graphs with K t -decompositions, respectively.

In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of $\eta$-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of $\eta$-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define $n+1$ types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss periodicity. We show that the mixed paths are periodic for any $\eta$. In addition, we provide a necessary and sufficient condition for a mixed cycle to be periodic and determine their periods., Comment: 22 pages, 12 figures

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi = (G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. A $ \mathbb{T} $-gain graph $\Phi$ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is $1$. Signed graphs are special cases of $\mathbb{T}$-gain graphs. The adjacency matrix of $\Phi$, denoted by $ \mathbf{A}(\Phi)$ is defined canonically. The gain Laplacian for $\Phi$ is defined as $\mathbf{L}(\Phi) = \mathbf{D}(\Phi) - \mathbf{A}(\Phi)$, where $\mathbf{D}(\Phi)$ is the diagonal matrix with diagonal entries are the degrees of the vertices of $G$. The minimum number of vertices (resp., edges) to be deleted from $\Phi$ in order to get a balanced gain graph the frustration number (resp, frustration index). We show that frustration number and frustration index are bounded below by the smallest eigenvalue of $\mathbf{L}(\Phi)$. We provide several lower and upper bounds for extremal eigenvalues of $\mathbf{L}(\Phi)$ in terms of different graph parameters such as the number of edges, vertex degrees, and average $2$-degrees. The signed graphs are particular cases of the $\mathbb{T}$-gain graphs, all the bounds established in paper hold for signed graphs. Most of the bounds established here are new for signed graphs. Finally, we perform comparative analysis for all the obtained bounds in the paper with the state-of-the-art bounds available in the literature for randomly generated Erd\H{o}s-Re\'yni graphs. Some of the major highlights of our paper are the gain-dependent bounds, limit convergence of the bounds to the extremal eigenvalues, and optimal extremal bounds obtained by posing optimization problems to achieve the best possible bounds., Comment: Preliminary version. 32 pages

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free., Slightly revised version

Given a d-dimensional tensor T:A ×…×A →F (where F is a field), the i-flattening rank of T is the rank of the matrix whose rows are indexed by A , columns are indexed by B =A ×…×A ×A ×…×A and whose entries are given by the corresponding values of T. The max-flattening rank of T is defined as mfrank(T)=max frank (T). A tensor T:A →F is called semi-diagonal, if T(a,…,a)≠0 for every a∈A, and T(a ,…,a )=0 for every a ,…,a ∈A that are all distinct. In this paper we prove that if T:A →F is semi-diagonal, then mfrank(T)≥[Formula presented], and this bound is the best possible. We give several applications of this result, including a generalization of the celebrated Frankl-Wilson theorem on forbidden intersections. Also, addressing a conjecture of Aharoni and Berger, we show that if the edges of an r-uniform multi-hypergraph H are colored with z colors such that each color class is a matching of size t, then H contains a rainbow matching of size t provided z>(t−1)(rtr). This improves previous results of Alon and Glebov, Sudakov, and Szabó., Linear Algebra and its Applications, 625, ISSN:0024-3795, ISSN:1873-1856

Let $V$ be a finite-dimensional vector space over the finite field ${\mathbb F}_q$ and suppose $W$ and $\widetilde{W}$ are subspaces of $V$. Two linear transformations $T:W\to V$ and $\widetilde{T}:\widetilde{W}\to V$ are said to be similar if there exists a linear isomorphism $S:V\to V$ with $SW=\widetilde{W}$ such that $S\circ T=\widetilde{T}\circ S $. Given a linear map $T$ defined on a subspace $W$ of $V$, we give an explicit formula for the number of linear maps that are similar to $T$. Our results extend a theorem of Philip Hall that settles the case $W=V$ where the above problem is equivalent to counting the number of square matrices over ${\mathbb F}_q$ in a conjugacy class., 15 pages, 3 figures

Identifying graphs with extremal properties is an extensively studied topic in spectral graph theory. In this paper, we study the log-concavity of a type of iteration sequence related to the $\alpha$-normal weighted incidence matrices which is presented by Lu and Man for computing the spectral radius of hypergraphs. By using results obtained about the sequence and the method of some edge operations, we will characterize completely extremal k-graphs with the smallest spectral radius among bicyclic hypergraphs with given size., Comment: 22 pages, 3 figures

Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With his note I hope to clarify the history of this bound and some of its generalizations.

In this paper, structural properties of chordal graphs are analysed, in order to establish a relationship between these structures and integer Laplacian eigenvalues. We present the characterization of chordal graphs with equal vertex and algebraic connectivities, by means of the vertices that compose the minimal vertex separators of the graph; we stablish a sufficient condition for the cardinality of a maximal clique to appear as an integer Laplacian eigenvalue. Finally, we review two subclasses of chordal graphs, showing for them some new properties., 15 pages, 5 figures

The distance energy of a simple connected graph $G$ is defined as the sum of absolute values of its distance eigenvalues. In this paper, we mainly give a positive answer to a conjecture of distance energy of clique trees proposed by Lin, Liu and Lu [H.~Q.~ Lin, R.~F.~Liu, X.~W.~Lu, The inertia and energy of the distance matrix of a connected graph, {\it Linear Algebra Appl.,} 467 (2015), 29-39.], Comment: 10 pages, 1 figure

Let $G$ be an arbitrary group. We define a gain-line graph for a gain graph $(\Gamma,\psi)$ through the choice of an incidence $G$-phase matrix inducing $\psi$. We prove that the switching equivalence class of the gain function on the line graph $L(\Gamma)$ does not change if one chooses a different $G$-phase inducing $\psi$ or a different representative of the switching equivalence class of $\psi$. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of $G$ on the set $\mathcal H_\Gamma$ of $G$-phases of $\Gamma$ allows us to characterize gain functions on $\Gamma$, gain functions on $L(\Gamma)$, their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph., Comment: 28 pages, 6 figures, 1 table

For a given graph $\mathcal{G}$ of order $n$ with $m$ edges, and a real symmetric matrix associated to the graph, $M\left(\mathcal{G}\right)\in\mathbb{R}^{n\times n}$, the interlacing graph reduction problem is to find a graph $\mathcal{G}_{r}$ of order $r

We study bases of the lattice generated by the cycles of an undirected graph, defined as the integer linear combinations of the 0/1-incidence vectors of cycles. We prove structural results for this lattice, including explicit formulas for its dimension and determinant, and we present efficient algorithms to construct lattice bases, using only cycles as generators, in quadratic time. By algebraic considerations, we relate these results to the more general setting with coefficients from an arbitrary Abelian group. Our results generalize classical results for the vector space of cycles of a graph over the binary field to the case of an arbitrary field., Final publication revision; 19 pages, 0 figures

Let $A$ be an $n \times n$ positive definite Hermitian matrix with all eigenvalues between 1 and 2. We represent the permanent of $A$ as the integral of some explicit log-concave function on ${\Bbb R}^{2n}$. Consequently, there is a fully polynomial randomized approximation scheme (FPRAS) for the permanent of $A$., 5 pages

In quantum information theory there is a construction for quantum channels, appropriately called a quantum graph, that generalizes the confusability graph construction for classical channels in classical information theory. In this paper, we provide a definition of connectedness for quantum graphs that generalizes the classical definition. This is used to prove a quantum version of a particular case of the classical tree-packing theorem from graph theory. Generalizations for the related notions of $k$-connectedness and of orthogonal representation are also proposed for quantum graphs, and it is shown that orthogonal representations have the same implications for connectedness as they do in the classical case., Comment: 15 pages

Let $\kappa'(G)$, $\kappa(G)$, $\mu_{n-1}(G)$ and $\mu_1(G)$ denote the edge-connectivity, vertex-connectivity, the algebraic connectivity and the Laplacian spectral radius of $G$, respectively. In this paper, we prove that for integers $k\geq 2$ and $r\geq 2$, and any simple graph $G$ of order $n$ with minimum degree $\delta\geq k$, girth $g\geq 3$ and clique number $\omega(G)\leq r$, the edge-connectivity $\kappa'(G)\geq k$ if $\mu_{n-1}(G) \geq \frac{(k-1)n}{N(\delta,g)(n-N(\delta,g))}$ or if $\mu_{n-1}(G) \geq \frac{(k-1)n}{\varphi(\delta,r)(n-\varphi(\delta,r))}$, where $N(\delta,g)$ is the Moore bound on the smallest possible number of vertices such that there exists a $\delta$-regular simple graph with girth $g$, and $\varphi(\delta,r) = \max\{\delta+1,\lfloor\frac{r\delta}{r-1}\rfloor\}$. Analogue results involving $\mu_{n-1}(G)$ and $\frac{\mu_1(G)}{\mu_{n-1}(G)}$ to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in [Linear Algebra Appl. 439 (2013) 3777--3784], [Linear Algebra Appl. 578 (2019) 411--424], [Linear Algebra Appl. 579 (2019) 72--88], [Appl. Math. Comput. 344-345 (2019) 141--149] and [Electronic J. Linear Algebra 34 (2018) 428--443] are improved or extended., Comment: 14 pages

A universal adjacency matrix of a graph G with adjacency matrix A is any matrix of the form U = α A + β I + γ J + δ D with α ≠ 0 , where I is the identity matrix, J is the all-ones matrix and D is the diagonal matrix with the vertex degrees. In the case that G is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix of G, and the signless Laplacian of the complement of G (i.e. the join of regular graphs). The main tool is a simple but useful lemma on equitable matrix partitions. With this note we also want to propagate this technique.

The almost-principal rank characteristic sequence (apr-sequence) of an n × n symmetric matrix is introduced, which is defined to be the string a 1 a 2 ⋯ a n − 1 , where a k is either A , S , or N , according as all, some but not all, or none of its almost-principal minors of order k are nonzero. In contrast to the other principal rank characteristic sequences in the literature, the apr-sequence of a matrix does not depend on principal minors. The almost-principal rank of a symmetric matrix B, denoted by ap-rank ( B ) , is defined as the size of a largest nonsingular almost-principal submatrix of B. A complete characterization of the sequences not containing an A that can be realized as the apr-sequence of a symmetric matrix over a field F is provided. A necessary condition for a sequence to be the apr-sequence of a symmetric matrix over a field F is presented. It is shown that if B ∈ F n × n is symmetric and non-diagonal, then rank ( B ) − 1 ≤ ap-rank ( B ) ≤ rank ( B ) , with both bounds being sharp. Moreover, it is shown that if B is symmetric, non-diagonal and singular, and does not contain a zero row, then rank ( B ) = ap-rank ( B ) .

Dual Seidel switching is a graph operation introduced by W. Haemers in 1984. This operation can change the graph, however it does not change its bipartite double, and because of this, the operation leaves the squares of the eigenvalues invariant. Thus, if a graph is integral then it is still integral after dual Seidel switching. In this paper two new infinite families of integral graphs are obtained by applying dual Seidel switching to the Star graphs and the Odd graphs. In particular, three new 4-regular integral graphs with their spectra are found.

We characterize the spectrum of the Laplacian of graphs composed of one or two finite or infinite chains connected to a complete graph. We show the existence of localized eigenvectors of two types, eigenvectors that vanish exactly outside the complete graph and eigenvectors that decrease exponentially outside the complete graph. Our results also imply gaps between the eigenvalues corresponding to localized and extended eigenvectors.

Quantum walks on undirected graphs have been studied using symmetric matrices, such as the adjacency or Laplacian matrix, and many results about perfect state transfer are known. We extend some of those results to oriented graphs. We also study the phenomena, unique to oriented graphs, of multiple state transfer, where there is a set of vertices such that perfect state transfer occurs between every pair in that set. We give a characterization of multiple state transfer, and a new example of a graph where it occurs., Comment: 17 pages, 3 figures

For a graph $G=(V, E)$ and $i, j\in V$, denote the distance between $i$ and $j$ in $G$ by $D(i, j)$ and the degrees of $i$, $j$ by $d_i$, $d_j$, respectively. Let $f(D(i, j), d_{i}, d_{j})$ be a function symmetric in $i$ and $j$. Define a matrix $W_f(G)$, called the weighted distance matrix, of $G$, with the $ij$-entry $W_f(G)(i, j)=f(D(i, j), d_{i}, d_{j})$ if $i\neq j$ and $W_f(G)(i, j)=0$ if $i=j$. In this paper, we prove that if the symmetric function $f$ satisfies that $f(D(i, j), (1+o(1))np, (1+o(1))np)=(1+o(1))f(D(i, j), np, np)$, then for almost all graphs $G_p$ in the $Erd\ddot{o}s$-$R\acute{e}nyi$ random graph model $\mathcal{G}_{n, p}$, the energy of $W_f(G_p)$ is $\{(\frac{8}{3\pi}\sqrt{p(1-p)}+o(1))\cdot|f(1, np, np)-f(2, np, np)|+o(|f(2, np, np)|)\}\cdot n^{3/2}$. As a consequence, we give the asymptotic values of energies of a variety of weighted distance matrices with function $f$ from distance-based only and mixed with degree-distance-based topological indices of chemical use. This generalizes our former result with only degree-based weights., Comment: 15 pages