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An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we present three new invariants for cospectral graphs, characterized by their arithmetic nature and apparent novelty. Specifically, let $G$ and $H$ be two graphs with adjacency matrices $A(G)$ and $A(H)$, respectively. We show, among other results, that if $G$ and $H$ are cospectral, then $e^{\rm T}A(G)^me\equiv e^{\rm T}A(H)^m e~({\rm mod}~4)$ for any integer $m\geq 0$, where $e$ is the all-one vector. As a simple consequence, we demonstrate that under certain conditions, every graph cospectral with $G$ is determined by its generalized spectrum., Comment: 13 pages, 1 figure

For a connected graph $G$ with order $n$, let $e(G)$ represent the number of its distinct eigenvalues, and let $d$ denote its diameter. We denote the eigenvalue multiplicity of $\mu$ in $G$ by $m_G(\mu)$. It is well established that the inequality $e(G) \geq d + 1$ implies that when $\mu$ is an eigenvalue of $P_{d+1}$, it follows that $m_G(\mu) \leq n - d$; otherwise, for any real number $\mu$, we have $m_G(\mu) \leq n - d - 1$. A graph is termed minimal if $e(G) = d + 1$. In 2013, Wong et al. characterized all minimal graphs for which $m_G(0) = n - d$. In this article, we provide a complete characterization of the graphs $G$ such that $m_G(0) = n - d - 1$., Comment: arXiv admin note: text overlap with arXiv:2410.15000

This paper investigates the $r$-th power energy, also known as $r$-Schatten energy, of connected graphs. We focus on addressing a question posed by Nikiforov regarding whether the $r$-th power energy of a connected graph is greater than or equal to that of the star graph $ S_n $ for $ 1 < r < 2 $. Our main result confirms that, for any connected graph $ G $ of order $ n $, $ \mathcal{E}_r(G) \geq \mathcal{E}_r(S_n) $ for $ 0 < r < 2 $, with equality if and only if $ G $ is the star graph $ S_n $. We derive a Coulson-Jacobs-type formula for the $r$-th power energy of connected graphs and obtain sharper upper bounds for the $r$-th power energy when $ r > 2 $. These results improve previous bounds given by Nikiforov and extend the understanding of graph energies, especially in the range $ 0 < r < 2 $., Comment: 9 pages. Fix a few typos

Let $G$ be a simple connected graph of order $n$ with diameter $d$. Let $m_G(-1)$ denote the multiplicity of the eigenvalue $-1$ of the adjacency matrix of $G$, and let $P = P_{d+1}$ be the diameter path of $G$. If $-1$ is not an eigenvalue of $P$, then by the interlacing theorem, we have $m_G(-1)\leq n - d - 1$. In this article, we characterize the extremal graphs where equality holds. Moreover, for the completeness of the results, we also characterize the graphs $G$ that achieve $m_G(-1) = n - d - 1$ when $-1$ is an eigenvalue of $P$. Thus, we provide a complete characterization of the graphs $G$ for which $m_G(-1) = n - d - 1$.

A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{regular} if each of its row sum equals one, i.e., $Qe = e$ where $e$ is the all-one vector. This paper presents a method for generating all regular rational orthogonal matrices using the classic Cayley transformation. Specifically, we demonstrate that for any regular rational orthogonal matrix $Q$, there exists a permutation matrix $P$ such that $QP$ does not possess an eigenvalue of $-1$. Consequently, $Q$ can be expressed in the form $Q = (I_n + S)^{-1}(I_n - S)P$, where $I_n$ is the identity matrix of order $n$, $S$ is a rational skew-symmetric matrix satisfying $Se = 0$, and $P$ is a permutation matrix. Central to our approach is a pivotal intermediate result, which holds independent interest: given a square matrix $M$, then $MP$ has $-1$ as an eigenvalue for every permutation matrix $P$ if and only if either every row sum of $M$ is $-1$ or every column sum of $M$ is $-1$.

Let $G$ be an Eulerian graph on $n$ vertices with adjacency matrix $A$ and characteristic polynomial $\phi(x)$. We show that when $n$ is even (resp. odd), the square-root of $\phi(x)$ (resp. $x\phi(x)$) is an annihilating polynomial of $A$, over $\mathbb{F}_2$. The result was achieved by applying the Jordan canonical form of $A$ over the algebraic closure $\bar{\mathbb{F}}_2$. Based on this, we show a family of Eulerian graphs are determined by their generalized spectrum among all Eulerian graphs, which significantly simplifies and strengthens the previous result.

In this paper, we investigate the positive and negative $r$-th power energy of graphs and their behavior under edge addition. Specifically, we extend the classical notions of positive and negative square energies to the $r$-th power energies, denoted as $s^{+}_r(G)$ and $s^{-}_r(G)$, respectively. We derive bounds for $s^{+}_r(G)$ and $s^{-}_r(G)$ under edge addition, which provide tighter bounds when $r=2$ compared to those of Abiad et al. Moreover, we address the problem of monotonicity of the positive $r$-th power energy under edge addition, providing a family of counterexamples for \( 1 \leq r < 3 \). Finally, three related conjectures are also proposed. One of them is a reformulation of Guo's conjecture, we believe the monotonicity property holds for $r\geq 3$; the other two generalize a conjecture of Elphick et al.

Spectral characterizations of graphs is an important topic in spectral graph theory which has been studied extensively by researchers in recent years. The study of oriented graphs, however, has received less attention so far. In Qiu et al.~\cite{QWW} (Linear Algebra Appl. 622 (2021) 316-332), the authors gave an arithmetic criterion for an oriented graph to be determined by its \emph{generalized skew spectrum} (DGSS for short). More precisely, let $\Sigma$ be an $n$-vertex oriented graph with skew adjacency matrix $S$ and $W(\Sigma)=[e,Se,\ldots,S^{n-1}e]$ be the \emph{walk-matrix} of $\Sigma$, where $e$ is the all-one vector. A theorem of Qiu et al.~\cite{QWW} shows that a self-converse oriented graph $\Sigma$ is DGSS, provided that the Smith normal form of $W(\Sigma)$ is ${\rm diag}(1,\ldots,1,2,\ldots,2,2d)$, where $d$ is an odd and square-free integer and the number of $1$'s appeared in the diagonal is precisely $\lceil \frac{n}{2}\rceil$. In this paper, we show that the above square-freeness assumptions on $d$ can actually be removed, which significantly improves upon the above theorem. Our new ingredient is a key intermediate result, which is of independent interest: for a self-converse oriented graphs $\Sigma$ and an odd prime $p$, if the rank of $W(\Sigma)$ is $n-1$ over $\mathbb{F}_p$, then the kernel of $W(\Sigma)^{\rm T}$ over $\mathbb{F}_p$ is \emph{anisotropic}, i.e., $v^{\rm T}v\neq 0$ for any $0\ne v\in{{\rm ker}\,W(\Sigma)^{\rm T}}$ over $\mathbb{F}_p$.

A switching method is a graph operation that results in cospectral graphs (graphs with the same spectrum). Work by Wang and Xu [Discrete Math. 310 (2010)] suggests that most cospectral graphs with cospectral complements can be constructed using regular orthogonal matrices of level 2, which has relevance for Haemers' conjecture. We present two new switching methods and several combinatorial and geometrical reformulations of existing switching operations of level 2. We also introduce the concept of reducibility and use it to classify all irreducible switching methods that correspond to a conjugation with a regular orthogonal matrix of level 2 with one nontrivial indecomposable block, up to switching sets of size 12, extending previous results.

Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1 \geq \cdots \geq\lambda_n$. Let \[s^+(G)=\sum_{\lambda_i>0} \lambda_i^2, \qquad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called the \emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] which implies that $s(G)\geq \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$., Comment: 5 pages, 1 figure

For a simple graph $G$, the complement and the line graph of $G$ are denoted by $G^c$ and $L(G)$, respectively. In this paper, we show that for every simple connected regular graph $G$ with at least $5$ vertices, the graph $\mathcal{R}(G):=L(L(G)^c)^c$ is a Ramanujan graph with diameter at most three.

Two graphs $G$ and $H$ are \emph{cospectral} if the adjacency matrices share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature, e.g. the famous GM-switching method. In this paper, we shall construct cospectral graphs via regular rational orthogonal matrix $Q$ with level two and three. We provide two straightforward algorithms to characterize with adjacency matrix $A$ of graph $G$ such that $Q^TAQ$ is again a (0,1)-matrix, and introduce two new switching methods to construct families of cospectral graphs which generalized the GM-switching to some extent., Comment: 19 pages,16 figures

For a graph $G$, let $\lambda_2(G)$ denote the second largest eigenvalue of the adjacency matrix of $G$. We determine the extremal trees with maximum/minimum adjacency eigenvalue $\lambda_2$ in the class $\mathcal{T}(n,d)$ of $n$-vertex trees with diameter $d$. This contributes to the literature on $\lambda_2$-extremization over different graph families. We also revisit the notion of the spectral center of a tree and the proof of $\lambda_2$ maximization over trees.

The quadratic unitary Cayley graph $\mathcal{G}_{\Zl_n}$ has vertex set $\mathbb{Z}_n: =\{0,1, \hdots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if and only if $u - v$ or $v-u$ is a square of some units in $\mathbb{Z}_n$. This paper explores the periodicity and perfect state transfer of Grover walks on quadratic unitary Cayley graphs. We determine all periodic quadratic unitary Cayley graphs. From our results, it follows that there are infinitely many integral as well as non-integral graphs that are periodic. Additionally, we also determine the values of $n$ for which the quadratic unitary Cayley graph $\mathcal{G}_{\Zl_n}$ exhibits perfect state transfer.

For a graph $G$, the spectral radius $\lambda_{1}(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. An odd wheel $W_{2k+1}$ with $k\geq2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all the vertices of the cycle. Let ${\rm SPEX}(n,W_{2k+1})$ be the set of $W_{2k+1}$-free graphs of order $n$ with the maximum spectral radius. Very recently, Cioab\u{a}, Desai and Tait \cite{CDT2} characterized the graphs in ${\rm SPEX}(n,W_{2k+1})$ for sufficiently large $n$, where $k\geq2$ and $k\neq4,5$. And they left the case $k=4,5$ as a problem. In this paper, we settle this problem. Moreover, we completely characterize the graphs in ${\rm SPEX}(n,W_{2k+1})$ when $k\geq4$ is even and $n\equiv2~(\mod4)$ is sufficiently large. Consequently, the graphs in ${\rm SPEX}(n,W_{2k+1})$ are characterized completely for any $k\geq2$ and sufficiently large $n$.

Extending a classic result of Johnson and Newman, this paper provides a matrix characterization for two generalized cospectral graphs with a pair of generalized cospectral vertex-deleted subgraphs. As an application, we present a new condition for the reconstructibility of a graph. In particular, we show that a graph with at least three vertices is reconstructible if there exists a vertex-deleted subgraph that is almost controllable and has a nontrivial automorphism., Comment: 9 pages

Let A be a graph type and B an equivalence relation on a group $G$. Let $[g]$ be the equivalence class of $g$ with respect to the equivalence relation B. The B superA graph of $G$ is an undirected graph whose vertex set is $G$ and two distinct vertices $g, h \in G$ are adjacent if $[g] = [h]$ or there exist $x \in [g]$ and $y \in [h]$ such that $x$ and $y$ are adjacent in the A graph of $G$. In this paper, we compute spectrum of equality/conjugacy supercommuting graphs of dihedral/dicyclic groups and show that these graphs are not integral.

Bollob\'as and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$ edges \[ \lambda_1^2+\lambda_2^2\le \bigg( 1-\frac{1}{\omega(G)}\bigg)2m\] where $\lambda_1$ and $\lambda_2$ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and $\omega$ denotes the clique number of $G$. This conjecture was recently verified for triangle-free graphs by Lin, Ning and Wu and for regular graphs by Zhang. Elphick, Wocjan and Linz proposed a generalization of this conjecture. In this note, we verify this generalized conjecture for the family of graphs on $m$ edges, which contain at most $O(m^{1.5-\varepsilon})$ triangles for some $\varepsilon > 0$. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs.

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we say that $X_1$ and $X_2$ are degree similar if there is an invertible real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs $X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian matrices, unsigned Laplacian matrices and normalized Laplacian matrices are similar. We first show that the converse is not true. Then, we provide a number of constructions of degree-similar graphs. Finally, we show that the matrices $A_1-\mu D_1$ and $A_2-\mu D_2$ are similar over the field of rational functions $\mathbb{Q}(\mu)$ if and only if the Smith normal forms of the matrices $tI-(A_1-\mu D_1)$ and $tI-(A_2-\mu D_2)$ are equal., Comment: 22 pages, 8 figures

For a graph G, the spectral radius \r{ho}(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we seek the relationship between \r{ho}(G) and the walks of the subgraphs of G. Especially, if G contains a complete multi-partite graph as a spanning subgraph, we give a formula for \r{ho}(G) by using an infinite series on walks of the subgraphs of G. These results are useful for the current popular spectral extremal problem., Comment: 13 pages,0 figures, normal article

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon.

In this paper, we investigate the Sombor index of the total graph and unit graph of $\mathbb{Z}_n$ which is denoted by $T_{\Gamma}(\mathbb{Z}_n)$ and $G(\mathbb{Z}_n)$ respectively for $n \in \{2k, p^{\alpha}, pq, p^2q\}$ where $p$ and $q$ are distinct odd prime numbers such that $p < q$. Moreover, we compute the Sombor index of any finite local ring., Comment: 11 pages

The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices $u$ and $v$ are adjacent if $\gcd(u - v, n) = 1$. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.

Transportation matrices are $m\times n$ non-negative matrices whose row sums and row columns are equal to, or dominated above with given integral vectors $R$ and $C$. Those matrices belong to a convex polytope whose extreme points have been previously characterized. In this article, a more general set of non-negative transportation matrices is considered, whose row sums are bounded by two integral non-negative vectors $R_{min}$ and $R_{max}$ and column sums are bounded by two integral non-negative vectors $C_{min}$ and $C_{max}$. It is shown that this set is also a convex polytope whose extreme points are then fully characterized., Comment: 10 pages

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on the average degrees of induced bipartite subgraphs and a more explicit one in terms of the chromatic number and the independence number of $G$. In particular, motivated by our bounds, we introduce two graph invariants that are of interest on their own. Finally, special attention goes to the investigation of the sharpness of our bounds in various classes of graphs as well as the comparison with an existing well-known upper bound., Comment: 10 pages, to appear in Archiv der Mathematik

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\geq tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. In this paper, it is verified that $G$ is a 1-tough graph unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $\rho_{\alpha}(G)\geq\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$, where $\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$ equals the largest root of $x^{3}-((\alpha+1)n+\alpha-3)x^{2}+(\alpha n^{2}+(\alpha^{2}-\alpha-1)n-2\alpha+1)x-\alpha^{2}n^{2}+(3\alpha^{2}-\alpha+1)n-4\alpha^{2}+5\alpha-3=0$. Further, we present an $A_{\alpha}$-spectral radius condition for a graph to be a $t$-tough graph., Comment: 11 pages

The spectral extremal problem of planar graphs has aroused a lot of interest over the past three decades. In 1991, Boots and Royle [Geogr. Anal. 23(3) (1991) 276--282] (and Cao and Vince [Linear Algebra Appl. 187 (1993) 251--257] independently) conjectured that $K_2 + P_{n-2}$ is the unique graph attaining the maximum spectral radius among all planar graphs on $n$ vertices, where $K_2 + P_{n-2}$ is the graph obtained from $K_2\cup P_{n-2}$ by adding all possible edges between $K_2$ and $P_{n-2}$. In 2017, Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] confirmed this conjecture for all sufficiently large $n$. In this paper, we consider the spectral extremal problem for planar graphs without specified subgraphs. For a fixed graph $F$, let $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ denote the set of graphs attaining the maximum spectral radius among all $F$-free planar graphs on $n$ vertices. We describe a rough sturcture for the connected extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,F)$ when $F$ is a planar graph not contained in $K_{2,n-2}$. As applications, we determine the extremal graphs in $\mathrm{SPEX}_{\mathcal{P}}(n,W_k)$, $\mathrm{SPEX}_{\mathcal{P}}(n,F_k)$ and $\mathrm{SPEX}_{\mathcal{P}}(n,(k+1)K_2)$ for all sufficiently large $n$, where $W_k$, $F_k$ and $(k+1)K_2$ are the wheel graph of order $k$, the friendship graph of order $2k+1$ and the disjoint union of $k+1$ copies of $K_2$, respectively., Comment: 22 pages

In this paper we establish connections between common neighborhood Laplacian and common neighborhood signless Laplacian energies and the first Zagreb index of a graph $\mathcal{G}$. We introduce the concepts of CNL-hyperenergetic and CNSL-hyperenergetic graphs and showed that $\mathcal{G}$ is neither CNL-hyperenergetic nor CNSL-hyperenergetic if $\mathcal{G}$ is a complete bipartite graph. We obtain certain relations between various energies of a graph. Finally, we conclude the paper with several bounds for common neighborhood Laplacian and signless Laplacian energies of a graph., Comment: 27 pages

Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of transmissions over all vertices of $G$, and $\partial^Q(G)$ is the spectral radius of the distance signless Laplacian matrix $D(G)+\mbox{diag}(D_1,D_2,\ldots,D_n)$. In this paper, we present a sharp lower bound of $2D_{\max}(G)-\partial^Q(G)$ among all $n$-vertex connected graphs, and characterize the extremal graphs. Furthermore, we give the minimum values of respective $D_{\max}(G)-\partial(G)$ and $2D_{\max}(G)-\partial^Q(G)$ on trees and characterize the extremal trees.

In this paper, we present two new matrices, namely the resistance Laplacian and resistance signless Laplacian matrix of a connected graph. We provide a generalized form of these matrices for different classes of graphs, including the complete graph, complete bipartite graph, and cycle. We investigate the spectral properties of these matrices, analyzing their eigenvalues and eigenvectors. Moreover, we introduce a concept similar to graph energy and define it as the resistance Laplacian energy of a graph and further discuss some bounds for this energy., Comment: 18 pages

In 2010, Butler introduced the unfolding operation on a bipartite graph to produce two bipartite graphs, which are cospectral for the adjacency and the normalized Laplacian matrices. In this article, we describe how the idea of unfolding a bipartite graph with respect to another bipartite graph can be extended to nonbipartite graphs. In particular, we describe how unfoldings involving reflexive bipartite, semi-reflexive bipartite, and multipartite graphs are used to obtain cospectral nonisomorphic graphs for the adjacency matrix., Comment: arXiv admin note: text overlap with arXiv:2110.09034

A graph in which all minimal zero forcing sets are in fact minimum size is called ``well-forced." This paper characterizes well-forced trees and presents an algorithm for determining which trees are well-forced. Additionally, we characterize which vertices in a tree are contained in no minimal zero forcing set.

A chord of a cycle $C$ is an edge joining two non-consecutive vertices of $C$. A cycle $C$ in a graph $G$ is chorded if the vertex set of $C$ induces at least one chord. In this paper, we prove that if $G$ is a graph with order $n\geq 6$ and $\rho(G)\geq \rho(K_{2,n-2})$, then $G$ contains a chorded cycle unless $G\cong K_{2,n-2}$. This gives one answer to a question posed by Gould [Results and problems on chorded cycles: A survey, Graphs Combin. 38 (2022) 189]., Comment: 9 pages

In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs with smallest eigenvalue at least $-3$, analogous to existing classification results available in the case that the smallest eigenvalue is at least $-2$.

We investigate whether there is a relationship between the discrete Bakry-\'{E}mery curvature of a graph and the smooth curvature of an ambient surface into which the graph is embedded geodesically. As we used weighted graphs as test objects, we developed a program for the calculation of the discrete curvature and with the help of this calculator, we observed some indications of such a relationship., Comment: 18 pages, 12 figures, 1 table

Let $G$ be an simple graph of order $n$ whose adjacency eigenvalues are $\lambda_1\ge\dots\ge\lambda_n$. The HL--index of $G$ is defined to be $R(G)= \max\{|\lambda_{h}|, |\lambda_{l}|\}$ with $h=\left\lfloor\frac{n+1}{2}\right\rfloor$ and $ l=\left\lceil\frac{n+1}{2}\right\rceil.$ Mohar conjectured that $R(G)\le 1$ for every planar subcubic graph $G$. In this note, we prove that Mohar's Conjecture holds for every $K_4$-minor-free subcubic graph. Note that a $K_4$-minor-free graph is also called a series--parallel graph. In addition, $R(G)\le 1$ for every subcubic graph $G$ which contains a subgraph $K_{2,3}$., Comment: 6 pages, 1figure

In this article we study two discrete curvature notions, Bakry-\'Emery curvature and Ollivier Ricci curvature, on Cayley graphs. We introduce Right Angled Artin-Coxeter Hybrids (RAACHs) generalizing Right Angled Artin and Coxeter groups (RAAGs and RACGs) and derive the curvatures of Cayley graphs of certain RAACHs. Moreover, we show for general finitely presented groups $\Gamma = \langle S \, \mid\, R \rangle$ that addition of relators does not lead to a decrease the weighted curvatures of their Cayley graphs with adapted weighting schemes.

It is well known that the spectrum and the Smith normal form of a matrix can be computed in polynomial time. Thus, it is interesting to explore how good are these parameters for distinguishing graphs. This is relevant since it is related to the Graph Isomorphism Problem (GIP), which asks to determine whether two graphs are isomorphic. In this paper, we explore the computational advantages of using the spectrum and the Smith normal form of two matrices associated with a graph. By considering the SNF or the spectrum of two matrices of a graph as a single parameter, we compute the number of non-isomorphic graphs with the same parameter with up to 9 vertices, and with up to 10 vertices when the number of graphs with 9 vertices with the same parameter is less than 1000. Focusing on the best 20 combinations of matrices for graphs with 10 vertices, we notice that the number of such graphs with a mate is less than 100 in any of these 20 cases. This computational result improves on similar previous explorations. Therefore, the use of the spectrum or the SNF of two matrices at the same time shows a substantial improvement in distinguishing graphs.

We consider two distinct $q$-analogues of the bipartite distance matrix, namely the $q$-bipartite distance matrix and the exponential distance matrix. We provide formulae of the inverse for these matrices, which extend the existing results for the bipartite distance matrix. These investigations lead us to introduce a $q$-analogue version of the bipartite Laplacian matrix.

In zero forcing, the focus is typically on finding the minimum cardinality of any zero forcing set in the graph; however, the number of cardinalities between $0$ and the number of vertices in the graph for which there are both zero forcing sets and sets that fail to be zero forcing sets is not well known. In this paper, we introduce the zero forcing span of a graph, which is the number of distinct cardinalities for which there are sets that are zero forcing sets and sets that are not. We introduce the span within the context of standard zero forcing and skew zero forcing as well as for standard zero forcing on directed graphs. We characterize graphs with high span and low span of each type, and also investigate graphs with special zero forcing polynomials.

Let $G$ be an $n$-vertex graph and $Q(G)$ be its signless Laplacian matrix. The $Q$-walk matrix of $G$, denoted by $W_Q(G)$, is $[e,Q(G)e,\ldots,Q^{n-1}(G)e]$, where $e$ is the all-one vector. Let $G\circ P_m$ be the graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with an endvertex of the $i$-th copy of $P_m$ for each $i$. We prove that, $$\det W_Q(G\circ P_m)=\pm (\det Q(G))^{m-1}(\det W_Q(G))^m$$ holds for any $m\ge 2$. This gives a signless Laplacian counterpart of the following recently established identity [17]: $$\det W_A(G\circ P_m)=\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W_A(G))^m,$$ where $A(G)$ is the adjacency matrix of $G$ and $W_A(G)=[e,A(G)e,\ldots,A^{n-1}(G)e]$. We also propose a conjecture to unify the above two equalities., Comment: 16 pages, 1 figure