Related papers: Some signed graphs whose eigenvalues are main

We consider signed graphs, i.e, graphs with positive or negative signs on their edges. We determine the admissible parameters for the $\{5,6,\ldots,10\}$-regular signed graphs which have only two distinct eigenvalues. For each obtained…

Let $\gamma'_s(G)$ be the signed edge domination number of G. In 2006, Xu conjectured that: for any $2$-connected graph G of order $ n (n \geq 2),$ $\gamma'_s(G)\geq 1$. In this article we show that this conjecture is not true. More…

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le…

Suppose that $G$ is a connected simple graph with the vertex set $V(G)=\{v_1, v_2,\cdots,v_n\}$. Then the adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$ if $v_i$ is adjacent to $v_j$, and otherwise $a_{ij}=0$. The…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha…

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable…

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp_S(G^\sigma)$.…

In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let $G=(V,E)$ be a primitive strongly regular graph with parameters $(v,k,\lambda,\mu)$. Let $S(G,H)$ be the graph from $G$ by switching with…

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible…

For a graph $G$ and a non-zero real number $\alpha$, the graph invariant $S_{\alpha}(G)$ is the sum of the $\alpha^{th}$ power of the non-zero signless Laplacian eigenvalues of $G$. In this paper, we obtain the sharp bounds of…

Let mu(G) and mu_min(G) be the largest and smallest eigenvalues of the adjacency matricx of a graph G. We refine quantitatively the following two results on graph spectra. (i) if H is a proper subgraph of a connected graph G, then…

A signed graph is a graph whose edges are labeled either as positive or negative. The concept of vector valued switching and balancing dimension of signed graphs were introduced by S. Hameed et al. In this paper, we deal with the balancing…

A signed graph $G$ is a graph where each edge is assigned a + (positive edge) or a - (negative edge). The signed degree of a vertex $v$ in a signed graph, denoted by $sdeg(v)$, is the number of positive edges incident to $v$ subtracted by…

Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \ldots, n.$ F. Ashraf et al.…

Let $G$ be a graph in which each edge is assigned one of the colours $1, 2, \ldots, m$, and let $\Gamma$ be a subgroup of $S_m$. The operation of switching at a vertex $x$ of $G$ with respect to an element $\pi$ of $\Gamma$ permutes the…

A signed graph is an ordered pair $\Sigma=(G,\sigma),$ where $G=(V,E)$ is the underlying graph of $\Sigma$ with a signature function $\sigma:E\rightarrow \{1,-1\}$. In this article, we define $n^{th}$ power of a signed graph and discuss…

Let $\Gamma=(G, \sigma)$ be a signed graph of order $n$ with eigenvalues $\mu_1,\mu_2,\ldots,\mu_n.$ We define the Estrada index of a signed graph $\Gamma$ as $EE(\Gamma)=\sum_{i=1}^ne^{\mu_i}$. We characterize the signed unicyclic graphs…

We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of parts is $\frac{\alpha}{1-\alpha}$ and the…

We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices…

Given any square matrix, $\mathbf{M}$, whose diagonal elements are negative, and which is multiplied by a variable, $\sigma$, we wish to find the minimal $\sigma$ such that the eigenvalue of $\mathbf{M}_{\sigma}$ is exactly zero. By…