We will discuss a few basic facts about the distribution of eigenvalues of the adjacency matrix, and some applications. Then we discuss the question of computing the eigenvalues of a symmetric matrix. 1 Eigenvalue distribution Let us consider a d-regular graph G on n vertices. Its adjacency matrix AG is an n× n symmetric matrix, with all of its eigenvalues lying in [−d, d]. How are the eigenval...

It is well known that spectral Turán type problem one of the most classical problems in graph theory. In this paper, we consider problem. Let G be a and let set graphs, say G-free if does not contain any element as subgraph. Denote by ?1 ?2 largest second eigenvalues adjacency matrix A(G) G, respectively. paper focus on characterization graphs without short odd cycles according to graphs. First...

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain another upper bound which is sharp on the spectral radius of the adjacency matrix and compare with some known upper bounds with the help of some examples of graphs. We also characterize graphs for which the bound is attained.

let $d$ be a digraph with skew-adjacency matrix $s(d)$‎. ‎the skew‎ ‎energy of $d$ is defined as the sum of the norms of all‎ ‎eigenvalues of $s(d)$‎. ‎two digraphs are said to be skew‎ ‎equienergetic if their skew energies are equal‎. ‎we establish an‎ ‎expression for the characteristic polynomial of the skew‎ ‎adjacency matrix of the join of two digraphs‎, ‎and for the‎ ‎respective skew energ...

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. For the Erdős-Rényi graph G(n, d/n), our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that d log n. Toget...

A uniform lift of a given network is a network with no loops and no multiple arrows that admits the first network as quotient. Given a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type) with loops or multiple arrows, we prove that it is always possible to construct a uniform lift whose adjacency matrix has only two pos...

a multicone graph is defined to be join of a clique and a regular graph. a graph $ g $ is  cospectral with  graph $ h $ if their adjacency matrices have the same eigenvalues. a graph $ g $ is  said to be  determined by its spectrum or ds for short, if for any graph $ h $ with $ spec(g)=spec(h)$, we conclude that $ g $ is isomorphic to $ h $. in this paper, we present new classes of  multicone g...