Introduction

Let G be a finite and undirected simple graph, with vertex set V (G) and edge set E(G). The number of vertices of G is n, and its vertices are labeled by v 1,v 2,…,v n . The adjacency matrix A(G) of the graph G is a square matrix of order n, whose (i,j)-entry is equal to 1 if the vertices v i and v j are adjacent and is equal to zero otherwise. The characteristic polynomial of the adjacency matrix, i.e., det(x I n −A(G)), where I n is the unit matrix of order n, is said to be the characteristic polynomial of the graph G and will be denoted by ϕ(G,x). The eigenvalues of a graph G are defined as the eigenvalues of its adjacency matrix A(G), and so they are just the roots of the equation ϕ(G,x)=0. Since A(G) is symmetric, its eigenvalues are all real. Denote them by λ1,λ2,…,λ n , and as a whole, they are called the spectrum of G and denoted by Spec(G).