Seidel-Estrada index

Abstract

Let G be a simple graph with n vertices and (0,1)(0,1)-adjacency matrix A. As usual, S(G)=J−2A−IS(G)=J−2A−I denotes the Seidel matrix of the graph G. Suppose θ1,θ2,…,θnθ1,θ2,…,θn and λ1,λ2,…,λnλ1,λ2,…,λn are the eigenvalues of the adjacency matrix and the Seidel matrix of G, respectively. The Estrada index of the graph G is defined as ∑ni=1eθi∑i=1neθi. We define and investigate the Seidel-Estrada index, SEE=SEE(G)=∑ni=1eλiSEE=SEE(G)=∑i=1neλi. In this paper the basic properties of the Seidel-Estrada index are investigated. Moreover, some lower and upper bounds for the Seidel-Estrada index in terms of the number of vertices are obtained. In addition, some relations between SEESEE and the Seidel energy Es(G)Es(G) are presented.