On the Laplacian Estrada Index of a Graph

Let G = (V,E) be a graph without loops and multiple edges. Let n and m be the number of vertices and edges of G, respectively. Such a graph will be referred to as an (n,m)-graph. For v ∈ V (G), let d(v) be the degree of v. In this paper, we are concerned only with undirected simple graphs (loops and multiple edges are not allowed). Let G be a graph with n vertices and the adjacency matrix A(G). Let D(G) be a diagonal matrix with degrees of the corresponding vertices of G on the main diagonal and zero elsewhere. The matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G. Since A(G) and L(G) are real symmetric matrices, their eigenvalues are real numbers. So we can assume that λ1 ≥ λ2 ≥ · · · ≥ λn, and μ1 ≥ μ2 ≥ · · · ≥ μn = 0 are the adjacency and the Laplacian eigenvalues of G, respectively. The multiset of eigenvalues of A(G) (L(G)) is called the adjacency (Laplacian) spectrum of G. Other undefined notations may be referred to [1]. The basic properties of the eigenvalues and Laplacian eigenvalues of the graph can be found in the book [2].