Abstract
Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 , ?2 , . . . , ?n be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as EE = EE(G)= ?n/i=1 e?i . We define and investigate the Laplacian Estrada index of the graph G, LEE=LEE(G)= ?n/i=1 e(?i - 2m/n). Bounds for LEE are obtained, as well as some relations between LEE and graph Laplacian energy.
Highlights
Let G = (V, E) be a graph without loops and multiple edges
Let G be a graph with n vertices and the adjacency matrix A(G)
We can assume that λ1 ≥ λ2 ≥ · · · ≥ λn, and μ1 ≥ μ2 ≥ · · · ≥ μn = 0 are the adjacency and the Laplacian eigenvalues of G, respectively
Summary
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. Much work on the Estrada index of the graph appeared in the mathematical literature (see, for instance, [3, 10]). A. de la Pena et al [3] established lower and upper bounds for EE in terms of the number of vertices and number of edges, and obtained some inequalities between EE and the energy of G.
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