New Bounds for Estrada Index

An (m, n)-graph G is a simple graph with n vertices and m edges. Let AI, lambda(1),...lambda(n) be the eigenvalues of its adjacent matrix. The Estrada index of G, denoted by EE(G), is the sum of the terms ear. The first Zagreb index, denoted by Z(g1)G), is the sum of the terms d(i)(2)where d(i) is the degree of v(i) i = 1,2...n. In this paper we prove that m + n <= EE(G) for every (m, n)-graph, and establish a new upper bound for EE(G) in terms of the first Zagreb index.