A Proof of a Conjecture on the Estrada Index

Abstract

Let G be a graph on n vertices, and λ1 ,λ 2, ··· ,λ n its eigenvalues. The Estrada index of G is a graph invariant, defined as EE(G )= � n=1 e λ i .I n this paper, it is shown that the path Pn and the star Sn have the minimum and the maximum Estrada indices among n-vertex trees, respectively; and the path Pn and the complete graph Kn have the minimum and the maximum Estrada indices among connected graphs of order n, respectively. This proves a conjecture of de la Pena, Gutman and Rada.