The Estrada index of trees

The Estrada index of a graph G is defined as , where λ1,λ2,…,λn are the eigenvalues of its adjacency matrix. We determine the unique tree with maximum Estrada index among the set of trees with given number of pendant vertices. As applications, we determine trees with maximum Estrada index among the set of trees with given matching number, independence number, and domination number, respectively. Finally, we give a proof of a conjecture in [J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010) 799–810] on trees with minimum Estrada index among the set of trees with two adjacent vertices of maximum degree.