On Estrada index of trees

Let G be a graph on n vertices, and let λ1,λ2,…,λn be its eigenvalues. The Estrada index is defined as . We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices with fixed diameter. For , we also determine the tree with maximum Estrada index among the trees on n vertices with maximum degree Δ. It gives a partial solution to the conjecture proposed by Ilić and Stevanović in Ref. [14].