More on the Estrada indices of trees

Abstract

Let [Formula: see text] be the set of trees with [Formula: see text] vertices. The first six trees with the maximal Estrada indices in [Formula: see text] with [Formula: see text] were previously reported. A new transformation for studying the Estrada index is introduced. Then, with the aid of the new transformation, we completely deduce the 7th to the 13th trees with the maximal Estrada indices in [Formula: see text] with [Formula: see text]. Furthermore, for the trees in [Formula: see text] with [Formula: see text], we obtain the relationship among three decreasing orders, namely, by their largest Estrada indices, by their largest eigenvalues, and by their largest resolvent Estrada indices, respectively.