On trees with smallest resolvent energy

Abstract

Let Pn-1(a) be the tree obtained by attaching a pendent vertex at position a of the (n-1)-vertex path Pn-1. We prove here a conjecture of Gutman et al. [MATCH Commun. Math. Comput. Chem. 73 (2015), 267{270] that Pn-1(a) has the a-th smallest resolvent Estrada index among all trees of order n, for 2 ≤ a ≤ 〈n/2〉, and show that the analogous result also holds for the resolvent energy.