Some new bounds on resolvent energy of a graph

Let G G be a simple graph of order n n with eigenvalues λ 1 ≥ λ 2 ≥ ... ≥ λ n. {\lambda }_{1}\ge {\lambda }_{2}\ge \ldots \ge {\lambda }_{n}. The resolvent energy of G G is a spectrum-based graph invariant defined as ER (G) = ∑ i = 1 n (n - λ i) - 1. {\rm{ER}}(G)={\sum }_{i=1}^{n}{(n-{\lambda }_{i})}^{-1}. In this work, we propose some new bounds for ER (G) {\rm{ER}}(G). As a direct consequence of these bounds, we present some (n, m) (n,m) -type results for triangle-free graphs.