Resolvent energy of graphs

Abstract

The resolvent energy of a graph \(G\) of order \(n\) is defined as \(ER = \sum_{i=1}^n (n - \lambda_i)^{-1}\), where \(\lambda_1, \lambda_2, \ldots\lambda_n\) are the eigenvalues of \(G\). We establish a number of properties of \(ER\). In particular, we establish lower and upper bounds for \(ER\) and characterize the trees, unicyclic, and bicyclic graphs with smallest and greatest \(ER\).