Resolvent Estrada index - computational and mathematical studies

Abstract

The resolvent Estrada index of a (non-complete) graph \(G\) of order \(n\) is defined as \(EE_r =\sum_{i=1}^n(1-\lambda_i/(n-1))^{-1}\), where \(\lambda_1, \lambda_2, \lambda_n\) are the eigenvalues of \(G\). Combining computational and mathematical approaches, we establish a number of properties of \(EE_r\). In particular, any tree has smaller \(EE_r\)-value than any unicyclic graph of the same order, and any unicyclic graph has smaller \(EE_r\)-value than any tricyclic graph of the same order. The trees, unicyclic, bicyclic, and tricyclic graphs with smallest and greatest \(EE_r\) are determined.