Pseudo-resolvent energy of weighted graphs

Abstract

Let G be a finite order graph. Its resolvent matrix and its resolvent energy are respectively given by R xi ( A ( G )) = (A(G) - xi I)-1 -1 and ER ( G ) = Sigma n n i = 1 (n - lambda i)-1. i ) -1 . For some classes of weighted graphs, we know that lambda 1 1 = n which means that the formula for ER ( G ) is not applicable. In this work, the pseudo-resolvent energy for those classes of graphs are obtained where lambda 1 1 = n.