The increase in the resolvent energy of a graph due to the addition of a new edge

Abstract

The resolvent energy ER ( G ) of a graph G on n vertices whose adjacency matrix has eigen- values λ1 , . . . , λn is the sum of the reciprocals of the numbers n −λ1 , . . . , n −λn . We in troduce the resolvent energy matrix R ( G ) and present an algorithm that produces this matrix. This algorithm may also be used to update R ( G ) when new edges are introduced to G . Using the resolvent energy matrix R ( G ), we determine the increase in the resolvent energy ER ( G ) of G caused by such edge additions made to G . Moreover, we express this increase in terms of the characteristic polynomial of G and the characteristic polynomials of three vertex-deleted subgraphs of G .