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

摘要

The resolvent energy ER(G) of a graph G on n vertices whose adjacency matrix has eigenvalues λ1,…,λn is the sum of the reciprocals of the numbers n−λ1,…,n−λn. We introduce 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.