An upper bound for difference of energies of a graph and its complement

Abstract

The A-energy of a graph G, denoted by EA(G), is defined as sum of the absolute values of eigenvalues of adjacency matrix of G. Nikiforov in Nikiforov (2016) proved that EA(G¯)−EA(G)≤2μ¯1and EA(G)−EA(G¯)≤2μ1for any graph G and posed a problem to find best possible upper bound for EA(G)−EA(G¯), where μ1and μ1¯are the largest adjacency eigenvalues of G and its complement G¯respectively. We attempt to provide an answer by giving an improved upper bound on a class of graphs where regular graphs become particular case. As a consequence, it is proved that there is no strongly regular graph with negative eigenvalues greater than −1. The obtained results also improves some of the other existing results.

Keywords

• Graph energy
• Strongly regular graph
• Equienergetic graphs