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

Abstract

The $A$-energy of a graph $G$, denoted by $E_A\left(G\right)$, is defined as sum of the absolute values of eigenvalues of adjacency matrix of $G$. Nikiforov in Nikiforov (2016) proved that $E_A\left(\overline{G}\right)-E_A\left(G\right)\le 2{\overline{\mu }}_1$ and $E_A\left(G\right)-E_A\left(\overline{G}\right)\le 2{\mu }_1$ for any graph $G$ and posed a problem to find best possible upper bound for $E_A\left(G\right)-E_A\left(\overline{G}\right)$, where ${\mu }_1$ and $\overline{{\mu }_1}$ are the largest adjacency eigenvalues of $G$ and its complement $\overline{G}$ 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.