Proof of a conjecture of V. Nikiforov

Abstract

Using analytical tools, we prove that for any simple graph G on n vertices and its complement $\bar G$ the inequality $\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4} {3}n - 1$ holds, where μ(G) and $\mu \left( {\bar G} \right)$ denote the greatest eigenvalue of adjacency matrix of the graphs G and $\bar G$ respectively.