A sharp upper bound for the spectral radius of the Nordhaus–Gaddum type

Abstract

Let G be a simple graph with n vertices and let G c be its complement. Let ρ( G) be the spectral radius of adjacency matrix A( G) of G. In this paper, a sharp upper bound of the Nordhaus–Gaddum type is obtained: ρ(G)+ρ(G c )⩽ 2− 1 k − 1 k ̄ n(n−1) , where k and k̄ are the chromatic numbers of G and G c, respectively. Equality holds if and only if G is a complete graph or an empty graph.