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

Let G be a simple graph with n vertices and let Gc 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(Gc)⩽2−1k−1k̄n(n−1),where k and k̄ are the chromatic numbers of G and Gc, respectively. Equality holds if and only if G is a complete graph or an empty graph.