On the Laplacian spectral radius of a graph

Let G be a simple graph with n vertices and m edges and Gc be its complement. Let δ(G)=δ and Δ(G)=Δ be the minimum degree and the maximum degree of vertices of G, respectively. In this paper, we present a sharp upper bound for the Laplacian spectral radius as follows: Equality holds if and only if G is a connected regular bipartite graph. Another result of the paper is an upper bound for the Laplacian spectral radius of the Nordhaus–Gaddum type. We prove that