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 �(G) = � 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: λ1(G) � ( � + δ − 1) + � ( � + δ − 1) 2 + 4(4m − 2δ(n − 1)) 2 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 λ1(G) + λ1(G c) � n − 2 + ( � + δ + 1 − n) 2 + n 2 + 4( � − δ)(n − 1)