On the Laplacian spectral radius of a graph

Let <f>G</f> be a simple graph with <f>n</f> vertices and <f>m</f> edges and <f>Gc</f> be its complement. Let <f>δ(G)=δ</f> and <f>Δ(G)=Δ</f> be the minimum degree and the maximum degree of vertices of <f>G</f>, respectively. In this paper, we present a sharp upper bound for the Laplacian spectral radius as follows:Equality holds if and only if <f>G</f> 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 [Copyright &y& Elsevier]