Sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph

Let G=(V,E)G=(V,E) be a simple connected graph. Denote by D(G)D(G) the diagonal matrix of its vertex degrees and by A(G)A(G) its adjacency matrix. Then the signless Laplacian matrix of G   is Q(G)=D(G)+A(G)Q(G)=D(G)+A(G). In this paper, we obtain some new and improved sharp upper bounds on the spectral radius q1(G)q1(G) of the signless Laplacian matrix of a graph G.