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

Let $G=(V\text{,}E)$ be a simple connected graph. Denote by $D(G)$ the diagonal matrix of its vertex degrees and by $A(G)$ its adjacency matrix. Then the signless Laplacian matrix of G is $Q(G)=D(G)+A(G)$ . In this paper, we obtain some new and improved sharp upper bounds on the spectral radius $q_1(G)$ of the signless Laplacian matrix of a graph G.