Sharp bounds for the largest eigenvalue of the signless Laplacian of a graph

Let G be a simple connected graph with n vertices and m edges. Denote the degree of vertex vi by d(vi). The matrix Q(G)=D(G)+A(G) is called the signless Laplacian of G, where D(G)=diag(d(v1),d(v2),…,d(vn)) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let q1(G) be the largest eigenvalue of Q(G). In this paper, we first present two sharp upper bounds for q1(G) involving the maximum degree and the minimum degree of the vertices of G and give a new proving method on another sharp upper bound for q1(G). Then we present three sharp lower bounds for q1(G) involving the maximum degree and the minimum degree of the vertices of G. Moreover, we determine all extremal graphs which attain these sharp bounds.