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 v(i) by d(v(i)). The matrix Q(G) = D(G) + A(G) is called the signless Laplacian of G, where D(G) = diag(d(v(1)), d(v(2)) ..... d(vn)) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let q(1) (G) be the largest eigenvalue of Q(C). In this paper, we first present two sharp upper bounds for q(1) (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 q(1) (G). Then we present three sharp lower bounds for q(1) (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. (C) 2010 Elsevier Inc. All rights reserved.