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 (v₁), d (v₂), ..., d (v_n)) 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.