Some bounds on the largest eigenvalues of graphs

Let G be a simple graph with n vertices. The matrix L(G) = D(G) - A(G) is called the Laplacian of G, while 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(v(n))) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let mu(1)(G) (resp. lambda(1)(G), q(1)(G)) be the largest eigenvalue of L(G) (resp. A(G), Q(G)). In this paper, we first present a new upper bound for lambda(1)(G) when each edge of G belongs to at least t (t >= 1) triangles. Some new upper and lower bounds on q(1)(G), q(1)(G) q(1)(G(C)) are determined, respectively. We also compare our results in this paper with some known results. (C) 2011 Elsevier Ltd. All rights reserved.