Some bounds on the largest eigenvalues of graphs

Let GG be a simple graph with nn vertices. The matrix L(G)=D(G)−A(G)L(G)=D(G)−A(G) is called the Laplacian of GG, while the matrix Q(G)=D(G)+A(G)Q(G)=D(G)+A(G) is called the signless Laplacian of GG, where D(G)=diag(d(v1),d(v2),…,d(vn))D(G)=diag(d(v1),d(v2),…,d(vn)) and A(G)A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of GG, respectively. Let μ1(G)μ1(G) (resp. λ1(G)λ1(G), q1(G)q1(G)) be the largest eigenvalue of L(G)L(G) (resp. A(G)A(G), Q(G)Q(G)). In this paper, we first present a new upper bound for λ1(G)λ1(G) when each edge of GG belongs to at least t(t≥1) triangles. Some new upper and lower bounds on q1(G)q1(G), q1(G)+q1(Gc)q1(G)+q1(Gc) are determined, respectively. We also compare our results in this paper with some known results.