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 μ 1 ( G ) (resp. λ 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 λ 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.