Some Relations Between the Eigenvalues of Adjacency, Laplacian and Signless Laplacian Matrix of a Graph

Abstract

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G) = D(G) ? A(G) and the signless Laplacian matrix of G is Q(G) = D(G) + A(G). Let ? 1(G), μ 1(G) and q 1(G) denote the largest eigenvalues of A(G), L(G) and Q(G), respectively. In this paper, we give upper and lower bounds on q 1(G) ? μ 1(G), q 1(G) ? ? 1(G) and μ 1(G) ? ? 1(G), moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, we give a sharp lower bound on q 1(G) + q 2(G).