On the spread of the distance signless Laplacian matrix of a graph

Abstract

Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix DQ(G) is defined as DQ(G) = Diag(Tr(G)) + D(G), where Diag(Tr(G)) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix of G. The distance signless Laplacian eigenvalues of G are the eigenvalues of DQ(G) and are denoted by δ1 Q(G), δ2 Q(G), ..., δn Q(G). δ1 Q is called the distance signless Laplacian spectral radius of DQ(G). In this paper, we obtain upper and lower bounds for SDQ (G) in terms of the Wiener index, the transmission degree and the order of the graph.