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

Let G be a connected graph with n vertices, m edges. The distance signless Laplacian matrix D^Q(G) is defined as D^Q(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 D^Q(G) and are denoted by δ₁^Q(G), δ₂^Q(G), ..., δ_n^Q(G). δ₁^Q is called the distance signless Laplacian spectral radius of D^Q(G). In this paper, we obtain upper and lower bounds for S_D^Q (G) in terms of the Wiener index, the transmission degree and the order of the graph.