Distance signless Laplacian eigenvalues, diameter, and clique number

Let G be a connected graph of order n. Let Diag(Tr) be the diagonal matrix of vertex transmissions and let D(G) be the distance matrix of G. The distance signless Laplacian matrix of G is defined as D-Q(G) = Diag(Tr) D(G) and the eigenvalues of D-Q(G) are called the distance signless Laplacian eigenvalues of G. Let partial derivative(Q)(1) (G) >= partial derivative(Q)(2)(G) >= ... >= partial derivative(Q)(n) (G) be the distance signless Laplacian eigenvalues of G. The largest eigenvalue partial derivative(Q)(1) (G) is called the distance signless Laplacian spectral radius. We obtain a lower bound for partial derivative(Q)(1) (G) in terms of the diameter and order of G. With a given interval I, denote by m(DQ (G)()) I the number of distance signless Laplacian eigenvalues of G which lie in I. For a given interval I, we also obtain several bounds on m(DQ(G)) I in terms of various structural parameters of the graph G, including diameter and clique number.