On the distribution of distance signless laplacian eigenvalues with given independence and chromatic number

For a connected graph G of order n, let D(G) be the distance matrix and Tr(G) be the diagonal matrix of vertex transmissions of G. The dis-tance signless Laplacian (dsL, for short) matrix of G is defined as DQ(G) = Tr(G) +D(G), and the corresponding eigenvalues are the dsL eigenvalues of G. For an interval I, let mDQ(G)I denote the number of dsL eigenvalues of G lying in the interval I. In this paper, for some prescribed interval I, we obtain bounds for mDQ(G)I in terms of the independence number alpha and the chromatic number chi of G. Furthermore, we provide lower bounds of partial differential 1Q(G), the dsL spectral radius, for certain families of graphs in terms of the order n and the independence number alpha, or the chromatic number chi.