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 α and the chromatic number χ of G. Furthermore, we provide lower bounds of @Q 1 (G), the dsL spectral radius, for certain families of graphs in terms of the order n and the independence number χ, or the chromatic number χ.