A note on distance signless Laplacian spectrum of graphs

Let G = (V(G); E(G)) be a connected graph with its vertex set V(G) = fv1; v2; : : : ; vng and its edge set E(G). The transmission Tr(vi) of vertex vi is defined to be the sum of the distances from vi to all other vertices. Let Tr(G) be the n n diagonal matrix with its (i; i)-entry equal to TrG(vi). The distance signless Laplacian matrix is defined as DQ(G) = Tr(G) + D(G), where D(G) is the distance matrix of G. The distance signless Laplacian spectral radius of a graph G is the largest eigenvalue of DQ(G). In this paper we first determine some upper and lower bounds on the distance signless Laplacian spectral radius of G based on its order and independence number, and characterize the extremal graph. In addition, we give an upper and lower bounds for the minimum covering distance signless Laplacian energy of G