On distance signless Laplacian spectrum and energy of graphs

The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G, defined as D-Q(G) = Tr(G) + D(G), where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of vertex transmissions of G. In this paper, we 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 graphs. In addition, we give an exact description of the distance signless Laplacian spectrum and the distance signless Laplacian energy of the join of regular graphs in terms of their adjacency spectrum.