On the distance signless Laplacian spectral radius and the distance signless Laplacian 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 DQ(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 bounds on the distance signless Laplacian spectral radius of G based on some graph invariants, and characterize the extremal graphs. In addition, we define distance signless Laplacian energy, similar to that in [J. Yang, L. You and I. Gutman, Bounds on the distance Laplacian energy of graphs, Kragujevac J. Math. 37 (2013) 245–255] and give some bounds on the distance signless Laplacian energy of graphs.