On the distance signless Laplacian of a graph

The distance signless Laplacian of a connected graph (Formula presented.) is defined by (Formula presented.) , where (Formula presented.) is the distance matrix of (Formula presented.) , and (Formula presented.) is the diagonal matrix whose main entries are the vertex transmissions in (Formula presented.). The spectrum of (Formula presented.) is called the distance signless Laplacian spectrum of (Formula presented.). In the present paper, we study some properties of the distance signless Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance signless Laplacian eigenvalues. We prove several bounds on (Formula presented.) eigenvalues and establish a relationship between (Formula presented.) being a distance signless Laplacian eigenvalue of (Formula presented.) and (Formula presented.) containing a bipartite component.