On the distance signless Laplacian of a graph

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Abstract
The distance signless Laplacian of a connected graph is defined by , where is the distance matrix of , and is the diagonal matrix whose main entries are the vertex transmissions in . The spectrum of is called the distance signless Laplacian spectrum of . 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 eigenvalues and establish a relationship between being a distance signless Laplacian eigenvalue of and containing a bipartite component.

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