Group for Research in Decision Analysis

# On the distance signless Laplacian of a graph

## Mustapha Aouchiche and Pierre Hansen

The distance signless Laplacian of a connected graph $$G$$ is defined by $$\mathcal{D}^\mathcal{Q} = Diag(Tr) + \mathcal{D}$$, where $$\mathcal{D}$$ is the distance matrix of $$G$$, and $$Diag(Tr)$$ is the diagonal matrix whose main entries are the vertex transmissions in $$G$$. The spectrum of $$\mathcal{D}^\mathcal{Q}$$ is called the distance signless Laplacian spectrum of $$G$$. 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 $$\mathcal{D}^\mathcal{Q}$$ eigenvalues and establish a relationship between $$n-2$$ being a distance signless Laplacian eigenvalue of $$G$$ and $$\overline{G}$$ containing a bipartite component.

, 12 pages