Groupe d’études et de recherche en analyse des décisions

# A Signless Laplacian for the Distance Matrix of a Graph

## Mustapha Aouchiche et Pierre Hansen

We introduce a signless Laplacian for the distance matrix of a connected graph, called the distance signless Laplacian. We study the distance signless Laplacian spectrum of a connected graph. We show the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. We also establish a relationship between the smallest eigenvalue of the distance signless Laplacian of a connected graph G and the existence of a bipartite component in the complement $\overline{G}$.

, 16 pages