Group for Research in Decision Analysis

# On distance Laplacian and distance signless Laplacian eigenvalues of graphs

## Kinkar Chandra Das, Mustapha Aouchiche, and Pierre Hansen

Let $${\mathcal D(G)}$$, $${\mathcal D}^L(G)={\mathcal Diag(Tr)} - {\mathcal D(G)}$$ and $${\mathcal D}^Q(G)={\mathcal Diag(Tr)} + {\mathcal D(G)}$$ be, respectively, the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix of graph $$G$$, where $${\mathcal Diag(Tr)}$$ denotes the diagonal matrix of the vertex transmissions in $$G$$. The eigenvalues of $${\mathcal D}^L(G)$$ and $${\mathcal D}^Q(G)$$ will be denoted by $$\partial^L_1 \geq \partial^L_2 \geq \cdots \geq \partial^L_{n-1} \geq \partial^L_n=0$$ and $$\partial^Q_1 \geq \partial^Q_2 \geq \cdots \geq \partial^Q_{n-1} \geq \partial^Q_n$$, respectively. In this paper we study the properties of the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues of graph $$G$$.

, 19 pages