Group for Research in Decision Analysis

G-2017-97

On distance Laplacian and distance signless Laplacian eigenvalues of graphs

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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