On Distance Signless Laplacian Estrada Index and Energy of Graphs

Abdolla Alhevaz ; Maryam Baghipur ; Shariefuddin Pirzada

Kragujevac Journal of Mathematics, Tome 45 (2021) no. 6, p. 837

Résumé

For a connected graph $G$, the distance signless Laplacian matrix is defined as $D^{Q}(G)=\Tr(G)+D(G)$, where $D(G)$ is the distance matrix of $G$ and $\Tr(G)$ is the diagonal matrix of vertex transmissions of $G$. The eigenvalues $\rho_{1}, \rho_{2}, \ldots , \rho_{n}$ of $D^{Q}(G)$ are the distance signless Laplacian eigenvalues of the graph $G$. In this paper, we define the distance signless Laplacian Estrada index of the graph $G$ as $D^{Q}_{E}E(G)=\sum_{i=1}^{n}e^{\big(\rho_{i}-\frac{2\sigma(G)}{n}\big)}$, where $\sigma(G)$ is the transmission of a graph $G$. We obtain upper and lower bounds for $D^{Q}_{E}E(G)$ and the distance signless Laplacian energy in terms of other graph invariants. Moreover, we derive some relations between $D^{Q}_{E}E(G)$ and the distance signless Laplacian energy of $G$.

Détail

Classification : 05C35, 05C50, 15A18
Keywords: distance signless Laplacian matrix, distance signless Laplacian Estrada index, distance Estrada index, transmission regular graph, distance signless Laplacian energy