On Distance Signless Laplacian Estrada Index and Energy of Graphs


Abdolla Alhevaz, Maryam Baghipur, Shariefuddin Pirzada




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 $G$.