Some bounds for distance signless Laplacian energy-like invariant of networks

Abstract

For a graph or network $G$, denote by $D(G)$ the distance matrix and $Tr(G)$ the diagonal matrix of vertex transmissions. The distance signless Laplacian matrix of $G$ is $D^{Q}(G)=Tr(G)+D(G)$. We introduce the distance signless Laplacian energy-like invariant as $DEL(G)=\sum_{i=1}^{n}\sqrt{\rho_{i}}$, where $\rho_{1}\geq\rho_{2}\geq \dots\geq \rho_{n}$ are the eigenvalues of distance signless Laplacian matrix. In this paper, we obtain new upper and lower bounds for $DEL(G)$. These bounds involve some important invariants including diameter, minimum and maximum transmission degree, distance signless Laplacian spectral radius and the Wiener index. Additionally, we characterize the extremal graphs attaining these bounds. Finally, we establish some relations between different versions of distance signless Laplacian energy of graphs.