Asymptotic Behavior of Laplacian-Energy-Like Invariant of Some Graphs

Abstract

Let $G$ be a connected graph of order $n$ with Laplacian eigenvalues ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_n=0$. The Laplacian-energy-like invariant ($LEL$ for short) of $G$ is defined as $\text{LEL}=\sum _{i=1}^{n-1}\sqrt{{\mu }_i}$. In this paper, we investigate the asymptotic behavior of the $LEL$ of iterated line graphs of regular graphs. Furthermore, we derive the exact formula and asymptotic formula for the $LEL$ of square, hexagonal, and triangular lattices with toroidal boundary conditions.