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

Abstract

Let \(G\) be a connected graph of order \(n\) with Laplacian eigenvalues \(\mu_1 \geq \mu_2 \geq \cdots \geq \mu_n = 0\). 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.