Asymptotic Laplacian-Energy-Like Invariant of Lattices

TL;DR

This paper investigates the Laplacian-energy-like invariant of lattice graphs, demonstrating its boundary-condition independence and deriving explicit asymptotic values for various lattice types.

Contribution

It provides the first explicit asymptotic formulas for the Laplacian-energy-like invariant of lattices and shows boundary-condition independence.

Findings

Laplacian-energy-like per vertex is boundary-condition independent for various lattices.

Explicit asymptotic values of the invariant are derived.

The approach suggests general boundary-condition independence for other lattices.

Abstract

Let $\mu_1\ge \mu_2\ge\cdots\ge\mu_n$ denote the Laplacian eigenvalues of $G$ with $n$ vertices. The Laplacian-energy-like invariant, denoted by $LEL(G)= \sum_{i=1}^{n-1}\sqrt{\mu_i}$, is a novel topological index. In this paper, we show that the Laplacian-energy-like per vertex of various lattices is independent of the toroidal, cylindrical, and free boundary conditions. Simultaneously, the explicit asymptotic values of the Laplacian-energy-like in these lattices are obtained. Moreover, our approach implies that in general the Laplacian-energy-like per vertex of other lattices is independent of the boundary conditions.