Asymptotic Laplacian-Energy-Like Invariant of 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.

6 pages, 2 figures