Asymptotic Laplacian-energy-like invariant of lattices

Let μ1⩾μ2⩾⋯⩾μnμ1⩾μ2⩾⋯⩾μn denote the Laplacian eigenvalues of a graph G with n   vertices. The Laplacian-energy-like invariant, denoted by LEL(G)=∑i=1n-1μi, 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.