## Asymptotic behavior of Laplacian-energy-like invariant of some graphs.(English)Zbl 1340.05180

Summary: Let $$G$$ be a connected graph of order $$n$$ with Laplacian eigenvalues $$\mu_1\geq \mu_2\geq \dots \geq \mu_n=0$$. The Laplacian-energy-like invariant (LEL for short) of $$G$$ is defined as $$\mathrm{LEL}=\sum_{i=1}^{n-1} \sqrt {\mu_i}$$. In this paper, we consider the asymptotic behavior of the LEL of iterated line graphs of regular graphs. In addition, the formula and asymptotic formula of the LEL of the square (resp. hexagonal, triangular) lattices with toroidal boundary condition are obtained.

### MSC:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C90 Applications of graph theory 15A18 Eigenvalues, singular values, and eigenvectors 05C76 Graph operations (line graphs, products, etc.)