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This paper mainly studies the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice. By utilizing the tensor product of matrices and the diagonalization of block circulant matrices, we derive closed-form formulas expressing the Laplacian-energy-like invariants of these lattices. In addition, we obtain explicit asymptotic values of these invariants with software-aided computations of some integrals.

Molecular structure descriptors or topological indices are used for modelling information of molecules, including toxicologic, chemical, and other properties of chemical compounds in theoretical chemistry. Topological indices play a very important role in mathematical chemistry, especially in the quantitative structure-property relationship (QSPR) and quantitative structure activity relationship (QSAR). Many topological indices have been introduced and investigated by mathematicians, chemists, and biologists, which contain energy [

Compared with adjacency matrix, the definition of Laplacian matrix added to all vertices degrees. As Mohar said, the Laplacian eigenvalues can reflect more the combination properties of graphs. Cvetković and Simić [

All the graphs discussed in this paper are simple, finite, and undirected. For a graph

The energy of a graph

Let

For two matrices

Let

Let

That is,

In what follows, we will explore the Laplacian-energy-like invariants formulas of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice.

The modified hexagon lattice with toroidal boundary condition is denoted by

Let

With the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix is

The matrix

Let

Therefore, there is a reversible matrix

In fact,

So

It is not difficult to find that

This means that the eigenvalues of the matrix

By formula (

So

The numerical integration value in last line is calculated with the software MATLAB [

By Theorems

For the modified hexagonal lattices

The modified Union Jack lattice with toroidal boundary condition is denoted by

Let

With a proper labelling of the vertices of the modified Union Jack lattice, its Laplacian matrix can be represented as

Based on Theorem

Let

Actually,

So

It is not difficult to find that

This means that the eigenvalues of the matrix

By formula (

So

By Theorems

For the modified Union Jack lattices

The honeycomb lattice with toroidal boundary condition, denoted by

Let

Similarly, the Laplacian matrix of the honeycomb lattice is

Based on Theorem

Let

Similarly,

hence,

So

It is not difficult to find that

By

Therefore, the

Let

By the definition of double integration, we arrive at

By Theorems

For the honeycomb lattices

In this paper, we mainly studied the Laplacian-energy-like invariants of the modified hexagonal lattice, modified Jack lattice, and honeycomb lattice. The Laplacian-energy-like invariants formulas of these lattices are obtained. The proposed results imply that the asymptotic Laplacian-energy-like invariants of those lattices are independent of the three boundary conditions.

The problems on the various topological indices of lattices have much important significance in the mathematical theory, chemical energy, statistical physics, and networks science. This paper investigated the Laplacian-energy-like invariants of some lattices. However, the other topological indices of the general lattices remain to be studied.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to express their sincere gratitude to the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China (nos. KJ2013A196, KJ2013B105, and KJ2015A331), the key project of the Outstanding Young Talent Support Program of the University of Anhui Province (gxyqZD2016367), Anhui Provincial Natural Science Foundation (no. 1408085QA03), NSF of Department of Education of Anhui Province (KJ2015ZD27), and Quality Engineering Projects of Anhui Province of China under Grant no. 2014msgzs168.