The Laplacian-Energy-Like Invariants of Three Types of Lattices

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.


Introduction
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 [1], the Laplacian-energy-like invariant [2][3][4][5], the Kirchhoff index [6][7][8][9][10][11][12][13], and so forth. The energy of the graph is an important invariant of the adjacency spectrum and is the sum of the absolute values of all the eigenvalues of a graph , which is studied in chemistry and used to approximate the total electron energy of a molecule [1]. During researching the character of the conjugated carbon oxides, chemists found that the "general electric" is closely related to the energy releasing from the formation progress of the conjugated carbon oxides and could be approximately calculated by Hückel molecular orbital theory. And in the method of HMO, the calculation of can be attributed to the sum of the absolute values of all the eigenvalues of its molecular graph [14][15][16][17][18][19][20].
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 [21][22][23] pointed out that, as molecular structure descriptors, the Laplacianenergy-like invariant not only well describes the properties of most of the descriptors which are indicated, such as entropy, molar volume, and molar refractivity, but also is able to describe some more difficult properties, such as boiling point and rub points. Due to the fact that Laplacian-energy-like invariant has a significant physical and chemical background [24,25], it has received wide attention to research it from many mathematical and chemical workers.

Journal of Analytical Methods in Chemistry
Definition 1 (see [1]). The energy of a graph is the sum of the absolute values of all the eigenvalues of ; that is, (1) Definition 2 (see [2]). Let be a graph of order . The Laplacian-energy-like invariant of , denoted by LEL( ), is defined as Definition 3 (see [35]). For two matrices = ( , ) × , = ( , ) × , the tensor product of and , denoted by ⊗ , is defined as Theorem 4 (see [35]). Let { } be a sequence of finite simple graphs with bounded average degree such that Let { } be a sequence of spanning subgraphs of { } such that That is, and have the same asymptotic Laplacianenergy-like invariant.
In what follows, we will explore the Laplacian-energylike invariants formulas of the modified hexagonal lattice, modified Union Jack lattice, and honeycomb lattice.

The Laplacian-Energy-Like Invariant of the Modified
Hexagonal Lattice. The modified hexagon lattice with toroidal boundary condition is denoted by MH ( 1 , 2 ).
Proof. With the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix is where 1 , 2 are the unit matrices and ⊗ is tensor product of matrices and . Consider Journal of Analytical Methods in Chemistry 3 The matrix (MH ( 1 , 2 )) can be defined as follows: Let {1 = 0 , 1 , . . . , −1 } be a cyclic group of order . Obviously, : → can express the group. The cyclic group of order has linear values of ( = 0, 1, . . . , − 1), ( ) = , where are said -times unit roots.
By formula (2), the Laplacian-energy-like invariant is Remark 6. The numerical integration value in last line is calculated with the software MATLAB [37]. As such computations would be possible on a computer with high memory and processing speed, we used Mac Pro with processor 2 × 2.93 GHz 6-core Intel Xeon (24 hyperthreads in total) and memory 24 GB 1333 MHz DDR3 to obtain the results.
By Theorems 4 and 5, we can immediately arrive at the following theorem.

The Laplacian-Energy-Like Invariant of the Modified
Union Jack Lattice. The modified Union Jack lattice with toroidal boundary condition is denoted by ( 1 , 2 ).

The Laplacian-Energy-Like Invariant of the Honeycomb
Lattice. The honeycomb lattice with toroidal boundary condition, denoted by HC ( 1 , 2 ), can be constructed by starting with an × square lattice and adding two diagonal edges to each square.   ) ) ) ) ) ) ) where represents the unit matrix of 1 × 1 and represents the unit matrix of × , respectively. Based on Theorem 5, the matrix can be written as such that Similarly, hence, So It is not difficult to find that 2 ⊗( 1 + −1 1 )+ −1 2 ⊗ 1 is a diagonal matrix whose diagonal elements are 1 + − 1 + − 2 , so matrix (HC ( 1 , 2 )) can be reduced to the following form: ) .

Conclusions
In this paper, we mainly studied the Laplacian-energylike invariants of the modified hexagonal lattice, modified Jack lattice, and honeycomb lattice. The Laplacian-energylike invariants formulas of these lattices are obtained. The proposed results imply that the asymptotic Laplacian-energylike 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.