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

Chu, Zheng-Qing; Liu, Jia-Bao; Li, Xiao-Xin

doi:https://doi.org/10.1155/2016/7320107

Journal of Analytical Methods in Chemistry

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Research Article | Open Access

Volume 2016 | Article ID 7320107 | https://doi.org/10.1155/2016/7320107

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

Zheng-Qing Chu,¹Jia-Bao Liu,¹and Xiao-Xin Li²

Academic Editor: Karoly Heberger

Received18 Nov 2015

Revised04 Mar 2016

Accepted09 Mar 2016

Published30 Mar 2016

Abstract

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.

1. 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 [2–5], the Kirchhoff index [6–13], 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 [14–20].

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 [21–23] pointed out that, as molecular structure descriptors, the Laplacian-energy-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.

All the graphs discussed in this paper are simple, finite, and undirected. For a graph , the vertex set and edge set of will be denoted by and , respectively [26]. The adjacency matrix and the diagonal matrix of are, respectively, and ; then the matrix is called the Laplacian matrix of the graph [27, 28]. The characteristic polynomials and Laplacian polynomials of the graph are and [29]. Both and are symmetric matrices; their eigenvalues are real numbers [30, 31]. Thus, we can order the eigenvalues of the graph as , and the Laplacian eigenvalues are [32, 33]. If is a connected graph, then , [34–36]. Next, we will recall some basic concepts.

Definition 1 (see [1]). The energy of a graph is the sum of the absolute values of all the eigenvalues of ; that is,

Definition 2 (see [2]). Let be a graph of order . The Laplacian-energy-like invariant of , denoted by , 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 then That is, and have the same asymptotic Laplacian-energy-like invariant.

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

2. Main Results

2.1. The Laplacian-Energy-Like Invariant of the Modified Hexagonal Lattice

The modified hexagon lattice with toroidal boundary condition is denoted by .

Theorem 5. Let , . Then

Proof. With the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix iswhere , are the unit matrices and is tensor product of matrices and . ConsiderThe matrix can be defined as follows: Let be a cyclic group of order . Obviously, can express the group. The cyclic group of order has linear values of , , where are said -times unit roots.
Therefore, there is a reversible matrix such that In fact, henceSo It is not difficult to find that is a diagonal matrix whose diagonal elements are where and .
This means that the eigenvalues of the matrix are , , and , where and .
By formula (2), the Laplacian-energy-like invariant is So

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 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.

Theorem 7. For the modified hexagonal lattices , , and with toroidal, cylindrical, and free boundary conditions, then,

2.2. The Laplacian-Energy-Like Invariant of the Modified Union Jack Lattice

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

Theorem 8. Let ; . Then

Proof. With a proper labelling of the vertices of the modified Union Jack lattice, its Laplacian matrix can be represented as Based on Theorem 5, we get Letsuch thatActually, consequently, So It is not difficult to find that is a diagonal matrix whose diagonal elements are where and .
This means that the eigenvalues of the matrix are , , and , where and .
By formula (2), the Laplacian-energy-like invariant is So

By Theorems 4 and 8, it is not difficult to arrive at the following theorem.

Theorem 9. For the modified Union Jack lattices , , and with toroidal, cylindrical, and free boundary conditions, then,

2.3. The Laplacian-Energy-Like Invariant of the Honeycomb Lattice

The honeycomb lattice with toroidal boundary condition, denoted by , can be constructed by starting with an square lattice and adding two diagonal edges to each square.

Theorem 10. Let and . Then

Proof. Similarly, the Laplacian matrix of the honeycomb lattice is , where and is an matrix. The matrix can be written in the following form: where represents the unit matrix of and represents the unit matrix of , respectively.
Based on Theorem 5, the matrix can be written as Let such that Similarly, hence, So It is not difficult to find that is a diagonal matrix whose diagonal elements are , so matrix can be reduced to the following form: By , we can get Therefore, the characteristic eigenvalues are where and .
Let and . By formula (2), we may obtain the Laplacian-energy-like invariant: By the definition of double integration, we arrive at

By Theorems 4 and 10, we can easily obtain the following theorem.

Theorem 11. For the honeycomb lattices , , and with toroidal, cylindrical, and free boundary conditions, then,

3. Conclusions

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.

Competing Interests

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

Acknowledgments

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.

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Copyright

Copyright © 2016 Zheng-Qing Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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