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

Authors: Zheng-Qing Chu; Jia-Bao Liu; Xiao-Xin Li
Journal: Journal of Analytical Methods in Chemistry
 (2016)

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## 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.


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## Body

## 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 G, 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” Eπ 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 Eπ 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 graphG, the vertex set and edge set of G will be denoted by VG=v1,v2,…,vn and EG={e1,e2,…,em}, respectively [26]. The adjacency matrix and the diagonal matrix of G are, respectively, A(G) and D(G); then the matrix L(G)=D(G)-A(G) is called the Laplacian matrix of the graph G [27, 28]. The characteristic polynomials and Laplacian polynomials of the graph G are χG(λ)=det⁡(λI-A(G)) and μG(λ)=det⁡(λI-L(G)) [29]. Both A(G) and L(G) are symmetric matrices; their eigenvalues are real numbers [30, 31]. Thus, we can order the eigenvalues of the graph G as λ1≥λ2≥⋯≥λn, and the Laplacian eigenvalues are μ1≥μ2≥⋯≥μn [32, 33]. If G is a connected graph, then μi>0,i=1,2,…,n-1, μn=0  [34–36]. Next, we will recall some basic concepts.Definition 1 (see [1]).
The energy of a graphG is the sum of the absolute values of all the eigenvalues of G; that is,(1)EG=∑i=1nλi.Definition 2 (see [2]).
LetG be a graph of order n. The Laplacian-energy-like invariant of G, denoted by LEL(G), is defined as(2)LELG=∑i=1nμi.Definition 3 (see [35]).
For two matricesA=(ai,j)m×n, B=(bi,j)s×t, the tensor product of A and B, denoted by A⊗B, is defined as(3)a11Ba12B⋯a1nB⋯⋯⋯⋯⋯⋯⋯⋯am1Bam2B⋯amnB.Theorem 4 (see [35]).
LetGn be a sequence of finite simple graphs with bounded average degree such that (4)limn→∞⁡VGn=∞,limn→∞⁡LELGnVGn=h≠0.
LetHn be a sequence of spanning subgraphs of Gn such that (5)limn→∞⁡v∈VHn:dHnv=dGnvVGn=1;then (6)limn→∞⁡LELHnVGn=h.
That is,Gn and Hn 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 byMHt(n1,n2).Theorem 5.
Letαi=2πi/n1, βj=2πj/n2. Then(7)1LELMHtn1,n2=∑i=0n1-1∑j=0n2-16-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj,2limn1→∞⁡limn2→∞⁡LELMHtn1,n2n1n2=14π2∬02π6-2cos⁡x-2cos⁡y-2cos⁡x-ydxdy≈2.3705.Proof.
With the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix is(8)LMHtn1,n2=C-P0⋯00-PT-PTC-P⋯0000-PTC⋯000⋯⋯⋯⋯⋯⋯⋯000⋯C-P0000⋯-PTC-P-P00⋯0-PTCn2×n2,C=6-10⋯00-1-16-1⋯0000-16⋯000⋯⋯⋯⋯⋯⋯⋯000⋯6-10000⋯-16-1-100⋯0-16n1×n1,P=100⋯001110⋯000011⋯000⋯⋯⋯⋯⋯⋯⋯000⋯100000⋯110000⋯011n1×n1,where In1, In2 are the unit matrices and M⊗N is tensor product of matrices M and N. Consider(9)Bn2=010⋯00001⋯00⋯⋯⋯⋯⋯⋯000⋯01100⋯00n2×n2.
The matrixL(MHt(n1,n2)) can be defined as follows: (10)LMHtn1,n2=In2⊗Cn1-Bn2⊗Pn1-Bn2T⊗Pn1T=In2⊗6In1-Bn1-Bn1T-Bn2⊗In1+Bn1T-Bn2T⊗In1+Bn1.
Let{1=g0,g1,…,gn-1} be a cyclic group of order n. Obviously, ρ:gi→Bni can express the group. The cyclic group of order n has linear values of nχi(i=0,1,…,n-1),  χi(g)=ωni, where ωn are said n-times unit roots.
Therefore, there is a reversible matrix(11)Qn=ωnijn0≤i,j≤n-1,such that (12)Qn-1BnQn=diag⁡1,ωn,…,ωnn-1≕Dn.
In fact,(13)BnT=Bn-1,QnT=Qn-1;hence(14)Qn-1BnTQn=diag⁡1,ω-1,…,ωn-n-1≕Dn-1.
So(15)Qn2-1⊗Qn1-1LMHtn1,n2Qn2⊗Qn1=Qn2-1⊗Qn1-1In2⊗6In1-Bn1-Bn1T-Bn2⊗In1+Bn1T-Bn2T⊗In1+Bn1Qn2⊗Qn1=In2⊗6In1-Dn1-Dn1-1-Dn2⊗In1+Dn1-1-Dn2-1⊗In1+Dn1.
It is not difficult to find thatIn2⊗(6In1-Dn1-Dn1-1)-Dn2⊗(In1+Dn1-1)-Dn2-1⊗(In1+Dn1) is a diagonal matrix whose diagonal elements are (16)6-ωn1i-ωn1-i-ωn2j-ωn2-j-ωn1iωn2-j+ωn1-iωn2j=6-2cos⁡2iπn1-2cos⁡2jπn2-2cos⁡2iπn1-cos⁡2jπn2,where 0≤i≤n1-1 and  0≤j≤n2-1.
This means that the eigenvalues of the matrixL are μ=6-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj, 0≤i≤n1-1, and  0≤j≤n2-1, where αi=2πi/n1 and βj=2πj/n2.
By formula (2), the Laplacian-energy-like invariant is (17)LELMHtn1,n2=∑i=0n1-1∑j=0n2-16-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj.
So(18)limn1→∞⁡limn2→∞⁡LELMHtn1,n2n1n2=limn1→∞limn2→∞⁡1n1n2∑i=0n1-1∑j=0n2-16-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj=∬016-2cos⁡2πx-2cos⁡2πy-2cos⁡2πx-ydxdy=14π2∬02π6-2cos⁡x-2cos⁡y-2cos⁡x-ydxdy≈2.3705.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 Theorems4 and 5, we can immediately arrive at the following theorem.Theorem 7.
For the modified hexagonal latticesMHt(n1,n2), MHc(n1,n2), and MHf(n1,n2) with toroidal, cylindrical, and free boundary conditions, then,(19)1limn1→∞⁡limn2→∞⁡LELMHtn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELMHcn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELMHfn1,n2n1n2≈2.3705,2LELMHtn1,n2=LELMHcn1,n2=LELMHfn1,n2≈2.3705n1n2.


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

The modified Union Jack lattice with toroidal boundary condition is denoted bySt(n1,n2).Theorem 8.
Letαi=2πi/n1;  βj=2πj/n2. Then (20)1LELStn1,n2=∑i=0n1-1∑j=0n2-18-2cos⁡αi-2cos⁡βj-4cos⁡αicos⁡βj,2limn1→∞⁡limn2→∞⁡LELStn1,n2n1n2=14π2∬02π8-2cos⁡x-2cos⁡y-4cos⁡xcos⁡ydxdy≈2.7586.Proof.
With a proper labelling of the vertices of the modified Union Jack lattice, its Laplacian matrix can be represented as(21)LStn1,n2=G-U0⋯00-UT-UTG-U⋯0000-UTG⋯000⋯⋯⋯⋯⋯⋯⋯000⋯G-U0000⋯-UTG-U-U00⋯0-UTGn2×n2,G=8-10⋯00-1-18-1⋯0000-18⋯000⋯⋯⋯⋯⋯⋯⋯000⋯8-10000⋯-18-1-100⋯0-18n1×n1,U=110⋯001111⋯000011⋯000⋯⋯⋯⋯⋯⋯⋯000⋯110000⋯111100⋯011n1×n1.
Based on Theorem5, we get (22)LStn1,n2=In2⊗Gn1-Bn2⊗Un1-Bn2T⊗Un1T=In2⊗8In1-Bn1-Bn1T-Bn2⊗In1+Bn1+Bn1T-Bn2T⊗In1+Bn1T+Bn1.
Let(23)Qn=ωnijn0≤i,j≤n-1,such that(24)Qn-1BnQn=diag⁡1,ωn,…,ωnn-1≕Dn.
Actually,(25)BnT=Bn-1,QnT=Qn-1;consequently, (26)Qn-1BnTQn=diag⁡1,ω-1,…,ωn-n-1≕Dn-1.
So(27)Qn2-1⊗Qn1-1LStn1,n2Qn2⊗Qn1=Qn2-1⊗Qn1-1In2⊗8In1-Bn1-Bn1T-Bn2⊗In1+Bn1+Bn1T-Bn2T⊗In1+Bn1T+Bn1Qn2⊗Qn1=In2⊗8In1-Dn1-Dn1-1-Dn2⊗In1+Dn1+Dn1-1-Dn2-1⊗In1+Dn1-1+Dn1.
It is not difficult to find that(28)In2⊗8In1-Dn1-Dn1-1-Dn2⊗In1+Dn1+Dn1-1-Dn2-1⊗In1+Dn1-1+Dn1is a diagonal matrix whose diagonal elements are (29)8-ωn1i-ωn1-i-ωn2j-ωn2-j-ωn1iωn2j+ωn1iωn2-j+ωn1-iωn2-j+ωn1-iωn2j=8-2cos⁡2iπn1-2cos⁡2jπn2-4cos⁡2iπn1cos⁡2jπn2,where 0≤i≤n1-1 and  0≤j≤n2-1.
This means that the eigenvalues of the matrixL are μ=8-2cos⁡αi-2cos⁡βj-4cos⁡αiβj, 0≤i≤n1-1, and  0≤j≤n2-1, where αi=2πi/n1 and βj=2πj/n2.
By formula (2), the Laplacian-energy-like invariant is (30)LELStn1,n2=∑i=0n1-1∑j=0n2-18-2cos⁡αi-2cos⁡βj-4cos⁡αicos⁡βj.
So(31)limn1→∞⁡limn2→∞⁡LELStn1,n2n1n2=limn1→∞⁡limn2→∞⁡1n1n2∑i=0n1-1∑j=0n2-18-2cos⁡αi-2cos⁡βj-4cos⁡αicos⁡βj=∬018-2cos⁡2πx-2cos⁡2πy-4cos⁡2πxcos⁡2πydxdy=14π2∬02π8-2cos⁡x-2cos⁡y-4cos⁡xcos⁡ydxdy≈2.7586.By Theorems4 and 8, it is not difficult to arrive at the following theorem.Theorem 9.
For the modified Union Jack latticesSt(n1,n2), Sc(n1,n2), and Sf(n1,n2) with toroidal, cylindrical, and free boundary conditions, then, (32)1limn1→∞⁡limn2→∞⁡LELStn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELScn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELSfn1,n2n1n2≈2.7586,2LELStn1,n2=LELScn1,n2=LELSfn1,n2≈2.7586n1n2.


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

The honeycomb lattice with toroidal boundary condition, denoted byHCt(n1,n2), can be constructed by starting with an m×n square lattice and adding two diagonal edges to each square.Theorem 10.
Letαi=2πi/n1 and βj=2πj/n2. Then (33)1LELHCtn1,n2=∑i=0n1-1∑j=0n2-13+3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj+∑i=0n1-1∑j=0n2-13-3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj,2limn1→∞⁡limn2→∞⁡LELHCtn1,n22n1n2=18π2∬02π3+3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy+18π2∬02π3-3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy≈1.6357.Proof.
Similarly, the Laplacian matrix of the honeycomb lattice isLHCtn1,n2=3IM-F-FT3IM, where M=n1n2 and F is an M×M matrix. The matrix F can be written in the following form: (34)F=W00⋯0IIIW0⋯0000IW⋯000⋯⋯⋯⋯⋯⋯⋯000⋯W00000⋯IW0000⋯0IWn2×n2,W=100⋯001110⋯000011⋯000⋯⋯⋯⋯⋯⋯⋯000⋯100000⋯110100⋯011n1×n1,where I represents the unit matrix of n1×n1 and IM represents the unit matrix of M×M, respectively.
Based on Theorem5, the matrix F can be written as (35)F=In2⊗Wn1+Bn2T⊗In1=In2⊗In1+Bn1T+Bn2T⊗In1.
Let(36)Qn=ωnijn0≤i,j≤n-1,such that (37)Qn-1BnQn=diag⁡1,ωn,…,ωnn-1≕Dn.
Similarly,(38)BnT=Bn-1,QnT=Qn-1;
hence,(39)Qn-1BnTQn=diag⁡1,ω-1,…,ωn-n-1≕Dn-1.
So(40)Qn2-1⊗Qn1-1FQn2⊗Qn1=Qn2-1⊗Qn1-1In2⊗In1+Bn1T+Bn2T⊗In1Qn2⊗Qn1=In2⊗In1+Dn1-1+Dn2-1⊗In1.
It is not difficult to find thatIn2⊗In1+Dn1-1+Dn2-1⊗In1 is a diagonal matrix whose diagonal elements are 1+ωn1-i+ωn2-j, so matrix L(HCt(n1,n2)) can be reduced to the following form: (41)LHCtn1,n2=3-1-ωn1-i-ωn2-j-1-ωn1-i-ωn2-j3.
Bydet⁡(μI-L(HCt(n1,n2)))=0, we can get (42)μ-32=-1-ωn1-i-ωn2-j-1-ωn1-i-ωn2-j=3+ωn1i+ωn1-i+ωn2j+ωn2-j+ωn1-iωn2j+ωn1iωn2-j=3+2cos⁡2πin1+2cos⁡2πjn2+2cos⁡cos⁡2πin1-cos⁡2πjn2.
Therefore, theL(HCt(n1,n2)) characteristic eigenvalues are (43)μ=3±3+2cos⁡2πin1+2cos⁡2πjn2+2cos⁡2πin1-2πjn2,where 0≤i≤n1-1 and 0≤j≤n2-1.
Letαi=2πi/n1 and βj=2πj/n2. By formula (2), we may obtain the Laplacian-energy-like invariant: (44)LELHCtn1,n2=∑i=0n1-1∑j=0n2-13+3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj+∑i=0n1-1∑j=0n2-13-3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj.
By the definition of double integration, we arrive at(45)limn1→∞⁡limn2→∞⁡LELHCtn1,n22n1n2=limn1→∞⁡limn2→∞⁡12n1n2∑i=0n1-1∑j=0n2-13+3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj+limn1→∞⁡limn2→∞⁡12n1n2∑i=0n1-1∑j=0n2-13-3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj=12∬013+3+2cos⁡2πx+2cos⁡2πy+2cos⁡2πx-ydxdy+12∬013-3+2cos⁡2πx+2cos⁡2πy+2cos⁡2πx-ydxdy=18π2∬02π3+3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy+18π2∬02π3-3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy≈1.6357.By Theorems4 and 10, we can easily obtain the following theorem.Theorem 11.
For the honeycomb latticesHCt(n1,n2), HCc(n1,n2), and HCf(n1,n2) with toroidal, cylindrical, and free boundary conditions, then, (46)1limn1→∞⁡limn2→∞⁡LELHCtn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELHCcn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELHCfn1,n2n1n2≈1.6357;2LELHCtn1,n2=LELHCcn1,n2=LELHCfn1,n2≈1.6357n1n2.


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

The modified hexagon lattice with toroidal boundary condition is denoted byMHt(n1,n2).Theorem 5.
Letαi=2πi/n1, βj=2πj/n2. Then(7)1LELMHtn1,n2=∑i=0n1-1∑j=0n2-16-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj,2limn1→∞⁡limn2→∞⁡LELMHtn1,n2n1n2=14π2∬02π6-2cos⁡x-2cos⁡y-2cos⁡x-ydxdy≈2.3705.Proof.
With the proper labelling of the vertices of the modified hexagonal lattice, its Laplacian matrix is(8)LMHtn1,n2=C-P0⋯00-PT-PTC-P⋯0000-PTC⋯000⋯⋯⋯⋯⋯⋯⋯000⋯C-P0000⋯-PTC-P-P00⋯0-PTCn2×n2,C=6-10⋯00-1-16-1⋯0000-16⋯000⋯⋯⋯⋯⋯⋯⋯000⋯6-10000⋯-16-1-100⋯0-16n1×n1,P=100⋯001110⋯000011⋯000⋯⋯⋯⋯⋯⋯⋯000⋯100000⋯110000⋯011n1×n1,where In1, In2 are the unit matrices and M⊗N is tensor product of matrices M and N. Consider(9)Bn2=010⋯00001⋯00⋯⋯⋯⋯⋯⋯000⋯01100⋯00n2×n2.
The matrixL(MHt(n1,n2)) can be defined as follows: (10)LMHtn1,n2=In2⊗Cn1-Bn2⊗Pn1-Bn2T⊗Pn1T=In2⊗6In1-Bn1-Bn1T-Bn2⊗In1+Bn1T-Bn2T⊗In1+Bn1.
Let{1=g0,g1,…,gn-1} be a cyclic group of order n. Obviously, ρ:gi→Bni can express the group. The cyclic group of order n has linear values of nχi(i=0,1,…,n-1),  χi(g)=ωni, where ωn are said n-times unit roots.
Therefore, there is a reversible matrix(11)Qn=ωnijn0≤i,j≤n-1,such that (12)Qn-1BnQn=diag⁡1,ωn,…,ωnn-1≕Dn.
In fact,(13)BnT=Bn-1,QnT=Qn-1;hence(14)Qn-1BnTQn=diag⁡1,ω-1,…,ωn-n-1≕Dn-1.
So(15)Qn2-1⊗Qn1-1LMHtn1,n2Qn2⊗Qn1=Qn2-1⊗Qn1-1In2⊗6In1-Bn1-Bn1T-Bn2⊗In1+Bn1T-Bn2T⊗In1+Bn1Qn2⊗Qn1=In2⊗6In1-Dn1-Dn1-1-Dn2⊗In1+Dn1-1-Dn2-1⊗In1+Dn1.
It is not difficult to find thatIn2⊗(6In1-Dn1-Dn1-1)-Dn2⊗(In1+Dn1-1)-Dn2-1⊗(In1+Dn1) is a diagonal matrix whose diagonal elements are (16)6-ωn1i-ωn1-i-ωn2j-ωn2-j-ωn1iωn2-j+ωn1-iωn2j=6-2cos⁡2iπn1-2cos⁡2jπn2-2cos⁡2iπn1-cos⁡2jπn2,where 0≤i≤n1-1 and  0≤j≤n2-1.
This means that the eigenvalues of the matrixL are μ=6-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj, 0≤i≤n1-1, and  0≤j≤n2-1, where αi=2πi/n1 and βj=2πj/n2.
By formula (2), the Laplacian-energy-like invariant is (17)LELMHtn1,n2=∑i=0n1-1∑j=0n2-16-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj.
So(18)limn1→∞⁡limn2→∞⁡LELMHtn1,n2n1n2=limn1→∞limn2→∞⁡1n1n2∑i=0n1-1∑j=0n2-16-2cos⁡αi-2cos⁡βj-2cos⁡αi-βj=∬016-2cos⁡2πx-2cos⁡2πy-2cos⁡2πx-ydxdy=14π2∬02π6-2cos⁡x-2cos⁡y-2cos⁡x-ydxdy≈2.3705.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 Theorems4 and 5, we can immediately arrive at the following theorem.Theorem 7.
For the modified hexagonal latticesMHt(n1,n2), MHc(n1,n2), and MHf(n1,n2) with toroidal, cylindrical, and free boundary conditions, then,(19)1limn1→∞⁡limn2→∞⁡LELMHtn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELMHcn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELMHfn1,n2n1n2≈2.3705,2LELMHtn1,n2=LELMHcn1,n2=LELMHfn1,n2≈2.3705n1n2.


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

The modified Union Jack lattice with toroidal boundary condition is denoted bySt(n1,n2).Theorem 8.
Letαi=2πi/n1;  βj=2πj/n2. Then (20)1LELStn1,n2=∑i=0n1-1∑j=0n2-18-2cos⁡αi-2cos⁡βj-4cos⁡αicos⁡βj,2limn1→∞⁡limn2→∞⁡LELStn1,n2n1n2=14π2∬02π8-2cos⁡x-2cos⁡y-4cos⁡xcos⁡ydxdy≈2.7586.Proof.
With a proper labelling of the vertices of the modified Union Jack lattice, its Laplacian matrix can be represented as(21)LStn1,n2=G-U0⋯00-UT-UTG-U⋯0000-UTG⋯000⋯⋯⋯⋯⋯⋯⋯000⋯G-U0000⋯-UTG-U-U00⋯0-UTGn2×n2,G=8-10⋯00-1-18-1⋯0000-18⋯000⋯⋯⋯⋯⋯⋯⋯000⋯8-10000⋯-18-1-100⋯0-18n1×n1,U=110⋯001111⋯000011⋯000⋯⋯⋯⋯⋯⋯⋯000⋯110000⋯111100⋯011n1×n1.
Based on Theorem5, we get (22)LStn1,n2=In2⊗Gn1-Bn2⊗Un1-Bn2T⊗Un1T=In2⊗8In1-Bn1-Bn1T-Bn2⊗In1+Bn1+Bn1T-Bn2T⊗In1+Bn1T+Bn1.
Let(23)Qn=ωnijn0≤i,j≤n-1,such that(24)Qn-1BnQn=diag⁡1,ωn,…,ωnn-1≕Dn.
Actually,(25)BnT=Bn-1,QnT=Qn-1;consequently, (26)Qn-1BnTQn=diag⁡1,ω-1,…,ωn-n-1≕Dn-1.
So(27)Qn2-1⊗Qn1-1LStn1,n2Qn2⊗Qn1=Qn2-1⊗Qn1-1In2⊗8In1-Bn1-Bn1T-Bn2⊗In1+Bn1+Bn1T-Bn2T⊗In1+Bn1T+Bn1Qn2⊗Qn1=In2⊗8In1-Dn1-Dn1-1-Dn2⊗In1+Dn1+Dn1-1-Dn2-1⊗In1+Dn1-1+Dn1.
It is not difficult to find that(28)In2⊗8In1-Dn1-Dn1-1-Dn2⊗In1+Dn1+Dn1-1-Dn2-1⊗In1+Dn1-1+Dn1is a diagonal matrix whose diagonal elements are (29)8-ωn1i-ωn1-i-ωn2j-ωn2-j-ωn1iωn2j+ωn1iωn2-j+ωn1-iωn2-j+ωn1-iωn2j=8-2cos⁡2iπn1-2cos⁡2jπn2-4cos⁡2iπn1cos⁡2jπn2,where 0≤i≤n1-1 and  0≤j≤n2-1.
This means that the eigenvalues of the matrixL are μ=8-2cos⁡αi-2cos⁡βj-4cos⁡αiβj, 0≤i≤n1-1, and  0≤j≤n2-1, where αi=2πi/n1 and βj=2πj/n2.
By formula (2), the Laplacian-energy-like invariant is (30)LELStn1,n2=∑i=0n1-1∑j=0n2-18-2cos⁡αi-2cos⁡βj-4cos⁡αicos⁡βj.
So(31)limn1→∞⁡limn2→∞⁡LELStn1,n2n1n2=limn1→∞⁡limn2→∞⁡1n1n2∑i=0n1-1∑j=0n2-18-2cos⁡αi-2cos⁡βj-4cos⁡αicos⁡βj=∬018-2cos⁡2πx-2cos⁡2πy-4cos⁡2πxcos⁡2πydxdy=14π2∬02π8-2cos⁡x-2cos⁡y-4cos⁡xcos⁡ydxdy≈2.7586.By Theorems4 and 8, it is not difficult to arrive at the following theorem.Theorem 9.
For the modified Union Jack latticesSt(n1,n2), Sc(n1,n2), and Sf(n1,n2) with toroidal, cylindrical, and free boundary conditions, then, (32)1limn1→∞⁡limn2→∞⁡LELStn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELScn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELSfn1,n2n1n2≈2.7586,2LELStn1,n2=LELScn1,n2=LELSfn1,n2≈2.7586n1n2.


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

The honeycomb lattice with toroidal boundary condition, denoted byHCt(n1,n2), can be constructed by starting with an m×n square lattice and adding two diagonal edges to each square.Theorem 10.
Letαi=2πi/n1 and βj=2πj/n2. Then (33)1LELHCtn1,n2=∑i=0n1-1∑j=0n2-13+3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj+∑i=0n1-1∑j=0n2-13-3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj,2limn1→∞⁡limn2→∞⁡LELHCtn1,n22n1n2=18π2∬02π3+3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy+18π2∬02π3-3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy≈1.6357.Proof.
Similarly, the Laplacian matrix of the honeycomb lattice isLHCtn1,n2=3IM-F-FT3IM, where M=n1n2 and F is an M×M matrix. The matrix F can be written in the following form: (34)F=W00⋯0IIIW0⋯0000IW⋯000⋯⋯⋯⋯⋯⋯⋯000⋯W00000⋯IW0000⋯0IWn2×n2,W=100⋯001110⋯000011⋯000⋯⋯⋯⋯⋯⋯⋯000⋯100000⋯110100⋯011n1×n1,where I represents the unit matrix of n1×n1 and IM represents the unit matrix of M×M, respectively.
Based on Theorem5, the matrix F can be written as (35)F=In2⊗Wn1+Bn2T⊗In1=In2⊗In1+Bn1T+Bn2T⊗In1.
Let(36)Qn=ωnijn0≤i,j≤n-1,such that (37)Qn-1BnQn=diag⁡1,ωn,…,ωnn-1≕Dn.
Similarly,(38)BnT=Bn-1,QnT=Qn-1;
hence,(39)Qn-1BnTQn=diag⁡1,ω-1,…,ωn-n-1≕Dn-1.
So(40)Qn2-1⊗Qn1-1FQn2⊗Qn1=Qn2-1⊗Qn1-1In2⊗In1+Bn1T+Bn2T⊗In1Qn2⊗Qn1=In2⊗In1+Dn1-1+Dn2-1⊗In1.
It is not difficult to find thatIn2⊗In1+Dn1-1+Dn2-1⊗In1 is a diagonal matrix whose diagonal elements are 1+ωn1-i+ωn2-j, so matrix L(HCt(n1,n2)) can be reduced to the following form: (41)LHCtn1,n2=3-1-ωn1-i-ωn2-j-1-ωn1-i-ωn2-j3.
Bydet⁡(μI-L(HCt(n1,n2)))=0, we can get (42)μ-32=-1-ωn1-i-ωn2-j-1-ωn1-i-ωn2-j=3+ωn1i+ωn1-i+ωn2j+ωn2-j+ωn1-iωn2j+ωn1iωn2-j=3+2cos⁡2πin1+2cos⁡2πjn2+2cos⁡cos⁡2πin1-cos⁡2πjn2.
Therefore, theL(HCt(n1,n2)) characteristic eigenvalues are (43)μ=3±3+2cos⁡2πin1+2cos⁡2πjn2+2cos⁡2πin1-2πjn2,where 0≤i≤n1-1 and 0≤j≤n2-1.
Letαi=2πi/n1 and βj=2πj/n2. By formula (2), we may obtain the Laplacian-energy-like invariant: (44)LELHCtn1,n2=∑i=0n1-1∑j=0n2-13+3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj+∑i=0n1-1∑j=0n2-13-3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj.
By the definition of double integration, we arrive at(45)limn1→∞⁡limn2→∞⁡LELHCtn1,n22n1n2=limn1→∞⁡limn2→∞⁡12n1n2∑i=0n1-1∑j=0n2-13+3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj+limn1→∞⁡limn2→∞⁡12n1n2∑i=0n1-1∑j=0n2-13-3+2cos⁡αi+2cos⁡βj+2cos⁡αi-βj=12∬013+3+2cos⁡2πx+2cos⁡2πy+2cos⁡2πx-ydxdy+12∬013-3+2cos⁡2πx+2cos⁡2πy+2cos⁡2πx-ydxdy=18π2∬02π3+3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy+18π2∬02π3-3+2cos⁡x+2cos⁡y+2cos⁡x-ydxdy≈1.6357.By Theorems4 and 10, we can easily obtain the following theorem.Theorem 11.
For the honeycomb latticesHCt(n1,n2), HCc(n1,n2), and HCf(n1,n2) with toroidal, cylindrical, and free boundary conditions, then, (46)1limn1→∞⁡limn2→∞⁡LELHCtn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELHCcn1,n2n1n2=limn1→∞⁡limn2→∞⁡LELHCfn1,n2n1n2≈1.6357;2LELHCtn1,n2=LELHCcn1,n2=LELHCfn1,n2≈1.6357n1n2.


## 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.



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