Asymptotic behavior of Laplacian-energy-like invariant of the 3.6.24 lattice with various boundary conditions

Let G be a connected graph of order n with Laplacian eigenvalues \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _1(G)\ge \mu _2(G)\ge \cdots \ge \mu _n(G)=0$$\end{document}μ1(G)≥μ2(G)≥⋯≥μn(G)=0. The Laplacian-energy-like invariant of G, is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr{L}}{\mathscr{E}}{\mathscr{L}}(G)=\sum _{i=1}^{n-1}\sqrt{\mu _i}$$\end{document}LEL(G)=∑i=1n-1μi. In this paper, we investigate the asymptotic behavior of the 3.6.24 lattice in terms of Laplacian-energy-like invariant as m, n approach infinity. Additionally, we derive that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M ^t(n,m)$$\end{document}Mt(n,m), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M ^c(n,m)$$\end{document}Mc(n,m) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M ^f(n,m)$$\end{document}Mf(n,m) have the same asymptotic Laplacian-energy-like invariants.

For more work on LEL(G), the readers are referred to the most recent papers (Liu and Pan 2015b;Liu et al. 2015Das and Gutman 2014).
Historically in lattice statistics, the hexagonal lattice, 3.12.12 lattice and 3.6.24 lattice have attracted the most attention (Liu and Yan 2013;Ye 2011b;Zhang 2013). Some topological indices of graphs were studied in Li et al. (2015), Yan and Zhang (2009), Ye (2011a), Liu et al. (2014bLiu et al. ( , 2016d and Liu and Pan (2016). In fact, Liu et al. have already studied the asymptotic incidence energy (Liu and Pan 2015a) and the Laplacian-energylike invariant of lattices (Liu et al. 2015).
It is an interesting problem to study the various energies of some lattices with various boundary conditions. W. Wang considered the behavior of Laplacian-energy-like invariant of some graphs in Wang (2014). In present paper, we derive the the Laplacianenergy-like invariant of 3.6.24 lattice via the graph spectrum of the line graph of the subdivision graph of a graph G with the help of computer calculation, which is different from the approach of Wang (2014). Yan et al. investigated the asymptotic behavior of some indices of iterated line graphs of regular graphs in Liu et al. (2016c). Motivated by the above results, in this paper we consider the problem of computations of the LEL(G) of the 3.6.24 lattice with various boundary conditions.

Preliminaries
We first recall some underlying definitions and lemmas in graph theory.

Some definitions and lemmas
The subdivision graph s(G) of a graph G is obtained from G by deleting every edge uv of G and replacing it by a vertex w of degree 2 that is joined to u and v (see p. 151 of Chartrand and Zhang 2004).
The line graph of a graph G, denoted by l(G), is the graph whose vertices correspond to the edges of G with two vertices of l(G) being adjacent if and only if the corresponding edges in G share a common vertex (Klein and Yi 2012).
Lemma 1 (Gao et al. 2012) Let G be an r-regular connected graph with n vertices and m edges, then where φ L l(G); x and φ L s(G); x are the characteristic polynomial for the Laplacian matrix of graphs l(G) and s(G), respectively.
Let a bipartite graph G with a bipartition V (G) = (U, V ) is called an (r, s)-semiregular graph if all vertices in U have degree r and all vertices in V have degree s.
Lemma 2 (Mohar and Alavi 1991) Let G be an (r, s)-semiregular connected graph with n vertices. Then where φ L l(G); x is the Laplacian characteristic polynomial of the line graph l(G) and m is the number of edges of G.
The 3.12.12 and 3.6.24 lattices The 3.12.12 lattice with toroidal boundary condition (Liu and Yan 2013), denoted J t (n, m), is illustrated in Fig. 1. Many problems related to the 3.12.12 lattice were considered by physicists (Liu and Yan 2013;Zhang 2013;Liu et al. 2014b). The 3.6.24 lattice with toroidal boundary condition (Zhang 2013), denoted M t (n, m), is illustrated in Fig. 2.
Based on the constructions of the 3-12-12 and 3.6.24 lattices, we notice that a very important and interesting relationship between 3-12-12 lattice J t (n, m) and 3-6-24 M t (n, m) lattice. The relationship is illustrated as follows.

Main results
In this section, we will explore the Laplacian spectrum of the 3.6.24 lattice with toroidal boundary condition. We begin with the adjacency spectrum of 3.12.12 lattice.
The following adjacency spectrum of 3.12.12 lattice is shown in Liu and Yan (2013).
Theorem 1 (Liu and Yan 2013) Let J t (n, m) be the 3.12.12 lattice with toroidal boundary condition. Then the adjacency spectrum is The Laplacian spectrum of the 3.12.12 lattice with toroidal boundary condition is given by the following theorem.
Theorem 2 Let J t (n, m) be the 3.12.12 lattice with toroidal boundary condition and

. Then the Laplacian spectrum is
Proof Consider that J t (n, m) is a 3-regular graph of order n, then D(G) = 3I n . Hence, Define the map ϕ( i ) = 3 − i maps the eigenvalues of A(J t (n, m)) to the eigenvalues of L(J t (n, m)) and can be considered as an isomorphism of the A-spectrum to the corresponding the L-spectrum for J t (n, m). Based on the fact that G is an r-regular graph with n vertices and Spec A (G) = { 1 , 2 , . . . , n }.
Next, we will deduce the Laplacian spectrum of the 3.6.24 lattice M t (n, m).
It follows from Eq. (6) that the Laplacian spectrum of M t (n, m) is where µ i are the Laplacian eigenvalues of the 3.12.12 lattice J t (n, m).  )) .
Proof Based on Theorems 2, 3 and the definition of the Laplacian-energy-like invariant, we can arrive at the statement 1 of Theorem 4. Note that the term A can decompose four terms Similarly,

Then
We consider that where The above numerical integration values are calculated by using the computer software Matlab. 18(m + 1)(n + 1) = √ 3 + √ 5 6  LEL M t (n, m) ≈ 18.1764(m + 1)(n + 1), That is, G n and H n have the same asymptotic Laplacian-energy-like invariant.
Remark 1 Theorem 5 provides a very effective approach to handle the asymptotic the Laplacian-energy-like invariant of a graph with bounded average degree.
Based on Theorem 5, the following result is straightforward.
Theorem 6 Let M t (n, m) (resp. M c (n, m), M f (n, m)) be the toroidal (resp. cylindrical, free) boundary condition of the 3.6.24 lattice. Then Remark 2 It follows from Theorems 5 and 6 that the growth rate of the LEL(G) of the 3.6.24 lattice M t (n, m) (resp. M c (n, m), M f (n, m)) with toroidal (resp. cylindrical, free) boundary condition is only dependent on the number of vertices of it.

Conclusions
In this paper, we deduced the formulae expressing the Laplacian-energy-like invariant of the 3.6.24 lattice with various boundary conditions. Moreover, we obtained the explicit asymptotic values of the Laplacian-energy-like invariant by utilizing the analysis methods with the help of software Matlab calculation. In addition, we showed that their growth rates are independent of the structure of M (n, m) and only dependent on the number of vertices of M (n, m). These and some other related issues are very good topics on lattices, which deserves further exploration.