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On the Incidence Energy of Some Toroidal Lattices
The incidence energy , defined as the sum of the singular values of the incidence matrix of , is a much studied quantity with well known applications in chemical physics. In this paper, we derived the closed-form formulae expressing the incidence energy of the 3.12.12 lattice, triangular kagomé lattice, and lattice, respectively. Simultaneously, the explicit asymptotic values of the incidence energy in these lattices are obtained by utilizing the applications of analysis method with the help of software calculation.
A general problem of interest in physics, chemistry, and mathematics is the calculations of the energy of graphs [1–3], which has now become a popular topic of research; however, almost all of literature deal with the energy of the finite graphs. Yan and Zhang  first considered the asymptotic energy of the infinite lattice graphs; they obtained the asymptotic formulae for energies of various lattices. Historically in lattice statistics, the hexagonal lattice, 3.12.12 lattice, triangular kagomé lattice, and lattice have attracted the most attention [4–9]. Ising spins and XXZ/Ising spins on the have been studied in [10, 11].
Let be a simple graph with vertices, let be the adjacency matrix, and let be the diagonal matrix of vertex degrees of , respectively. The Laplacian eigenvalues of are and the signless Laplacian matrix is . The characteristic polynomial , of is called the characteristic polynomial or polynomial of and is denoted by . The spectrum of which consists of the eigenvalues is also called the spectrum of , respectively. It is well known that , , and are symmetric and positive semidefinite; then we denote the eigenvalues of , , and by , , and , respectively. Details on its theory can be found in recent papers [12–14] and the references cited therein.
The famous graph energy for a simple graph , introduced by Gutman , is defined as . The quantity can be used to estimate the total -electron energy in conjugated hydrocarbons. As an analogue of , the incidence energy , is a novel topological index, inspired by Nikiforov idea , Jooyandeh et al.  introduced the concept of a graph as , which is the sum of the singular values of the incidence matrix . The index has attracted extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth; for more work on , the readers are referred to papers [15–18].
In [4, 19] the energy and Kirchhoff index of toroidal lattices were studied. It is an interesting problem to study the incidence energy of some lattices with toroidal boundary condition. Motivated by results above, we consider the problem of computations of the of the 3.12.12 lattice, triangular kagomé lattice, and lattice with toroidal condition in this paper.
2. Main Results
2.1. The 3.12.12 Lattice
Recently, the adjacency spectrum of 3.12.12 lattice has been proposed in  as follows.
Theorem 1 (see ). Let be the 3.12.12 lattice with toroidal boundary condition. Then the adjacency spectrum is where , , , .
The following result is an important relationship between and .
Consider that if is an -regular graph of order , then Consequently, One can conclude that Define the mapping maps the eigenvalues of to the eigenvalues of and can be considered as an isomorphism of the -spectrum to the corresponding the -spectrum for regular graphs.
Suppose that is an -regular graph with vertices and . Then
Note that is the line graph of the subdivision of which is a -regular graph with vertices, and has vertices. Hence, we get the following theorem.
Theorem 2. Let be the 3.12.12 lattice with toroidal boundary condition and , , , . Then the signless Laplacian spectrum is
By the definition of the incidence energy, we can easily get the incidence energy of .
Theorem 3. Let , , , . Then the incidence energy of can be expressed as
From theorem above, we consider that Consequently, one can easily arrive to the asymptotic value of incidence energy
The numerical integration value in last line is calculated with MATLAB software calculation.
Hence has the asymptotic incidence energy .
2.2. The Triangular Kagomé Lattice
In order to obtain the of toroidal boundary condition, we recall the spectrum and the Laplacian spectrum of .
Theorem 4 (see ). The spectrum and the Laplacian spectrum of are where , , , .
Note that the triangular kagomé lattice is the line graph of the lattice and is a -regular graph with vertices.
Consequently, we can easily get the signless Laplacian spectrum of : where , , , .
Theorem 5. Let , , , . Then the incidence energy of can be expressed as
The above numerical integration value implies that has the asymptotic incidence energy .
Remark 6. In comparison to , the authors have derived the formulae of the number of spanning trees, the energy, and the Kirchhoff index of the triangular kagomé lattice with toroidal boundary condition in , while we have handled the of the 3.12.12 lattice, triangular kagomé lattice, which enriches and extends the earlier results by Liu and Yan .
2.3. The Lattice
The eigenvalues of have been obtained in .
Lemma 7. The eigenvalues of are
Notice that is a -regular graph. Let be the signless Laplacian matrix of , and then the signless Laplacian eigenvalues of are
Based on Lemma 7 and the definition of the incidence energy, it is easy to deduce the following.
Theorem 8. Let , , , , and then the incidence energy of can be expressed as
Similarly, one can readily derive that The above numerical integration value implies that has the asymptotic incidence energy . Summing up, we complete the proof.
3. Remarking Conclusions
In this paper, we deduced the formulae and asymptotic formulae expressing the incidence energy of the 3.12.12 lattice, triangular kagomé lattice, and lattice with toroidal boundary condition, respectively.
It is well known that dealing with the problem of the asymptotic incidence energy of various lattices with the free boundary is not an easy task; however, we can convert the more difficult problems to relatively simple ones via the applications of analysis approach with the help of calculational software. In fact, our approach can be used widely to handle the asymptotic behavior of other lattices and can obtain some useful results simultaneously.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work of J. B. Liu is partly supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B105, and the National Science Foundation of China under Grant nos. 11471016, and 11401004. The work of J. Xie was funded by the Natural Science Foundation of Anhui Province of China under Grant no. 1208085MA15 and the Key Project Foundation of Scientific Research, Education Department of Anhui Province, under Grant no. KJ2014ZD30 and the Key Construction Disciplines Foundation of Hefei University under Grant no. 2014XK08.
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