Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 286876, 8 pages

http://dx.doi.org/10.1155/2014/286876

Research Article

The Kirchhoff Index of Toroidal Meshes and Variant Networks

Jia-Bao Liu,1,2,3 Xiang-Feng Pan,¹ Jinde Cao,2,4 and Xia Huang⁵

¹School of Mathematical Sciences, Anhui University, Hefei 230601, China

²Department of Mathematics, Southeast University, Nanjing 210096, China

³Department of Public Courses, Anhui Xinhua University, Hefei 230088, China

⁴Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

⁵College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China

Received 14 March 2014; Accepted 20 May 2014; Published 3 June 2014

Academic Editor: He Huang

Copyright © 2014 Jia-Bao Liu 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.

Abstract.

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf() is the sum of resistance distances between all pairs of vertices in . In this paper, we established the relationships between the toroidal meshes network and its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of , , , and were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.

1. Introduction

Throughout this paper we are concerned with finite undirected connected simple graphs (networks). Let be a graph with vertices labelled . The adjacency matrix of is an matrix with the -entry equal to 1 if vertices and are adjacent and 0 otherwise. Suppose is the degree diagonal matrix of , where is the degree of the vertex , . Let be called the Laplacian matrix of . Then, the eigenvalues of and are called eigenvalues and Laplacian eigenvalues of , respectively.

Given graphs and with vertex sets and , the Cartesian product of graphs and is a graph such that the vertex set of is the Cartesian product ; and any two vertices and are adjacent in if and only if either and is adjacent with in or and is adjacent with in [1]. It is well known that many of the graphs (networks) operations can produce a great deal of novel types of graphs (networks), for example, Cartesian product of graphs, line graph, subdivision graph, and so on. The clique-inserted graph, denoted by , is defined as a line graph of the subdivision graph [2, 3]. The subdivision graph of an -regular graph is -semiregular graph. Consequently, the clique-inserted graph of an -regular graph is the line graph of an -semiregular graph.

The resistance distances between vertices and , denoted by , are defined as the effective electrical resistance between them if each edge of is replaced by a unit resistor [4]. A famous distance-based topological index, the Kirchhoff index , is defined as the sum of resistance distances between all pairs of vertices in ; that is, , known as the Kirchhoff index of [4]; recently, this classical index has also been interpreted as a measure of vulnerability of complex networks [5].

The Kirchhoff index attracted extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [6–13]. Besides, the Kirchhoff index also is a structure descriptor [14]. Unfortunately, it is rather hard to directly design some algorithms [15–17] to calculate resistance distances and the Kirchhoff indexes of graphs. So, many researchers investigated some special classes of graphs [18–21]. In addition, many efforts were also made to obtain the Kirchhoff index bounds for some graphs [17, 22]. Details on its theory can be found in recent papers [17, 22] and the references cited therein.

Motivated by the above results, we present the corresponding calculating formulae for the Kirchhoff index of , , , and in this paper. The rest of this paper is organized as follows. Section 2 presents some underlying notations and preliminaries in our discussion. The proofs of our main results and some asymptotic behavior of Kirchhoff index are proposed in Sections 3 and 4, respectively.

2. Notations and Some Preliminaries

In this section, we introduced some basic properties which we need to use in the proofs of our main results. Suppose that stands for the graphs for the convenience of description. It is trivial for are 1, 2, without loss of generality, we discuss the situations for any positive integer .

Zhu et al. [15] and Gutman and Mohar [8] proved the relations between Kirchhoff index of a graph and Laplacian eigenvalues of the graph as follows.

Lemma 1 (see [8, 15]). Let be a connected graph with vertices and let be the Laplacian eigenvalues of graph ; then

The line graph of a graph , denoted by , is the graph whose vertices correspond to the edges of with two vertices of being adjacent if and only if the corresponding edges in share a common vertex. The subdivision graph of a graph , denoted by , is the graph obtained by replacing every edge in with a copy of (path of length two). The total graph of a graph , denoted by , is the graph whose vertices correspond to the union of the set of vertices and edges of , with two vertices of being adjacent if and only if the corresponding elements are adjacent or incident in . Let be the characteristic polynomial of the Laplacian matrix of a graph ; the following results were shown in [23].

Lemma 2 (see [23]). Let be an -regular connected graph with vertices and edges; then

where , , and are the characteristic polynomials for the Laplacian matrix of graphs , , and , respectively.

A bipartite graph with a bipartition is called an -semiregular graph if all vertices in have degree and all vertices in have degree . Apparently, the subdivision graph of an -regular-graph is -semiregular graph.

Lemma 3 (see [24]). Let be an -semiregular connected graph with vertices. Then

where is the Laplacian characteristic polynomial of the line graph and is the number of edges of .

Lemma 4 (see [23]). Let be a connected simple r-regular graph with vertices and edges and let be the line graph of . Then

Lemma 5 (see [23]). Let be a connected simple -regular graph with vertices; then

The following lemma gives an expression on and of a regular graph .

Lemma 6 (see [25]). Let be a -regular connected graph with vertices and edges, and ; then

Lemma 7 presents the formula for calculating Kirchhoff index of ; in the following proof, some techniques in [26] are referred to.

Lemma 7 (see [26]). For the toroidal networks with any positive integer ,

Proof. Suppose the Laplacian eigenvalues of and are and , ; ; then the Cartesian product and the Laplacian eigenvalues of are

According to Lemma 1, the Kirchhoff index of the toroidal networks is

Since , (10) in the last line holds.

The following consequence was presented in [26]. Here we give a short proof.

Lemma 8 (see [26]). For the toroidal networks with any positive integer ,

Proof. By virtue of (9), one can derive that

Hence,

3. Main Results

3.1. The Kirchhoff Index of

In the following theorem, we proposed the formula for calculating the Kirchhoff index of the line graph of , denoted by .

Theorem 9. Let be line graphs of with any positive integer ; then

Proof. Apparently the toroidal networks are 4-regular graphs which have vertices and edges, respectively.

We clearly obtained the following relationship and from Lemma 4:

Substituting the results of Lemma 7 into (15), we can get the formula for the Kirchhoff index of ,

which completes the proof.

3.2. The Kirchhoff Index of

In an almost identical way as Theorem 9, we derived the formula for the Kirchhoff index on the subdivision graph of , denoted by .

Theorem 10. Let be subdivision graphs of with any positive integer ; then

Proof. Noting that are 4-regular graphs which have vertices, we clearly obtained from Lemma 5

Together with the results of Lemma 7 and (18), we can get the formula for the Kirchhoff index on the subdivision graph of :

The proof is completed.

3.3. The Kirchhoff Index of

Now we proved the formula for estimating the Kirchhoff index in the total graph of , denoted by .

Theorem 11. Let be total graphs of with any positive integer ; then

Proof. Supposing that the Laplacian eigenvalues of are , one can readily see that

Applying Lemma 6, the following result is straightforward:

Notice that have nonzero Laplacian eigenvalues, and

where and .

Consequently, the relationships between and its variant networks for Kirchhoff index are as follows:

According to the results of Lemma 7, we can verify the formula for the Kirchhoff index of the total graph of from (24). Consider

This completes the proof of Theorem 11.

3.4. The Kirchhoff Index of

We will explore the formula for estimating the Kirchhoff index in the clique-inserted graph of , denoted by .

Theorem 12. Let be clique-inserted graphs of with any positive integer ; then

where the first summation ; , and .

Proof. Noting that is -semiregular graphs and supposing that has   vertices and edges, then obviously , , and , respectively.

By virtue of Lemma 3,

Let be the graph ; that is,

From the definition of clique-inserted graph, one can immediately obtain that

Obviously, it follows from Lemma 2,

Replace with in (30); moreover, have vertices and edges; we have that

Based on (29) and (31),

Since the roots of are

where are the Laplacian eigenvalues of and , ; .

It follows from (32) that the Laplacian spectrum of is

where , , , and .

Employing Lemma 1, (33), and the Laplacian spectrum of , the following result is straightforward:

Hence Theorem 12 holds.

Remark 13. The consequences of Lemma 7 and Theorems 9–12 above present closed-form formulae for immediately obtaining its Kirchhoff indexes in terms of finite various networks; however, the quantities are rather difficult to calculate directly.

4. The Asymptotic Behavior of Related Kirchhoff Index

We explore the asymptotic behavior of Kirchhoff index for the investigated networks above as tend to infinity. It is interesting and surprising that the quantity tends to a constant even though , as tend to infinity; that is,

Moreover, one can employ the applications of analysis approach to obtain the explicit approximate values of Kirchhoff index for the related networks.

Theorem 14. Let be line graphs of with any positive integer ; then

Proof. According to (15) and the result of Lemma 8, we can derive that

Consequently,

The result is equivalent to having asymptotic Kirchhoff index,

Theorem 15. Let be subdivision graph of with any positive integer ; then

Proof. Similarly, according to (18), we can easily verify that

Hence,

Theorem 16. Let be total graph of with any positive integer ; then

Proof. Consider the summation term .

Since

The value in last line via the mathematic software MATLAB, which can obtain the result above.

Combining with (22), we can obtain that

So

Theorem 17. Let be clique-inserted graph of with any positive integer ; then

Proof. From the proof of Theorem 12, we know that

where the first summation ; , and .

As tend to infinity, it follows from the first summation term:

Similarly, it holds from the second summation term when tend to infinity,

Combining with the consequences of Theorem 12 and (50) and (51), it follows that

Summing up, we complete the proof.

5. Conclusions

Resistance distance was introduced by Klein and Randi as a generalization of the classical distance. In this paper, we have deduced the relationships between the toroidal meshes network and its variant networks in terms of the Kirchhoff index via spectral graph theory. The explicit formulae for calculating the Kirchhoff indexes of , , , and were proposed for any positive integer , respectively.

The asymptotic behavior of Kirchhoff indexes has been investigated with the applications of analysis approach, and the explicit approximate values are obtained by calculations for the related networks. The values of Kirchhoff indexes with respect to various networks can be immediately obtained via this approach; however, the quantities are rather difficult to calculate directly.

Conflict of Interests

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

Acknowledgments

The work of Jia-Bao Liu is partly supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B105. The work of Xiang-Feng Pan is partly supported by the National Science Foundation of China under Grant nos. 10901001, 11171097, and 11371028.

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