Discrete Dynamics in Nature and Society

Volume 2013, Article ID 543189, 7 pages

http://dx.doi.org/10.1155/2013/543189

## The Kirchhoff Index of Hypercubes and Related Complex Networks

^{1}Department of Mathematics, Southeast University, Nanjing 210096, China^{2}School of Mathematical Sciences, Anhui University, Hefei 230601, China^{3}Anhui Xinhua University, Hefei 230088, China^{4}Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

Received 14 October 2013; Accepted 31 October 2013

Academic Editor: Guanghui Wen

Copyright © 2013 Jiabao 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 between any two vertices of is defined as the network effective resistance between them if each edge of is replaced by a unit resistor. The Kirchhoff index Kf() is the sum of resistance distances between all the pairs of vertices in . We firstly provided an exact formula for the Kirchhoff index of the hypercubes networks by utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networks and its three variant networks , , by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes of , , and were proposed, respectively.

#### 1. Introduction

In this work we are concerned with finite undirected connected simple graphs. For the graph theoretical definitions and notations we follow [1]. A network is usually modelled by connected graphs , where denotes the set of processors and denotes the set of communication links between processors. It is well known that the standard distance between two vertices of , denoted by , is the shortest path connecting the two vertices. The Wiener index [2], denoted by , is a famous distance-based topological index and is defined as the sum of distances between all the pairs of vertices in :

As an analogue to the Wiener index , another novel distance function named resistance distance was firstly proposed by Klein and Randić [3]. The resistance distance between two arbitrary in an electrical networks, many properties over resistance distances have been actually proved [2, 4–9]. The resistance distance between any two vertices of is defined as the networks effective resistance between them if each edge of is replaced by a unit resistor. They also defined the Kirchhoff index of as the sum of resistance distances between all pairs of vertices in ; that is,

Klein and Randić [3] proved that and with equality if and only if is a tree; it is shown that the Kirchhoff index has very nice purely mathematical and physical interpretations.

The Kirchhoff index has wide applications in physical interpretations, electric circuit, graph theory, and chemistry [10–15]. For example, Zhu et al. [16] and Gutman and Mohar [17] proved that the Kirchhoff index of a graph or networks is the sum of reciprocal nonzero Laplacian eigenvalues of the graph or networks multiplied by the number of the vertices. The Kirchhoff index also is a structure descriptor like the Wiener index [9]. The Kirchhoff index has been computed for cycles [4], geodetic graphs [5], and some fullerenes including buckminsterfullerenes [6]. The Kirchhoff index of some product graphs, join graphs, and corona graphs was studied [8]. More results of the applications on Kirchhoff index were explored in [2, 7, 10, 14].

The hypercubes network obtained considerable attention due to its perfect properties, such as symmetry, regular structure, strong connectivity, and small diameter [18, 19]. For more results on the hypercubes network and its applications, see [18–25]. As the importance the hypercubes networks , many variants of it were presented, among which, for instance, are generalized hypercubes, folded hypercubes, the line graphs of hypercubes , the subdivision graphs of hypercubes , and the total graphs of hypercubes [19, 20].

The hypercubes networks may be constructed from the family of subsets of a set with a binary string of length , by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single binary string. The hypercubes networks admits several definitions of which two are stated as follows [26].

*Definition 1 (see [26]). *The hypercubes networks has vertices each labelled with a binary string of length . Two vertices and are adjacent if and only if there exists an , , such that , where denoted the complement of binary digit and , for all , and .

*Definition 2 (recursive construction [26]). *The hypercubes network is recursively constructed by taking two copies of , denoted by and , and adding edges as follows: let and . A vertex of is joined to a vertex of if and only if for every .

At the end of [10], the authors presented a problem to consider the Kirchhoff index derived from a single graph, such as the line graph, the subdivision graph and the total graph Gao et al. [27] obtained special formulae for the Kirchhoff index of , , and , where is a regular graph. Motivated by the previous results, we present the corresponding formulae for the Kirchhoff index of the hypercubes network and its three variant networks , , and in this paper.

The remainder of this paper is organized as follows. Section 2 gives some basic notations and some preliminaries in our discussion. The proofs of our main results are in Section 3. Finally, some conclusions are given in Section 4.

#### 2. Notations and Some Preliminaries

In this section, we recall some basic notations and results in graphs theory. The adjacency matrix of is an matrix with the -entry equal to 1 if vertices and are adjacent and 0 otherwise. Suppose that 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. For more details the readers may refer to [1].

Yin and Wang [28] proved the following Lemma.

Lemma 3 (see [28]). *For the hypercubes networks with ,
**
where the are the eigenvalues of the Laplacian matrix of hypercubes networks, and are the multiplicities of the eigenvalues .*

Gutman and Mohar [17] and Zhu et al. [16] presented the Kirchhoff index of a graph in terms of Laplacian eigenvalues as follows.

Lemma 4 (see [16, 17]). *Let be a connected graph with vertices; then
*

Let be the characteristic polynomial of the Laplacian matrix of a graph , the following results were shown in [27].

Lemma 5 (see [27]). *Let be an r-regular connected graph with vertices and edges; then
**
where , , and are the characteristic polynomials for the Laplacian matrix of graphs , , and , respectively.*

It is worthwhile to note that the conclusion of Lemma 5 is not completely correct, the authors [29] recently show the Laplacian characteristic polynomial of t(G), where G is a regular graph, which correct the Lemma 5 in Gao et al. [27] (2012) as follows.

Lemma 6 (see [29]). *Let be a r-regular connected graph with vertices and edges, then
**
where , are the characteristic polynomial for the Laplacian matrix of graphs and , respectively.*

The following Lemma give an expression on and of a regular graph .

Lemma 7 (see [29]). *Let be a r-regular connected graph with vertices and edges, and , then
*

For proving the formula of the Kirchhoff index on the subdivision graph of hypercubes, we prove the following Lemma by utilizing Vieta’s Theorem; in our proof, some techniques in [27] are referred to.

Lemma 8. *Let be the characteristic polynomial of the Laplacian matrix of the hypercubes networks with and
**
then
**
where are the coefficients of and in the characteristic polynomial, respectively.*

*Proof. *Let Spec. Then satisfy the following equation:
it is not difficult to check that are the roots of equation

Note that is connected graph and the multiplicity of 0 as an eigenvalue of is equal to the number of the connected components in . So, ; by Lemma 4 and Vieta’s Theorem
where are the coefficients of and in the characteristic polynomial of the Laplacian matrix of the hypercubes networks .

#### 3. Main Results

##### 3.1. The Kirchhoff Index in Hypercubes Networks

In this section, we firstly give formula for the Kirchhoff index in the hypercubes with .

Theorem 9. *For the hypercubes networks with ,
**
where the are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients are the multiplicities of the eigenvalues .*

*Proof. *Since the hypercubes networks have vertices and edges, then by Lemma 3,
then
The proof of Theorem 9 is completed.

*Remark 10. *Palacios and Renom studied the Kirchhoff index of the d-dimensional hypercube in [5] by using probabilistic tools, where they obtained a closed-form formula for the Kirchhoff index and found the asymptotic value . We present the formula for Kirchhoff index by directly calculating the eigenvalues of the Laplacian matrix of hypercubes networks, which is different from their technique and results.

##### 3.2. The Kirchhoff Index in the Line Graph of Hypercubes Networks

The line graph of a graph , denoted by , is the graph whose vertices correspond to the edges of , in which two vertices are adjacent if and only if the corresponding edges of share a common vertex. In the following theorem, we proposed a formula for the Kirchhoff index in the line graph of hypercubes networks , denoted by , where the eigenvalues of are .

Theorem 11. *Let be line graphs of hypercubes with ; then
**
where the are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients are the multiplicities of the eigenvalues .*

*Proof. *Now, for convenience, we denoted the numbers of vertices and edges in the hypercubes networks by and , respectively. Obviously, , so the line graphs of hypercubes have vertices.

By Lemma 5,

Comparing the spectrum of ,

We can easily obtain the spectrum of as follows:
where are the eigenvalues of and .

Notice that the line graphs of hypercubes have vertices; by Lemma 4,

Substituting the results of Theorem 9 (15) into (22), we can get the formula for the Kirchhoff index on the line graph of hypercubes as follows:
This completes the proof.

##### 3.3. The Kirchhoff Index in the Subdivision Graph of Hypercubes Networks

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). In an almost identical way as Theorem 11, we required the formula for the Kirchhoff index in the subdivision graph of hypercubes , denoted by .

Theorem 12. *Let be subdivision graphs of the hypercubes with ; then
**
where the are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients are the multiplicities of the eigenvalues .*

*Proof. *Now supposing that , is the characteristic polynomial of the Laplacian matrix of the hypercubes , , and
where are the Laplacian eigenvalues of Then by Lemma 8,
By Lemma 5, we have

Consequently, the coefficient of in is
and the coefficient of in is
Note that has vertices. By Lemma 6 and substituting the coefficients into (26),
By substituting and the results of Lemma 5 into (31), we have

Simplifying (33) and substituting the results of Theorem 9 (15), we can get the formula of the Kirchhoff index on the subdivision graph of the hypercubes Kf as follows:
Thus, the results of the Theorem 12 hold.

##### 3.4. The Kirchhoff Index in the Total Graph of Hypercubes Networks

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 . We now proved the formula of the Kirchhoff index in the total graph of the hypercubes networks , denoted by .

Theorem 13. *Let be the total graphs of the hypercubes networks with ; then
*

*Proof. *Let
Then by Lemma 8,
Applying Lemma 6, the Laplacian characteristic polynomial of is
Consider that has vertices and , by Lemma 7, we can get the following equality,
Consequently, the relationships between the hypercubes networks and its variant networks for Kirchhoff index is
Substitutes the result of Theorem 9 and simplifies about equation, we can get the formula for the Kirchhoff index on the total graph of hypercubes networks as follows.
This completes the proof of the Theorem.

#### 4. Conclusions

In this paper, we focused on the Kirchhoff index of the hypercubes networks and related networks, which are important networks topology indexes for parallel processing computer systems. We obtained some exact formulae for the Kirchhoff index of the hypercubes networks and related networks by utilizing spectral graph theory, such as , where the are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients are the multiplicities of the eigenvalues .

We also obtained the relationship for Kirchhoff index between hypercubes networks and its three variant networks , , and , respectively, by deducing the characteristic polynomial of the Laplacian matrix related networks.

Finally, the special formulae for the Kirchhoff indexes of , , and were proposed, respectively, by making use of spectral graph theory and Vieta’s Theorem.

#### Acknowledgments

The work of Jinde Cao and Ahmed Elaiw was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under Grant 3–130/1434/HiCi. The authors, therefore, acknowledge technical and financial support of KAU; the work of Jiabao Liu was supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B015; the work of Xiangfeng Pan was supported by the National Science Foundation of China under Grant nos. 10901001, 11171097, and 11371028.

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