The Kirchhoff Index of Folded Hypercubes and Some Variant Networks

Abstract

The -dimensional folded hypercube is an important and attractive variant of the -dimensional hypercube , which is obtained from by adding an edge between any pair of vertices complementary edges. is superior to in many measurements, such as the diameter of which is , about a half of the diameter in terms of . The Kirchhoff index is the sum of resistance distances between all pairs of vertices in . In this paper, we established the relationships between the folded hypercubes networks and its three variant networks , , and on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of , , , and were proposed, respectively.

1. Introduction

It is well known that interconnection networks play an important role in parallel communication systems. An interconnection network is usually modelled by connected graphs , where denotes the set of processors and denotes the set of communication links between processors in networks. In this work, we are concerned with finite undirected connected simple graphs (networks). For the graph theoretical definitions and notations, we follow [1].

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, we refer to [1].

Let be a graph with vertices labelled . The resistance distances between vertices and , denoted by , are defined to be the effective electrical resistance between them if each edge of is replaced by a unit resistor [2]. A famous distance-based topological index as the Kirchhoff index Kf(), is defined as the sum of resistance distances between all pairs of vertices in . Define known as the Kirchhoff index of [2].

The Kirchhoff index attracted extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [3–6]. For example, Zhu et al. [7] and Gutman and Mohar [8] proved that relations between Kirchhoff index of a graph and Laplacian eigenvalues of the graph. The Kirchhoff index also is a structure descriptor [9].

However, it is difficult to design some algorithms [7, 10, 11] to calculate resistance distances and the Kirchhoff indexes of graphs. Hence, it makes sense to find explicit closed form for some special classes of graphs, for instance, the Kirchhoff index for cycles and complete graphs [12] which has been computed, geodetic graphs [13], some composite graphs [14], and composite networks [15]. Besides, many efforts were also made to obtain the Kirchhoff index bounds for some graphs [11, 16] and characterize extremal graphs as well, such as bicyclic graphs and cacti graphs [6, 17]. Details on its theory can be found in recent papers [11, 16, 17] and the references cited therein.

The hypercube is one of the most popular and efficient interconnection networks due to its excellent performance for some practical applications. There is a large amount of literature on the properties of hypercubes networks [18–20].

As an important variant of , the folded hypercube networks , proposed by El-Amawy and Latifi [18], are the graph obtained from by adding an edge between any pair of vertices complementary addresses. The folded hypercube , in which diameter of is , about half the diameter of , has the same number of vertices as a hypercube and edges more than hypercube; at the same time, the folded hypercubes preserve the symmetric properties of the hypercubes. The folded hypercubes obtained considerable attention due to its perfect properties, such as symmetry, regular structure, strong connectivity, and small diameter, and many of its properties have been explored [21–26].

But few works appear on the Kirchhoff index for the combinatorial networks, such as hypercubes and folded hypercubes , except that Palacios and Renom [11] studied the bounds of the Kirchhoff index of hypercubes by using probabilistic tools in 2010. In present paper, we established the relationships between the folded hypercubes networks and three variant networks , , and on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory.

Recall the definitions of n-dimensional folded hypercubes networks as follows [18].

Definition 1 (see [18]). The folded hypercubes can be constructed from by adding an edge to every pair of vertices with complementary addresses. Two vertices and are adjacent in the folded hypercubes .

From above definition of , it is easy to get that the folded hypercubes have vertices and edges, respectively.

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 .

Gao et al. [27] obtained special formulae for the Kirchhoff index of , , and , where is a regular graph. Motivated by above results, we present the corresponding calculated formulae for the Kirchhoff index of the hypercubes networks and its three-variant networks , , and in this paper.

The remainder of present 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 and some conclusions are given in Section 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.

M. Chen and B. X. Chen have studied the Laplacian spectra of folded hypercubes networks in 2011 [21].

Lemma 2 (see [21]). For the folded hypercubes networks with , the spectrum of Laplacian matrix in terms of hypercubes networks is as follows.
If ,
If , where are the binomial coefficients; the elements in the first and second rows are the eigenvalues of the Laplacian matrix of folded hypercubes networks and the multiplicities of the corresponding eigenvalues.

Lemma 3 (see [7, 8]). 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 4 (see [27]). Let be a r-regular connected graph with vertices and edges; then, where , , and are the characteristic polynomial for the Laplacian matrix of graphs , , and , respectively.

It is worthwhile to note that the conclusion of Lemma 4 is not completely correct; the authors [28] recently show the Laplacian characteristic polynomial of , where is a regular graph, which corrects Lemma 3 in Gao et al. [27] as follows.

Lemma 5 (see [28]). 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 gives an expression on and of a regular graph .

Lemma 6 (see [28]). Let be a r-regular connected graph with vertices and edges and ; then,

For proving the formula for the Kirchhoff index on the subdivision graph of hypercubes , we prove the following lemma utilizing Vieta's theorem; in our proof, some techniques in [27] are referred.

Lemma 7. Let be the characteristic polynomial of the Laplacian matrix of the folded hypercubes with and then, where are the coefficient of and in the characteristic polynomial, respectively.

Proof. Let . 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 3 and Vieta's theorem, where are the coefficient of and in the characteristic polynomial of the Laplacian matrix of the hypercubes .

3. Main Results

3.1. The Kirchhoff Index in Folded Hypercubes Networks

In this section, we firstly give formula for the Kirchhoff index of the folded hypercubes with any positive integer .

Theorem 8. For the folded hypercubes networks with any positive integer , where the , or are the eigenvalues of the Laplacian matrix of hypercubes networks and are the binomial coefficients,

Proof. The hypercubes networks have vertices and edges.
By Lemma 3, if , then
By Lemma 3, if , then The proof of Theorem 8 is completed.

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

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

Theorem 9. Let be line graphs of hypercubes with any positive integer ; then where the , or are the eigenvalues of the Laplacian matrix of folded hypercubes networks and are the binomial coefficients.

Proof. Now for convenience, we denote the numbers of vertices and edges in the folded hypercubes networks by and , respectively.
By Lemma 4, Notice that folded hypercubes networks are regular graphs with the degree of any vertex being ; compare the spectrum of as follows.
If , If ,
We can easily obtain the spectrum of as follows.
If , If , where are the eigenvalues of and .
By Lemma 4, Therefore, we clearly obtained Substituting the results of Theorem 8 into (25), we can get the formula for the Kirchhoff index on the line graph of folded hypercubes : which completes the proof.

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

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

Theorem 10. Let be subdivision graphs of the folded hypercubes with any positive integer ; then,

Proof. Now suppose that is the characteristic polynomial of the Laplacian matrix of the ; let and where , are the Laplacian eigenvalues of .
Then, by Lemma 7, By Lemma 5,
Consequently, the coefficient of in is and the coefficient of in is Since has vertices. For convenience, denote , , and , respectively. By Lemma 7 and substituting the coefficients into (12), we get Combining the results of Lemma 7 and (34), Hence,
Simplifying (36) by substituting , and , we can get
Note that the results of Theorem 8 and (37), we can get the formulae for the Kirchhoff index of the subdivision graph of the folded hypercubes : which completes the proof.

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

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

Theorem 11. Let be the total graphs of the folded hypercubes networks with any positive integer ; then, where the or , are the eigenvalues of the Laplacian matrix of folded hypercubes networks and are the binomial coefficients.

Proof. Let and where , are the Laplacian eigenvalues of .
Applying Lemma 5, the Laplacian characteristic polynomial of is
Notice that has vertices, that is, vertices. By Lemma 6, we can get the following: Consider that has vertices and , we have
From (43), the relationships between the folded hypercubes networks and its variant networks for Kirchhoff index are as follows:
Substituting results of Theorem 8 into (44) and simplifying (44), we can get the formula for the Kirchhoff index on the total graph of folded hypercubes ; we consider two cases in terms of if and , where the , or , are the eigenvalues of the Laplacian matrix of folded hypercubes networks and are the binomial coefficients.
This completes the proof of Theorem 11.

Remark 12. The authors [27] gave the formulae for the Kirchhoff index of , , and , where is a regular graph. In this paper, we proposed the special formulae for calculating the Kirchhoff indexes of the folded hypercubes networks and its three variant networks , , and , respectively, which enhanced related results on the Kirchhoff index.

4. Conclusions

The main purpose of this paper is to investigate the Kirchhoff index of the folded hypercubes networks and related networks, which are important networks topology indexes for parallel processing computer systems. We established the relationships between the folded hypercubes networks and its three variant networks , , and for their Kirchhoff index, respectively, by deducing the characteristic polynomial of the Laplacian matrix related networks.

Moreover, the explicit formulae were proposed for calculating the Kirchhoff indexes of , , , and , respectively, by making use of the characteristic polynomial of the Laplacian matrix in spectral graph theory and Vieta's theorem.

Conflict of Interests

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

Acknowledgments

The work of Jinde Cao was funded by the National Natural Science Foundation of China under Grant 61272530, the Natural Science Foundation of Jiangsu Province of China under Grant BK2012741, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20110092110017. The work of Jiabao Liu was supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B105. The work of Xiangfeng Pan was supported by the National Science Foundation of China under Grant nos. 11171097 and 11371028.