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The Kirchhoff Index of Some Combinatorial Networks
The Kirchhoff index Kf() is the sum of the effective resistance distances between all pairs of vertices in . The hypercube and the folded hypercube are well known networks due to their perfect properties. The graph , constructed from , is the line graph of the subdivision graph . In this paper, explicit formulae expressing the Kirchhoff index of and are found by deducing the characteristic polynomial of the Laplacian matrix of in terms of that of .
It is well known that interconnection networks play an important role in parallel communication systems. An interconnection network is usually modelled by a connected graph , where denotes the set of processors and denotes the set of communication links between processors in networks. The hypercube and the folded hypercube are two very popular and efficient interconnection networks due to their excellent performance in some practical applications. The symmetry, regular structure, strong connectivity, small diameter, and many of their properties have been explored [1–5].
The adjacency matrix of is an matrix with the -entry equal to 1 if vertices and are adjacent and to 0 if otherwise. Let be the degree diagonal matrix of , and is called the Laplacian matrix of . Denote the Laplacian characteristic polynomial of by , where are the coefficients of the Laplacian characteristic polynomial . The eigenvalues of and are called eigenvalues and Laplacian eigenvalues of , respectively. In this paper we are concerned with some finite undirected connected simple graphs (networks). For the underlying graph, theoretical definitions, and notations, we follow .
Let be a graph with vertices labelled . It is well known that the standard distance between two vertices of , denoted by , is the shortest path connecting the two vertices. A novel distance function named resistance distance was firstly proposed by Klein and Randić . The resistance distance between vertices and , denoted by , is defined to be the effective electrical resistance between them if each edge of is replaced by a unit resistor . A famous distance-based topological index as the Kirchhoff index, , is defined as the sum of resistance distances between all pairs of vertices in .
The Kirchhoff index has been attracting extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [9–18]. Details on its theory can be found in recent papers [19, 20] and the references cited therein. But there are only few works appearing on the Kirchhoff index in combinatorial networks. In the present paper, we establish the closed-form formulae expressing the Kirchhoff index of and , where the graph , constructed from , is the line graph of the subdivision graph .
The main purpose of this paper is to investigate the Kirchhoff index of some combinatorial networks. The graph , constructed from , is the line graph of the subdivision graph . We have established the relationships between , and their variant networks , , in terms of Kirchhoff index, respectively. Moreover, explicit formulae have been proposed for expressing the Kirchhoff index of and by making use of the characteristic polynomial of the Laplacian matrix in spectral graph theory.
The remainder of the paper is organized as follows. Section 2 provides some underlying definitions and preliminaries in our discussion. The proofs of main results and some examples are given in Sections 3 and 4, respectively.
2. Definitions and Preliminaries
In this section, we recall some underlying definitions and properties which we need to use in the proofs of our main results as follows.
Definition 1 (hypercube ). The hypercube 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 (folded hypercube ). The folded hypercube can be constructed from by adding an edge to every pair of vertices with complementary addresses. Two vertices and are adjacent in the folded hypercube .
Definition 3 (construction of ). Define the following operation of , constructing from , as follows :(i)Replace each vertex by , the complete graph on vertices.(ii)There is an edge joining a vertex of and a vertex of in if and only if there is an edge joining and in .(iii)For each vertex of , .
Recall the following two underlying conceptions that related to the above construction of . The subdivision graph of a graph is obtained from by deleting every edge of and replacing it by a vertex of degree 2 that is joined to and (see page 151 of ). 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 .
It is amazing and interesting that , constructed from as the graph operation above, is equivalent to the line graph of the subdivision graph ; that is, .
Remark 4. Note that there is an elementary and important property: if is an -regular graph (combinatorial network), then is also an -regular graph (combinatorial network); however, the topological structure of is quite more complicated than ; consequently, dealing with the problems of calculating Kirchhoff index of and is not easy, even though we have handled the formulas for calculating the Kirchhoff index of and in [24, 25].
Yin and Wang  have proved the following Lemma.
Lemma 5 (see ). For with any integer , the spectrum of Laplacian matrix of is where , , are the eigenvalues of the Laplacian matrix of and are the multiplicities of the eigenvalues .
M. Chen and B. X. Chen have studied the Laplacian spectra of in .
Lemma 6 (see ). For with any integer , the spectra of Laplacian matrix of are as follows:(1)If , then (2)If , thenwhere are the binomial coefficients and the elements in the first and second rows are the Laplacian eigenvalues of and the multiplicities of the corresponding eigenvalues, respectively.
Let be the characteristic polynomial of the Laplacian matrix of a graph ; the following results were shown in .
Lemma 8 (see ). Let be an r-regular connected graph with vertices and edges; thenwhere and are the characteristic polynomials for the Laplacian matrix of graphs and , respectively.
Let be a bipartite graph with a bipartition; is called an -semiregular graph if all vertices in have degree and all vertices in have degree .
Lemma 9 (see ). Let be an -semiregular connected graph with vertices and edges, and is the Laplacian characteristic polynomial of the line graph . Then
3. Main Results
Theorem 10. For with any integer , one has
Proof. Notice that is -regular graph with vertices and edges. Suppose that has vertices and edges, for convenience, and denote the degree of vertices in by . Obviously, and , respectively.
From Lemma 9, we can getBy virtue of Lemma 8, it follows thatReplacing with in (9), we haveSubstituting (8) with (10), the Laplacian characteristic polynomial of isFrom the definition graph and (11), one can immediately obtainCombining (12), , , and , it holds thatSince the roots of arewhere are the Laplacian eigenvalues of .
It follows from (12) that the Laplacian spectrum of isNoticing that has vertices, we get the following result from Lemmas 5 and 7 and (15). Therefore,This completes the proof.
The following theorem  provided the closed-form formula expressing the Kirchhoff index of with any integer .
Theorem 11 (see ). Let be the binomial coefficients for with any integer . Then
Theorem 12. Let be the binomial coefficients for with any integer . Then
Remark 13. Theorem 11 gives the value of in a nice closed-form formula. In  a similar, slightly more involved, closed-form formula was given, and, moreover, an asymptotic value of was given for . Comparing the asymptotic relative sizes of and in the present article, the latter is much larger than the former.
In the following, we will further address the Kirchhoff index of . Primarily, notice that is a regular graph with degree for any vertex and the Laplacian spectrum of is as follows:(1)If , then (2)If , then
Theorem 14. For with any integer , one has
In , the authors have proposed the following Kirchhoff index of with any integer .
Theorem 15 (see ). Let denote the binomial coefficients for with any integer . Then(1), , if ,(2), , if .
Theorem 16. Let denote the binomial coefficients for with any integer . Then(1), , if ,(2) + , , if .
Remark 17. Theorems 12 and 16 have presented a method to calculate the Kirchhoff index of and , which is difficult to calculate directly. We found a relationship between the graph and by deducing the characteristic polynomial of the Laplacian matrix and obtained the Laplacian spectrum of and . If we can compute the Kirchhoff index readily, then, by Laplacian spectrum of , we can also obtain the Kirchhoff index which is hard to calculate immediately. Furthermore, utilizing this approach one can also formulate the Kirchhoff index of other general graphs.
4. Some Examples
To demonstrate the theoretical analysis, we provide some examples in this subsection, which are an application of our results. Without loss of generality, we suppose that the case is for simplicity. Obviously, , and the eigenvalues of the Laplacian matrix of are , , and . Based on Lemma 7, it is easy to obtain that According to the consequence of Theorem 10, one can readily derive thatOn the other hand, we use another approach to calculate . For a circulant graph , the authors of  showed thatThe first equality holds if and only if is and the second does if and only if is .
By virtue of the definition of , it is not difficult to get that .
Consequently, the same Kirchhooff index can be drawn as follows:
As the application of Theorem 14, we proceed to derive that .
Summing up the examples, the results above coincide the fact, which show our theorems are correct and effective.
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 was supported by Anhui Provincial Natural Science Foundation under Grant no. KJ2013B105 and the National Science Foundation of China under Grant nos. 11471016 and 11401004. The work of F. T. Hu was supported by Anhui Provincial Natural Science Foundation (1408085QA03) and the National Science Foundation of China under Grant no. 11401004. The authors would like to express their sincere gratitude to the anonymous referees for their valuable suggestions, which led to a significant improvement of the original paper.
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