The Kirchhoff Index of Some Combinatorial Networks

The Kirchhoff index Kf(G) is the sum of the effective resistance distances between all pairs of vertices in G. The hypercube Q n and the folded hypercube FQ n are well known networks due to their perfect properties. The graph G, constructed from G, is the line graph of the subdivision graph S(G). In this paper, explicit formulae expressing the Kirchhoff index of (Q n ) ∗ and (FQ n ) ∗ are found by deducing the characteristic polynomial of the Laplacian matrix of G∗ in terms of that of G.


Introduction
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][2][3][4][5].
Let  be a graph with vertices labelled 1, 2, . . ., .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 [8].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 [8].A famous distance-based topological index as the Kirchhoff index, Kf() = (1/2) ∑  =1 ∑  =1   (), is defined as the sum of resistance distances between all pairs of vertices in  [8].
The Kirchhoff index has been attracting extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [9][10][11][12][13][14][15][16][17][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.

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 2 (folded hypercube   [2]).The folded hypercube   can be constructed from   by adding an edge to every pair of vertices with complementary addresses.Two vertices  =  1  2 ⋅ ⋅ ⋅   and  =  1  2 ⋅ ⋅ ⋅   are adjacent in the folded hypercube   .
(ii) There is an edge joining a vertex of  ( 1 ) and a vertex of  ( 2 ) in  * if and only if there is an edge joining  1 and  2 in .
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 V of  and replacing it by a vertex  of degree 2 that is joined to  and V (see page 151 of [23]).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 [22].
It is amazing and interesting that  * , constructed from  as the graph operation above, is equivalent to the line graph of the subdivision graph () [22]; 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 [26] have proved the following Lemma.
Lemma 5 (see [26]).For   with any integer  ≥ 2, the spectrum of Laplacian matrix of   is where 2,  = 0, 1, . . ., , are the eigenvalues of the Laplacian matrix of   and    are the multiplicities of the eigenvalues 2.
M. Chen and B. X. Chen have studied the Laplacian spectra of   in [3].
Lemma 6 (see [3]).For   with any integer  ≥ 2, the spectra of Laplacian matrix of   are as follows: (1) If  ≡ 0 (mod2), then (2) where    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.Lemma 7 (see [11,27]).Let  be a connected graph, with  ≥ 2 vertices, and  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   = 0 are the Laplacian eigenvalues of ; then Let  () () be the characteristic polynomial of the Laplacian matrix of a graph ; the following results were shown in [28].
Lemma 8 (see [28]).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.
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 .

Main Results
Theorem 10.For (  ) * with any integer  ≥ 2, one has Proof.Notice that   is -regular graph with 2  vertices and 2 −1 edges.Suppose that (  ) has  vertices and  edges, for convenience, and denote the degree of vertices in   by .
The following theorem [24] provided the closed-form formula expressing the Kirchhoff index of   with any integer  ≥ 2.
Theorem 11 (see [24]).Let    be the binomial coefficients for   with any integer  ≥ 2. Then Theorem 12. Let    be the binomial coefficients for (  ) * with any integer  ≥ 2. Then Proof.From Theorems 10 and 11 one can immediately arrive at the explicit formula expressing the Kirchhoff index of (  ) * with any integer .
Remark 13.Theorem 11 gives the value of Kf(  ) in a nice closed-form formula.In [30] a similar, slightly more involved, closed-form formula was given, and, moreover, an asymptotic value of 2 2 / was given for Kf(  ).Comparing the asymptotic relative sizes of Kf(  ) and Kf((  ) * ) 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  ≡ 0 (mod2), then (2) If  ≡ 1 (mod2), then In an almost identical way as Theorem 10, we derive the following formula expressing the Kirchhoff index of (  ) * .The proof is omitted here for the completely similar deduction to Theorem 10.
In [25], the authors have proposed the following Kirchhoff index of   with any integer  ≥ 2.
Theorem 15 (see [25]).Let    denote the binomial coefficients for   with any integer  ≥ 2. Then Proof.From Theorems 14 and 15, it is not difficult to deduce the above formula expressing the Kirchhoff index of (  ) * with any integer .
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 Kf((  ) * ) and Kf((  ) * ).If we can compute the Kirchhoff index Kf() readily, then, by Laplacian spectrum of * , we can also obtain the Kirchhoff index Kf( * ) which is hard to calculate immediately.Furthermore, utilizing this approach one can also formulate the Kirchhoff index of other general graphs.

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  = 2 for simplicity.Obviously, |( 2 )| = 4, and the eigenvalues of the Laplacian matrix of  2 are  1 = 4,  2 =  3 = 2, and  4 = 0. Based on Lemma 7, it is easy to obtain that Kf ( 2 ) = 4 ⋅ According to the consequence of Theorem 10, one can readily derive that Kf (( 2 ) * ) = 8Kf ( 2 ) + 2 = 42.
On the other hand, we use another approach to calculate Kf(( 2 ) * ).For a circulant graph , the authors of [31] showed that The As the application of Theorem 14, we proceed to derive that Kf(( 2 ) * ).
Note that the eigenvalues of the Laplacian matrix of  2 are  1 =  2 =  3 = 4, and  4 = 0. Based on Lemma 7, Summing up the examples, the results above coincide the fact, which show our theorems are correct and effective.

Theorem 16 .
Let    denote the binomial coefficients for (  ) * with any integer  ≥ 2. Then (1) ((  ) 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 ( 2 ) * ≅  8 .Consequently, the same Kirchhooff index can be drawn as follows: we have Kf ( 2 ) = 4 ⋅