# The Kirchhoff Index of Some Combinatorial Networks

Authors: Jia-Bao Liu; Xiang-Feng Pan; Jinde Cao; Fu-Tao Hu
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

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 F
Q
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 (
F
Q
n
)
∗ are found by deducing the characteristic polynomial of the Laplacian matrix of G
∗ in terms of that of G.


---

## Body

## 1. 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 graphG
=
(
V
(
G
)
,
E
(
G
)
), where V
(
G
) denotes the set of processors and E
(
G
) denotes the set of communication links between processors in networks. The hypercube Q
n and the folded hypercube F
Q
n 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 matrixA
(
G
) of G is an n
×
n matrix with the (
i
,
j
)-entry equal to 1 if vertices i and j are adjacent and to 0 if otherwise. Let D
(
G
) be the degree diagonal matrix of G, and L
(
G
)
=
D
(
G
)
-
A
(
G
) is called the Laplacian matrix of G. Denote the Laplacian characteristic polynomial of G by σ
L
(
G
)
(
x
)
=
det
⁡
(
x
I
-
L
(
G
)
)
=
x
n
+
q
1
(
G
)
x
n
-
1
+
q
2
(
G
)
x
n
-
2
+
⋯
+
q
n
-
1
(
G
)
x
+
q
n
(
G
), where q
i
(
i
=
1,2
,
…
,
n
) are the coefficients of the Laplacian characteristic polynomial [6]. The eigenvalues of A
(
G
) and L
(
G
) are called eigenvalues and Laplacian eigenvalues of G, 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 [7].LetG be a graph with vertices labelled 1,2
,
…
,
n. It is well known that the standard distance between two vertices of G, denoted by d
i
j, 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 i and j, denoted by r
i
j, is defined to be the effective electrical resistance between them if each edge of G is replaced by a unit resistor [8]. A famous distance-based topological index as the Kirchhoff index, Kf
(
G
)
=
(
1
/
2
)
∑
i
=
1
n
‍
∑
j
=
1
n
‍
r
i
j
(
G
), is defined as the sum of resistance distances between all pairs of vertices in G [8].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 (
Q
n
)
∗ and (
F
Q
n
)
∗, where the graph G
∗, constructed from G, is the line graph of the subdivision graph S
(
G
).The main purpose of this paper is to investigate the Kirchhoff index of some combinatorial networks. The graphG
∗, constructed from G, is the line graph of the subdivision graph S
(
G
). We have established the relationships between Q
n, F
Q
n and their variant networks (
Q
n
)
∗, (
F
Q
n
)
∗, in terms of Kirchhoff index, respectively. Moreover, explicit formulae have been proposed for expressing the Kirchhoff index of (
Q
n
)
∗ and (
F
Q
n
)
∗ by making use of the characteristic polynomial of the Laplacian matrix in spectral graph theory.The remainder of the paper is organized as follows. Section2 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 (hypercubeQ
n [21]).
The hypercubeQ
n has 2
n vertices each labelled with a binary string of length n. Two vertices X
=
x
1
x
2
⋯
x
n and Y
=
y
1
y
2
⋯
y
n are adjacent if and only if there exists an i, 1
≤
i
≤
n, such that x
i
=
y
i
¯, where y
i
¯ denoted the complement of binary digit y
i and x
j
=
y
j, for all j
≠
i, and 1
≤
j
≤
n.Definition 2 (folded hypercubeF
Q
n [2]).
The folded hypercubeF
Q
n can be constructed from Q
n by adding an edge to every pair of vertices with complementary addresses. Two vertices X
=
x
1
x
2
⋯
x
n and X
¯
=
x
1
¯
x
2
¯
⋯
x
n
¯ are adjacent in the folded hypercube F
Q
n.Definition 3 (construction ofG
∗ [22]).
Define the following operation ofG, constructing G
∗ from G, as follows [22]:(i)
Replace each vertexu
∈
V
(
G
) by K
(
u
), the complete graph on de
g
G
(
u
) vertices.
(ii)
There is an edge joining a vertex ofK
(
u
1
) and a vertex of K
(
u
2
) in G
∗ if and only if there is an edge joining u
1 and u
2 in G.
(iii)
For each vertexv of K
(
u
), de
g
G
∗
(
v
)
=
de
g
G
(
u
).
Recall the following two underlying conceptions that related to the above construction ofG
∗. The subdivision graph S
(
G
) of a graph G is obtained from G by deleting every edge u
v of G and replacing it by a vertex w of degree 2 that is joined to u and v (see page 151 of [23]). The line graph of a graph G, denoted by L
(
G
), is the graph whose vertices correspond to the edges of G with two vertices of L
(
G
) being adjacent if and only if the corresponding edges in G share a common vertex [22].
It is amazing and interesting thatG
∗, constructed from G as the graph operation above, is equivalent to the line graph of the subdivision graph S
(
G
) [22]; that is, G
∗
≅
L
(
S
(
G
)
).Remark 4.
Note that there is an elementary and important property: ifG is an r-regular graph (combinatorial network), then G
∗ is also an r-regular graph (combinatorial network); however, the topological structure of G
∗ is quite more complicated than G; consequently, dealing with the problems of calculating Kirchhoff index of (
Q
n
)
∗ and (
F
Q
n
)
∗ is not easy, even though we have handled the formulas for calculating the Kirchhoff index of (
Q
n
) and (
F
Q
n
) in [24, 25].Yin and Wang [26] have proved the following Lemma.Lemma 5 (see [26]).
ForQ
n with any integer n
≥
2, the spectrum of Laplacian matrix of Q
n is (1)
S
p
e
c
L
Q
n
=
0
2
⋯
2
i
⋯
2
n
C
n
0
C
n
1
⋯
C
n
i
⋯
C
n
n
,where 2
i, i
=
0,1
,
…
,
n, are the eigenvalues of the Laplacian matrix of Q
n and C
n
i are the multiplicities of the eigenvalues 2
i.M. Chen and B. X. Chen have studied the Laplacian spectra ofF
Q
n in [3].Lemma 6 (see [3]).
ForF
Q
n with any integer n
≥
2, the spectra of Laplacian matrix of F
Q
n are as follows:(1)
Ifn
≡
0
(
mod
⁡
⁡
2
), then (2)
S
p
e
c
L
F
Q
n
=
0
4
8
⋯
2
n
-
8
2
n
-
4
2
n
C
n
n
C
n
n
-
1
+
C
n
n
-
2
C
n
n
-
3
+
C
n
n
-
4
⋯
C
n
5
+
C
n
4
C
n
3
+
C
n
2
C
n
1
+
C
n
0
.
(2)
Ifn
≡
1
(
mod
⁡
⁡
2
), then(3)
S
p
e
c
L
F
Q
n
=
0
4
8
⋯
2
n
-
6
2
n
-
2
2
n
+
2
C
n
0
C
n
1
+
C
n
2
C
n
3
+
C
n
4
⋯
C
n
n
-
4
+
C
n
n
-
3
C
n
n
-
2
+
C
n
n
-
1
C
n
n
,where C
n
i are the binomial coefficients and the elements in the first and second rows are the Laplacian eigenvalues of F
Q
n and the multiplicities of the corresponding eigenvalues, respectively.Lemma 7 (see [11, 27]).
LetG be a connected graph, with n
≥
2 vertices, and μ
1
≥
μ
2
≥
⋯
≥
μ
n
=
0 are the Laplacian eigenvalues of G; then(4)
K
f
G
=
n
∑
i
=
1
n
-
1
1
λ
i
.Letσ
(
G
)
(
x
) be the characteristic polynomial of the Laplacian matrix of a graph G; the following results were shown in [28].Lemma 8 (see [28]).
LetG be an r-regular connected graph with n vertices and m edges; then(5)
σ
L
G
x
=
x
-
2
r
m
-
n
σ
G
x
,
σ
S
G
x
=
-
1
m
2
-
x
m
-
n
σ
G
x
r
+
2
-
x
,where σ
L
(
G
)
(
x
) and σ
S
(
G
)
(
x
) are the characteristic polynomials for the Laplacian matrix of graphs L
(
G
) and S
(
G
), respectively.LetG be a bipartite graph with a bipartition; V
(
G
)
=
(
U
,
V
) is called an (
r
,
s
)-semiregular graph if all vertices in U have degree r and all vertices in V have degree s.Lemma 9 (see [29]).
LetG be an (
r
,
s
)-semiregular connected graph with n vertices and m edges, and σ
L
(
G
)
(
x
) is the Laplacian characteristic polynomial of the line graph L
(
G
). Then(6)
σ
L
G
x
=
-
1
n
x
-
r
+
s
m
-
n
σ
G
r
+
s
-
x
.


## 3. Main Results

Theorem 10.
For(
Q
n
)
∗ with any integer n
≥
2, one has(7)
K
f
Q
n
∗
=
n
2
+
2
n
K
f
Q
n
+
n
2
-
n
-
2
4
n
+
2
n
n
n
+
2
.Proof.
Notice thatQ
n is n-regular graph with 2
n vertices and n
2
n
-
1 edges. Suppose that S
(
Q
n
) has u vertices and w edges, for convenience, and denote the degree of vertices in Q
n by d. Obviously, u
=
2
n
+
2
n
-
1
d and w
=
2
n
d, respectively.
From Lemma9, we can get(8)
σ
L
S
Q
n
x
=
-
1
u
x
-
d
+
2
w
-
u
σ
S
Q
n
d
+
2
-
x
.
By virtue of Lemma8, it follows that(9)
σ
S
Q
n
x
=
-
1
2
n
-
1
d
2
-
x
2
n
-
1
d
-
2
n
σ
Q
n
x
d
+
2
-
x
.
Replacingx with d
+
2
-
x in (9), we have(10)
σ
S
Q
n
d
+
2
-
x
=
-
1
2
n
-
1
d
x
-
d
2
n
-
1
d
-
2
n
σ
Q
n
x
d
+
2
-
x
.
Substituting (8) with (10), the Laplacian characteristic polynomial of L
(
S
(
Q
n
)
) is(11)
σ
L
S
Q
n
x
=
-
1
u
+
2
n
-
1
d
x
-
d
2
n
-
1
d
-
2
n
·
x
-
d
+
2
w
-
u
σ
Q
n
x
d
+
2
-
x
.
From the definition graph(
Q
n
)
∗ and (11), one can immediately obtain(12)
σ
Q
n
∗
x
=
-
1
u
+
2
n
-
1
d
x
-
d
2
n
-
1
d
-
2
n
x
-
d
+
2
w
-
u
·
σ
Q
n
x
d
+
2
-
x
.
Combining (12), u
=
2
n
+
2
n
-
1
d, w
=
2
n
d, and d
=
n, it holds that(13)
σ
Q
n
∗
x
=
-
1
2
n
d
+
1
x
-
d
2
n
-
1
n
-
2
·
x
-
d
+
2
2
n
-
1
n
-
2
σ
Q
n
x
d
+
2
-
x
=
-
1
2
n
n
+
1
x
-
n
2
n
-
1
n
-
2
x
-
n
+
2
2
n
-
1
n
-
2
·
σ
Q
n
x
n
+
2
-
x
.Since the roots of x
(
n
+
2
-
x
)
=
μ
i are(14)
x
1
,
i
=
n
+
2
-
n
+
2
2
-
4
μ
i
2
,
x
2
,
i
=
n
+
2
+
n
+
2
2
-
4
μ
i
2
,where μ
1
≥
μ
2
⋯
≥
μ
2
n
=
0 are the Laplacian eigenvalues of Q
n.
It follows from (12) that the Laplacian spectrum of (
Q
n
)
∗ is(15)
Spe
c
L
Q
n
∗
=
n
,
n
,
…
,
n
︸
2
n
-
1
n
-
2
,
n
+
2
,
n
+
2
,
…
,
n
+
2
︸
2
n
-
1
n
-
2
·
⋃
i
=
1
2
n
n
+
2
-
n
+
2
2
-
4
μ
i
2
·
⋃
i
=
1
2
n
n
+
2
+
n
+
2
2
-
4
μ
i
2
.Noticing that (
Q
n
)
∗ has 2
n
n vertices, we get the following result from Lemmas 5 and 7 and (15). Therefore,(16)
Kf
Q
n
∗
=
2
n
n
2
n
-
1
n
-
2
n
+
2
n
-
1
n
-
2
n
+
2
+
∑
i
=
1
2
n
-
1
1
x
1
,
i
+
∑
i
=
1
2
n
1
x
2
,
i
=
4
n
n
2
-
n
-
2
n
+
2
+
2
n
n
∑
i
=
1
2
n
-
1
2
n
+
2
-
n
+
2
2
-
4
μ
i
+
∑
i
=
1
2
n
2
n
+
2
+
n
+
2
2
-
4
μ
i
=
4
n
n
2
-
n
-
2
n
+
2
+
2
n
n
∑
i
=
1
2
n
-
1
2
n
+
2
-
n
+
2
2
-
4
μ
i
+
∑
i
=
1
2
n
-
1
2
n
+
2
+
n
+
2
2
-
4
μ
i
+
1
n
+
2
=
4
n
n
2
-
n
-
2
n
+
2
+
2
n
n
n
+
2
+
2
n
n
∑
i
=
1
2
n
-
1
n
+
2
μ
i
=
n
2
-
n
-
2
4
n
+
2
n
n
n
+
2
+
n
n
+
2
2
n
∑
i
=
1
2
n
-
1
1
μ
i
=
n
2
+
2
n
Kf
Q
n
+
n
2
-
n
-
2
4
n
+
2
n
n
n
+
2
.
This completes the proof.The following theorem [24] provided the closed-form formula expressing the Kirchhoff index of Q
n with any integer n
≥
2.Theorem 11 (see [24]).
LetC
n
i be the binomial coefficients for Q
n with any integer n
≥
2. Then(17)
K
f
Q
n
=
2
n
∑
i
=
1
n
C
n
i
2
i
.Theorem 12.
LetC
n
i be the binomial coefficients for (
Q
n
)
∗ with any integer n
≥
2. Then(18)
K
f
Q
n
∗
=
n
2
+
2
n
2
n
∑
i
=
1
n
C
n
i
2
i
+
n
2
-
n
-
2
4
n
+
n
2
n
n
+
2
.Proof.
From Theorems10 and 11 one can immediately arrive at the explicit formula expressing the Kirchhoff index of (
Q
n
)
∗ with any integer n.Remark 13.
Theorem11 gives the value of Kf
(
Q
n
) 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
d
/
d was given for Kf
(
Q
n
). Comparing the asymptotic relative sizes of Kf
(
Q
n
) and Kf
(
(
Q
n
)
∗
) in the present article, the latter is much larger than the former.In the following, we will further address the Kirchhoff index of(
F
Q
n
)
∗. Primarily, notice that F
Q
n is a regular graph with degree for any vertex and the Laplacian spectrum of F
Q
n is as follows:(1)
Ifn
≡
0
(
mod
⁡
⁡
2
), then (19)
Spe
c
L
F
Q
n
=
0
4
8
⋯
2
n
-
8
2
n
-
4
2
n
C
n
n
C
n
n
-
1
+
C
n
n
-
2
C
n
n
-
3
+
C
n
n
-
4
⋯
C
n
5
+
C
n
4
C
n
3
+
C
n
2
C
n
1
+
C
n
0
.
(2)
Ifn
≡
1
(
mod
⁡
⁡
2
), then (20)
Spe
c
L
F
Q
n
=
0
4
8
⋯
2
n
-
6
2
n
-
2
2
n
+
2
C
n
0
C
n
1
+
C
n
2
C
n
3
+
C
n
4
⋯
C
n
n
-
4
+
C
n
n
-
3
C
n
n
-
2
+
C
n
n
-
1
C
n
n
.In an almost identical way as Theorem10, we derive the following formula expressing the Kirchhoff index of (
F
Q
n
)
∗. The proof is omitted here for the completely similar deduction to Theorem 10.Theorem 14.
For(
F
Q
n
)
∗ with any integer n
≥
2, one has(21)
K
f
F
Q
n
∗
=
n
2
+
4
n
+
3
K
f
F
Q
n
+
n
2
+
n
-
2
4
n
+
n
+
1
2
n
n
+
3
.In [25], the authors have proposed the following Kirchhoff index of F
Q
n with any integer n
≥
2.Theorem 15 (see [25]).
LetC
n
i denote the binomial coefficients for F
Q
n with any integer n
≥
2. Then(1)
K
f
(
F
Q
n
)
=
2
n
∑
i
=
1
n
/
2
‍
(
(
C
n
n
-
i
+
C
n
n
-
i
-
1
)
/
4
i
), i
=
1,2
,
…
,
n
/
2, if n
≡
0
mod
⁡
⁡
2,
(2)
K
f
(
F
Q
n
)
=
2
n
∑
i
=
1
n
-
1
/
2
‍
(
(
C
n
2
i
-
1
+
C
n
2
i
)
/
4
i
)
+
2
n
-
1
/
(
n
+
1
), i
=
1,2
,
…
,
(
n
-
1
)
/
2, if n
≡
1
(
mod
⁡
⁡
2
).Theorem 16.
LetC
n
i denote the binomial coefficients for (
F
Q
n
)
∗ with any integer n
≥
2. Then(1)
K
f
(
(
F
Q
n
)
∗
)
=
(
n
2
+
4
n
+
3
)
2
n
∑
i
=
1
n
/
2
‍
(
(
C
n
n
-
i
+
C
n
n
-
i
-
1
)
/
4
i
)
+
(
n
2
+
n
-
2
4
n
+
n
+
1
2
n
)
/
(
n
+
3
), i
=
1,2
,
…
,
n
/
2, if n
≡
0
mod
⁡
⁡
2,
(2)
K
f
(
(
F
Q
n
)
∗
)
=
(
n
2
+
4
n
+
3
)
2
n
∑
i
=
1
n
-
1
/
2
‍
(
(
C
n
2
i
-
1
+
C
n
2
i
)
/
4
i
)
+
(
n
2
+
n
-
2
4
n
+
n
+
1
2
n
)
/
(
n
+
3
) + (
n
+
3
)
2
n
-
1, i
=
1,2
,
…
,
(
n
-
1
)
/
2, if n
≡
1
(
mod
⁡
⁡
2
).Proof.
From Theorems14 and 15, it is not difficult to deduce the above formula expressing the Kirchhoff index of (
Q
n
)
∗ with any integer n.Remark 17.
Theorems12 and 16 have presented a method to calculate the Kirchhoff index of (
Q
n
)
∗ and (
F
Q
n
)
∗, which is difficult to calculate directly. We found a relationship between the graph G and G
∗ by deducing the characteristic polynomial of the Laplacian matrix and obtained the Laplacian spectrum of Kf
(
(
Q
n
)
∗
) and Kf
(
(
F
Q
n
)
∗
). If we can compute the Kirchhoff index Kf
(
G
) readily, then, by Laplacian spectrum of G
∗, we can also obtain the Kirchhoff index Kf
(
G
∗
) 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 isn
=
2 for simplicity. Obviously, |
V
(
Q
2
)
|
=
4, and the eigenvalues of the Laplacian matrix of Q
2 are λ
1
=
4, λ
2
=
λ
3
=
2, and λ
4
=
0. Based on Lemma 7, it is easy to obtain that (22)
Kf
Q
2
=
4
·
∑
i
=
1
3
1
λ
i
=
4
·
1
λ
1
+
1
λ
2
+
1
λ
3
=
5
.According to the consequence of Theorem 10, one can readily derive that(23)
Kf
Q
2
∗
=
8
Kf
Q
2
+
2
=
42
.On the other hand, we use another approach to calculate Kf
(
(
Q
2
)
∗
). For a circulant graph G, the authors of [31] showed that(24)
n
-
1
≤
Kf
G
≤
n
3
-
n
12
.The first equality holds if and only if G is K
n and the second does if and only if G is C
n.By virtue of the definition ofG
∗, it is not difficult to get that (
Q
2
)
∗
≅
C
8.Consequently, the same Kirchhooff index can be drawn as follows:(25)
Kf
Q
2
∗
=
n
3
-
n
12
=
8
3
-
8
12
=
42
.As the application of Theorem14, we proceed to derive that Kf
(
(
F
Q
2
)
∗
).Note that the eigenvalues of the Laplacian matrix ofF
Q
2 are λ
1
=
λ
2
=
λ
3
=
4, and λ
4
=
0. Based on Lemma 7, we have(26)
Kf
F
Q
2
=
4
·
∑
i
=
1
3
1
λ
i
=
4
·
1
λ
1
+
1
λ
2
+
1
λ
3
=
3
.Similarly, according to the consequence of Theorem 14, it holds that(27)
Kf
F
Q
2
∗
=
15
Kf
F
Q
n
+
76
5
=
60.2
.Summing up the examples, the results above coincide the fact, which show our theorems are correct and effective.



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*Source: 340793-2015-10-01.xml*