The Kirchhoff Index of Hypercubes and Related Complex Networks

Abstract

The resistance distance between any two vertices of 𝐺 is defined as the network effective resistance between them if each edge of 𝐺 is replaced by a unit resistor. The Kirchhoff index Kf(𝐺) is the sum of resistance distances between all the pairs of vertices in 𝐺. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networks 𝑄_𝑛 by utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networks 𝑄_𝑛 and its three variant networks 𝑙(𝑄_𝑛), 𝑠(𝑄_𝑛), 𝑡(𝑄_𝑛) by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes of 𝑙(𝑄_𝑛), 𝑠(𝑄_𝑛), and 𝑡(𝑄_𝑛) 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 𝐺=(𝑉,𝐸), where 𝑉 denotes the set of processors and 𝐸 denotes the set of communication links between processors. It is well known that the standard distance between two vertices of 𝐺, denoted by 𝑑_𝑖𝑗, is the shortest path connecting the two vertices. The Wiener index [2], denoted by 𝑊(𝐺), is a famous distance-based topological index and is defined as the sum of distances between all the pairs of vertices in 𝐺: 𝑊(𝐺)=_𝑖<𝑗𝑑_𝑖𝑗(𝐺). (1)

As an analogue to the Wiener index 𝑊(𝐺), 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 𝐺 is defined as the networks effective resistance between them if each edge of 𝐺 is replaced by a unit resistor. They also defined the Kirchhoff index Kf(𝐺) of 𝐺 as the sum of resistance distances between all pairs of vertices in 𝐺; that is, Kf(𝐺)=_𝑖<𝑗𝑟_𝑖𝑗(𝐺). (2)

Klein and Randić [3] proved that 𝑟_𝑖𝑗≤𝑑_𝑖𝑗 and Kf(𝐺)≤𝑊(𝐺) with equality if and only if 𝐺 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 𝑄_𝑛 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 𝑄_𝑛, many variants of it were presented, among which, for instance, are generalized hypercubes, folded hypercubes, the line graphs of hypercubes 𝑙(𝑄_𝑛), the subdivision graphs of hypercubes 𝑠(𝑄_𝑛), and the total graphs of hypercubes 𝑡(𝑄_𝑛) [19, 20].

The hypercubes networks 𝑄_𝑛 may be constructed from the family of subsets of a set with a binary string of length 𝑛, 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 𝑄_𝑛 admits several definitions of which two are stated as follows [26].

Definition 1 (see [26]). The hypercubes networks 𝑄_𝑛 has 2^𝑛 vertices each labelled with a binary string of length 𝑛. Two vertices 𝑋=𝑥₁𝑥₂⋯𝑥_𝑛 and 𝑌=𝑦₁𝑦₂⋯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 (recursive construction [26]). The hypercubes network 𝑄_𝑛 is recursively constructed by taking two copies of 𝑄_𝑛−1, denoted by 𝑄⁰_𝑛−1 and 𝑄¹_𝑛−1, and adding 2^𝑛−1 edges as follows: let 𝑉(𝑄⁰_𝑛−1)={0𝑈=0𝑢₂𝑢₃⋯𝑢_𝑛∶𝑢_𝑖=0or1} and 𝑉(𝑄¹_𝑛−1)={1𝑉=1𝑣₂𝑣₃⋯𝑣_𝑛∶𝑣_𝑖=0or1}. A vertex 0𝑈=0𝑢₂𝑢₃⋯𝑢_𝑛 of 𝑄⁰_𝑛−1 is joined to a vertex 1𝑉=1𝑣₂𝑣₃⋯𝑣_𝑛 of 𝑄¹_𝑛−1 if and only if 𝑢_𝑖=𝑣_𝑖 for every 𝑖,2≤𝑖≤𝑛.

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 𝑙(𝐺), 𝑠(𝐺), and 𝑡(𝐺), where 𝐺 is a regular graph. Motivated by the previous results, we present the corresponding formulae for the Kirchhoff index of the hypercubes network 𝑄_𝑛 and its three variant networks 𝑙(𝑄_𝑛), 𝑠(𝑄_𝑛), and 𝑡(𝑄_𝑛) 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 𝐴(𝐺) of 𝐺 is an 𝑛×𝑛 matrix with the (𝑖,𝑗)-entry equal to 1 if vertices 𝑖 and 𝑗 are adjacent and 0 otherwise. Suppose that 𝐷(𝐺)=diag(𝑑₁(𝐺),𝑑₂(𝐺),…,𝑑_𝑛(𝐺)) is the degree diagonal matrix of 𝐺, where 𝑑_𝑖(𝐺) is the degree of the vertex 𝑖, 𝑖=1,2,…,𝑛. Let 𝐿(𝐺)=𝐷(𝐺)−𝐴(𝐺) be called the Laplacian matrix of 𝐺. Then the eigenvalues of 𝐴(𝐺) and 𝐿(𝐺) are called eigenvalues and Laplacian eigenvalues of 𝐺, 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 𝑄_𝑛 with 𝑛≥2, Spec𝑄_𝑛=𝐶02⋯2𝑖⋯2𝑛_0𝑛𝐶¹_𝑛⋯𝐶^𝑖_𝑛⋯𝐶^𝑛_𝑛, (3) where the 2𝑖(𝑖=0,1,…,𝑛) are the eigenvalues of the Laplacian matrix of hypercubes networks, and 𝐶^𝑖_𝑛 are the multiplicities of the eigenvalues 2𝑖.

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 𝐺 be a connected graph with 𝑛≥2 vertices; then Kf(𝐺)=𝑛^𝑛−1_𝑖=11𝜆_𝑖. (4)

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

Lemma 5 (see [27]). Let 𝐺 be an r-regular connected graph with 𝑛 vertices and 𝑚 edges; then 𝑃_𝑙(𝐺)(𝑥)=(𝑥−2𝑟)^𝑚−𝑛𝑃_𝐺(𝑥),𝑃_𝑠(𝐺)(𝑥)=(−1)^𝑚(2−𝑥)^𝑚−𝑛𝑃_𝐺(𝑥(,𝑃𝑟+2−𝑥))_𝑡(𝐺)(𝑥)=(−1)^𝑚(𝑟+1−𝑥)^𝑛×(2𝑟+2−𝑥)^𝑚−𝑛𝑃_𝐺𝑥(𝑟+2−𝑥),𝑟−𝑥+1 (5) where 𝑃_𝑙(𝐺)(𝑥), 𝑃_𝑠(𝐺)(𝑥), and 𝑃_𝑡(𝐺)(𝑥) are the characteristic polynomials for the Laplacian matrix of graphs 𝑙(𝐺), 𝑠(𝐺), and 𝑡(𝐺), 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 𝐺 be a r-regular connected graph with 𝑛 vertices and 𝑚 edges, then 𝑃_𝑠(𝐺)(𝑥)=(−1)^𝑛(2−𝑥)^𝑚−𝑛𝑃_𝐺(𝑥(,𝑃𝑟+2−𝑥))_𝑡(𝐺)(𝑥)=𝑥(𝑥−𝑟−2)(𝑥−2𝑟−2)^𝑚−𝑛×^𝑛−1_𝑖=1𝑥²−2𝑥−𝑟𝑥+(3−2𝑥+𝑟)𝜇_𝑖+𝜇²_𝑖. (6) where 𝑃_𝑠(𝐺)(𝑥), 𝑃_𝑡(𝐺)(𝑥) are the characteristic polynomial for the Laplacian matrix of graphs 𝑠(𝐺) and 𝑡(𝐺), respectively.

The following Lemma give an expression on Kf(𝑡(𝐺)) and Kf(𝐺) of a regular graph 𝐺.

Lemma 7 (see [29]). Let 𝐺 be a r-regular connected graph with 𝑛 vertices and 𝑚 edges, and 𝑟≥2, then (𝑡(𝐺=()Kf))𝑟+2²2()(𝐺)+𝑛𝑟+3Kf²𝑟²−48()+𝑛𝑟+12+𝑛(𝑟+2)(𝑟+4)2(𝑟+3)^𝑛−1_𝑖=11𝜇_𝑖.+3+𝑟 (7)

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 𝑃_𝑄𝑛(𝑥) be the characteristic polynomial of the Laplacian matrix of the hypercubes networks 𝑄_𝑛 with 𝑛≥2 and 𝑃_𝑄𝑛(𝑥)=𝑥^2𝑛+𝑎₁𝑥^2𝑛−1+𝑎₂𝑥^2𝑛−2+⋯+𝑎₂^𝑛₋₁𝑥; (8) then Kf𝑄_𝑛2^𝑛𝑎=−^2𝑛−2𝑎₂^𝑛₋₁, (9) where 𝑎₂^𝑛₋₁,𝑎₂^𝑛₋₂ are the coefficients of 𝑥 and 𝑥² in the characteristic polynomial, respectively.

Proof. Let Spec(𝑄_𝑛)=(𝜆₀,𝜆₁,𝜆₂,…𝜆_𝑛,𝜆_𝑛+1,…,𝜆₂^𝑛₋₁). Then 𝜆_𝑖,𝑖=1,2,…,2^𝑛−1 satisfy the following equation: 𝑥^2𝑛−1+𝑎₁𝑥^2𝑛−2+⋯+𝑎₂^𝑛₋₁=0; (10) it is not difficult to check that 1/𝜆_𝑖,𝑖=1,2,…,2^𝑛−1 are the roots of equation 𝑎₂^𝑛₋₁𝑥^2𝑛−1+𝑎₂^𝑛₋₂𝑥^2𝑛−2+⋯+𝑎₁𝑥+1=0. (11)
Note that 𝑄_𝑛 is connected graph and the multiplicity of 0 as an eigenvalue of 𝐿(𝑄_𝑛) is equal to the number of the connected components in 𝑄_𝑛. So, 𝑎₂^𝑛₋₁≠0; by Lemma 4 and Vieta’s Theorem Kf𝑄_𝑛2^𝑛=^2𝑛−1_𝑖=11𝜆_𝑖𝑎=−^2𝑛−2𝑎₂^𝑛₋₁, (12) where 𝑎₂^𝑛₋₁,𝑎₂^𝑛₋₂ are the coefficients of 𝑥 and 𝑥² in the characteristic polynomial of the Laplacian matrix of the hypercubes networks 𝑄_𝑛.

3. Main Results

3.1. The Kirchhoff Index in Hypercubes Networks 𝑄_𝑛

In this section, we firstly give formula for the Kirchhoff index in the hypercubes 𝑄_𝑛 with 𝑛≥2.

Theorem 9. For the hypercubes networks 𝑄_𝑛 with 𝑛≥2, Kf𝑄_𝑛=2^𝑛𝑛_𝑖=1𝐶^𝑖_𝑛,2𝑖 (13) where the 2𝑖(𝑖=1,…,𝑛) are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients 𝐶^𝑖_𝑛 are the multiplicities of the eigenvalues 2𝑖.

Proof. Since the hypercubes networks 𝑄_𝑛 have 2^𝑛 vertices and 𝑛2^𝑛−1 edges, then by Lemma 3, Spec𝑄_𝑛=𝐶02⋯2𝑖⋯2𝑛_0𝑛𝐶¹_𝑛⋯𝐶^𝑖_𝑛⋯𝐶^𝑛_𝑛; (14) then Kf𝑄_𝑛=2^{𝑛2𝑛−1}_𝑖=11𝜆_𝑖=2^𝑛𝐶¹_𝑛+𝐶2×1^2𝑛+𝐶2×2+⋯_𝑖𝑛𝐶2×𝑛−1+⋯+^𝑛𝑛2×𝑛=2^𝑛𝑛_𝑖=1𝐶^𝑖_𝑛.2𝑖 (15) 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 2^2𝑑/𝑑. 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 𝑙(𝑄_𝑛)

The line graph of a graph 𝐺, denoted by 𝑙(𝐺), is the graph whose vertices correspond to the edges of 𝐺, in which two vertices are adjacent if and only if the corresponding edges of 𝐺 share a common vertex. In the following theorem, we proposed a formula for the Kirchhoff index in the line graph of hypercubes networks 𝑙(𝑄_𝑛), denoted by Kf(𝑙(𝑄_𝑛)), where the eigenvalues of 𝑄_𝑛 are 𝜆₀≤𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛≤𝜆_𝑛+1≤⋯≤𝜆₂^𝑛₋₁.

Theorem 11. Let 𝑙(𝑄_𝑛) be line graphs of hypercubes 𝑄_𝑛 with 𝑛≥2; then Kf𝑙𝑄_𝑛=𝑛2^𝑛𝑛−1_𝑖=1𝐶^𝑖_𝑛2𝑖++𝑛2^2𝑛−3−2^2𝑛−2, (16) where the 2𝑖(𝑖=1,…,𝑛) are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients 𝐶^𝑖_𝑛 are the multiplicities of the eigenvalues 2𝑖.

Proof. Now, for convenience, we denoted the numbers of vertices and edges in the hypercubes networks 𝑄_𝑛 by 𝑝 and 𝑞, respectively. Obviously, 𝑝=2^𝑛,𝑞=𝑛2^𝑛−1, so the line graphs of hypercubes 𝑙(𝑄_𝑛) have 𝑞=𝑛2^𝑛−1 vertices.
By Lemma 5, 𝑃_{𝑙(𝑄𝑛)}(𝑥)=(𝑥−2𝑟)^𝑞−𝑝𝑃_𝐺(𝑥). (17)
Comparing the spectrum of 𝑄_𝑛, Spec𝑄_𝑛=𝐶02⋯2𝑖⋯2𝑛_0𝑛𝐶¹_𝑛⋯𝐶^𝑖_𝑛⋯𝐶^𝑛_𝑛. (18)
We can easily obtain the spectrum of 𝑙(𝑄_𝑛) as follows: Spec𝑙𝑄_𝑛=2𝑛2𝑛⋯2𝑛𝜆₀⋯𝜆_2𝑛−1, (19) where 𝜆₀,𝜆₁,…𝜆_𝑛,𝜆_𝑛+1,…,𝜆₂^𝑛₋₁ are the eigenvalues of 𝑄_𝑛 and 𝜆₀≤𝜆₁≤𝜆₂≤⋯𝜆_𝑛≤𝜆_𝑛+1≤⋯≤𝜆₂^𝑛₋₁.
Notice that the line graphs of hypercubes 𝑙(𝑄_𝑛) have 𝑞=𝑛2^𝑛−1 vertices; by Lemma 4, Kf𝑙𝑄_𝑛=𝑞^2𝑛−1_𝑖=11𝜆_𝑖+𝑞−𝑝=2𝑛𝑛2^𝑛−12^𝑛Kf𝑄_𝑛+𝑞²−𝑝𝑞=𝑛2𝑛2Kf𝑄_𝑛+𝑛2^2𝑛−3−2^2𝑛−2. (20)(21)(22)
Substituting the results of Theorem 9 (15) into (22), we can get the formula for the Kirchhoff index on the line graph of hypercubes Kf(𝑙(𝑄_𝑛)) as follows: Kf𝑙𝑄_𝑛=𝑛2^𝑛𝑛−1_𝑖=1𝐶^𝑖_𝑛2𝑖+𝑛2²ⁿ⁻³−2^2𝑛−2. (23) This completes the proof.

3.3. The Kirchhoff Index in the Subdivision Graph of Hypercubes Networks 𝑠(𝑄_𝑛)

The subdivision graph of a graph 𝐺, denoted by 𝑠(𝐺), is the graph obtained by replacing every edge in 𝐺 with a copy of 𝑃₂ (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 𝑠(𝑄_𝑛), denoted by Kf(𝑠(𝑄_𝑛)).

Theorem 12. Let 𝑠(𝑄_𝑛) be subdivision graphs of the hypercubes 𝑄_𝑛 with 𝑛≥2; then Kf𝑠𝑄_𝑛=(𝑛+2)2^𝑛+𝑛2^𝑛−1×^𝑛_𝑖=1𝐶^𝑖_n2𝑖+2^𝑛−1+𝑛²2^2𝑛−3−2^2𝑛−1, (24) where the 2𝑖(𝑖=1,…,𝑛) are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients 𝐶^𝑖_𝑛 are the multiplicities of the eigenvalues 2𝑖.

Proof. Now supposing that 𝑛≥2, 𝑃_(𝑄𝑛)(𝑥) is the characteristic polynomial of the Laplacian matrix of the hypercubes 𝑄_𝑛, 𝑃_𝑄𝑛(𝑥)=𝑥^2𝑛+𝑎₁𝑥^2𝑛−1+𝑎₂𝑥^2𝑛−2+⋯+𝑎₂^𝑛₋₁𝑥, and Spec𝑄_𝑛=𝜆₀,𝜆₁,𝜆₂,…𝜆_𝑛,𝜆_𝑛+1,…,𝜆₂^𝑛₋₁, (25) where 𝜆_𝑖,𝑖=0,1,2,…,2^𝑛−1 are the Laplacian eigenvalues of 𝑄_𝑛 Then by Lemma 8, Kf𝑄_𝑛2^𝑛𝑎=−^2𝑛−2𝑎₂^𝑛₋₁. (26) By Lemma 5, we have 𝑃_{𝑠(𝑄𝑛)}(𝑥)=(−1)^𝑝(2−𝑥)^𝑞−𝑝×𝑥^2𝑛(𝑛+2−𝑥)^2𝑛+𝑎₁𝑥^2𝑛−1(𝑛+2−𝑥)^2𝑛−1+⋯+𝑎₂^𝑛₋₂𝑥²(𝑛+2−𝑥)²+𝑎₂^𝑛₋₁𝑥(𝑛+2−𝑥). (27)
Consequently, the coefficient of 𝑥² in 𝑃_{𝑠(𝑄𝑛)}(𝑥) is 𝑎₂^𝑛₋₂=(−1)^𝑝2^𝑞−𝑝(𝑛+2)²𝑎₂^𝑛₋₂−2^𝑞−𝑝𝑎₂^𝑛₋₁−𝑞−𝑝2^{𝑞−𝑝−1}(𝑛+2)𝑎₂^𝑛₋₁, (28) and the coefficient of 𝑥 in 𝑃_{𝑠(𝑄𝑛)}(𝑥) is (−1)^𝑝2^𝑞−𝑝(𝑛+2)𝑎₂^𝑛₋₁. (29) Note that 𝑠(𝑄_𝑛) has 2^𝑛+𝑛2^𝑛−1 vertices. By Lemma 6 and substituting the coefficients into (26), Kf(𝑠(𝐺))2^𝑛+𝑛2^𝑛−1=−2^𝑞−𝑝(𝑛+2)²𝑎₂^𝑛₋₂−2^𝑞−𝑝𝑎₂^𝑛₋₁−𝑞−𝑝2^{𝑞−𝑝−1}𝑎₂^𝑛₋₁(×2𝑛+2)_𝑞−𝑝𝑎₂^𝑛₋₁(𝑛+2)⁻¹𝑎=−^2𝑛−2(𝑛+2)𝑎₂^𝑛₋₁+1+𝑛+2𝑞−𝑝2. (30)(31) By substituting 𝑞=𝑛2^𝑛−1,𝑝=2^𝑛 and the results of Lemma 5 into (31), we have Kf𝑠𝑄_𝑛2^𝑛+𝑛2^𝑛−1𝑎=−^2𝑛−2(𝑛+2)𝑎₂^𝑛₋₁+1+𝑛+2𝑛2^𝑛−1−2^𝑛2=(𝑛+2)Kf𝑄_𝑛2^𝑛+1+𝑛+2𝑛2^𝑛−1−2^𝑛2. (32)(33)
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(𝑠(𝑄_𝑛)) as follows: Kf𝑠𝑄_𝑛=(𝑛+2)2^𝑛+𝑛2^𝑛−12^𝑛Kf𝑄_𝑛+2^𝑛+𝑛2^𝑛−1+2𝑛+2^𝑛+𝑛2^𝑛−1𝑛2^𝑛−1−2^𝑛2=(𝑛+2)2^𝑛+𝑛2^𝑛−12^𝑛Kf𝑄_𝑛+2^𝑛−1+𝑛²2^2𝑛−3−2^2𝑛−1=(𝑛+2)2^𝑛+𝑛2^𝑛−1×^𝑛_𝑖=1𝐶^𝑖_𝑛2𝑖+2^𝑛−1+𝑛²2^2𝑛−3−2^2𝑛−1. (34) Thus, the results of the Theorem 12 hold.

3.4. The Kirchhoff Index in the Total Graph of Hypercubes Networks 𝑡(𝑄_𝑛)

The total graph of a graph 𝐺, denoted by 𝑡(𝐺), is the graph whose vertices correspond to the union of the set of vertices and edges of 𝐺, with two vertices of 𝑡(𝐺) being adjacent if and only if the corresponding elements are adjacent or incident in 𝐺. We now proved the formula of the Kirchhoff index in the total graph of the hypercubes networks 𝑡(𝑄_𝑛), denoted by Kf(𝑡(𝑄_𝑛)).

Theorem 13. Let 𝑡(𝑄_𝑛) be the total graphs of the hypercubes networks 𝑄_𝑛 with 𝑛≥2; then Kf𝑡𝑄_𝑛=2^𝑛−1(𝑛+2)²𝑛+3^𝑛_𝑖=1𝐶^𝑖_𝑛+𝑛2𝑖⁴−4𝑛²8()+𝑛𝑛+12+𝑛²+2𝑛(𝑛+4)2(𝑛+3)^𝑛_𝑖=1𝐶^𝑖_𝑛,2𝑖+3+𝑛 (35) where the 2𝑖(𝑖=0,1,…,𝑛) are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients 𝐶^𝑖_𝑛 are the multiplicities of the eigenvalues 2𝑖.

Proof. Let 𝑃_{𝑡(𝑄𝑛)}(𝑥)=𝑥^2𝑛+𝑎₁𝑥^2𝑛−1+𝑎₂𝑥^2𝑛−2+…+𝑎₂^𝑛₋₁𝑥. (36) Then by Lemma 8, Kf𝑡𝑄_𝑛2^𝑛𝑎=−^2𝑛−2𝑎₂^𝑛₋₁. (37) Applying Lemma 6, the Laplacian characteristic polynomial of 𝑡(𝑄_𝑛) is 𝑃_{𝑡(𝑄𝑛)}(𝑥)=𝑥(𝑥−𝑛−2)(𝑥−2𝑛−2)^𝑞−𝑝×^2𝑛−1_𝑖=1𝑥²−2𝑥−𝑛𝑥+[3−2𝑥+𝑛]𝜆_𝑖+𝜆²_𝑖=𝑥(𝑥−𝑛−2)(𝑥−2𝑛−2)^𝑞−𝑝×^2𝑛−1_𝑖=1𝑥²−2+𝑛+2𝜆_𝑖𝑥+𝜆²_𝑖+(𝑛+3)𝜆_𝑖. (38) Consider that 𝑄_𝑛 has 2^𝑛 vertices and 𝑟=𝑛, by Lemma 7, we can get the following equality, Kf𝑡𝑄_𝑛=(𝑟+2)²2()𝑄𝑟+3Kf_𝑛+𝑛²𝑟²−48()+𝑛𝑟+12+𝑛(𝑟+2)(𝑟+4)2(𝑟+3)^2𝑛−1_𝑖=11𝜆_𝑖=()+3+𝑟𝑛+2²2()𝑄𝑛+3Kf_𝑛+𝑛⁴−4𝑛²8()+𝑛𝑛+12+𝑛²+2𝑛(𝑛+4)2(𝑛+3)^2𝑛−1_𝑖=11𝜆_𝑖.+3+𝑛 (39) Consequently, the relationships between the hypercubes networks 𝑄_𝑛 and its variant networks 𝑡(𝑄_𝑛) for Kirchhoff index is Kf𝑡𝑄_𝑛=(𝑛+2)²2()𝑄𝑛+3Kf_𝑛+𝑛⁴−4𝑛²8()+𝑛𝑛+12+𝑛²+2𝑛(𝑛+4)2(𝑛+3)^2𝑛−1_𝑖=11𝜆_𝑖.+3+𝑛 (40) 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 Kf(𝑡(𝑄_𝑛)) as follows. Kf𝑡𝑄_𝑛=(𝑛+2)²2()𝑄𝑛+3Kf_𝑛+𝑛⁴−4𝑛²8()+𝑛𝑛+12+𝑛²+2𝑛(𝑛+4)2(𝑛+3)^2𝑛−1_𝑖=11𝜆_𝑖=2+3+𝑛_𝑛−1(𝑛+2)²𝑛+3^𝑛_𝑖=1𝐶^𝑖_𝑛+𝑛2𝑖⁴−4𝑛²8()+𝑛𝑛+12+𝑛²+2𝑛(𝑛+4)2(𝑛+3)^𝑛_𝑖=1𝐶^𝑖_𝑛.2𝑖+3+𝑛 (41) 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 𝑄_𝑛 and related networks by utilizing spectral graph theory, such as Kf(𝑄_𝑛)=2^𝑛∑^𝑛_𝑖=1(𝐶^𝑖_𝑛/2𝑖), where the 2𝑖(𝑖=1,…,𝑛) are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients 𝐶^𝑖_𝑛 are the multiplicities of the eigenvalues 2𝑖.

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

Finally, the special formulae for the Kirchhoff indexes of 𝑙(𝑄_𝑛), 𝑠(𝑄_𝑛), and 𝑡(𝑄_𝑛) 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.