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 dij, 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:
(1)W(G)=∑i<jdij(G).
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,
(2)Kf(G)=∑i<jrij(G).
Klein and Randić [3] proved that rij≤dij 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 Qn 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 Qn, many variants of it were presented, among which, for instance, are generalized hypercubes, folded hypercubes, the line graphs of hypercubes l(Qn), the subdivision graphs of hypercubes s(Qn), and the total graphs of hypercubes t(Qn) [19, 20].
The hypercubes networks Qn 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 Qn admits several definitions of which two are stated as follows [26].
Definition 1 (see [26]).
The hypercubes networks Qn has 2n vertices each labelled with a binary string of length n. Two vertices X=x1x2⋯xn and Y=y1y2⋯yn are adjacent if and only if there exists an i, 1≤i≤n, such that xi=yi¯, where yi¯ denoted the complement of binary digit yi and xj=yj, for all j≠i, and 1≤j≤n.
Definition 2 (recursive construction [26]).
The hypercubes network Qn is recursively constructed by taking two copies of Qn-1, denoted by Qn-10 and Qn-11, and adding 2n-1 edges as follows: let V(Qn-10)={0U=0u2u3⋯un:ui=0 or 1} and V(Qn-11)={1V=1v2v3⋯vn:vi=0 or 1}. A vertex 0U=0u2u3⋯un of Qn-10 is joined to a vertex 1V=1v2v3⋯vn of Qn-11 if and only if ui=vi 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 Qn and its three variant networks l(Qn), s(Qn), and t(Qn) 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),…,dn(G)) is the degree diagonal matrix of G, where di(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 Qn with n≥2,
(3)
Spec
(Qn)=(02⋯2i⋯2nCn0Cn1⋯Cni⋯Cnn),
where the 2i (i=0,1,…,n) are the eigenvalues of the Laplacian matrix of hypercubes networks, and Cni 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
(4)
Kf
(G)=n∑i=1n-11λi.
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
(5)Pl(G)(x)=(x-2r)m-nPG(x),Ps(G)(x)=(-1)m(2-x)m-nPG(x(r+2-x)),Pt(G)(x)=(-1)m(r+1-x)nPt(G)(x)=×(2r+2-x)m-nPG(x(r+2-x)r-x+1),
where Pl(G)(x), Ps(G)(x), and Pt(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
(6)Ps(G)(x)=(-1)n(2-x)m-nPG(x(r+2-x)),Pt(G)(x)=x(x-r-2)(x-2r-2)m-n Pt(G)(x)=×∏i=1n-1[(x2-2x-rx)+(3-2x+r)μi+μi2].
where Ps(G)(x), Pt(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
(7)
Kf
(t(G))=(r+2)22(r+3)
Kf
(G)+n2(r2-4)8(r+1)+n2 +n(r+2)(r+4)2(r+3)∑i=1n-11μi+3+r.
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 PQn(x) be the characteristic polynomial of the Laplacian matrix of the hypercubes networks Qn with n≥2 and
(8)PQn(x)=x2n+a1x2n-1+a2x2n-2+⋯+a2n-1x;
then
(9)
Kf
(Qn)2n=-a2n-2a2n-1,
where a2n-1,a2n-2 are the coefficients of x and x2 in the characteristic polynomial, respectively.
Proof.
Let Spec(Qn)=(λ0,λ1,λ2,…λn,λn+1,…,λ2n-1). Then λi,i=1,2,…,2n-1 satisfy the following equation:
(10)x2n-1+a1x2n-2+⋯+a2n-1=0;
it is not difficult to check that 1/λi,i=1,2,…,2n-1 are the roots of equation
(11)a2n-1x2n-1+a2n-2x2n-2+⋯+a1x+1=0.
Note that Qn is connected graph and the multiplicity of 0 as an eigenvalue of L(Qn) is equal to the number of the connected components in Qn. So, a2n-1≠0; by Lemma 4 and Vieta’s Theorem
(12)Kf(Qn)2n=∑i=12n-11λi=-a2n-2a2n-1,
where a2n-1,a2n-2 are the coefficients of x and x2 in the characteristic polynomial of the Laplacian matrix of the hypercubes networks Qn.
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 Qn and related networks by utilizing spectral graph theory, such as Kf(Qn)=2n∑i=1n(Cni/2i), where the 2i (i=1,…,n) are the eigenvalues of the Laplacian matrix of hypercubes networks and the binomial coefficients Cni are the multiplicities of the eigenvalues 2i.
We also obtained the relationship for Kirchhoff index between hypercubes networks Qn and its three variant networks l(Qn), s(Qn), and t(Qn), respectively, by deducing the characteristic polynomial of the Laplacian matrix related networks.
Finally, the special formulae for the Kirchhoff indexes of l(Qn), s(Qn), and t(Qn) were proposed, respectively, by making use of spectral graph theory and Vieta’s Theorem.