The hypercube Qn is one of the most admirable and efficient interconnection network due to its excellent performance for some practical applications. The Kirchhoff index KfG is equal to the sum of resistance distances between any pairs of vertices in networks. In this paper, we deduce some bounds with respect to Kirchhoff index of hypercube network Qn.

China Postdoctoral Science Foundation2017M621579Postdoctoral Science Foundation of Jiangsu Province1701081BProject of Anhui Jianzhu University2016QD1162017dc031. Introduction

Network is usually modelled by a connected graph G=VG,EG with order n, labeled as VG=v1,v2,…,vn and EG=e1,e2,…,em. The adjacency matrix AG of G is a square matrix with n vertices, in which elements aij are 1 or 0, depending on whether there is an edge or not between vertices i and j. The degree diagonal matrix of G is denoted by DG=diagd1,d2,…,dn, where d1,d2,…,dn are the degree of vertices v1,v2,…,vn, respectively. Together with the adjacency and degree matrix, one arrives at the Laplacian matrix, whose expression can be written as LG=DG−AG. For other notations and graph theoretical terminologies that not state here, we follow [1].

Various parameters are always used to characterize and describe the complex networks of which the fundamental one is named as the distance dij, concerned as the shortest path between the vertices i and j in networks. Similarly considering the distance dij, Klein and Randić in 1993 presented a novel distance function, named as resistance distance [2]. Denote rij the resistance distance between two arbitrary vertices i and j in electrical networks by replacing every edge by a unit resistor [3–7]. The Kirchhoff index KfG of networks is defined as(1)KfG=∑i<jrijG.

The Kirchhoff index has attracted more and more attentions due to its practical applications in the fields of physical interpretations, electric circuit, and so on [8–11]. The Kirchhoff index of some product graphs, join graphs, and corona graphs were studied [5, 7]. The more results of the applications on the Kirchhoff index were explored in [12–14].

In what follows, the rest of the context is summarized. Section 2 proposes the main definition and preliminaries in our discussion. Some bounds on the Kirchhoff index of hypercubes Qn are deduced in Section 3. We conclude the paper in Section 4.

2. Definition and Preliminaries

In this section, we recall some basic definition in graph theory. The hypercube network 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 hypercube network Qn admits several definitions of which one is stated as below [15].

The hypercube network Qn is repeatedly constructed by making two copies of Qn−1, written as Qn−10 and Qn−11, respectively. Meanwhile, adding repeatedly 2n−1 edges as below, let VQn−10=0U=0u2u3…un:ui=0 or 1 and VQn−11=1V=1v2v3…vn:vi=0 or 1. A node 0U=0u2u3…un of Qn−10 is linked to another node 1V=1v2v3…vn of Qn−11 if and only if ui=vi for each i,2≤i≤n.

The hypercube network Qn obtained more and more admirable concentrations due to its surprising properties, for instance, symmetry, regular structure, strong connectivity, small diameter, and so on [16, 17]. For more results on the hypercube network and its applications, see [18–21].

Next, we recall the formula for the Kirchhoff index in the hypercube Qn with n≥2.

Theorem 1 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

For the hypercube network Qn with n≥2, (2)KfQn=2n∑i=1nni12i,where 2ii=1,…,n is the eigenvalue of the Laplacian matrix of the hypercube network and the binomial coefficients niare the multiplicities of the eigenvalues 2i.

Theorem 2 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

(3)limn⟶∞∑i=0n−1n2in−i=2.

The authors of [23] obtained a closed-form formula for the Kirchhoff index of the d-dimensional hypercube and found the asymptotic value 22d/d by using probabilistic tools. The result of Theorem 3 is obtained by directly calculating the eigenvalues of the Laplacian matrix of the hypercube network, which is different from the technique in [23].

3. Main Results

In this section, one will estimate the Kirchhoff index of n-dimensional hypercube, i.e., our goal is to estimate the quantity:(4)2n∑i=1nni12i.

Theorem 3.

For the hypercube network Qn with n≥2, then(5)4nnnn+1−nn+22n+1n+1≤KfQn.

Simply, from the left of the above inequality, we obtain(11)4nnnn+1−nn+22n+1n+1≤KfQn.

Apparently, the left of the above inequality converges to the asymptotic value 22d/d for large enough n. The proof of lower bound is completed.

For the upper bound, we have similar theorem to consider as follows.

Theorem 4.

For the hypercube networks Qn with n≥2, then(12)KfQn≤4nn2nn+1−n+22nn+1,(13)∑i=1n12iCni≤∑i=1n1i+1Cni=12Cn1+13Cn2+…+1n+1Cnn=12n!1!n−1!+13n!2!n−2!+…+1n+1=1n+1n+1!2!n−1!+1n+1n+1!3!n−2!+…+1n+1=∑i=1n1n+1Cn+1i+1.

Based on equation (8), we can obtain that(14)2n∑i=1nCni2i=KfQn≤2n2n+1−n−2n+1.

Hence,(15)KfQn≤2n2n+1−n−2n+1=4nn2nn+1−n+22nn+1.

The above estimate looks a little complicated. The upper bound is roughly twice the asymptotic value. Hence, a new upper bound is explored as follows.

Theorem 5.

For the hypercube network Qn with n≥2, (16)KfQn≤4nn.

Following the identity which is obtained in [24],(17)∑i=1nxii=1n+1n−1+⋯+1+∑i=1nnix−1ii.

Fixing x=2, one arrives at(18)∑i=1nni1i=∑i=1n2ii−1n+1n−1+⋯+1.

Namely,(19)∑i=1nni1i=∑i=1n2i−1i.

According to equation (19) and Theorem 2, one obtains(20)KfQn=2n−1⋅∑i=1n2i−1i.

Using equation (20), one has(21)KfQn≤2n−1⋅∑i=1n2ii.

On the contrary,(22)2n−1⋅∑i=1n2ii=1n⋅22n−1⋅∑i=0n−1n2in−i.

Using Theorem 2 and substituting equations (22) to (21), one obtains the desired result:(23)KfQn≤4nn.

This has completed the proof.

4. Further Discussion

We, at this place, try another way to estimate the Kirchhoff index of n-dimensional hypercubes.

Theorem 6.

For the hypercube networks Qn with n≥2, then(24)KfQn=2n−1∑i=1n2i−1i.

Let Sn=∑i=1nCni/i, then(25)Sn−Sn−1=∑i=1nCnii−∑i=1n−1Cn−1ii=1n+∑i=1n−1Cnii−∑i=1n−1Cn−1ii=1n+∑i=1n−11iCni−Cn−1i=1n+∑i=1n−11iCn−1i−1.

One can easily check that S1=1. Hence, Sn−Sn−1=2n/n−1/n.

By virtue of the above equality, we obtain(27)Sn=∑i=1n2ii−∑i=1n1i.

Therefore,(28)KfQn=2n−1∑i=1n2i−1i.

The proof of Theorem 6 is completed.

Data Availability

The data used to support the findings of this study are available within paper.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

Data curation was carried out by J-B.L.; J-B.L. and J.Cao helped with the methodology; J.Z., Z-Y.S., and F.E. Alsaadi wrote the original draft. All authors read and approved the final manuscript.

Acknowledgments

The work of was partly supported by the China Postdoctoral Science Foundation under Grant no. 2017M621579, Postdoctoral Science Foundation of Jiangsu Province under Grant no. 1701081B, and Project of Anhui Jianzhu University under Grant nos. 2016QD116 and 2017dc03.

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