COMPLEXITYComplexity1099-05261076-2787Hindawi10.1155/2020/53076705307670Research ArticleUpper and Lower Bounds for the Kirchhoff Index of the n-Dimensional Hypercube Networkhttps://orcid.org/0000-0002-9620-7692LiuJia-Bao12ZhaoJing2ShiZhi-Yu2https://orcid.org/0000-0003-3133-7119CaoJinde1https://orcid.org/0000-0001-6420-3948AlsaadiFuad E.3CamposEric1School of MathematicsSoutheast UniversityNanjing 210096Chinaseu.edu.bd2School of Mathematics and PhysicsAnhui Jianzhu UniversityHefei 230601Chinaahjzu.edu.cn3Department of Electrical and Computer EngineeringFaculty of EngineeringKing Abdulaziz UniversityJeddah 21589Saudi Arabiakau.edu.sa20201662020202017032020150520202605202016620202020Copyright © 2020 Jia-Bao Liu et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 University2016QD1162017dc03
1. 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=DGAG. For other notations and graph theoretical terminologies that not state here, we follow .

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 . Denote rij the resistance distance between two arbitrary vertices i and j in electrical networks by replacing every edge by a unit resistor . 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 . 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 .

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 .

The hypercube network Qn is repeatedly constructed by making two copies of Qn1, written as Qn10 and Qn11, respectively. Meanwhile, adding repeatedly 2n1 edges as below, let VQn10=0U=0u2u3un:ui=0 or 1 and VQn11=1V=1v2v3vn:vi=0 or 1. A node 0U=0u2u3un of Qn10 is linked to another node 1V=1v2v3vn of Qn11 if and only if ui=vi for each i,2in.

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 .

Next, we recall the formula for the Kirchhoff index in the hypercube Qn with n2.

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

For the hypercube network Qn with n2, (2)KfQn=2ni=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)limni=0n1n2ini=2.

The authors of  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 .

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)2ni=1nni12i.

Theorem 3.

For the hypercube network Qn with n2, then(5)4nnnn+1nn+22n+1n+1KfQn.

Consider that(6)i=1nni1ii=1nni1i+1=n112+n213++nn1n+1=12n!1!n1!+13n!2!n2!++1n+1=1n+1n+1!2!n1!+1n+1n+1!3!n2!++1n+1=i=1nn+1i+11n+1.

By virtue of(7)i=1nn+1i+1=2n+1n2.

By means of calculating the right of equation (6), one can establish the following identity:(8)i=1nn+1i+11n+1=2n+1n2n+1.

Since(9)2n+1n2n+1i=1nni1i=2i=1nni12i,2n+1n2n+12i=1nni12i.

Hence,(10)2n2nn+12n1n+2n+1=2n12n+1n2n+1KfQn=2ni=1nni12i.

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

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 n2, then(12)KfQn4nn2nn+1n+22nn+1,(13)i=1n12iCnii=1n1i+1Cni=12Cn1+13Cn2++1n+1Cnn=12n!1!n1!+13n!2!n2!++1n+1=1n+1n+1!2!n1!+1n+1n+1!3!n2!++1n+1=i=1n1n+1Cn+1i+1.

Based on equation (8), we can obtain that(14)2ni=1nCni2i=KfQn2n2n+1n2n+1.

Hence,(15)KfQn2n2n+1n2n+1=4nn2nn+1n+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 n2, (16)KfQn4nn.

Following the identity which is obtained in ,(17)i=1nxii=1n+1n1++1+i=1nnix1ii.

Fixing x=2, one arrives at(18)i=1nni1i=i=1n2ii1n+1n1++1.

Namely,(19)i=1nni1i=i=1n2i1i.

According to equation (19) and Theorem 2, one obtains(20)KfQn=2n1i=1n2i1i.

Using equation (20), one has(21)KfQn2n1i=1n2ii.

On the contrary,(22)2n1i=1n2ii=1n22n1i=0n1n2ini.

Using Theorem 2 and substituting equations (22) to (21), one obtains the desired result:(23)KfQn4nn.

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 n2, then(24)KfQn=2n1i=1n2i1i.

Let Sn=i=1nCni/i, then(25)SnSn1=i=1nCniii=1n1Cn1ii=1n+i=1n1Cniii=1n1Cn1ii=1n+i=1n11iCniCn1i=1n+i=1n11iCn1i1.

Consequently,(26)nSnSn1=1+i=1n1n!i!ni!=1+i=1n1Cni=2n1.

One can easily check that S1=1. Hence, SnSn1=2n/n1/n.

By virtue of the above equality, we obtain(27)Sn=i=1n2iii=1n1i.

Therefore,(28)KfQn=2n1i=1n2i1i.

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.

XuJ. M.Topological Structure and Analysis of Interconnection Networks2001London, UKKluwer Academic PublishersKleinD. J.RandićM.Resistance distancesJournal of Mathematical Chemistry1993121819510.1007/bf011646272-s2.0-21144484097LiuJ. -B.CaoJ.PanX.-F.ElaiwA.The Kirchhoff index of hypercubes and related complex networksDiscrete Dynamics in Nature and Society20132013754318910.1155/2013/5431892-s2.0-84893710256EstradaE.HatanoN.Topological atomic displacements Kirchhoff and Wiener indices of moleculesChemical Physics Letters20104864–616617010.1016/j.cplett.2009.12.0902-s2.0-75349091543FowlerP. W.Resistance distance in fullerene graphsCroatica Chemica Acta2002752401408ZhangH. P.JiangX. Y.YangY.Bicyclic graphs with extremal Kirchhoff indexMatch Communications in Mathematical and in Computer Chemistry2009613697712ArauzC.The Kirchhoff indexes of some composite networksDiscrete Applied Mathematics2009160101429144010.1016/j.dam.2012.02.0082-s2.0-84859885847ZhangH.YangY.Kirchhoff index of composite graphsDiscrete Applied Mathematics2009157112918292710.1016/j.dam.2009.03.0072-s2.0-67349106443BianchiM.CornaroA.PalaciosJ. L.TorrieroA.Bounds for the Kirchhoff index via majorization techniquesJournal of Mathematical Chemistry201351256958710.1007/s10910-012-0103-x2-s2.0-84872320496WangH. H.HuaH.WangD.Cacti with minimum, second-minimum, and third-minimum Kirchhoff indicesMathematical Communications2010152347358HongM.SunW.LiuS.XuanT.Coherence analysis and Laplacian energy of recursive trees with controlled initial statesFrontiers of Information Technology Electronic Engineering20202193193810.1631/fitee.19001332-s2.0-85071771552LiuJ.-B.PanX.-F.CaoJ.CaoJ.Some properties on Estrada index of folded hypercubes networksAbstract and Applied Analysis20142014638087410.1155/2014/1676232-s2.0-84894648554LiuJ.-B.PanX.-F.Minimizing Kirchhoff index among graphs with a given vertex bipartitenessApplied Mathematics and Computation2016291848810.1016/j.amc.2016.06.0172-s2.0-84977619595LiuJ.-B.PanX.-F.YuL.LiD.Complete characterization of bicyclic graphs with minimal Kirchhoff indexDiscrete Applied Mathematics20162009510710.1016/j.dam.2015.07.0012-s2.0-84959321264RamanI.ChoudumS. A.Embedding certain height-balanced trees and complete Pm-ary trees into hypercubesJournal of Discrete Algorithms2013221536510.1016/j.jda.2013.07.0052-s2.0-84884139607FinkJ.Perfect matchings extend to Hamilton cycles in hypercubesJournal of Combinatorial Theory, Series B20079761074107610.1016/j.jctb.2007.02.0072-s2.0-34548442909BossardA.KanekoK.k-pairwise disjoint paths routing in perfect hierarchical hypercubesThe Journal of Supercomputing2013143111ParkJ.-H.LimH.-S.KimH.-C.Panconnectivity and pancyclicity of hypercube-like interconnection networks with faulty elementsTheoretical Computer Science20073771–317018010.1016/j.tcs.2007.02.0292-s2.0-34247608120WangD.LuM.Edge fault tolerance of super edge connectivity for three families of interconnection networksInformation Sciences2012188326026810.1016/j.ins.2011.11.0062-s2.0-84855450638XuJ. M.WangJ. W.WangW. W.On super and restricted connectivity of some interconnection networksArs Combinatoria20109462532LiX. J.XuM.Edge fault tolerance of hypercube-like networksInformation Processing Letters201311319–2176076310.1016/j.ipl.2013.07.0102-s2.0-84881113983ZhangJ.XiangY.SunW.A discrete random walk on the hypercubePhysica A: Statistical Mechanics and Its Applications20184941710.1016/j.physa.2017.12.0052-s2.0-85037976271PalaciosJ. L.RenomJ. M.Bounds for the Kirchhoff index of regular graphs via the spectra of their random walksInternational Journal of Quantum Chemistry201011091637164110.1002/qua.223232-s2.0-77953413439KnuthD. E.The art of computer programmingFundamental Algorithms199713rdBoston, MA, USAAddison-Wesley