, ,

(1. School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, China;2. Department of Mathematics, Central South University, Changsha 410083, China)

On the characteristic polynomials of graphs with nullityn-4

WUTingzeng1,FENGLihua2,MAHaicheng1

(1. School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, China;2. Department of Mathematics, Central South University, Changsha 410083, China)

The nullity of a graph is the multiplicity of zeroes in its adjacency spectrum. And the nullity of a graphGwithnverticesequalstonminustherankofadjacencymatrixofG.Thecharacteristicpolynomialsofgraphswithnullityn-4iscomputed.Inparticular,itisshownthatsomegraphswithnullityn-4aredeterminedbytheirspectra.Andsomepairsofcospectralgraphswithnullityn-4arepresented.

characteristic polynomial; rank; nullity; cospectral graphs

1 Introduction

TheadjacencymatrixofGisdenotedbyA(G).ThecharacteristicpolynomialofagraphG,bydefinition,is

whereIistheidentitymatrixofordern.

ThespectrumofGconsistsoftheeigenvaluestogetherwiththeirmultiplicitiesofA(G).Twographsaresaidtobecospectralif they share the same spectrum or characteristic polynomial.

A graph is said to bedeterminedbythespectrum(DS for short) if there is no other non-isomorphic graph with the same spectrum. The nullity ofG,denotedbyη(G),isthemultiplicityofzeroesinthespectrumofG.Letr(G)betherankofA(G).Obviously,η(G)=n-r(G).Whenη(G)=0,thegraphGiscallednonsingular.

TheproblemcharacterizingallgraphsGwithη(G)>0isofgreatinterestinbothchemistryandmathematics.ForabipartitegraphGwhichcorrespondstoanalteranthydrocarboninchemistry,ifη(G)>0,itisindicatedthatthecorrespondingmoleculeisunstable.Thenullityofagraphisalsomeaningfulinmathematicssinceitisrelatedtothesingularityofadjacentmatrix.Theproblemhasnotyetbeensolvedcompletely.Anditreceivedalotofattentionfromresearchersinrecentyears,see[2-3].Herewehighlighttworesultswhichinvolveextremalnullityofgraphs.Thatis,ChengandLiu[4]characterizedthegraphswhosenullityreachtheupperboundsn-2andn-3.Changetal. [5-6]masterlyusedthedefinitionofmultiplicationofvertices(seep.53of[7])tocharacterizeallconnectedgraphswithrank4or5.Moreinformationsee[8-9].

BythedefinitionofG°m[m1,m2,…,mn],wecanobtainsomeprimarypropertiesaboutthestructureofG°m as follows.

Property 3 For any triangle inG°m, there exactly exists a complete 3-partite induced subgraph obtained by multiplying vertices on a triangle inGwhichcontainsit.Then

whereuvwdenotesatriangleinG,andthesumistakenoveralltrianglesinG.

Property4Every4-cycleinG°m must be contained in an induced subgraph obtained by multiplying every vertex on an edge, oneP3oroneC4inG.Directcomputationyields

wherethefirstsumistakenoveralledgesinG,thesecondsumistakenoverallP3’sinG,andthethirdsumistakenoverallC4’sinG.

Thispaperisorganizedasfollows.InSection2,wepresentsomelemmasandcharacterizeallgraphswithnullityn-4.InSection3,wecomputethecharacteristicpolynomialsofgraphswithnullityn-4.InSection4,weinvestigatewhichgraphswithnullityn-4areDS.Precisely,weshowthattwoclassesofregulargraphsandoneclassofnon-bipartitegraphsareDS.Andwepresentsomepairsofcospectralbipartitegraphs.

2 Preliminaries

Inthissection,wewillpresentsomeresultswhichplayakeyroleintheproofsofthemaintheorems.

Lemma 1[1]LetGbeagraphwithpverticesandqedges,andlet(d1,d2,…,dp)bethedegreesequenceofG.ThecoefficientsofthecharacteristicpolynomialofagraphGsatisfy:

(i)a0=1;

(ii)a1=0;

(iii)a2=-q;

(iv)a3=-2c3(G);

Lemma 2[10]LetGbeagraph.Fortheadjacencymatrix,thefollowingcanbeobtainedfromthespectrum.

(i) The number of vertices.

(ii) The number of edges.

(iii) WhetherGisregular.

(iv) WhetherGisregularwithanyfixedgirth.

(v) The number of closed walk of any length.

(vi) WhetherGisbipartite.

Theorem 1[4]Suppose thatGisasimplegraphonnverticesandn≥2.Thenη(G)=n-2ifandonlyifGisisomorphictoKn1,n2∪kK1,wheren1+n2+k=n,n1andn2>0,andk≥0.

Theorem 2[5]LetGbeaconnectedgraph.Thenr(G)=4ifandonlyifG∈M(G1,G2,G3,G4,G5,G6,G7,G8),whereGi(i=1,2,…,8)isdepictedinFig.1.

CombiningTheorems1and2,wecanobtainthefollowingresult.

Fig.1 The graphs G1,G2,G3,G4,G5,G6,G7,G8,G9

3 The characteristic polynomials of graphs with nullity n-4

Inthissection,wewillcomputethecharacteristicpolynomialsofgraphswithrank4.

Proof

cd)xa+b+c+d-2-2bcdxa+b+c+d-3+abcdxa+b+c+d-4

bd+ce)x2-2abcx+abdc+abec+dbec]

xa+b+c+d+e-4[x4-(ab+ae+be+bc+cd+de)·

x2-2abex+abed+adec+abcd+abec]

cf+cd+de+ef)x2-2bcfx+abed+

abcd+bdec+abcf+abef+bcef+fbcd+fbed]

3abcd4-cycles.ByLemma1,wehave

bc+bd+cd)x2-

2(abc+abd+acd+bcd)x-3abcd]

abef+acdf+bcde4-cycles.ByLemma1,wehave

[x4-(ab+ac+af+bc+be+cd+

df+de+fe)x2-2(abc+def)x+abed+

abdf+acbe+aced+acef+afed+abcd+fcde+

abcf+bcef+fbcd+fbed]

[x4-(ab+bc+cd)x2+abcd]

[x4-(ab+bc+cd+ed)x2+abcd]

[x4-(ab+cd)x2+abcd]

Bytheaboveargument,weobtainthedesiredresult.

4 The spectral characterization of graphs with nullity n-4

MaandRen[15]investigatedwhichkindofcompletemultipartitegraphisDSsinceitiswellknownthatnotallcompletemultipartitegraphsareDS.Andtheyprovedthefollowingresults.

ByTheorems5and6,weobtaindirectlyacorollaryasfollows.

Theorem 7

(ii) The complete regular 4-partite graph is DS.

Additionally, by the definition ofG5。 m[m1,m2,m3,m4],wenotethatG5°m is a complete 4-partite graphs. Observing Theorem 4, it is easy to see that only the fifth coefficient of characteristic polynomial ofG5°m is negative, which implies that the following proposition.

Theorem 9 2Kt,tisDS.

Fromtheargumentabove,weobtainthatGisDS.

Proof. By Theorem 4 (vii) and (viii), we obtain that

Therefore,G∪kK1andH∪kK1arecospectral,wherea,k≥0.

5 Conclusion

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2015-11-28

國家自然基金資助項(xiàng)目 (11101245, 11271208, 11301302, 11561056)；青海省自然基金資助項(xiàng)目(2016-ZJ-947Q)；山東省自然基金資助項(xiàng)目(BS2013SF009)；中南大學(xué)數(shù)學(xué)與交叉科學(xué)項(xiàng)目資助項(xiàng)目；青海民族大學(xué)自然科學(xué)基金資助項(xiàng)目(2015XJZ12)

吳廷增(1978年生)，男；研究方向：數(shù)學(xué)化學(xué)、圖的匹配理論和計(jì)算機(jī)網(wǎng)絡(luò)；E-mail:mathtzwu@163.com

零度為n-4的圖的特征多項(xiàng)式*

吳廷增1, 馮立華2, 馬海成1

(1. 青海民族大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 青海 西寧 810007;2. 中南大學(xué)數(shù)學(xué)系, 湖南 長沙 410083)

圖的零度是指圖的鄰接譜中零特征根的重數(shù)。顯然,n個頂點(diǎn)的圖G的零度等于n減去其鄰接矩陣的秩。計(jì)算了零度為n-4的所有圖的特征多項(xiàng)式。特別地, 證明了許多零度為n-4的圖是譜唯一確定的, 并構(gòu)造了許多對零度為n-4的同譜圖。

特征多項(xiàng)式; 秩; 零度; 同譜圖

O157.6；O

A

0529-6579(2016)06-0057-07

10.13471/j.cnki.acta.snus.2016.06.008