Enumeration of cospectral and coinvariant graphs

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Applied Mathematics and Computation

journalhomepage:www.elsevier.com/locate/amc

Enumeration of cospectral and coinvariant graphs

Aida Abiad

^a,b,c,∗

, Carlos A. Alfaro

^d

aDepartment of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, The Netherlands

bDepartment of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

cDepartment of Mathematics and Data Science, Vrije Universiteit Brussels, Brussels, Belgium

dBanco de México, Mexico City, Mexico

a rt i c l e i nf o

Article history:

Received 28 August 2020 Revised 29 April 2021 Accepted 2 May 2021

Keywords:

Graph invariant Eigenvalues Invariant factors Smith normal form Enumeration

a b s t ra c t

We present enumeration results onthe number ofconnected graphs up to10 vertices for whichthere isatleast one othergraphwiththe samespectrum (cospectral mate), orat leastone othergraphwith thesame Smith normalform(coinvariant mate)with respecttoseveralmatricesassociatedtoagraph.Thepresentednumericaldatagivesome indicationthatpossiblytheSmithnormalformofthedistanceLaplacianandthesignless distance Laplacianmatrices couldbe afiner invariant thanthespectrum todistinguish graphs. Finally, weprove agraphcharacterization usingthe Smith normal form ofthe distancesignlessLaplacianmatrix.

© 2021TheAuthor(s).PublishedbyElsevierInc.

ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Spectral graph theory aims tounderstand to what extentgraphs are characterized by their spectra.Starting fromthe eigenvaluesofamatrixassociatedtoagraph,itseekstodeducecombinatorialpropertiesofthegraph.Forthis,weassociate agraphGtoamatrixMandanalyzetheeigenvalues ofM.TheseeigenvaluesarecalledthespectrumofGwithrespectto thematrixM,anditsmultisetisdenotedbyM-spectrum(G).M-cospectralgraphsaregraphsthatshareM-spectrum.Agraph GisdeterminedbyitsM-spectrum,M-DS,ifonlyisomorphicgraphsarecospectralwithG.

Motivatedbythegraphisomorphismproblem,itisofinterestwhatfractionofallgraphsisuniquelydeterminedbyits spectrum.HaemersconjecturedthatthefractionofgraphsonnverticeswithaM-cospectralmatetendstozeroasntends toinfinity.Anumericalstudyforn≤9wasgivenbyGodsilandMcKay[16],forn=10,11byHaemersandSpence[18]and forn=12by BrouwerandSpence[12].AouchicheandHansen[6]presentedcomputationalresultsinwhichtheystudied cospectrality forthe distance,distanceLaplacian anddistancesignless Laplacian matricesofall the connected graphson up to10 vertices.Recently,Pinheiro, SouzaandTrevisan [26] providedsome numericalevidencethat thecomplementary spectrum ofagraphdistinguishesmoregraphsthanother standardgraphspectra,buttheyalsoshowedthatitishardto computethecomplementaryspectrum.

The mainquestioniswhetherit ispossibletodefine amatrixMofGsuchthat everygraphbecomesM-DS.In[14]it wasshownthattheanswertothisquestionispositive.However,inthiscaseitismoreworktocheckcospectralityofthe matricesthan testingisomorphism.Iftherewouldbe aneasily computable matrixMforwhichevery graphbecomesM-

∗Corresponding author.

E-mail addresses: [email protected] (A. Abiad), [email protected] (C.A. Alfaro).

https://doi.org/10.1016/j.amc.2021.126348

0 096-30 03/© 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Table 1

Number of connected graphs with at least one cospectral mate for A , L , Qand D .

n 5 6 7 8 9 10

|^G n| ^{21 112 853 11,117 261,080} 11,716,571

|G ^spn(A )| 0 2 63 1353 46,930 2,462,141

|G ^spn(L )| 0 4 115 1611 40,560 1,367,215

|G ^spn(^Q)| 2 10 80 1047 17,627 615,919

|G ^spn(D )| 0 0 22 658 25,058 1,389,986

DS,thegraphisomorphismproblemwouldbesolved.Hence,whenMisoneofthecommonlyusedmatricesassociatedto graphs(adjacency,Laplacian,distancematrices,signlessLaplacian,normalizedLaplacian),onecansaythatthereisnotsuch amatrixMforwhichallgraphsareM-DS,sincethereexistmanyexamplesofnon-isomorphic graphsthatsharethesame M-spectrum.Thisleavesopenthepossibilityofamplifyingorreplacingspectrawiththeuseofmorerefinedrepresentations forobtainingmorefaithfulgraphinformation.

The main goalofthis articleisto propose a newwayofrepresenting agraph usingthe Smithnormalform(SNF) of certain distance matrices. We provide some numerical evidence that this new algebraicgraph representation may do a better job in distinguishing graphs. Forthiswe first need to recall some definitions.Two matricesM,N areequivalent if thereexistunimodularmatricesPandQ withentriesinZsatisfyingM=PNQ.TheSmithnormalformofaintegermatrix M,denoted by SNF(^M)^{,isthe uniquediagonal matrixdiag}(^f1,...,fr,0,...,0)^{equivalentto Msuch thatr}=rank(^M)^and f_i

|

^fjfori<j.Theinvariantfactors (orelementarydivisors)ofMaretheintegers inthediagonaloftheSNF(^M)^.IfMisan integersymmetricmatrixassociatedtoagraph,thenwesaythatthegraphsGandHareM-coinvariantiftheSNFsofM(^G) andM(^H)^{,computedover}Z,arethesame.Coinvariantgraphswereintroducedin[28].NotethatrelatedtotheSNFthere isthe p-rank,i.e.,therankofthematrixconsidered overthefinitefieldFp.Weshouldnotethatthe p-rankhasalsobeen usedintheliteraturetodistinguishgraphs;forinstance,the2-rankandthespectrumcharacterizesymplecticgraphsover F2 [24],the 2-rankcan beused todistinguishstronglyregular graphswiththesame parametersasthesymplectic graph [1],andsome p-ranksandthespectrumwereusedtocharacterizedistance-regulargraphs[25].

Inparticular,inthisworkwestudyifthereisamatrixM(sayadjacency,Laplacian,signlessLaplacian,etc.)whoseSNF distinguishesmoregraphs.Broadlyspeaking,theideaistoverifywhethertheportionofgraphsthathaveaM-coinvariant mateissmallerthantheportionofgraphshavingaM-cospectralmateforaparticularmatrixM.Cospectralityandcoinvari- ancyboth playan importantroleinthe famousgraphisomorphismproblem. Whileitis unknownwhethertestinggraph isomorphismisahardproblemornot,determiningwhethertwographsarecospectralorcoinvariantcanbedoneinpoly- nomial time[20,27].Itisalsoknownthat testingcoinvariancyisexperimentallyfasterthantestingcospectrality[2].Thus, onecanfocusontestingisomorphismamongcoinvariantgraphs.

OurresultsshowthattheinvariantfactorsofthedistancesignlessLaplacianmatrixprovideawayofrepresentinggraphs whichdoesabetterjobthanthespectrumindistinguishingthem.ThedistanceLaplacianandthedistancesignlessLaplacian matriceshavereceived quitea lotofattentionover thelast years[7,8,10,15,22,23].Thisarticleisa sequelto theworkby AouchicheandHansen[6].Numericaldataonthenumberofcospectralandcoinvariantgraphsisgivenforseveralmatrices, andwealsotaketheopportunitytocorrectanearliervalue.ThispaperalsocomplementstheworkbyHaemersandSpence [18],LepovicitePLXBIB0022andGodsilandMcKay[17]onenumeratingcospectralgraphs.

In particular,we extendthecomputer enumerationforcospectral graphsof[17,18,21]and[6]to allconnected graphs onatmost10verticesthat haveatleastacospectralmatewithrespecttothedistanceLaplacianmatrixandthedistance signless Laplacianmatrix.We alsoenumerategraphswithatmost10vertices whichhaveatleastacoinvariant matefor several associated matrices. Finally, we present a novel method to show a graph characterization using the SNF of the distanceLaplaciananddistancesignlessLaplacianmatrix,illustratingthepoweroftheproposedgraphinvariant.

2. Enumeration

Since we will use severalgraph distance matrices, we focus on connected graphs such that ourenumeration results are comparable.Denote by Gn thesetof connectedgraphs withn vertices.Givena connectedgraphG,we willstudythe followingassociatedmatrices:theadjacencymatrixA(^G)^{,theLaplacianmatrixL}(^G)^{,thedistancematrixD}(^G)^,thesignless LaplacianmatrixQ(^G)^{,thedistanceLaplacianmatrixDL}(^G)^{andthedistancesignlessLaplacianmatrixDQ}(^G)^.

LetGn^sp(^M)^{bethesetofgraphsin}GnwhichhaveatleastonecospectralmateinGnwithrespecttothematrixM.Table1 providesthenumberofcospectralmatesofconnectedgraphswithrespecttoseveralassociatedmatrices.

Analogously, let Gⁱⁿn(^M)^{be thesetof graphsin}Gn which haveatleastonecoinvariant mateinGn withrespectto the matrixM.Table2showstheenumerationofGⁱⁿn(^M)^{forseveralassociatedmatrices.}

Extensive researchhasbeendevotedtounderstandcospectralgraphs,butmuchlesshasbeendedicatedtounderstand coinvariant mates and its potential to characterize graphs.The reason forthis could be that for matricesA, L,Q and D, thereisalargeproportionofconnectedgraphshavingaM-coinvariantmate,asFig.1shows.Wefollow[13]indefiningthe

Table 2

Number of connected graphs with at least one coinvariant mate for A , L , Q and D .

n 4 5 6 7 8 9 10

|^G n| ^{6 21 112 853 11,117 261,080} 11,716,571

|G _nⁱⁿ(A )| 4 20 112 853 11,117 261,080 11,716,571

|G nⁱⁿ(L )| 2 8 57 526 8027 221,834 11,036,261

|G nⁱⁿ(^Q)| 2 11 78 620 7962 201,282 10,086,812

|G nⁱⁿ(D )| 2 15 102 835 11,080 260,991 11,716,249

Fig. 1. The fraction of graphs on n vertices that have at least one cospectral mate with respect to a certain associated matrix is denoted as sp. The fraction of graphs on n vertices with respect to a certain associated matrix that have at least one coinvariant mate is denoted as in .

Table 3

Number of connected graphs with a cospectral or a coinvariant mate for D ^Land D ^Q.

n 5 6 7 8 9 10

|^G n^sp(D ^L)| ^{0 0 43 745 20,455 787,851}

|G nⁱⁿ(D ^L)| 0 0 18 455 16,505 642,002

|G n^sp−in(D ^L)| 0 0 14 435 16,006 611,987

|G n^sp(D ^Q)| 2 6 38 453 8168 319,324

|G nⁱⁿ(D ^Q)| 2 4 20 259 7444 264,955

|G n^sp−in(D ^Q)| 2 4 20 243 6676 255,964

spectraluncertaintysp_n(^M)^{withrespecttoMastheratio}

|

Gn^sp(^M)

|

/

|

Gn

|

^{,andtheinvariant}uncertaintyinn(^M)^withrespect toMastheratio

|

Gnⁱⁿ(^M)

|

/

|

Gn

|

^.

Table 3 presentsthe number ofconnected graphs havingat leastone cospectral or atleast one coinvariantmate for D^L andD^Q.Here,

|

G^spn⁻ⁱⁿ(^M)

|

^{denotes theamountofgraphswitha cospectralmatewhichisalsoa}coinvariantmatewith respect to thecorresponding matrix M. Aouchiche andHansen [6]enumeratedcospectral graphsfor D,D^L andD^Q ofall connected graphs with at most10 vertices. While most of their results are consistent with ours, in Table 3 we obtain 20,455cospectralgraphswith9verticeswithrespecttoD^L,whiletheyreportedthatthereare19,778ofsuchgraphs.

Fig.2displaysthespectralandtheinvariant uncertaintyforD^L andD^Q.WealsoincludethespectraluncertaintyforQ, sinceaccordingtoTable1,thiswouldbethebestinvariant fordistinguishinggraphsusingthespectrum.Accordingtoour results,theSNFofD^Q performsbetterthanthespectrum fordistinguishinggraphs forallconsidered matrices.Weshould alsonotethatthereisnosignificantimprovementwhenboththespectrumandtheSNFareusedtogether,astheparameter G^sp−inn (^M)^{indicatesinTable3,thusthishasnotbeenaddedinFig.2.}

Inthisworkwealsotestedthediscriminationpowerofthe p-rankondistinguishinggraphs.However,sincethep-rank can take values from0 to n, in general, it seems not such a good graph invariant. Thus we performedan enumeration of graphswiththesameSNF forthematricesintroduced aboveover F_p with p∈2,3,5,7.We usedthissincethe p-rank followsfromtheSNFofamatrixMoverFp,butnot viceversa.Theenumerationresultsshowedacleartendencytoclaim that,foranymatrixM∈

{

^A,L,Q,D,D^L,D^Q

}

^{,almostallgraphsonnverticeshaveanothergraphwiththesameSNFofMover}

Fp.Thus,wedecidednottoincludethesenumericalresultsonthetables.

Fig. 2. The fraction of graphs on n vertices that have at least one cospectral mate with respect to a certain associated matrix is denoted as sp. The fraction of graphs on n vertices with respect to a certain associated matrix that have at least one coinvariant mate is denoted as in .

Table 4

Number of connected graphs with a cospectral and a coinvariant mate for D ^Land D ^Q.

n 7 8 9 10

|G ^spn(D ^L, D ^Q)| 0 90 1965 61,909

|G ⁱⁿn(D ^L, D ^Q)| 0 44 1447 46,239

|^{G sp}n(D, D ^L)| ^{0 0 32 9449}

|G ⁱⁿ_n(D, D ^L)| 0 32 1770 92,915

|G ^spn(D, D ^Q)| 0 0 0 7712

|G ⁱⁿn(D, D ^Q)| 0 20 432 24,517

|G ^spn(D, D ^L, D ^Q)| 0 0 0 7622

|G ⁱⁿn(D, D ^L, D ^Q)| 0 0 138 12,246

Fig. 3. The fraction of graphs on n vertices that have at least one cospectral mate with respect to a certain associated matrix is denoted as sp. The fraction of graphs on n vertices with respect to a certain associated matrix that have at least one coinvariant mate is denoted as in.

AouchicheandHansen[6]alsoexploredhowtoimprovethespectraluncertaintybyconsideringtwoandthreematrices together.Analogouslyasitwasdonein[6],inTable4weshowthenumberofcospectralandcoinvariantgraphswhentwo matricesare considered.LetGn^sp(^M,N) ^{bethe setofgraphsin}Gn which haveacospectralmateinGn withrespectto the matricesM andN,andlet Gnⁱⁿ(^M,N) ^{be thesetofgraphs in}Gn whichhavea coinvariantmatein Gn withrespectto the matricesMandN.Thus,sp_n(^M,N)=

|

Gn^sp(^M,N)

|

/

|

Gn

|

^andinn(^M,N)=

|

Gnⁱⁿ(^M,N)

|

/

|

Gn

|

^.

Fig.3showsthespectralandinvariantuncertaintyofthepairsofmatricesobtainedinTable4.Forthepair(^DL,D^Q)^,we see andadvantage inconsideringthe SNF.Butfortheother pairs,we observethatthe spectrumismuch better.There is

Table 5

|G ₁₀^sp−in(M, N)|.

M\^{N A L Q D D L D Q}

A 2,151,957 24,021 22,764 1,113,103 9253 7688 L 521,200 1,059,992 121,708 192,455 562,943 44,398 Q 136,347 84,058 486,524 48,413 44,848 250,068 D 1,073,185 15,176 13,496 1,145,275 8935 7646 D ^L 300,596 563,219 52,757 110,574 611,989 47,004 D ^Q 65,627 44,475 245,529 28,061 46,941 255,964

Table 6

Number of connected graphs with a cospec- tral and a coinvariant mate for several matri- ces combination.

n 8 9 10

|^{G sp−in}n (A, D ^L)| ^{0 32 9,253}

|G ^sp−inn (D, D ^L)| 0 32 8,935

|G ^sp−inn (A, D ^Q)| 0 2 7,688

|^{G sp−in}n (D, D ^Q)| ^{0 0 7,646}

|G ^sp−sp−inn (A, D, D ^L)| 0 32 8,743

|G ^sp−sp−inn (A, D, D ^Q)| 0 0 7,550

|G ^sp−inn ⁻ⁱⁿ(A, D ^L, D ^Q)| 0 0 7,490

|^{G sp−in}n ⁻ⁱⁿ(D, D ^L, D ^Q)| ^{0 0 7,510}

a clearimprovementby takingthe spectrumofthedistancematrixtogetherwiththespectrumofeitherD^L orD^Q to the sameobtainedbytheSNF.Thiswillalsostandinthefollowinganalysis.

Let

|

Gn^sp−in(^M,N)

|

^{denote the number of graphs witha cospectral mate with respect to the matrix M that is also a}

coinvariant matewith respectto the matrixN.In Table 5,we compute

|

G₁₀^sp−in(^M,N)

|

^{forall possiblepairs of associated}

matrices.Thelowestvaluesare

|

G₁₀^sp−in(^A,D^L)

|

^,

|

G₁₀^sp−in(^D,D^L)

|

^,

|

G₁₀^sp−in(^A,D^Q)

|

^and

|

G^sp−in₁₀ (^A,D^Q)

|

^.Itisinterestingtoobserve that the combinationof thespectrum ofD withthe SNF ofD^Q givesbetter resultsthan usingonly the spectrumof the two matrices.Therefore,thissuggeststhatwhendistinguishing graphs,weshould computefirst theSNFoftheir distance signlessLaplacianmatricesandthenthespectrumoftheirdistancematrices.

In order to improve the value obtained for

|

G₁₀^sp(^D,D^L,D^Q)

|

^{, we explore the use of the following} parameters. Let

|

G^spn^−sp−in(^M1,M2,M3)

|

^{bethenumberofgraphswithacospectralmateforthematrixM1whichisalsoacospectralmate}

withrespect toM₂,andthat isalsoa coinvariantmatewithrespect tothematrix M₃.Let

|

G^sp−inn ⁻ⁱⁿ(^M1,M₂,M₃)

|

^{be the}

numberofgraphswithacospectralmateforthematrixM₁ whichisalsoacoinvariantmatewithrespecttoM₂,andisalso acoinvariantmatewithrespecttothematrixM₃.TheresultsfromthisanalysisareshowninTable6.

From Table 6, we can see that

|

G₁₀^sp−sp−in(^A,D,D^Q)

|

^,

|

G₁₀^sp−in−in(^A,D^L,D^Q)

|

^and

|

G₁₀^sp−in−in(^D,D^L,D^Q)

|

^{are better than}

|

G^sp10(^D,D^L,D^Q)

|

=7,622. Actually,the best performance is obtainedwith

|

G10^sp−in−in(^A,D^L,D^Q)

|

=7,490. Notethat the or- derincomputingeachparametermattersonlyintheabilityoftheparametertodistinguishgraphs.

Tosumup,fromtheabovecomputationalresultsone canconcludethatthebestproceduretodistinguishgraphsusing the spectrumandtheSNF isfirsttocompute theSNF oftheir D^Q matrices, sincein_n(^DQ)^{hasthe best}performance over the spectralandinvariant uncertaintyofall matrices.Then, ifnecessary,compute thespectrumoftheir A matrices,since

|

G^spn⁻ⁱⁿ(^A,D^L)

|

^islowerthan

|

Gnⁱⁿ(^DL,D^Q)

|

^forn≤10.FinallycomputetheSNFoftheD^Lmatrices.

2.1. Coinvarianttrees

WeendupSection2withan observationoncoinvarianttrees.AouchicheandHansen[8]reportedenumerationresults oncospectraltreeswithatmost20verticeswithrespecttoD,D^LandD^Q.ForD,theyfoundthatamongthe123,867trees on 18 vertices,there are twopairs of D-cospectralmates. Among the 317,955 trees on19 vertices,there are sixpairs of D-cospectralmates. There are 14 pairsof D-cospectral mates overall the 823,065 trees on 20vertices. And surprisingly, aftertheenumerationofall1,346,023treesonatmost20vertices,theyfoundnoD^L-cospectralmatesandnoD^Q-cospectral mates.ThisfactleadAouchicheandHansentoconjecturethateverytreeisdeterminedbyitsdistanceLaplacianspectrum, andbyitsdistancesignlessLaplacianspectrum.

Analogously, forthe SNFofD,D^L andD^Q oftrees onecan obtainsome similar insights.Butforthat, firstwe need to state a resultby HouandWoo[19],who extendedtheGrahamandPollak celebratedformuladet(^D(^Tn+1))=(−1)ⁿⁿ²ⁿ⁻¹ foranytreeT_n+1withn+1verticestotheSNFofthedistancematrix.

Theorem1([19]). LetTn+1 beatreewithn+1vertices,thenSNF(^D(^Tn+1))=I22In−2(²ⁿ)^.

Fig. 4. A graph and its generalized distance matrix.

ThefollowingisastraightforwardconsequencefromTheorem1,sincethisimpliesthatalltreesonnverticeshavethe sameSNFofitsdistancematrixD.

Corollary2. AlltreeswithnverticesareD-coinvariantmates.

After enumeratingcoinvariant treeswith atmost20 verticeswith respectto D^L andD^Q, we found noD^L-coinvariant mates andnoD^Q-coinvariantmates amongall treeswithupto20vertices.Thisfactlead ustoconjecturethat almostall treesaredeterminedbytheSNFofitsD^L,andanalogously,bytheSNFofitsD^Q.

3. GraphsdeterminedbytheSNF

NotmuchisknownregardinggraphcharacterizationsusingtheSNF.AfewexamplesofgraphscharacterizedbytheSNF oftheadjacencyandtheLaplacianmatricesappearin[5]or[11].However,ourcomputationalresultsfromSection2provide anindicationthatpossiblyalmostnographhasacoinvariantmatewhenn→∞forthematricesD^LandD^Q.WhiletheSNF of D^L has beenrecentlyusedto characterizecomplete graphsandstargraphs [2],toour knowledgethere isnot yetany graphcharacterizationresultusingtheSNFofD^Q.Inthissectionwewillshowthatcompletegraphscanbedeterminedby consideringtheSNFofD^Q.

Asmentionedbefore,itisknownthatcompletegraphsandstargraphsaredeterminedbytheSNFoftheD^L matrix[2]. Theorem3([2]). CompletegraphsaredeterminedbytheSNFoftheD^Lmatrix.

Theorem4([2]). StargraphsaredeterminedbytheSNFoftheD^Lmatrix.

Inthissectionweshow ananalogousresulttoTheorem3,butusingtheSNFofthedistancesignlessLaplacianmatrix.

Inordertodosoweneedtodefinethedistanceidealsofagraph,whichwerefirstintroducedin[4].

LetG=(^V,E)^{beaconnectedgraphandletX}G=

{

^{xu :u}∈V(^G)

}

^beasetofindeterminatesassociatedwiththevertices ofG.Wedenotebydiag(^XG)^{,thediagonalmatrixwhoseentriesarethe}indeterminatesinX_G.Thematrixdiag(^XG)+D(^G)^is knownasgeneralizeddistancematrixofG.FromthismatrixwecanrecoverthematricesD,D^LandD^Q justbyevaluatingthe indeterminatesX_Gatthezeroortransmissionvectors. Recallthatthetransmissiontr(^u)^ofavertexuisv∈Vd_G(^u,

v

₎,and the transmissionvectortr(^G) ^{isthevector whoseentries areassociatedwiththe}transmission oftheverticesofG.Then, D(^G)=D(^G,0)^,DL(^G)=−D(^G,−tr(^G))^andDQ(^G)=D(^G,tr(^G))^.

LetnbethenumberofverticesofG.Fork∈

{

¹,...,n

}

^{,thek-thdistanceideal I}k(^G,X_G)^{ofthegraphGisdefinedasthe} idealgeneratedbythek-minorsofD(^G,X_G)ⁱⁿZ[X_G],thatis,

^minorsk(^D(^G,X_G))

^,whereminorsk(^D(^G,X_G))^{isthesetofthe} determinantsofthek×ksubmatricesofD(^G,XG)^{.Adistanceidealissaidtobeunitortrivialiftheidealisequalto}Z[XG], equivalently,theidealisgeneratedbytheunit.

Example 5. LetGbe thegraphinFig. 4.Thesecond distanceideal I₂(^G,X_G) ^{hasasgeneratingset thefollowingGröbner} basis

^{x0+1,x1+1,x2+1,x3+1,x4+1,3}

ThethirddistanceidealisgeneratedbythefollowingGröbnerbasis

^x0x₁−2x₀−2x₁+4,x₀x₂−2x₀−2x₂+4,x₀x₃−2x₀−2x₃+4, x0x4−2x0−2x4+4,3x0−6,x1x2−2x1−2x2+4,x1x3−2x1−2x3+4, x1x4−2x1−2x4+4,3x1−6,x2x3−2x2−2x3+4, x2x4−2x2−2x4+4,3x2−6,2x3x4−x3−x4−4

^.

ThefourthdistanceidealisgeneratedbythefollowingGröbnerbasis

^2x0x₁x₂−x₀x₁−x₀x₂−4x₀−x₁x₂−4x₁−4x₂+20, x0x1x3−2x0x1−2x0x3+4x0−2x1x3+4x1+4x3−8, x0x1x4−2x0x1−2x0x4+4x0−2x1x4+4x1+4x4−8, x0x2x3−2x0x2−2x0x3+4x0−2x2x3+4x2+4x3−8,

x0x2x4−2x0x2−2x0x4+4x0−2x2x4+4x2+4x4−8, 2x0x3x4−x0x3−x0x4−4x0−4x3x4+2x3+2x4+8, x1x2x3−2x1x2−2x1x3+4x1−2x2x3+4x2+4x3−8, x₁x₂x₄−2x₁x₂−2x₁x₄+4x₁−2x₂x₄+4x₂+4x₄−8, 2x₁x₃x₄−x₁x₃−x₁x₄−4x₁−4x₃x₄+2x₃+2x₄+8, 2x2x3x4−x2x3−x2x4−4x2−4x3x4+2x3+2x4+8

. ThelastdistanceidealisgeneratedbythefollowingGröbnerbasis

^{x0x1x2x3x4−4x0x1x2−x0x1x3−x0x1x4+4x0x1−x0x2x3−x0x2x4}

+4x0x2−4x0x3x4+4x0x3+4x0x4−x1x2x3−x1x2x4+4x1x2−4x1x3x4

+4x₁x₃+4x₁x₄−4x₂x₃x₄+4x₂x₃+4x₂x₄+16x₃x₄−12x₃−12x₄−16

.

AnalternativewayofcomputingtheSNFofa matrixMisthefollowing.Letusdenotebyk(^M)^{,thegreatestcommon} divisor ofall minorsofsize k ofthe matrixM.It isknown [2]that thek-thinvariant factorof thematrixM isequal to k(^M)/k−1(^M)^,where0(^M)=1.

ThefollowingresultshowsarelationbetweentheSNFofthedistancematrixandthedistanceideals.

Proposition6([4]). Letd∈Z^V(G).If f₁

|

· · ·

|

^{frarethenon-zeroinvariantfactorsoftheSNFofthematrixD}(^G,d)^,then I_k

(

^G,d

)

=

_k

j=1

f_j

=

k

(

^D

(

^G,d

))

^{for all1}≤k≤r.

Inthisway,distanceidealscanberegardedasageneralizationoftheSNFofthedistance,distanceLaplaciananddistance signlessLaplacianmatrices.Thus,wecanrecovertheSNFofthematricesD(^G)^,DL(^G)^andDQ(^G)^{byevaluatingthedistance} idealsatX_Gequalto0,−tr(^G)^andtr(^G)^,respectively.

Inordertoshowourmainresultofthissectionweneedthefollowingtheorem.

Theorem7([4]). LetKnbethecompletegraphwithnvertices.Thek-thdistanceidealofKnisgeneratedby

n

j=1

(

^xj−1

)

+n i=1

j =i

(

^xj−1

)

^ifk=n,

j∈I

(

^xj−1

)

^:I⊂[n]and

|

^I

|

^{=i−1 ifk<n.}

FromTheorem7wecanrecovertheSNFsofthedistance(D),distanceLaplacian(D^L)anddistancesignlessLaplacian(D^Q) matricesofthecomplete graphbyevaluating theirdistance ideals.Anevaluationatx_v=0foreach

v

_∈V,we obtainthat k(^D(^Kn))=1,fork∈[n−1],andn(^D(^Kn))=

₍₋1)ⁿ+n(−1)ⁿ⁻¹

₌n−1.From whichfollowsthat theSNFofD(^Kn)^is I_n−1(ⁿ−1)^{.Byevaluatingthedistanceidealsatx}v=−n+1foreach

v

∈V,weobtaintheSNFofD^L(^Kn)^is1nI_n−20. And byevaluating thedistanceidealsatx_v=n−1foreach

v

∈V,weobtaintheSNF ofD^Q(^Kn)^is1(ⁿ−2)^In−22(ⁿ− 1)(ⁿ−2)^.

OtherconsequenceofProposition6isthatifk-thinvariantfactorofamatrixobtainedofanevaluationofthegeneralized distancematrixofG,thenthek-thdistanceidealofGisnottrivial.Thus,foranygraphGandanyd∈Z^V(G),thenumberof invariantfactorsequalto1ofthematrixD(^G,d)^{isatleastthenumberoftrivialdistanceidealsofG.Therefore,thefamily} ofgraphswithatmostktrivialdistanceidealscontainsthefamiliesofgraphswhosematricesD,D^LandD^Q haveatmostk invariantfactorsequalto1.WearegoingtousethispropertytoobtainacharacterizationofthegraphswhoseD^Q matrices haveatleastoneinvariantfactorequalto1.Forthis,weneedthefollowingresult.

Theorem8([4]). LetGbeaconnectedgraph.GhasonlyonetrivialdistanceidealifandonlyifGiseitheracompletegraphor acompletebipartitegraph.

Therefore, tofind familyofgraphs such that their D^Q matriceshave atmostone invariant factorequal to1,then we only needtoconsider thefamiliesofcompletegraphsandcomplete bipartitegraphs. Wehavealreadyseenthat theSNF ofD^Q(^Kn) ^is1(ⁿ−2)^In−22(ⁿ−1)(ⁿ−2)^{.Fromwhichfollowsthat completegraphswithatleast4 verticeshaveone} invariant factor ofSNF ofD^Q equal to one.Now we are going toprove that thisfamilyis containsall thegraphs whose distancesignlessLaplacianmatrixhaveonlyoneinvariantfactorequaltoone.

Theorem9. LetGbeaconnectedgraph.TheSNFofD^Q(^G)^{hasonlyoneinvariantfactorequalto1ifandonlyifGisacomplete} graphwithn=3vertices.

Proof. It only remains to verifythat the second invariant factorof completebipartite graphs is equalto one. In [4], the second distanceidealof completebipartite graphs were computed. Let D(^Km,n,

{

^x1,...,xm,y₁,...,yn

}

) ^{be the}generalized distancematrixofK_m,n,whichisequaltothefollowingmatrix

diag

(

^x1,...,xm

)

−2Im+2Jm Jm,n

Jn,m diag

(

^y1,...,yn

)

^−2In+2Jn

.

Ifm≥2andn=1,then

I2

(

^Km,1,

{

^x1,...,xm,y1

} )

=

^x1−2,...,xm−2,2y1−1

.

Afterevaluatingtheidealatx_i=2m−1,andy₁=m,weobtaintheideal

^2m−3,2m−1

⊆Z.Sincethegcd(^2m−3,2m− 1)=1,itfollowsthatthesecondinvariantfactorofSNF(^DQ(^Km,1))^is1.Ifm≥2andn≥2,then

I2

(

^Km,n,

{

^x1,...,xm,y1,...,yn

} )

=

^x1−2,...,xm−2,y1−2,...,yn−2,3

.

After evaluatingtheidealatx_i=2m+n−2andy_i=2n+m−2,weobtaintheideal

^2m+n−4,m+2n−4,3

⊆Z.Since thegcd(^2m+n−4,m+2n−4,3)=1,itfollowsthatthesecondinvariantfactorofSNF(^DQ(^Km,n))^is1.

Nowwearereadytostatethemainresultofthissection.

Corollary10. CompletegraphsaredeterminedbytheSNFofthedistancesignlessLaplacianmatrix.

ItwouldbeinterestingtoobtainacharacterizationofgraphswhoseSNFofD^L andD^Q hastwoinvariantfactorsequalto 1.Thiscouldbe obtainedaftershowingacharacterizationofgraphshavingtwotrivialdistanceideals.However, thelatter problemseemstobedifficult,sincethereexistinfinitelymanyminimalinducedforbiddengraphs[3](mostofthemarethe sameneededinthecharacterizationofthewell-knownStrongPerfectGraphTheorem).

4. Concludingremarks

While theadjacency,LaplacianandsignlessLaplacianmatriceshaveattractedalotofattentioninthefield ofspectral characterizations ofgraphs,forsuchmatrices, theSNFdoesnotseemusefultodistinguish graphs,since almostallgraphs on10verticeshaveacoinvariantmate.However,ourenumerationresultssuggestthattheinvariantfactorsofthedistance LaplacianandthedistancesignlessLaplacianmatricescouldbeafinerinvarianttodistinguishgraphsincaseswhereother algebraicinvariants,suchasthosederivedfromthespectrum,fail.ThisconfirmswhatwassuggestedbyBiggs[9].Another argument to considerthe SNF asa parameter to distinguish graphs isthat this isa finer invariant than the p-rank: the p-rankisjustthenumberofinvariantfactorsnotdivisiblebyp.

In this work we show that the results by Aouchiche and Hansen [6]can be extended evenfurther. Inparticular, we providenumericalevidencethatusingtheinvariant factorsoftheSNFofcertain distancematricesonecanimprovesome oftheresultsbyAouchicheandHansen.Inthisregard,ourcomputationalresultssuggestthatpossiblyalmostnographhas acoinvariantmatewhenn→∞forthematricesD^L andD^Q.

Acknowledgements

TheauthorswouldliketothankWillemHaemersforacarefulreadingofthemanuscriptandforusefulcomments.The research ofA.AbiadispartiallysupportedbytheFWOgrant1285921N.Theresearch ofC.Alfaroispartiallysupportedby CONACyTandSNI.

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