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AUSTRALASIANJOURNALOFCOMBINATORICSVolume2003Pages239244Adirectedversio


ThisresearchwassupportedbytheYouthScienceFoundationofHebeiNormalUniversitytheYouthScienceFoundationofBeijingNormalUniversity105107197andNationalNaturalScienceFoundationofChina10171006ZHANGGENGSHENGAND

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Document on Subject : "AUSTRALASIANJOURNALOFCOMBINATORICSVolume2003Pages239244Adirectedversio"— Transcript:

1 AUSTRALASIANJOURNALOFCOMBINATORICSVolume
AUSTRALASIANJOURNALOFCOMBINATORICSVolume(2003),Pages239–244AdirectedversionofDezagraphs—DezadigraphsZhangGengshengCollegeofMathematicsandInformationScienceHebeiNormalUniversityShijiazhuang,050016WangKaishunDepartmentofMathematicsBeijingNormalUniversityBeijing100875AsageneralizationofDezagraphs,weintroduceDezadigraphsanddescribethebasictheoryofthesegraphs.Wealsoprovethenecessaryandsucientconditionswhenaweaklydistance-regulardigraphisaDeza1IntroductionIn[2],Erickson,Fernando,Haemers,HardyandHemmeterintroducedDezagraphsasageneralizationofstronglyregulargraphs.TheyintroducedseveralwaystoconstructDezagraphs,anddevelopedsomebasictheory.DeŢnition1.1Supposeisanundirectedgraphwithvertices,andisitsadjacencymatrix.iscalledan(n,k,b,cDezagraphkI,kJ,forsome(0suchthat,theallonesmatrix. ThisresearchwassupportedbytheYouthScienceFoundationofHebeiNormalUniversity,theYouthScienceFoundatio

2 nofBeijingNormalUniversity(105107197)and
nofBeijingNormalUniversity(105107197)andNationalNaturalScienceFoundationofChina(10171006) ZHANGGENGSHENGANDWANGKAISHUNNotethatisastronglyregulargraphifandonlyifInthispaper,weconsiderthedirectedversionofDezagraphsanddevelopsomebasictheory.Moreover,wediscusstheconnectionstoweaklydistance-regulardi-DeŢnition1.2Letbeadigraphwithverticesandletbetheadjacencymatrixof.issaidtobean(n,k,b,c,tDezadigraphtI,forsome(0suchthat,theallonesmatrix.Notethatif,thenan(n,k,b,c,t)-Dezadigraphisan(n,k,b,c)-Dezagraph.Itiseasytoseethatwecangetanequivalentde“nitionofDezadigraphsfromacombinatorialviewpoint.DeŢnition1.3Adigraphwithverticesisan(n,k,b,c,tDezadigraphifforu,vu,ww,vWenextgivesomeelementaryconstraintsontheparameters.Proposition1.1Letbeann,k,b,c,t-Dezadigraph.DeŢne,foravertexdonotdependon b,=2Št+cŠ Šc,=k2Št b,=2Št+bŠ Proof.Letbethenumberoforderedtriples(u,w,v)withu,ww,v.Thatis,u,w,vu

3 ,ww,vw,u=1,thenthenumberoftriples(u,w,v)
,ww,vw,u=1,thenthenumberoftriples(u,w,v)is(,whileifw,u)=1,thenthenumberoftriples(u,w,v)is(.Thus DEZADIGRAPHS,thenc.Otherwise,bythede“nitionof,wehavec.Ifweequatethesetwoexpressionsforanduse1,then Corollary1.2Supposebc.Thenthefollowinghold:bk2ConstructionsFirstlywewillgiveaconstructionofDezadigraphsusingCayleydigraphs.Proposition2.1LetbeaŢnitegroupoforderandletbea-subsetofcontainingtheidentityelement.IfwhereB,Cpartition,thentheCayleydigraphCayG,Sisann,k,b,c,tDezadigraph.Proof.Theproofisobviousandwillbeomitted. Letandbedigraphs.Thelexicographicproductduct2]ofandisadigraphwithvertexset)andadjacencyde“nedby))=1ifandonlyif)=1orLetbeadigraphwithadjacencymatrixvertices.iscalledastronglyregulardigraphwithparametersn,k,µ,,t),ifkJ.Theparametersarerelatedbytheequation))=Thesegraphswere“rstinvestigatedbyDuvalin[1].Formoreinformationaboutstronglyregulardigraphs,see[3],[4].Note

4 thatif,thenaDezadigraphisastronglyregula
thatif,thenaDezadigraphisastronglyregulardigraph.ThenexttheoremtellsushowtoderiveaDezadigraphusingastronglyregular ZHANGGENGSHENGANDWANGKAISHUNTheorem2.2Letbeastronglyregulardigraphwithparametersn,k,,µ,tbean,b,c,t-Dezadigraph.ThenThen2]isaDezadigraphifandonly,µn,n}|Proof.Let)and)beverticesof2].ThenHence2]isaDezadigraphifandonlyifthesenumberstakeonatmosttwovalues. Twocorollariesfolloweasilyfromthetheorem.Corollary2.3Letbeastronglyregulardigraphwithparametersn,k,,µ,,andbeacompletegraphonvertices.ThenThenKm]isaDezadigraphifandonly=1andCorollary2.4Letbeastronglyregulardigraphwithparametersn,k,,,tandlet beacocliqueonvertices.Then isan,kn,n,n,tnDezadigraph.Theorem2.5Letbetwodigraphs.TheproductaDezadigraphifandonlyifitisinthelistbelow. forsomeisan,k,b,c,t-Dezadigraphwithisann,k,b,c,t-Dezadigraphandisan-Dezadigraph,whereb,cProof.Firstnotethatisregularifandonlyifbotho

5 fandareregular.Moreover,thedegreeofis
fandareregular.Moreover,thedegreeofisthesumofthedegreesofandNowsuppose)and)aretwodistinctverticesof.ItiseasytocheckthatCase1. forsome2.Thenthethirdcaseforthesizeofgivenabovedoesnotoccur.SoisaDezadigraphifandonlyifisan,k,b,c,t)-Dezadigraphwith=0.Thus(i)occurs.Case2,Both0and2appearasthevalueof,so(ii)follows. DEZADIGRAPHS3Connectiontoweaklydistance-regulardigraphsInthissection,wewilldiscusstheconnectionstoweaklydistance-regulardigraphs.Foranytwoverticesx,y(),de“nex,yx,yy,xDeŢnition3.1Aconnecteddigraphissaidtobeweaklydistance-regularx,yx,zz,ydependsonlyonanddoesnotdependonthechoicesofx,yAsanaturalgeneralizationofdistance-regulargraphs,weaklydistance-regulardigraphswereintroducedin[5].Theorem3.1Letbeaweaklydistance-regulardigraphofdiameter.Letx,y)=(1)forsomex,yisaDezadigraphifandonlyiftakesonatmosttwovaluesasrangesoverx,yx,yProof.Letbetwoverticesofwithu,v.ThenHence

6 ,isaDezadigraphonlywhenthesenumberstake
,isaDezadigraphonlywhenthesenumberstakeonatmosttwovalues. Weknowthat=Cay()isaweaklydistance-regulardigraph.Bytheabovetheorem,itisaDezadigraph.Notethataweaklydistance-regulardigraphisdistance-regularx,yy,x)forallx,y,xCorollary3.2Adistance-regulardigraphofdiameterisaDezadigraphifandonlyifoneofthefollowingholds. ZHANGGENGSHENGANDWANGKAISHUN[1]A.Duval,Adirectedversionofstronglyregulargraphs,J.Combin.TheorySer.(1988),71…100.[2]M.Erickson,S.Fernando,W.H.Haemers,D.HardyandJ.Hemmeter,Dezagraphs:ageneralizationofstronglyregulargraphs,J.Combin.Designs[3]S.A.HobartandT.J.Shaw,AnoteonafamilyofdirectedstronglyregularEuropeanJ.Combin.(1999),819…820.[4]M.Klin,A.Munemasa,M.MuzychukandP.-H.Zieschang,Directedstronglyregulargraphsviacoherent(cellular)algebras,Preprint,Kyushu-MPS-1997-12,KyushuUniversity,1997.[5]H.SuzukiandK.S.Wang,Weaklydistance-regulardigraphs,preprint.(Received24June200