DOC

A_11

By Kathryn Ward,2014-09-20 14:47
11 views 0
A_11A,a,A_11,a_11

    A

aminimalgeneratingsetofS,where?5.ItisprovedthatCayleygraphCay(S,

    ;M:UM;)isHamittonianandedgesymmetric.

    ;LetGbeafinitegroup.andSbeageneratingsetofGwith1SandS=S.The

    ;CayleygraphX=Cay(G?S)onGwithrespecttoSisdefinedby

    ;V(x)G.g(x)((g.gs)g?G,s?S}.

    ;I_etSbethesymmetriegroupwithidentity(1).Letg=(123i).g(1i32)andMt

    ;

    ;{;4??),thenitiseasytoprovethatisaminimalgeneratingsetofS,where

    ;?5.Let={g.:4?i--_<n}andMMUM2,thenCay(S,U)Cay(S,).

    ;LetS,]{d?S:()).

    ;Inthispaper,permutationisdenotedbyproductofcycles.Theproductofcycles口口

    ;meansthatisfollowedbyot.Forexample(12)(13)(132).

    ;Agraphissaidtoheedgesymmetricifforeverypairofedges,sayaandb,thereex ;istsanautomorphismotthegraphthatmapsnintob.

    ;Thenotationsanddefinitionsnotdefinedherecanbefoundinr1.2]. ;Manyinterconnectionnetworksproposedintheliteraturearesyrnmetric.Aintercon

    ;nectionnetworkissymmetricifitlooksthesamefromeachofitsvertices.Cayleygraph ;canheusedasagrouptheoreticmodelforsymmetricinterconnectionnetworks.In[3], ;Aker,eta1.definedthesocalledtranspositiontreeandstudiedvariouspropertiesofits ;Cayleygraph,whichis,inourterminology,aCayleygraphofSIn-4,j],weprovedthat ;CayleygraphCay(S,S)isHamiltonian.whereSisgeneratingsetofS,eachelementof ;whichisatranspositionofS.In[63,jwoJungSing,eta1.provedthatCayleygraphCay

    ;(A,n)isHamiltonian,whereAisthealternatinggroupandn{(12i):3?i?)U

    ;{(1i2):3??n}.Inthispaper,weobtainthatCay(S.)isedgesymmetricandHamil

    ;Received{1998O622.Revised:19981l20

    emnos.

    ;Lemnm1.LetCandPbeaHamiltoniancircuitandpathinCay(5,M),respectvely,and ;if(1)andor(1)andareadjacentinCandP,where6??.Then

    ;(a)thereares1,s2,s3andssuchthatC(orP),sts8C,s3CandCJoinalargecir

    ;cult(orpath)inCayS…M1),wheres.?S1[+1”;

    ;(b)therearesand”suchthatC(orP),siCand”Cjoinalargecircuit(orpath)in

    ;Cay(S…+1).

    ;Lemma2.IetCandPbeaHamihoniancircuitandpathinCay(5…M)trespectively.If

    ;珥偿?anda-iglgor.gFand..gTg+areadjacentinCandP,where4?i??,then

    ;thereare.and.suchthatC(orP),.vie.ands,Cjoinalargecircuit(orpath)inCay ;(s+…M1),wheres.?SHl+1,andi,JareinvariableundertofS.

    ;Lemma3.LetandPbeaHamiltoniancircuitandpathinCay(S,).respectively.If ;‘5(15)and(12345jareadjacentinCandP,thentherearesands5suchthatG(or

    ;P)candssCjoinalargecircuit(orpath)inCay(5l,+1),where.?S一】+1,]

;and4,5areinvariableunder/r45ofS.

    ;Lemma4.IJetPbeaHamiltonianpathIoining/rand/r(35)(124)inCay(5…M),then

    ;thereare,andsuchthatP,P,53PandPjoinalargecircuitinCay(5+J,+),

    ;wheresES+[+1,:and2,3and5areinvariableunderofS.

    ;InCay(S5,)weobtainthefollowingHami[toniancircuitC5.

    ;(1),(1432),(345),(1254),(15234),(245)(13),(14532),(153)(24),(12534),(35), ;(125).(1435),(12543),(1354),(45)(23),(1542),(234),(1324),(35)(14),(2453), ;(145),(15)(243).(143).(12),(235),(1452),(15243),(25)(134),(5324),(1453),(45) ;(12),(2354),(13425),(1235),(25)(1g),(235)(147,(13254),(1543),(12)(34),(i3), ;(35)(12),(2534.(15)(24),(125)(34),(135),(145)(237,(45)(13),(1j4)(23),(134), ;(23),(142),(1243),(14352),(25),(15)(32),(1343),(12453),(35)(142),(254),(354) ;(12),(25)(34),(1352),(253),(1534).(124),(34),(12435),(1425),(15423),(45)(132), ;(345),(35)(124).(25)(14),(2435).(13452),(1523),(132),(1423),(13542),(45)(123), ;(154).(2543),(15)(34),(1245).(14523),(1524),(14)(23),(24),(14235),(1325), ;(153).(253)(14),(35)(24),(345)(12).(245),(152)(34).(243),(1342),(123),(14), ;(12354),(254)(13),(15342),(1253),(152),(2345).(13245),(15)(234),(13)(24), ;(1234),(13j24).(25)(143),(15432),(45),(12345),(15),(14325),(13j)(24),(14253), ;r???I

    ;r

    ;

    ;494App1.Math.J.ChineseUniv.Ser.BVol_L4.No.4

    ;(1532),(1).

    ;InCay(S5,M5)weobtainthe{ollowingHami|tonianpathPs.

    ;(1j,(1234),(13524),(25)(143),(152),(2345),(13245),(15)(234),(13)(24), ;(1432),(354),(1254),(15234),(245)(13),(14532),(153)(24).(12534),(35),(125), ;(1435),(12543),(1354),(45)(23).(1542),(234),(1324),(35)(14),(1253),(15342), ;(2453).(145).(15)(243),(143).(12),(235).(1452),(15243),(25)(134),(15324), ;(1453),(45)(12).(2354),(13425),(1235),(25)(13).(235)(14),(13254),(1543),(12) ;(34),(24),(14235),(1245),(14523),(1524),(14)(23),(13),(35)(12),(2534),(15) ;(24).(125)(34),(135),(145)(23).(45)(13),(154)(23),(134),(23),(142),(1243), ;(14352),(25).(15)(23).(1345),(12453),(35)(142).(254),(354)(12),(25)(34), ;(1352j,(253),(1534).(124),(34),(12435),(1425),(15423),(45)(132),(345),(1523), ;(14253),(35)(24),(4325),(5),(12345),(45),(15432),(254)(J3),(12354),(14),

    ;‘123),(1342),(243),(152)(34),(245)(345)(12),(35)(24),(253)(14),(153),(1325),

    ;(15)(34).(2543),(154),(45)(123),(13542),(1423),(132),(1523),(13452),(2435), ;(25)(14).(35)(124).’

    ;Ingeneralweobtainthefollowingresultbyinductionon.

    ;Theorem3.Gay(S,)?5)isHamihonian.

    ;References

    ;1HungeHord,ThomasW.,Algebra.SpringerVerlag.NewYork,1974.

    ;2Bondy?J.AaadMurty,U.S.R..GraphTheorywithApplications,London,MacmillanPressLtd ;l976.

    ;3Akers,S.B.

    KrishnamurthyB.,Agrouptheoreticmodelforsymmetricinterconnectionnetworks.

;IEEETrens.Comput.,1989,38(4).555666.

    ;4WangShiyingtHamiltonianpropertyofCayleygraphsonsymmetricgroups(I),JournalofXinjiang ;Unversity,L994,ll(3):1618.

    ;6WangShiying,HamiltonianpropettyofCayleygraphsonsymmetricgroups(?)1ourr~lofX1.nii~ng

    ;University.1994,(4)j2535.

    ;6woJungSing,Lakshmivarahan,S.andDhall,SK.Anewclassot1Ilterc0nnectlonnetworksbased ;o13thea]ternatinggroup,Networks,1993,23:315~326.

    ;Dept.ofMath?,ZhengzhouUniv.,Zhengzhou450052;InstituteofEconomics.XinjiangUniv.,Urumqi ;830046

    ;Dept.ofFoundation,ShanxiElectricPowerProfessionalUniv.,Taiyuan030013. ;Dept.ofMath.,ZhengzhouUniv.,Zhengzhou450052.

    ;

    ;

Report this document

For any questions or suggestions please email
cust-service@docsford.com