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1、HindawiPublishingCorporationJournalofAppliedMathematicsVolume2012,ArticleID520156,8pagesdoi:10.1155/2012/520156ResearchArticleTheMerrifield-SimmonsIndexandHosoyaIndexofCn,k,λGraphsShaojunDai1andRuihaiZhang21DepartmentofMathematics,TianjinPolytechnicUniversity,N
2、o.399BinshuixiRoad,XiqingDistrict,Tianjin300387,China2DepartmentofMathematics,TianjinUniversityofScienceandTechnology,Tianjin300457,ChinaCorrespondenceshouldbeaddressedtoShaojunDai,daishaojun1978@sina.comReceived29May2012;Revised18June2012;Accepted27June2012Acad
3、emicEditor:FerencHartungCopyrightq2012S.DaiandR.Zhang.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.TheMerrifield-Simmonsind
4、exiGofagraphGisdefinedasthenumberofsubsetsofthevertexset,inwhichanytwoverticesarenonadjacent,thatis,thenumberofindependentvertexsetsofGTheHosoyaindexzGofagraphGisdefinedasthetotalnumberofindependentedgesubsets,thatis,thetotalnumberofitsmatchings.ByCn,k,λwede
5、notethesetofgraphswithnvertices,kcycles,thelengthofeverycycleisλ,andalltheedgesnotonthecyclesarependantedgeswhichareattachedtothesamevertex.Inthispaper,weinvestigatetheMerrifield-SimmonsindexiGandtheHosoyaindexzGforagraphGinCn,k,λ.1.IntroductionLetGVG,E
6、GdenoteagraphwhosesetofverticesandsetofedgesareVGandEG,respectively.Foranyv∈VG,wedenotetheneighborsofvasNGv,andvNGv∪{v}.Byn,wedenotethenumberofverticesofG.Allgraphsconsideredherearebothfiniteandsimple.Wedenote,respectively,bySn,Pn,andCnthestar,pat
7、h,andcyclewithnvertices.Forothergraph-theoreticalterminologyandnotation,wereferto1.ByCn,k,λwedenotethesetofgraphswithnvertices,kcycles,thelengthofeverycycleisλandalltheedgesnotonthecyclesarependantedgeswhichareattachedtothesamevertex,wheren1n−λ−1k1≥0an
8、dthevertexvdenotesthecentralvertexofthegraphs,asshowninFigure1.TheMerrifield-SimmonsindexiGofagraphGisdefinedasthenumberofsubsetsofthevertexset,inwhichanytwoverticesa
