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1、GraphicalEnumerationofConformationsofChainsMILANRANDIc*AmesLaboratory,US.DepartmentofEnergyandDepartmentofChemistry.IowaStateUniversity,Ames,Iowa5001I.U.S.A.AbstractApictorialschemewithanequivalentnumericalinterpretationfortheenumerationofconfor-mationsoflongchainsisoutlined.Itex
2、ploitsaI:Icorrespondencebetweenspeciallyconstructedmoleculargraphsuniquelycharacterizingaconformationandcombinatorialobjectsinvolvingthreesymbols.Notallcombinationsofsymbolsarepermitted;neverthelesstheymaybeviewedasnumbersinaternarysystem.whichpermitstheirsimpleordcringandsubsequ
3、entlytheassignnentofanordinalnumbertoeachacceptableconformation.Analgorithmfortheenumerationofconformationsinalongchaincanhencebebasedontheconstructionofternarynumbershaving(n-2)digits,whicharesubjecttosomeconstraints,asindicatedinthetext.IntroductionTheenumerationofstructuresofp
4、rescribedformsdatesbacktomid1800whenCayleyopenedthefieldwithapaperconcernedwiththeenumerationofacyclicstructures:“Onthetheoryoftheanalyticalformscalledtrees”[I].Theapplicationsandimportanceofsuchenumerationssoonbecamerecognizedinchemistry.Equallyrecognizedweretheinherentdifficult
5、iesaccompanyingtheenumerations,suchasthepossibilityofomissionsandthepossibilityofnotrecognizingduplicates.Eventodayanumberofimportantproblemsinenu-merationremainunsolved,suchasthecountofstructuresbuiltfromfusedhexagons,knowninthemathematicalliteratureasoneexampleofthecellgrowthpr
6、oblem123,andthequestionofthenumberofpolyhedraofagivennumberofvertices[3].Theformerproblemisofconsiderableinterestinthechemistryofbenzenoidsystems,sincemanysuchstructurescorrespondtostablechemicalcompounds.Thedifficultyofsuchproblemsiswellillustratedbythefactthatin1900anattemptwas
7、madetoenumeratepolyhedraon16vertices,butdespitethepresentavailabilityofcomputers,theresult,whichisinsomedoubt,couldnotyetbec0nfirmed.tEnumerationofpositionalisomersandconformationsis,strictlyspeaking,beyondthecapabilityofgraphtheory,whichisfundamentallyconcernedwithbinaryrelation
8、sortheconnectivity.Allpositionalisomerso
