wiener index versus maximum degree in trees

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1、Wiener
ndexVersusaximumDegreeinTreesiran aFis hermannehrstuhl
furathematik,RWTH-Aa hen,52056Aa hen,Germany,s hermmath2.rwth-aa hen.de,ArneHomannehrstuhlCfurathematik,RWTH-Aa hen,52056Aa hen,Germany,homannmath .rwth-aa hen.de,1DieterRautenba hEquipeCombinatoire,UFR9

2、21,175ruedeChevaleret,75013aris,Fran e,rautenmath2.rwth-aa hen.de2aszloSzekelyDepartmentofathemati s,UniversityofSouthCarolina,Columbia,SC29208,USA,szekelymath.s .eduandutzVolkmann,ehrstuhl
furathematik,RWTH-Aa hen,52056Aa hen,Germany,volkmmath2.rwth-aa hen.deAbst

3、ra t.TheWienerindexofagraphisthesumofallpairwisedistan esofverti esofthegraph.
nthispaperwehara terizethetreeswhi hminimizetheWienerindexamongalltreesofgivenorderandmaximumdegreeandthetreeswhi hmaximizetheWienerindexamongalltreesofgivenorderthathaveonlyverti esoftwodierentde

4、grees.eywords.Tree;Wiener
ndex;AverageDistan e;Degree;Valen y1Terminologyand
ntrodu tionAllgraphsinthispaperwillbenite,simpleandundire tedandwewillusestandardgraph-theoreti alterminology.ForagraphG=(V(G);E(G)),theorderwillbedenotedbyn(G)=jV(G)jandtheneighbourhoodofavertexv2

5、V(G)willbedenotedby(v).GThedegreed(v)ofavertexv2V(G)inthegraphGisj(v)j.AvertexofdegreeoneGGisanendvertex.Themaximumdegreemaxd(v)ofagraphGisdenotedby
(G).Gv2V(G)ThesubgraphofGindu edbyasetXV(G)isdenotedbyG[X℄.Thedistan ed(u;v)betweentwoverti esu;v2V(G)inthegraphGistheminimu

6、mGnumberofedgesonapathinGfromutovor1ifnosu hpathexists.Thedistan esum1Correspondingauthor.Supportedbyapost-do toralDETgrant.2Thisresear hwassupportedinpartbytheSFgrantDS0072187.1(u)ofGwithrespe ttoavertexu2V(G)isdenedas(u)=d(u;v)andGGGv2V(G)thedistan esum(G)ofGorWien

7、erindexofGisdenedasXXX11(G)=d(u;v)=(u):GG22u2V(G)v2V(G)u2V(G)(G)Theaveragedistan ed(G)ofagraphGis.n(G)()2TheWienerindexandtheaveragedistan erankamongthosegraph-theoreti alparametersthatareofmostinteresttoothers ien es.
nfa t,itwasthehemistH.Wienerwhoin1947proposedin[19℄

8、and[20℄asameasureforthedegreeofmole ularbran hingwhi hseemstoberelat