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1、WienerndexVersusaximumDegreeinTreesiran
aFis
hermannehrstuhlfurathematik,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,ehrstuhlfurathematik,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;Wienerndex;AverageDistan
e;Degree;Valen
y1Terminologyandntrodu
tionAllgraphsinthispaperwillbenite,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)isdenedas(u)=d(u;v)andGGGv2V(G)thedistan
esum(G)ofGorWien
7、erindexofGisdenedasXXX11(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