The Topological Indices of some Dendrimer Graphs

Transcription

1 Iraa J Math Chem 8 March Iraa Joral of Mathematcal Chemstry Joral homepage: wwwjmckashaacr The Topologcal Ices of some Dermer Graphs M R DARASHEH a M NAMDARI b AND S SHOKROLAHI b a School of Mathematcs Statstcs a Compter Scece College of Scece Uersty of Tehra Tehra Ira b Departmet of Mathematcs Shah Chamra Uersty of Ahaz Ahaz Ira ARTICLE INO Artcle Hstory: Recee 6 Jaary 5 Accepte Jaary 6 Pblshe ole 5 September 6 Acaemc Etor: Hassa YosefAzar Keywors: Topologcal ex Dermer Weer ex Hyper Weer ex ABSTRACT I ths paper the Weer a hyper Weer ces of two ks of ermer graphs are compte Usg the Weer ex formla the Szege Schltz PI a Gtma ces of these graphs are also eterme 7 Uersty of Kasha Press All rghts resere INTRODUCTION Let G = E be a smple coecte graph wth ertex set a ege set E A topologcal ex of a smple coecte graph G s a graph arat whch s relate to the strctre of the graph The Weer ex s oe of the best kow topologcal ex of a smple coecte graph whch s ste both mathematcal a chemcal lteratre a t's efto s terms of staces betwee arbtrary pars of ertces see for example [ ] The Weer ex of G s eote by W G a t s efe by: W G G where s the stace betwee ertces a a Correspog Athor: Emal aress: shokrolahsara@yahoocom DOI: 5/jmc75

2 6 DARASHEH NAMDARI AND SHOKROLAHI The Szege ex [5 6] s aother arat of a graph whch s base o the strbto of the ertces a troce by Ia Gtma a t s the same wth the Weer ex the case that G s a tree The set of ertces of graph G whch are closer to resp tha resp s eote by N e G resp e G Ths ex s efe as the smmato of e G e G where e G resp e G s the mber of ertces of graph G closer to resp tha resp oer all eges of graph Now the Szege ex of G whch s eote by Sz G s efe as: Sz G e G e G e E N e The Pamaker-Ia PI ex [7 8] s aother topologcal ex of a smple coecte graph that takes to accot the strbto of eges so s closely relate to Szege ex The PI ex of G s efe by e E PI G e G e G where e e G resp e G s the mber of eges of the sbgraph of G whch has e the ertex set N e G resp N e G The moleclar topologcal ex Schltz ex was troce by Schltz a Schltz [9 ] I ato to the chemcal applcatos the Schltz ex attracte some atteto that the case of a tree t s relate to the Weer ex [] It s eote by SG a efe as follows: e S G where resp s the egree of ertex resp The Gtma ex whch attracts more atteto recetly s efe by Klažar a Gtma [ ] Ths ex s also kow as the Schltz ex of the seco k bt ths paper the frst ame s se Gtma [] has proe that f G s a tree the there s a relato betwee Weer a Gtma ces of G that we wll meto ths Secto The Gtma ex of G s eote by GtG a s efe as follows: Gt G The hyper Weer ex s oe of the graph arats se as a strctre escrptor relate to physcochemcal propertes of compos Ths ex was troce by Rać 99 as exteso of Weer ex [] a t has come to be kow as the hyper Weer ex by Kle [] The hyper Weer ex of G s eote by WW G a s efe as follows: WW G W G e

3 The Topologcal Ices of some Dermer Graphs 7 Here we maly try to eterme the Weer hyper Weer a PI ces of two ks of ermer graphs explae Secto the the Schltz Szege a Gtma ces are obtae as reslts of the relato betwee the Weer ex wth both the Schltz a Gtma ces + - gre The frst ermer graph G CALCULATING THE WIENER HYPER-WIENER AND PI INDICES O THE G IRST DENDRIMER GRAPH Let G = E be the graph wth ertex set a ege set E as gre Ths graph begs wth oe ertex whch coects to two other ertces sch that each oe of these two ertces coects to two other ertces a so o The ertces whch hae the same stace from are locate o a brach Let G hae braches so there are ertces the '-th brach We eote ths graph by G Proposto Let G E be the ermer graph gre the: W G Proof rom eftos we hae: W G G Ths graph has braches a there are eote the ertex set of ths brach by symmetrc strctre of the graph ertces the ' th brach so we hece we hae: Becase of the G gre l for eery ertex the 'th brach s costat a oes't epe o So we choose as represetate of the ' th brach

4 8 DARASHEH NAMDARI AND SHOKROLAHI ertces whch are lower brach of gre l are of the same stace from a ths ale eqals to: Also = ertces are of the same stace from a ths ale eqals to: ally cotg ths way the stace betwee to the last ertex the ' th brach s eqals to: So we hae: or comptg the seco part of the smmato ote that becase the graph G s a tree for eery ertex we hae: Coserg a : Becase Hece : o 6 By mltplyg cosere twce so f the Weer ex of W resp W we hae: the stace betwee ertces the ' th brach s G wth resp brach s eote by W W

5 The Topologcal Ices of some Dermer Graphs 9 So k k + + W k 5 k Corollary Sz G Proof The graph G s a tree so by [] the reslt s obtae + + Corollary S G Proof Becase G s a tree by [] we hae: S G W G - - where s the mber of ertces of G Now by replacg the close form of W G whch was obtae from proposto the proof s complete + + Corollary Gt Proof Becase G s the mber of ertces of G s a tree by [] we hae Gt G W G where G a by proposto t s oe + + Corollary 5 PI - - Proof Becase G Sbgraphs of G s a tree so for eery ege e G e G e of G we hae: G wth ertex sets N e G a N e G both are trees a whose mber of eges are e a e respectely The we hae: G G e e G E PI G + - e e G E e G e G Proposto 6 The hyper Weer ex of WW G - G gre s: Proof By efto we hae: WW G W G 5

6 DARASHEH NAMDARI AND SHOKROLAHI Becase of the symmetrc strctre of the graph G gre l for eery ertex the ' th brach s costat a oes't epe o so we choose as represetate of the ' th brach 6 The graph G s a tree so for eery ertex : Now by 6 we hae: 7 ertces are the ' th brach a by symmetrc strctre of the graph G we hae : 8 By the proof of the proposto a coserg 7 8: 8 9 Therefore: 6 Now let j So

7 The Topologcal Ices of some Dermer Graphs Now coserg 9 a the formla of W G whch was compte proposto a replacg those 5 the proof s oe U gre The seco ermer graph H CALCULATING THE WIENER HYPERWIENER AND PI INDICES O THE SECOND DENDRIMER GRAPH H Let G = E be the graph wth ertex set a ege set E that begs wth oe ertex gre that coects to three ertces whch form the frst brach a each oe of these three ertces coects to two other ertces seco brach a so o It meas that ay ertex bt the ' th brach jos to the two ertces the +' th brach so the ertces whch hae the same stace from are locate o oe brach Let G hae braches therefore there are ertces the ' th brach The graphg s aother k of ermer graph whch hae braches whch s eote by H Proposto Let H E be the ermer graph gre the: W H Proof The graph H cossts of a startg ertex a + braches sch that the ertex set of the ' th brach has ertces a s eote by a So we hae:

8 DARASHEH NAMDARI AND SHOKROLAHI Becase of the symmetrc strctre of the graph G gre for eery ertex the ' th brach s costat a oes't epe o so we choose as represetate of the ' th brach / ertces ' th brach hae the same stace from whch s: A the stace of / of the rest ertces ths brach from s: By cotg ths way we hae: Now becase H s a tree the path betwee ay two ertces s qe a for eery ertex we hae: So: By a we hae: If the Weer ex of H wth braches s eote by W we hae: W W Therefore W A the proof s complete

9 The Topologcal Ices of some Dermer Graphs Corollary Sz H Proof The graph H s a tree so by [] the reslt s obtae Corollary S Proof Becase H H s a tree by [] we hae S G W G - - where s the mber of ertces of H Now by replacg the close form of W H whch was obtae from proposto the proof s complete Corollary Gt Proof Becase G s the mber of ertces of H s a tree by [] we hae Gt G W G whch H a by proposto t s oe Corollary 5 PI - - Proof Becase H Sbgraphs of H s a tree so for eery ege e H e H e of H we hae: H wth ertex sets N e H a N e H both are trees so the mber of eges of them are e H a e H respectely The we hae: e PI H e H e e H E e H e H Proposto 6 The hyper Weer ex of H s: WW H Proof By the efto we hae: WW H W H Becase of the symmetrc strctre of the graph H gre for eery ertex the ' th brach s costat a oes't epe o so we choose as represetate of the ' th brach 5

10 DARASHEH NAMDARI AND SHOKROLAHI The graph H s a tree so for ay ertex : Now by 5 we hae: 6 / ertces of the ' th brach hae the same stace from whch s: a the stace of / of the rest ertces ths brach from s: 6 7 By the proof of the proposto a coserg 6 7: 8 Therefore: Now let j j So we hae: Now coserg 8 a the formla of H W whch was compte Proposto a replacg those the proof s oe

11 The Topologcal Ices of some Dermer Graphs 5 REERENCES R Etrger Dstace graphs: Trees J Comb Math Comb Compt I Gtma S Klažar B Mohar Es fty years of the Weer ex MATCH Comm Math Compt Chem H Hosoya Topologcal Iex A Newly Propose Qatty Characterzg the Topologcal Natre of Strctral Isomers of Satrate Hyrocarbos Bll Chem Soc Jp 97 9 H Weer Strctral Determato of Paraff Bolg Pots J Am Chem Soc I Gtma A formla for the Weer mber of trees a ts exteso to graphs cotag cycles Graph Theory Notes NY S Klažar A Rajapakse I Gtma The Szege a the Weer ex of graphs Appl Math Lett P Khakar O a Noel strctral escrptor PI Nat Aca Sc Lett 8 8 P Khakar S Karmakar a K Agrawal Relatoshp a relate correcto potetal of the Weer Szege a PI Ices Nat Aca Sc Lett H P Schltz T P Schltz Topologcal orgac chemstry Graph theory Bary a Decmal Ajacecy Matrces a Topologcal ces of Alkaes J Chem If Compt Sc 99 7 H P Schltz T P Schltz Topologcal orgac chemstry 6 Theory a topologcal ces of cycloalkaes J Chem If Compt Sc 99 I Gtma Selecte propertes of the Schltz moleclar topologcal ex J Chem If Compt Sc I Gtma Y N Yeh S L Lee Y L Lo Some recet reslts the theory of the Weer mber Ia J Chem M Rać Noel moleclar escrptor for strctre property stes Chem Phys Lett D J Kle I Lkots I Gtma O the efto of the hyper Weer ex for cycle-cotag strctres J Chem If Compt Sc