Iranian Journal of Mathematical Chemistry, Vol. 2, No. 2, December 2011, pp (Received September 10, 2011) ABSTRACT

Transcription

1 Iraa Joral of Mathatcal Chstry Vol No Dcbr IJMC Two Tys of Gotrc Arthtc dx of V hylc Naotb S MORADI S BABARAHIM AND M GHORBANI Dartt of Mathatcs Faclty of Scc Arak Ursty Arak I R Ira Dartt of Mathatcs Faclty of Scc Shahd Raja Tachr Trag Ursty Thra I R Ira Rcd Stbr 0 0 ABSTRACT Th coct of gotrc-arthtc dcs was trodcd th chcal grah thory Ths dcs ar dfd by th followg gral forla: G Q Q GA G Q Q whr Q s so atty that a ar ca b assocatd wth th rtx of grah G I ths ar th xact forla for two tys of gotrc-arthtc dx of V- hylc aotb ar g Kywords: GA dx V hylc aotb INTRODUCTION Throghot ths scto G s a sl coctd grah wth rtx ad dg sts VG ad G rsctly A toologcal dx s a rc atty fro th strctr of a grah whch s arat dr atoorhss of th grah dr cosdrato A toologcal dx s a rc atty fro th strctral grah of a olcl Usag of toologcal dcs chstry bga 97 wh chst Harold Wr dlod th ost wdly kow toologcal dscrtor th Wr dx ad sd t to dtr hyscal rorts of tys of alkas kow as araff Th coct of gotrc-arthtc dcs was trodcd th chcal grah thory Ths dcs Arch of SID Athor to who corrsodc shold b addrssd al: ghorba@srttd

2 0 S MORADI S BABARAHIM AND M GHORBANI QQ grally ar dfd as GA gral GAgral G whr Q G s so Q Q atty that a ar ca b assocatd wth th rtx of grah G Th frst ty of gotrc-arthtc dx s dotd by GA ad dfd as d d GA GA G G d d whr s a dg of th olclar grah G ad d stad for th dgr of th rtx s [] Th scod ty of gotrc-arthtc dx s dotd by GA ad dfd as GA GA G G whr s th br of rtcs of G lyg closr to tha to ad s th br of rtcs of G lyg closr to tha to s [] For G lt s th br of dgs of G lyg closr to tha to ad of G lyg closr to tha to Th thrd br of th class of GA gral by sttg Arch of SID s th br of dgs Q Q to b th br for th dg of th grah G s dfd as GA GA G G t has b trodcd th ar [] A V-hylc t s a tralt dcorato ad by altratg sars C ad hxagos C 6 ad octagos C 8 I rct yars so rsarchrs ar trstd to toologcal dcs of V-hylc aotb s [] for dtals Throghot ths ar = [ ] dots a arbtrary V-hylc aotb trs of th br of hxagos a fxd row ad th br of hxagos a fxd col Fgr Fgr V-hylc Naotb wth = ad =

3 Two Tys of Gotrc Arthtc dx of V hylc Naotb MAIN RSULTS I ths scto GA dx of th olclar grah of V-hylc aotb s cotd It s asy to s that V V [ ] 6 ad [ ] 9 I th followg thor th GA dx of V-hylc aotb s obtad Thor Th GA dx of = [ ] s cotd as follows: GA T k k s k V k k V k Arch of SID k k s Proof O ca s that thr ar thr sarat tys of dgs of V-hylc aotb ad th br of dgs s dffrt Sos ad ar rrstat dgs for ths tys Fgr Th St Th dgs of Ty W artto th dgs of V-hylc aotb to thr sbsts ad as follows: = { s th ty of } = { s th ty of k for k } = { s th ty of }

4 S MORADI S BABARAHIM AND M GHORBANI Th sts ad ar show by dashd ls Fgrs ad rsctly Thrfor by dfto of GA dx GA T Fgr Th St Th dgs of Ty Fgr Th St Th dgs of Ty Arch of SID W alat ach sato saratly For alatg th frst s w kow that V for f s w ha f s V Also th

5 Arch of SID Two Tys of Gotrc Arthtc dx of V hylc Naotb For ach w ha Sos s a ost tgr sch that f s w ha Sos s a ost tgr sch that f s w ha 6 Sos s a ost tgr sch that s s s s s For all cas f s w ha

6 S MORADI S BABARAHIM AND M GHORBANI V T V T If s w ha V T Fally for cotg th thrd s w attd for ach -th row = ad = 6- ad th br of dgs of thrd ty ach row s Sc Vhylc aotb s bartt th for ach w ha V Th 6 6 k k s k k s k k s k k s Thor Th GA dx of = [ ] s g by: k V T k k V T k k V T k k V T k GA 6 8 k k 0 whr th lts of ar show Fgr 6k Proof Th sts of ad ar dfd th sa way as s th ros thor Thrfor by dfto of GA dx GA [ ] Arch of SID ha: For ach f s w ha: ad f s w Th W ca artto to sbsts sch as sch that k = { s th ty of k } for k Thrfor

7 Arch of SID Two Tys of Gotrc Arthtc dx of V hylc Naotb 5 k Sos s a ost tgr ad sch that for ach By calclato w ha th followg rslts: 8 8 s s s s s s s s Sos s a ost tgr ad sch that for ach

8 Arch of SID 6 S MORADI S BABARAHIM AND M GHORBANI s s s s ad s s s s Sos s a ost tgr ad sch that for ach s s ad

9 Two Tys of Gotrc Arthtc dx of V hylc Naotb 7 s s Sos s a ost tgr ad ad sch that for ach Arch of SID s s s s for UV For O octago O hxago rtcal rtcal dgs Hc Ths colts th roof RFRNCS D Vkčć B Frtla Toologcal dx basd o th ratos of gotrcal ad arthtcal as of d- rtx dgrs of dgs J Math Ch G Fath-Tabar B Frtla I Gta A w gotrc-arthtc dx J Math Ch B Zho I Gta B Frtla Z D O two tys of gotrcarthtc dx Ch Phys Ltt B Zho I Gta B Frtla Z D O two tys of gotrcarthtc dx Ch Phys Ltt ds