Two Types of Geometric-Arithmetic Indices of Nanotubes and Nanotori

Transcription

1 MACH Concatons n Mathatcal and n Cotr Chstry MACH Con Math Cot Ch IN wo ys of Gotrc-Arthtc Indcs of Nanotbs and Nanotor ros Morad, oraya Baba-Rah Dartnt of Mathatcs, Faclty of cnc, Arak nrsty, Arak, Iran Rcd Jn 0, 00 Abstract h conct of gotrc-arthtc ndcs was ntrodcd n th chcal grah thory hs ndcs ar dfnd as G Q / G Q Q Q, whr Q s so antty that n a n annr can b assocatd wth th rtx of grah G In ths ar, xact forlas for two tys of gotrc-arthtc ndx of UC 4 C 8 nanotb and C 4 C 8 nanotors ar gn Introdcton hroghot ths scton G s a sl connctd grah wth rtx and dg sts VG and G, rsctly A toologcal ndx s a nrc antty fro th strctr of a grah whch s narant ndr atoorhss of th grah ndr consdraton A toologcal ndx s a nrc antty fro th strctral grah of a olcl Usag of toologcal ndcs n chstry bgan n 947 whn chst Harold Wnr dlod th ost wdly known toologcal dscrtor, th Wnr ndx, and sd t to dtrn hyscal rorts of tys of alkans known as araffn h Wnr ndx of G s th s of dstanc btwn all nordrd ar of rtcs of G, W G d,, {, } V G G whr d G, and s dfnd as th nbr of dgs n a nal ath connctng th rtcs and, s [] h conct of gotrc-arthtc ndcs was ntrodcd n th chcal grah thory hs ndcs gnrally ar dfnd as QQ gnral gnral G, G Q Q Corrsondng athor al addrss: s-orad@arakacr

2 -- whr grah G Q s so antty that n a n annr can b assocatd wth th rtx of h frst ty of gotrc-arthtc ndx s dnotd by and dfnd as d d G G d d, whr s an dg of th olclar grah G and d stand for th dgr of th rtx, s [] h ndx has bn ntrodcd lss than a yar ago [] Howr, a fw ars ar aard dalng wth ths antty, s [-5] h scond ty of gotrc-arthtc ndx s dnotd by and dfnd as nn G, whr n s th nbr of rtcs of G lyng closr to than G n n to and n s th nbr of rtcs of G lyng closr to than to, s [] For G, lt s th nbr of dgs of G lyng closr to than to and s th nbr of dgs of G lyng closr to than to h thrd br of th class of by sttng Q gnral Q to b th nbr for th dg of th grah G s dfnd as G, t has bn G ntrodcd n th ar [7] A C 4 C 8 nt s a tralnt dcoraton ad by altrnatng sars C 4 and octagons C 8 In rcnt yars, so rsarchrs ar ntrstd to toologcal ndcs of C 4 C 8 nanotbs and nanotor, s [8-] for dtals hy cotd so dstanc basd toologcal ndcs of ths nanotbs and nanotor h UC 4 C 8 nanotb s a athatcally batfl objct constrctd fro sars and octagons, Fgr a h a of ths artcl s to cot and ndcs of UC 4 C 8 nanotb and C 4 C 8 nanotors that obtand fro UC 4 C 8 nanotb by glng ts nds, Fgr b a Fgr a UC 4 C 8 nanotb, b C 4 C 8 nanotors b

3 -7- hroghot ths ar = [, ] dnots an arbtrary UC 4 C 8 nanotb n trs of th nbr of octagons n a fxd row and th nbr of octagons n a fxd coln, n th two-dnsonal lattc of, Fgr W also dnot a C 4 C 8 nanotors, Fgr, by = [, ] Fgr wo Dnsonal Lattc UC 4 C 8 nanotb, wth =5 and = Fgr wo Dnsonal Lattc C 4 C 8 nanotors, wth =5 and = Man Rslts In ths scton, and ndcs of th olclar grah of UC 4 C 8 nanotb and C 4 C 8 nanotors ar cotd It s asy to s that V V [, ] 8 and [, ], V V [, ] 8 and [, ] In th followng thor th ndx of UC 4 C 8 nanotb s obtand

4 -8- hor h ndx of = [, ] s cotd as follows: k k k k 4 k k V k k 4 k V 4 k 4k V 4 4 k V 4 k 4k Proof On can s that thr ar thr sarat tys of dgs of UC 4 C 8 nanotb and th nbr of dgs s dffrnt os, and ar rrsntat dgs for ths tys Fgr 4 h st h dgs of ty W artton th dgs of UC 4 C 8 nanotb nto thr sbsts, and, as follows: = { s th ty of }, = { s th ty of,k for k }, = { s th ty of }

5 -9- h sts, and ar shown by dashd lns n Fgrs 4, 5 and, rsctly,,, Fgr 5 h st h dgs of ty Fgr h st h dgs of ty hrfor, by dfnton of ndx, nn nn nn n n n n n n W alat ach ss saratly For alatng th frst s, w know that for, w ha n V nn n Also 4, thn 4 n n For ach, w ha n n 8 Obosly, for ry,, w ha n, n 8, for ry,, w ha n, n 8,, for ry,, w ha

6 -70- n, n 8 o n n 4 n n [8 ] By th sa thod whn nn, w can cot n n Fnally for cotng th thrd s, w attnd, for ach n k-th row, n = 4k and n = 8-4k and th nbr of dgs of thrd ty n ach row s nc UC 4 C 8 nanotb s bartt thn for ach, w ha n n V hn n n n n 4 hs colts th roof nn 4 hor h ndx of = [, ] s gn by: k k 4 k V 4 k 4 k V 4 k 4 4 k 8 k 4 k 0 whr th lnts of ar shown n Fgr 4 Proof W can now stat th analog of hor hn [, ] For ach,, thn 4 W can artton nto sbsts sch as,,,,,,, sch that,k = { s th ty of,k }, for k hrfor, k, By calclaton w ha th followng rslts os s an odd ost ntgr, sch that, for ach,,

7 4 and 4 os s an n ost ntgr, sch that, for ach,, 4 and 4 For, n k-th row, + =-4 and thn, =4+k, =- 8-k Hnc k k k hs colts th roof hor h and ndcs of = [, ] ar al and cotd as follows: Proof nc C 4 C 8 nanotor s bartt thn for ach, w ha V n n Moror for ach, V n n, thrfor by dfnton of -7-

8 -7- nn ndx w concld that, Now for obtanng th n n ndx, t s sffcnt to show that for ach, = ch as hor, w artton th dg st of = [, ] nto thr sbsts, and, ths sbsts ar shown n Fgr 7 by dashd lns For ach,, ach, For scond ty of dgs for ach 0 r r, whr r = n{,} Fnally for By abo argnt, w concld that, and ths colt th roof Fgr 7 h artton of nto, and Acknowldgnts h athors wold lk to xrss thr sncr thanks to th rfrs for thr alabl sggstons and conts

9 -7- Rfrncs [] H Wnr, trctral dtrnaton of th araffn bolng onts, J A Ch oc [] D Vkčć, B Frtla, oologcal ndx basd on th ratos of gotrcal and arthtcal ans of nd- rtx dgrs of dgs, J Math Ch [] M Ghorban, M Jalal, Cotng a nw toologcal ndx of nano strctrs, Dg J Nanoatr Bos [4] M Zraatkar, A Iranansh, Cotng ndx for so nanotbs, Otolctron Ad Matr Rad Con [5] A Iranansh, M Zraatkar, Cotng ndx of HAC 5 C 7 [,] and HAC 5 C C 7 [,] nanotbs, Otolctron Ad Matr Rad Con [] G Fath-abar, B Frtla, I Gtan, A nw gotrc-arthtc ndx, J Math Ch [7] B Zho, I Gtan, B Frtla, Z D, On two tys of gotrcarthtc ndx, Ch Phys Ltt [8] M Arzooand, B ar, A athatcal odl for UC 4 C 8 nanotbs and thrsytry gros, J Go Phys [9] A Hydar, B ar, zgd ndx of UC 4 C 8 nanotbs, r J Cob [0] A Iranansh, A R Ashraf, Balaban ndx of an archar olyhx, UC 4 C 8 R and UC 4 C 8 nanotors, J Cot hor Nanosc [] A Iranaansh, A Kafran, Cotaton of th frst dg-wnr ndx of UC 4 C 8 nanotb, MACH Con Math Cot Ch [] L X, H Dng, PI ndcs of tor,[c 4,C 8 ] corng by C 4 and C 8, MACH Con Math Cot Ch [] X, H Zhang, Hosoya olynoals of UC 4 C 8 nanotbs, J Math Ch [4] Yosf, A R Ashraf, An xact xrsson for th Wnr ndx of a UC 4 C 8 R nanotors, J Math Ch [5] Yosf, A R Ashraf, An algorth for constrctng Wnr atrx of UC 4 C 8 R nanotbs, Crrnt Nanosc [] A R Ashraf, M ahl, M Ghorban, h ccntrc connctty ndx of nanotbs and nanotor, J Cot Al Math

10 -74- [7] A Hydar, B ar, zgd ndx of UC 4 C 8 R nanotbs, MACH Con Math Cot Ch [8] A Hydar, B ar, Wnr and chltz ndcs of UC 4 C 8 nanotbs, MACH Con Math Cot Ch [9] A Kar, A Iranansh, A hranan, Cotng a nw dg-wnr ndx of UC 4 C 8 s nanotbs and UC 4 C 8 R nanotors, Otolctron Ad Matr Rad Con [0] A Mahan, O Khoral, A Iranansh, M Yosfdaz, h cotaton of total zgd ndx of UC 4 C 8 R nanotb Otolctron Ad Matr Rad Con [] F Hasan, O Khoral, A Iranansh, Cotaton of th frst rtx of CO_PI ndx of UC 4 C 8 nanotbs, Otolctron Ad Matr Rad Con