Computing PI and Szeged indices of multiple phenylenes and cyclic hexagonal square chain consisting of mutually isomorphic hexagonal chains

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1 J. Srb. h. Soc UD : JSS 40 Orgal sctfc papr optg PI ad Szgd dcs of ltpl phyls ad cyclc hxagoal sqar cha cosstg of tally soorphc hxagoal chas H. YOUSFI ZRI *, J. YZDI,. HRMI ad. R. SHRFI School of Mathatcs, Statstcs ad optr Scc, Ursty of Thra, Thra, Dpartt of Mathatcs, Faclty of Scc,Ursty of Kasha, Kasha , Ira Rcd 8 J, rsd 9 Dcbr 00 bstract: PI ad Szgd dcs ar two of th ost portat topologcal dcs dfd chstry. I ths stdy, th PI ad Szgd dcs of lar []-phyls ad a cyclc hxagoal-sqar cha cosstg of tally soorphc hxagoal chas wr coptd. Th PI ad Szgd dcs of a ltpl phyl, whch s th -dsoal cas of a phylc aotb ad aotors, wr dtrd. Kywords: PI dx, Szgd dx, lar []-phyls, ltpl phyls aotb. ITRODUTIO Lt G b a spl olclar graph wthot drctd ad ltpl dgs ad wthot loops, th rtx ad dg-sts of whch ar rprstd by VG ad G, rspctly. topologcal dx of a graph G s a rc qatty rlatd to G. Th oldst topologcal dx s th Wr dx. ros of ts chcal applcatos ha b rportd ad ts athatcal proprts ar wll drstood. 4 Khadkar ad co-athors 5 9 dfd a w topologcal dx ad ad t th Padakar Ia dx. Hr Padakar cos fro Padakar Khadkar, ad Ia fro Ia Gta. Thy abbratd ths w topologcal dx as PI. Ths wly proposd topologcal dx dos ot cocd wth th Wr dx for acyclc olcls. It s dfd as PIG Σ G [ G G], whr G s th br of dgs of G lyg closr to tha to ad G s th br of dgs of G lyg closr to tha to. Th Szgd dx s aothr topologcal dx whch was trodcd by Ia Gta. 0 To df th Szgd dx of a graph G, t s assd that s a dg coctg th rtcs ad. Sppos G s th br of rtcs of G lyg closr to ad G s th br of rtcs of G lyg closr to. dgs qdstac fro ad ar ot tak to accot. Th th * orrspodg athor. -al: hyosf@t.ac.r do: 0.998JS070Y 0

2 04 YOUSFI ZRI t al. Szgd dx of th graph G s dfd as: SzG G G G, s also Rf. 4. Th PI ad Szgd dcs of so hxagoal graphs cotag aotbs ad aotors ha b coptd. 5 9 I ths stdy, ths work s cotd wth th coptato of th PI ad Szgd dcs of so othr portat classs of chcal graphs. Th otato s stadard ad aly tak fro th ltratr. 0, RSULTS D DISUSSIO PI dx of ltpl phyls ad cyclc hxagoal sqar chas I ths scto, th PI dx of lar []-phyls, ltpl phyls ad a cyclc hxagoal sqar cha cosstg of tally soorphc hxagoal chas ar coptd., To do ths, sppos that G s a graph, G ad [ G G]. Th PIG Σ. Thrfor, for coptg th PI dx of G, t s sffct to calclat for ry. osdr th olclar graph of a lar []-phyl, Fg.. Th T h h 8h ad, hc, PIT 4h h 4 Σ. To calclat, thr cass ar cosdrd,.., that s rtcal, horzotal or oblq. If s horzotal or oblq, th ad for rtcal dgs, o has h. Ths, PIT 0h 44h 8. Fg.. Th olclar graph of a lar []-phyl. Sppos X, s th chcal graph of a ltpl phyl, Fg.. pplyg or thod for coptg PIX,, sppos α M{,}. It s clar that X, ad, hc, o obtas: X s horzotal s oblq s rtcal, α 8 4α 4 4 α 8α α 0 α Thrfor, PIX, 8 8 8α 8α α α 4. Lt, dot th olclar graph of a cyclc hxagoal-sqar cha cosstg of tally soorphc hxagoal chas H, H,, H, cyclcally cocatatd by crcts α of lgth 4, whch th H s ar chas cotag hxagos, Fg.. So xact forlas for th algbrac strctr cot S of th lar []-phyl, Fg., th ltpl phyls, Fg. ad th olc-

3 PI D SZGD IDIS OF MULTIPL PHYLS 05 lar graph of,, Fg. wr coptd. 0, t th d of ths scto, PI, s coptd. Obosly,, 5. Hc, o obtas: PI 5,, 5 s horzotal s oblq s rtcal. Fg.. Th olclar graph of a ltpl phyl. Fg.. Th olclar graph of a cyclc hxagoal-sqar cha,. Szgd dx of ltpl phyls ad cyclc hxagoal-sqar chas I th ltratr, thr s a papr by Dda whch th Szgd dx s tstd QSPR. 4 I ths papr, th s of th Szgd dx ad sral othr

4 0 YOUSFI ZRI t al. dstac-basd dcs o th corrlato wth th bolg pots of 45 cycloalkas wr xad. Moror, Khadkar t al. 5 dscrbd aros applcatos of th Szgd dx for odlg physcochcal proprts, as wll as th physologcal actts of orgac copods actg as drgs or possssg pharacologcal actty. Th athors of ths papr rwd 75 paprs pblshd o th sbjct of th Szgd dx. Ths shows that th sbjct of th Szgd dx s growg or ad or chstry, physcs ad also bology. I ths scto, th Szgd dx of th lar []-phyl X,, th ltpl phyl X, ad, wr coptd. Frst t shold b otcd that VX,, VX, ad V, 4. Th Szgd dx of X, was frst calclatd. To do ths, t was assd that, ad to b th st of all rtcal, oblq ad horzotal dgs, rspctly. Th o obtas: 0, 0 8 } { Sz X Th th Szgd dx of a ltpl phyl X, was calclatd. Slarly,,. } { } { Sz X To copt th last sato, t was spposd that U ad two sparat cass wr cosdrd, as follows: as :. I ths cas o obtas: }, { S S U whr S 95 ad,. Hc, th Szgd dx of X, s qal to as :. I ths cas o has U S S ad hc th Szgd dx of X, s qal to I th d of ths scto, th Szgd dx of, s coptd. Obosly,, V. If, ad ar st as bfor, th Sz,

5 PI D SZGD IDIS OF MULTIPL PHYLS 07 ИЗВОД ИЗРАЧУНАВАЊЕ PI И SZGD ИНДЕКСА ВИШЕСТРУКИХ ФЕНИЛЕНА И ЦИКЛИЧНОГ ХЕКСАГОНАЛНО КВАДРАТНОГ ЛАНЦА САЧИЊЕНОГ ОД МЕЂУСОБНО ИЗОМОРФНИХ ХЕКСАГОНАЛНИХ ЛАНАЦА H. YOUSFI ZRI, J. YZDI,. HRMI и. R. SHRFI School of Mathatcs, Statstcs ad optr Scc, Ursty of Thra, Thra, Dpartt of Mathatcs, Faclty of Scc,Ursty of Kasha, Kasha , Ira PI и Szgd индекси су два најважнија тополошка индекса дефинисана у хемији. У овом раду су израчунати PI и Szgd индекси линеарних []-фенилена и цикличног хексагонално квадратног ланца сачињеног од међусобно изоморфних хексагоналних ланаца. Одређени су PI и Szgd индекси вишеструког фенилена, као дводимензионог облика фениленских наноцеви и нанопрстена. Примљено 8. јуна, ревидирано 9. децембра 00 RFRS. H. Wr, J.. h. Soc R. Todsch, V. oso, Hadbook of Molclar Dscrptors, Wly, Wh, 000. D.. dha, I.. W, P. G. Sybold, J.. h. Soc G. Rckr,. Rckr, J. h. If. opt. Sc P. V. Khadkar, at. cad. Sc. Ltt P. V. Khadkar, P. P. Kal,. V. Dshpad, S. Kararkar, V. K. grawal, J. Math. h P. V. Khadkar, S. Kararkar, J. h. If. opt. Sc P. V. Khadkar, S. Kararkar, R. G. Vara, cta h. Slo P.. Joh, P. V. Khadkar, J. Sgh, J. Math. h I. Gta, Graph Thory ots of Y Das, G. Dootor, I. Gta, S. Josh, S. Kararkar, D. Khaddar, T. Khaddar, P. V. Khadkar, L. Popoc,. S. Sapr,. Sapr,. Shrhatt, J. Srb. h. Soc O. M. Malc, G. Katoa, M. V. Dda, M. Strj,. Graoac, I. Gta, roat. h. cta M. V. Dda, I. Gta, roat. h. cta S. Kararkar, S. Kararkar, S. Josh,. Das, P. V. Khadkar, J. Srb. h. Soc H. Yosf zar,. Maoochhra,. R. shraf, rr. ppl. Phys. 007, do:0.0j.cap R. shraf,. Logha, MTH o. Math. opt. h R. shraf,. Logha, J. opt. Thor. aosc R. shraf, F. Rza, MTH o. Math. opt. h H. Yosf zar,. R. shraf,. ahra, J. Yazda, sa J. h P. J. aro, obatorcs: Topcs, Tchqs, lgorths, abrdg Ursty Prss, abrdg, Trajstc, hcal Graph Thory, R Prss7, oca Rato, FL, 99. O. odroža Patc, R. Dorosloack, J. Math. h I. Gta, Z. atrforsch M. Dda, J. h. If. opt. Sc P. V. Khadkar, S. Kararkar, V. K. grawal, J. Sgh,. Shrastaa, I. Lkots, M. V. Dda, Ltt. Drg. Ds. Dsco