The Schultz Polynomial of Zigzag Polyhex Nanotubes

Transcription

1 Asan Journal of Chemstry Vol, No (9, 9-9 The Schultz Polynomal of Zgzag Polyhe Nanotubes MEHDI ELIASI and BIJAN TAERI* Deartment of Mathematcal Scences, Isfahan Unversty of Technology, Isfahan 86-8, Iran E-mal: The Schultz olynomal, S(G,, of a molecular grah G has the roerty that ts frst dervatve at s equal to the Schultz nde of grah Ivan Gutman dscovered that n the case of G s a tree, S(G,, has closely related to the Wener olynomal of G In ths aer, e fnd the eact eresson for Schultz olynomal of TUHC 6 [; q], the zgzag olyhe nanotubes, and obtan a relaton beteen Schultz and Wener olynomals of TUHC 6 [; q] Key Words: Schultz nde, Schultz olynomal, Nanotube INTRODUCTION Toologcal ndces are numercal descrtors that are derved from molecular grahs of chemcal comounds The Wener nde s the oldest toologcal ndces In 97 chemst Harold Wener develoed the most dely non toologcal descrtor, the Wener nde and used t to determne hyscal roertes of tyes of alanes non as araffns Numerous chemcal alcatons of Wener nde are reorted and ts mathematcal roertes are ell understood In the chemcal language, the Wener nde s equal to the sum of all shortest carbon carbon bond aths n a molecule In a grah theoretcal language, the Wener nde s equal to the count of all shortest dstances n a grah For a thorough survey n ths toc e encourage the reader to consult the reorted ors - Haruo Hosoya ntroduced a dstance-based olynomal, that he called t the Wener olynomal, related to each connected grah G as: H(G, { : u,v V(G, u v} The frst dervatve of H(G, at s equal to Wener nde of G Schultz 6 ntroduced the follong toologcal nde (Schultz nde S(G {(deg(u + deg(v : u,v V(G, u v}, here deg(u s degree of verte u, e the number of the vertces onng to the verte u These ndces have many chemcal alcatons 7,8 Smlar to Hosoya, Gutman ntroduced ne olynomal S(G, {(deg(u + deg(v : u, V(G, u v},

2 9 Elas et al Asan J Chem that the frst ts dervatve at s equal to the Schultz nde 9 He obtaned some relatonshs beteen ths olynomal and Wener olynomal of trees Sen-Peng et al also dd smlar or for heagonal chans RESULTS AND DISCUSSION In 99 Ima dscovered carbon nanotubes as mult alled structures Carbon nanotubes sho remarable mechancal roertes Eermental studes have shon that they belong to the stffest and elastc non materals These mechancal characterstcs clearly redestnate nanotubes for advanced comostes Dudea -6 as the frst chemst hch consdered the roblem of comutng toologcal ndces of nanostructures Recently comutng toologcal ndces of nanostructures has been the obect of many aers 7-7 In ths aer, e gve eact eresson for Schultz olynomal of TUHC 6 [; q], zgzag olyhe nanotube (Fg and obtan a relaton beteen Schultz and Wener olynomals of TUHC 6 [; q] Fg A TUHC 6 [; q] nanotube EXPERIMENTAL Throughout ths aer G: TUHC [; q], (Fg, denotes an arbtrary zg-zag olyhe nanotube n terms of the crcumference and the length q We also choose a coordnate label for vertces of TUHC [; q], as shon n Fg At frst e gve an mortant result on G Result : For a hte verte of level e have ( M level level d( d( + ( ( + f < f,

3 Vol, No (9 Schultz Polynomal of Zgzag Polyhe Nanotubes Fg A zg-zag olyhe nanotube lattce th 8 and q6 and for a blac verte of level e have b ( M level level d( d( + ( ( + f < f Proof: We comute b ( It suffces to consder For other blacs vertces the argument s smlar At frst note that the lattce s symmetrc (th resect to the lne onng to We dstngush three cases: Case : and s even In ths case for, here +, e have f s even d(, f s odd and obtan vertces havng dstance - from and vertces havng dstance b ( level d( + + s even d(, s odd Case : and s odd In ths case for, here +, e have d(,

4 9 Elas et al Asan J Chem f s even d(, - f s odd and e obtan vertces havng dstance - from, and vertces havng dstance from So b ( level d( + + s even d( Case : < For all + and > +, d(, +, s odd Thus the summaton of d (, s (for all s such that + and > + and symmetrc of s s S Also f +, then ( d(, f - s odd d(, - f - s even By consderng these vertces and ther symmetrc e obtan + vertces havng dstance and vertces havng - dstance from Therefore the summaton of d(, s (for all s such that + and symmetrc of s s S ( + + Hence b ( S + S ( ( ( In a smlar manner e can comute ( Result : (a (b d( d(, d(, d( Proof: By Result, e have d(, L d(, L d( ( + (,, ( + ( + +, d(u, d(u, + u level u level q +, ( + (, b ( + b ( b u level q- d(u, q q ( (

5 Vol, No (9 Schultz Polynomal of Zgzag Polyhe Nanotubes 9 Thus d(, d(, L d(,, ( + ( q( The roof of (b s smlar For a verte u of grah G e let d G (v, or sometmes d (v,, be d G (u, V(G By ths notaton e have Result : If q be an odd number, then (a d(, d( (b d(, d(, L d(, L d(,,, ( + (, b ( + b ( b q(+ ( + b ( b ( ( + ( ( Proof: Frst suose We consder the tube that can be bult u from to halves collasng at level The bottom art s the grah G TUHC 6 [; q-] and e can consder as one of the hte edges n the frst ro of the grah G Accordng to Result, e have d (, dg (, L dg (,, ( + ( q ( G The to art s the grah TUHC 6 [; q-] Ĝ and level of grah G s the frst ts ro and s such a blac verte of Ĝ Therefore by Result, d Ĝ b ( + b ( and (, d (, L d (, b ( + b ( d Ĝ Ĝ Ĝ, Snce ( b ( and dg (, dg (, + d (, b( Ĝ, then dg (, ( + ( q( + b( and such as t Smlarly for e can see that dg (, dg(, L dg (,, b( + b( bq( + ( By reetton of ths argument e obtan the result Result : If q be an even number, then (a (b d(, d(, L d( d(, d(, L d(,,, b ( + b ( b, ( + ( q(+ ( + ( ( ( + b ( b ( Proof: Frst suose that We consder the tube can be bult u from to halves collasng at level The bottom art s the grah G TUHC 6 [; q-] and the level of G s the frst level of G and e can consder one of the blac edges n the frst ro of grah G Accordng to result, e have d (, dg (, L dg (,, b( + b( bq ( G The to art s the grah TUHC 6 [; ] Ĝ and level of grah G s the frst level of Ĝ and s such a hte verte of Ĝ Therefore by result, e have (, ( + ( ( and d + Ĝ

6 96 Elas et al Asan J Chem d (, d (, L d (,, ( + ( + ( Ĝ Ĝ Ĝ ( b ( dg (, dg (, + d (, Ĝ Snce and (, then dg (, b( + ( bq( + ( + ( and smlarly dg (, L dg (,, b( + b( bq( + ( + ( We can reeat smlar ths rocess for and see that dg (, dg(, L dg(,, ( + ( q( + b( + b( By reetton of ths argument e can obtan the result For all q, ut and f(, g(, + ( + ( q( ( + b( + b ( + b( bq( ( + ( b ( ( In the follong result e fnd a relaton beeen Schultz and Wener olynomals of TUHC 6 [; q] Result : Let GTUHC 6 [; q] Then S(G, 6H(G, ( Proof: Let A and A q- be the set of vertces on the levels and q-, resectvely We have S(G, {u,v} V(G u V(G V(G u V(G u V(G (deg(u + deg(v u V(G V(G deg(u u V(G \ (A deg(u deg(ud(u, (deg(u + deg(v V(G deg(u deg(ud(u, + + u V(G V(G V(G deg(v deg(v deg(ud(u, + Aq q + u V(G Note that f u V(G \ (A Aq, then deg(u Suose that A, b {u A u s blac} and A, {u A u s the} Then f,b deg(u f, q deg(u deg(ud(u,

7 Vol, No (9 Schultz Polynomal of Zgzag Polyhe Nanotubes 97 The grah G can be obtaned from the grah G such that lne q - s ts frst ro and lne s ts last ro Let Aq -, b q Then Hence {u A u s blac n G } and {u A u s hte n G } u V(G \ (A Aq deg(ud(u, ( ( u V(G u V(G d(u, d(u, deg(u d(u, u V(G \ (A Aq d(u, u V(G \ (A Aq,b d(u, d(u, Aq -, f f q, q-,b q-, d(u, d(u, q,b d(u, d(u,( Accordng to result f q u A, b then (u, b ( d and for, e have d (u, u So q ( q A q,b e have (u, b ( u V(G \ (A Aq Also deg(ud(u, ( deg(ud(u, Also by consderng G and usng result, for all q d and for u A q,, d (u, ( u V(G u V(G d(u, d(u, 6 q q deg(ud(u, +,b q- d(u, +,b b ( + b ( b ( 6 q-, (, q d(u,, q ( ( q deg(ud(u, b ( q b ( q ( ( (

8 98 Elas et al Asan J Chem and smlarly q So by (-( e have S(G, deg(ud(u, u V(G d(u, q q- b ( + q- ( ( 6H(G, Theorem: The Schultz olynomal of G s gven by hen q q ( ( S(G, ( + ( + (q + and hen q > + ( + (6((( q + ( q( + q + ( S(G, 6((( q + ( ( + ( q q q + ( q + + q + + ( q q + (q + q, + q q + ( + + (q q Proof: We comute d(u, and ( Let u V(G A { and s evan}, A { and and u V(G A { q - and s evan}, A { Then e have d(u, B B [ q- d( V(G B A B A f(,, d(, + f(, + f(, + B A + B B A B (f(, + g(, + (f(, + g(, B A g(, g(, + d(, + g(, + B A B B + B B A g(, + A g(, + g(, B (f(, + g(,] (q q - and d(, + A B + f(, B ( + ( q s odd} A B q + s odd} A g(, f(, A d( q,

9 Vol, No (9 Schultz Polynomal of Zgzag Polyhe Nanotubes 99 and So u V(G q Suose that q Then by Lemma, for each q e have d(u, ( b( ( ( ( ( + + ( + q+ (( + (q + q ((q q q + q q+ q+ q+ q+ ( ( + (q + ( + q ( (+ and the result follos No suose that q > Let C : { - q - ( + -}, C C : { q - q - ( + -}, C q q q, : { - q - ( + q -}, : { q - q - ( + q -} We note that f C then > q also f C, then < q Therefore frst suose that C, so C and > q Thus by result, C f(, C f(, C f(, C g(, ( ( ( ( + ( ( ( ( ( + ( ( ( ( ( + ( + + ( ( ( +

10 9 Elas et al Asan J Chem C g(, C g(, So u V(G q ( ( ( ( ( + d(u, ((( q + q ( Also n ths case snce ( ( + ( ( + ( - Therefore ( + + q ( ( q + ( ( + ( ( ( + ( + (q + q, then by result e have + ( + S(G, (6((( q + ( ( + q q + + ( + + ( + q q ( + q + q+ + + ( + + (q ( + + ( q + q q ( + + (q + + q + + q + We can roof other condtons by ths method No dfferentaton of Schultz olynomal of G and lettng e obtan the Schultz nde of G Corollary: The Schultz nde of G s gven by hen q and hen S(G q( + q q 6 6 6q q + 8 q + q < q S(G ( + q + q q q Concluson We develoed a method hch s usually useful for calculatng Schultz olynomals of C 6 nanotubes As a consequence of calculatng Schultz olynomal of zgzag olyhe nanotubes e comuted Schultz nde of such nanotubes,

11 Vol, No (9 Schultz Polynomal of Zgzag Polyhe Nanotubes 9 ACKNOWLEDGEMENT One of the authors (BT as artally suorted by Center of Ecellence of Mathematcs of Isfahan Unversty of Technology (CEAMA, Iran REFERENCES H Wener, J Am Chem Soc, 69, 7 (97 AA Dobrynn, R Entrnger and I Gutman, Acta Al Math, 66, ( AA Dobrynn, I Gutman, S Klavzar and P Zgert, Acta Al Math, 7, 7 ( I Gutman and OE Polansy, Mathematcal Concets n Organc Chemstry, Srnger, Berln (986 H Hosoya, Dscrete Al Math, 9, 9 (988 6 HP Schultz, J Chem Inf Comut Sc, 9, 7 (989 7 Sen-Peng Eu, Bo-Yn Yang and Yeong-Nan Yeh, Int J Quantum Chem, 6, 6 (6 8 M Randc, J Chem Inf Comut Sc,, 67 ( 9 AA Dobrynn, Croat Chem Acta, 7, 869 (999 I Gutman, Bull lasse des Scences Math Naturelles Sc Math, CXXXI, ( S Ima, Nature,, 6 (99 MV Dudea, I Slagh-Dumtrescu and B Parv, MATCH Commun Math Comut Chem,, 7 ( MV Dudea and PE John, MATCH Commun Math Comut Chem,, ( MV Dudea, Bull Chem Soc (Jaan, 7, 87 ( MV Dudea, MATCH Commun Math Comut Chem,, 9 ( 6 MV Dudea, M Stefu, B Parv and PE John, Croat Chem Acta, 77, ( 7 AR Ashraf and A Loghman, MATCH Commun Math Comut Chem,, 7 (6 8 AR Ashraf and A Loghman, J Comut Theor Nanosc,, (6 9 S Yousef and AR Ashraf, MATCH Commun Math Comut Chem, 6, 69 (6 S Yousef and AR Ashraf, J Math Chem, (DOI: 7/s H Yousef-Azar, AR Ashraf, A Bahram and J Yazdan, Asan J Chem,, (8 M Elas and B Taer, MATCH Commun Math Comut Chem, 6, 8 (6 M Elas and B Taer, J Comut Theor Nano Sc,, (7 M Elas and B Taer, MATCH Commun Math Comut Chem, 9, 7 (8 A Heydar and B Taer, MATCH Commun Math Comut Chem, 7, 66 (7 6 A Heydar and B Taer, J Comut Theor Nano Sc,, 8 (7 7 A Heydar and B Taer, MATCH Commun Math Comut Chem, 7, 66 (7 (Receved: Setember 7; Acceted: Setember 8 AJC-686