MOLECULAR TOPOLOGICAL INDEX OF C 4 C 8 (R) AND C 4 C 8 (S) NANOTORUS

Transcription

1 Dgest Journal of Nanomaterals an Bostructures ol No December MOLECULAR TOPOLOICAL INDEX OF C C 8 R AND C C 8 S NANOTORUS ABBAS HEYDARI Deartment of Mathematcs Islamc Aza Unverst of Ara Ara I R Iran Let be a smle connecte grah an enote stance between vertces an of the grah The Molecular toologcal Schultz ne of the grah s efne b v ] where v s the verte egree of verte an A s the entr of aacenc matrce of In ths aer we comute the molecular toologcal ne of C C an C C 8 R nanotorus 8 S Receve Setember 9; accete October 9 Kewors: Toologcal Inces Molecular toologcal ne Nanotorus Introucton A real number that escrbes a molecular grah s calle a toologcal ne The frst use of a toologcal ne for the correlaton of the measure roertes of molecules wth ther structural features was mae n 97 b the chemst Harol Wener In that ear he ntrouce the noton of ath number of a grah as the sum of the stances between an two carbon atoms n the molecules n terms of carbon carbon bons Wener orgnall efne hs ne on trees an stue ts use for correlatons of hsco-chemcal roertes of alanes alcohols amnes an ther analogous comouns [] Let be a connecte grah the set of vertces an eges of wll be enote b an E resectvel The stance between a ar of vertces an of s enote b The molecular toologcal ne of the grah was ntrouce b Schultz n 989 an s efne as follows: v ] Snce n the efnton of MTI aears the stance between vertces of whch s base the Wener ne t has been foun that the Schultz ne an the Wener ne are closel relate uanttes for trees an ccles [-] The Schultz ne has been shown to be a useful molecular escrtor n the esgn of molecules wth esre roertes [-8]Therefore further stues on mathematcal an comutatonal roertes of the Schultz ne an on ts relaton to other molecular escrtors are esrable

2 69 A C C 8 S naotorus a Se vew an b To vew Recentl comutng toologcal nces of nano structures have been the obect of man aers [-9] Ashraf an coauthor comute the wener ne of C C 8 R notorus [] In ths aer we use of them results an comute the molecular toologcal ne of ths grah Also we comute the molecular toologcal ne of C C 8 S nanotorus b usng results such that obtane n [] Man results In ths secton we erve eact formulas for the molecular toologcal ne of grah C C 8 S an C C 8 R nanotorus At frst we conser the grah of C C 8 S nanotorus For ths urose frst we choose a coornate label for vertces of ths grah as shown n Fgure Let ths grah has rows an columns of vertces an are ostve even nteger To comute MTI at frst the summaton of stance between all of the ar vertces of the grah most be comute For ths urose we conser vertces an n the frst row of the grah an obtan summaton of stances between these two vertces an other vertces of the grah The obtane results n ths comutatons can be use for calculaton summaton of stances between each two vertces t an t for t /- an other vertces of the grah other b smmetr of the grah Let enotes the summaton of stances between verte an all of the vertces lace n th row of the grah Thus Smlarl we e_ne as follow: t t t t

3 695 In Lemma of Ref [] we comute an as follow: Lemma Let < then > an > Now we can comute uantt of eresson for grah of C C 8 S nanotorus Let b usng of Lemma we have 6

4 696 A C C 8 R naotorus a To vew an b Se vew The last result whch obtane for verte can be use for all of the vertces of grahs Therefore 6 Now suose > Thus 6 So n ths case we have Now we can comute the molecular toologcal ne of C C 8 S nanotorus Theorem The molecular toologcal ne of 8 S C C nanotorus gven b > MTI Proof: Snce egree of each verte of the grah s tree so v for all of the vetces of the grah Therefore

5 697 MTI v ] In the han snce s connecte so A E Thus A b usng [ 6 ] 6 Now suose > B usng we have A Therefore - 8 [ 6 ] The So the roof s comlete Now we comute the molecular toologcal ne of C C 8 R nanotorus For ths urose at frst most be comute for ths grah We use from calculaton of Ref[] an obtan followng results m mn[ m mn m n ] m n In whch enotes the greatest common vsor of ntegers an n Ma{ } m Mn{ } an 5 m n m m n n m n m n n n Thus the molecular toologcal ne of grah C C 8 R can be comute as same as the grah of C C 8 S b usng ervous notatons Theorem Let b If then The molecular toologcal ne of C C nanotorus gven - 9 MTI S

6 698 mnm nm 9n m - m 9 m n mnm 6nm 8n m - 7m 8 m n mnm nm 9n m - n - m 9 m n mnm 6nm 8n m 6n - m 8 m n Proof: Snce A E so the grah So b usng we have A Also v for all of the vertces of 5 MTI v ] 6 Thus s an even nteger b usng Also MTI 6 - then the results can be obtane b relacng n 5 9 Thus the roof s comlete References [] A R Ashraf_ S Yousef J Math Chem 7 [] A R Ashraf PNza Dgest Journal of Nanomaterals an Bostructures 69 9 [] M Duea J Chem Inf Comut Sc [] A Hear an B Taer MATCH Commun Math Comut Chem [5] A Hear an B Taer J Comut Theor Nano Sc In ress [6] A Hear an B Taer MATCH Commun Math Comut Chem [7] A Hear an B Taer J Comut Theor Nano Sc 7 [8] S Yousef an AR Ashraf MATCH Commun Math Comut Chem [9] H Yousef A R Ashraf M H Khaleh Dgest Journal of Nanomaterals an Bostructures 5 8 [] S Yousef an AR Ashraf J Math Chem 7 [] H Wener J Am Chem Soc [] H Hosoa Bull Chem Soc Jn 97 [] Z Mhalc D elan D Amc S Nolc D Plavšc an N Trnastc J Math Chem 99 [] D J Klen Z Mhalc D Plavšc an N Trnastc J Chem Inf Comut Sc 99 [5] D Plavšc S Nolc N Trnastc an D J Klen Croat Chem Acta [6] S Klavžar an I utman JChem Inf Comut Sc [7] I utman J Chem Inf Comut Sc [8] I utman an S Klavžar J Chem Inf Comut Sc