A new algorithm for computing distance matrix and Wiener index of zig-zag polyhex nanotubes

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1 Nanoscale Res Lett 00) :0 0 DOI /s y NANO EXPRESS A new algorthm for comutng dstance matrx and Wener ndex of zg-zag olyhex nanotubes Al Reza Ashraf Æ Shahram Yousef Receved: 1 December 00 / Acceted: 0 February 00 / Publshed onlne: 10 Arl 00 to the authors 00 Abstract The Wener ndex of a grah G s defned as the sum of all dstances between dstnct vertces of G. In ths aer an algorthm for constructng dstance matrx of a zg-zag olyhex nanotube s ntroduced. As a consequence, the Wener ndex of ths nanotube s comuted. Keywords Zg-zag olyhex nanotube Dstance matrx Wener ndex Introducton Carbon nanotubes form an nterestng class of carbon nanomaterals. These can be magned as rolled sheets of grahte about dfferent axes. These are three tyes of nanotubes: armchar, chral and zgzag structures. Further nanotubes can be categorzed as sngle-walled and multwalled nanotubes and t s very dffcult to roduce the former. Grah theory has found consderable use n chemstry, artcularly n modelng chemcal structure. Grah theory has rovded the chemst wth a varety of very useful tools, namely, the toologcal ndex. A toologcal ndex s a numerc quantty that s mathematcally derved n a drect and unambguous manner from the structural grah of a molecule. It has been found that many roertes of a A. R. Ashraf &) Insttute for Nanoscence and Nanotechnology, Unversty of Kashan, Kashan, Iran e-mal: ashraf@kashanu.ac.r S. Yousef Center for Sace Studes, Malek-Ashtar Unversty of Technology, Tehran, Iran chemcal comound are closely related to some toologcal ndces of ts molecular grah [1, ]. Among toologcal ndces, the Wener ndex [] s robably the most mortant one. Ths ndex was ntroduced by the chemst H. Wener, about 0 years ago to demonstrate correlatons between hysco-chemcal roertes of organc comounds and the toologcal structure of ther molecular grahs. Wener defned hs ndex as the sum of dstances between two carbon atoms n the molecules, n terms of carbon carbon bonds. Next Hosoya named such grah nvarants, toologcal ndex []. We encourage the reader to consult Refs. [5 ] and references theren, for further study on the toc. The fact that there are good correlatons between and a varety of hysco-chemcal roertes of chemcal comounds contanng bolng ont, heat of evaoraton, heat of formaton, chromatograhc retenton tmes, surface tenson, vaor ressure and artton coeffcents could be ratonalzed by the assumton that Wener ndex s roughly roortonal to the van der Waals surface area of the resectve molecule [8]. Dudea was the frst chemst whch consdered the roblem of comutng toologcal ndces of nanostructures [9 15]. The resented authors comuted the Wener ndex of a olyhex and TUC C 8 R/S) nanotor [1 18]. In ths aer, we contnue ths rogram to fnd an algorthm for comutng dstance matrx of a zg-zag olyhex nanotube. As an easy consequence, the Wener ndex of ths nanotube s comuted. John and Dudea [9] comuted the Wener ndex of zgzag olyhex nanotube T = T, q) = TUHC [,q], for the frst tmes. In ths aer, dstance matrx of these nanotubes are comuted. As an easy consequence of our results, a matrx method for comutng the Wener ndex of a zg-zag olyhex nanotube s ntroduced. We also reare an 1

2 Nanoscale Res Lett 00) :0 0 0 algorthm for comutng dstance matrx of these nanotubes. Throughout ths aer, our notaton s standard. They are aearng as n the same way as n the followng [, 19]. a) Base x 1,1) 1, Man results and dscusson 1,1) x, In ths secton, dstance matrx and Wener ndex of the grah T = TUHC [m,n], Fg. 1, were comuted. Here m s the number of horzontal zg-zags and n s the number of columns. It s obvous that n s even and VT) = mn. An algorthm for constructng dstance matrx of TUHC [m,n] b) Base We frst choose a base vertex b from the -dmensonal lattce of T and assume that x j s the,j)th vertex of T, Fg.. Defne D ð11þ mn ¼½d ð11þ j where d ð11þ j s dstance between 1,1) and,j), = 1,,..., m and j = 1,,..., n. By Fg., there are two searates cases for the 1,1)th vertex. For examle n the case a) of Fg., d ð11þ 11 ¼ 0 d ð11þ 1 ¼ d ð11þ 1 ¼ 1 and n case b), d ð11þ 11 ¼ 0 d ð11þ 1 ¼ 1 d ð11þ 1 ¼ : In general, we assume that D ðqþ mn s dstance matrx of T related to the vertex,q) and s ðqþ s the sum of th row of D ðqþ mn: Then there are two dstance matrx related to,q) such that s ðk1þ ¼ s ð1þ s ðkþ ¼ s ðþ 1 k n= 1 m 1 m: By Fg. and revous notatons, f b vares on a column of T then the sum of entres n the row contanng base vertex s equal to the sum of entres n the frst row of D ð11þ mn: On the other hand, one can comute the sum of entres n other rows by dstance from the oston of base vertex. Therefore, s ðjþ k ¼ sð11þ s ð1þ s ðjþ k ¼ sð1þ s ð11þ kþ1 1 k m 1 j n If kþ1 1 k m 1 j n j+j) kþ1 1 k m 1 j n If - + j) kþ1 1 k m 1 j n Fg. Two bascally dfferent cases for the vertex b We now descrbe our algorthm to comute dstance matrx of a zg-zag olyhex nanotube. To do ths, we defne matrces A ðaþ mðn=þ1þ ¼½a j B mðn=þ1þ ¼½b j and mðn=þ1þ ¼½c j as follows: a 1,1 =0 a 1, =1 a j ¼ a 1 -j a jj c 1,1 =0 c 1, =1 c j ¼ c 1 -j c jj b,1 = 1 b,j =b,j 1 +1 a 1 ¼ a 11 þ 1 a ¼ a 1 þ 1 j a ¼ a 1 þ 1 a 1 ¼ a þ 1 - c ¼ c 1 þ 1 c 1 ¼ c þ 1 j c 1 ¼ c 11 þ 1 c ¼ c 1 þ 1 - For comutng dstance matrx of ths nanotube we must comute matrces D ðaþ mn ¼½d a j and DðbÞ mn ¼½d b j: But by our calculatons, we can see that d a j ¼ Maxfa j b j g 1 j n= and d njþ j[n= þ 1 d b j ¼ Maxfa j c j g 1 j n= d njþ j[n= þ 1 Ths comletes calculaton of dstance matrx. Comutng Wener ndex of TUHC [m,n] Fg. 1 The zg-zag olyhex nanotube TUHC [0,n] In revous secton, dstance matrx D ðqþ mn related to vertex,q) s comuted. Suose s ðqþ s the sum of th row of D ðqþ mn: Then s ðk1þ ¼ s ð1þ and s ðkþ ¼ s ðþ where 1 k n= 1 m 1 m: On the other hand, by our calculatons n secton An algorthm for constructng dstance matrx of TUHC [m,n], 1

3 0 Nanoscale Res Lett 00) :0 0 s ð1k1þ s ð1kþ ¼ ¼ n þ n þ Þð 1Þ n þ 1 n 5Þ n þ 1 n þ n þ Þð 1Þ n þ 1 n Þ n þ 1 1 m 1 k n : Suose S ðaþ and S ðbþ are the sum of all entres of dstance matrx Dmn ðqþ n two cases a) and b). Then 1 ¼ mn/)mþnþþðm=þðm1þðmþmn=þ1 mn/)mþþðn=þðnþþðnþþ mn=þ1 1 ¼ mn/)mþnþþðm=þ ð m 1Þ mn=þ1 mn/)m1þþðn=þðn : Þ mn=þ1 If s arbtrary then one can see that: ¼ 1 þ X ¼ ¼ 1 þ X ¼ s ð1þ Xm ¼mþ s ð11þ Xm ¼mþ s ð11þ s ð1þ Thus t s enough to comute S ðaþ m n/, one can see that: ¼mn/)m þ n þ Þþðm=Þ m 1 m þ mn þ n þ m þ n) and S ðbþ: When ¼mn/)m þ n þ Þþðm=Þðm þ 1Þðm þ Þ m þ mn þ n þ m þ m þ n) To comlete our argument, we must nvestgate the case of m > n/ + 1. To do ths, we consder three cases that n/ + 1, m n/ + 1 m n + 1, m n/ + 1 < n/ + 1 and m > n + 1, > n/ + 1. I) n þ 1 and m n þ 1: In ths case we have: S ðaþ ¼mn ðmþ1þþ n n þ 1 ðn mn1nþþ n þ S ðbþ ¼mn ðmþþþ n ðnþðnþ þ 1 n mnnþ8 þ ðnþþ II) m n + 1 and m n þ 1\ n þ 1: Therefore, III) ¼ mn m þ n þ Þþm ðm 1Þ m þ mn þ n þ m þ n) ¼ mn m þ n þ Þþm m þ 1Þðm þ Þ m þ mn þ n þ m þ m þ n) m > n + 1 and [ n þ 1: In ths case, ¼ n n 1 n þ m Þðm þ 1Þþn ¼ n n 1 n þ 8 þ m Þðm þ Þþn Therefore, W mn ¼ 8 " P ðn=þ ðn=þ Pm= >: ¼1 ðm1þ= ¼1 þ þ >< # : þð1=þ ðmþ1þ= þsðbþ ðmþ1þ= -m We now substtute the values of Wener ndex of T, as follows: jm to comute the mn W mn ¼ ðm þmnþþ n m 1 ðm 1Þ m n þ1 mn ð8m þn Þ n 19ð n Þ m[ n þ1 : Constructng dstance matrces of some nanotubes In ths secton, dstance matrces of TUHC [8,10] and TUHC [8,1] together wth ther Wener ndces are comuted. To construct dstance matrces of TUHC [8,10], we must comute matrces A ðaþ 8 AðbÞ 8 and B 8. By defnton of these matrces, we have: A ðaþ 8 ¼

4 Nanoscale Res Lett 00) : ¼ B 8 ¼ : We now comute matrces D ðaþ 810 and DðbÞ 810 : By defnton, entres of the frst n/ + 1 columns of these matrces are maxmum values of fa ðaþ 8 B 8 } and f 8 B 8g resectvely. Thus, D ðaþ 810 ¼ D ðbþ 810 ¼ Ths mles that WTUHC [8,10]) = 19,00. To construct dstance matrces of TUHC [8,1], we must comute matrces A ðaþ 89 and AðbÞ 89 : Usng a smlar argument as above, we have: A ðaþ 89 ¼ ¼ On the other hand, B 89 ¼ Therefore, D ðaþ 81 ¼ D ðbþ 81 ¼ : By our calculatons, t s easy to see that WTUHC [8,1]) = 59,8. Acknowledgements We would lke to thank from referees for ther helful remarks and suggestons. Ths work was artally suorted by the Center of Excellence of Algebrac Methods and Alcatons of the Isfahan Unversty of Technology. References 1. R. Todeschn, V. Consonn, Handbook of Molecular Descrtors, Wley, Wenhem, 000). N. Trnajstc, Chemcal Grah Theory CRC Press, Boca Raton, FL, 198) 1

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