CROATICA CHEMICA ACTA CCACAA 77 (1 ) 111 115 (004) ISSN-0011-1643 CCA-907 Original Scientific Paer Wiener Index of Archair Polyhex Nanotubes* Mircea V. Diudea, a, ** Monica Stefu, a Basil Pârv, b and Peter E. John c a Babes-Bolyai University, Faculty of Cheistry and Cheical Engineering, Babes-Bolyai University, 3400 Cluj, Roania b Babes-Bolyai University, Faculty of Matheatics and Couter Science, Deartent of Couter Science, Babes-Bolyai University, 3400 Cluj, Roania c Technical University Ilenau, Institute of Matheatics, PSF 100565, D-98684 Ilenau, Gerany RECEIVED FEBRUARY 14, 003; REVISED MAY 14, 003; ACCEPTED JULY 1, 003 Key words Wiener index archair olyhex nanotubes Forulas for calculating the su of all distances, known as the Wiener index, in the»archair«nanotubes are given. The sae ethod was alied in the case of»zig-zag«tubes. INTRODUCTION Carbon nanotubes, the one-diensional carbon allotroes, are intensively studied, with resect to their roise to exhibit uniue hysical roerties: echanical, 1, otical, 3,4 electronic, 5 etc. The diaeter of single walled nanotubes, SWNTs, is distributed on a large allet fro less than 1 n to 10 n or ore. Thinner tubes show zero helicity 6 while those with diaeters larger than n usually exhibit defects, kinks, and twists. Wall defects and oen ends ay undergo cheical reactions, resulting in functionalized nanotubes. 7 Endohedral functionalization with fullerenes, etals or inorganic salts, enetrating by the caillarity effect the oen ends of SWNTs, has also been reorted. 8 10 This aer resents a ethod for calculating a toological roerty, naely the su of all distances, also known as the Wiener index, 11 in»archair«swnts. Note that in the constructive version of Diudea et al., 1 15 this class of non-twisted tubes is naed TUVC 6 c,n (see Figure 1). Wiener index forulas for various classes of tori (i.e., nanotubes the two ends of which are identified) have been resented elsewhere. 16 Figure 1. An»archair«TUVC 6 0,n. METHOD (A) Let us consider a hexagonal crenellated (i.e., archair) lattice, as illustrated in Figure. We choose a reference vertex v, fro which the toological distances to all other vertices are evaluated. The su of such distances, on each level, is given in the figure as S i. * Dedicated to Professor Nenad Trinajsti} on the occasion of his 65 th birthday for his ioneering activity in Cheical Grah Theory. ** Author to who corresondence should be addressed. (E-ail: diudea@che.ubbcluj.ro)
11 M. V. DIUDEA et al. The subtraction of the last ter in the above euation is reasoned as follows: the reference vertex v ay be located at any level 1 < <, each tie considering TUV as being obtained by two saller tubes sharing a coon level, naely that containing vertex v. Itisobvious that the actual level of v is counted twice. Exansion of functions in (9) leads to the final forula for calculating W, in the case : Figure. An»archair«olyhex lattice. The su fro v to all vertices lying at level =1is given by: s 1 (,z) = z (1) where z = od(,). Note that in our notation, 1 16 c =, n =, and a tube TUVC 6 c,n is euivalent to a (c/, c/) archair tube. For levels in the range 1 < k, (see the dashed line in Figure ) the increent to s 1 is calculated as: and the distance su: s n (k,z) =k +z 3 ( 1) k () sz s (,s,z) = sn ( k, z) ( z) k1 The total distance su u to level is given by: st (,,z) =( z) + s (, s, z) s The distance su at level = is: z s (,z) = sn ( k, z) ( z) k1 Now calculate the su at levels > as: and the total su u to : (3) (4) (5) s c (,s,z) =s (,z) +(s ) (6) st c (,,z) = s (, s, z) c s1 (7) The total su fro v located at level = 1 to all vertices in TUVC 6 will be: s v (,,z) =st (,,z) +st c (,,z) (8) We are now ready to calculate the Wiener index of a TUVC 6, as: W(,,z) = s v ( z,, ) + st ( z,, ) + 1 s 1 (,z) (/) s 1 (,z) (9) W TUVC 6 ( z) ( z1 ) 1( 1) 3( 1) z 3( 1) 1 z 1 z ( z) 1( 1) 1z ( z,, ) ( z 1) 3 6( 1) 8 1 3 8 6 188 ( z) 1 1( 1) ( z 1) 6( 1) 4z1 1 z 4 14 6( 1) ( ) Keeing in ind the following: (i) (10) ( 1) ( z) = 1, because: if is even, z = 0 and ( 1) ( z) = ( 1) =1 if is odd, z =1, 1 is even and ( 1) ( z) = ( 1) 1 =1 (ii) ( 1) ( z+1) = ( 1) ( 1) ( z) = ( 1) since, as calculated above, ( 1) z =1 (iii) ( 1) z = ( 1) because: if is even and z = 0, ( 1) z = ( 1) 0 = 1 = ( 1) if is odd and z = 1, ( 1) z = ( 1) 1 = 1 = ( 1) (iv) z = (1 ( 1) )/ relation (10) becoes: W TUVC 6 (, ) 1 [ ( 1 8) 8 ( ) 3( 1( 1) ) ] (11) In the case, (i.e., short tubes, stuv), W is calculated by the forula: W stuvc6 (,,z) = st (,, z) s 1(, z) ( / ) s 1(, z) (1) Exansion of functions in (1) leads to the final forula for calculating W, in the case of short tubes: W stuvc6 (,,z) = 1 3 1 z ( ) 4 4 8 3 ( 1 ) ( z 1) (13) Croat. Che. Acta 77 (1 ) 111 115 (004)
WIENER INDEX OF ARMCHAIR POLYHEX NANOTUBES 113 A siilar rocedure as used for relation (10) leads to the final forula: W stuvc6 (,) = 1 4 4 8 3( 1) ( 1( 1) ) (14) For =,, the forula for sile cycles on 4 vertices is recovered: W(C 4P )= 4 ( ) z ( )( 1 z 1 3 1 ) =8 3 (15) METHOD (B) The su fro v to all vertices lying at level»1«is: where z = od(,). s 1 (,z) = z For levels in the range 1 < k, the su is: sv k (,k,z) = + z ( 1) k +(k 1) od((k 1),) (16) The total distance su u to level n is given by: st n (,n,z) = z + sv (, k, z) and, after calculations: n k k (17) st n (,n,z) = n 1 z 1 ( 1)n + 1 3 n3 1 n 1 3 n + 1 4 1 ( 1)n (18) The distance su at level n = is: st () = 7 3 1 1 1 3 3 4 1 1 ( ( ) ) (19) Calculate now the sus at levels k > as follows: svn (,k) =3 +(k 1) (0) The total distance su fro v located at level»1«to all vertices in TUV, for n>, is: stn (,n) =st () + svn (, k) and, after calculations: n k1 (1) stn (,n) =st () +( + 1)( ) () The Wiener index of a TUVC 6,, is given by: W TUVC6 (,,z) = st n(, n, z) stn(, n) s1 (, z) (3) n 1 n1 In the case (i.e., short tubes, stuv), the forula is: W stuvc6 (,,z) = stn ( nz,, ) s 1( z, ) (4) n 1 Exansion of the above functions leads to the final forulas for calculating W. Case (long tubes, TUVC 6 ): W TUVC6 (,,z) = 13 4 ( 1 z ) ( 1 ( 1 ) ) 6 3 4 1 3 ( )[ 14 10 4 6 833( 1) 6z (5) With z = (1 ( 1) ) /, relation (5) transfors into relation (11). Case (short tubes, stuv); exansion of relation (4) leads to: W stuvc6 (,,z) = 1 4 1 6 3 4 1 1 1 z 1 1 ( ( ) ) ( ( ) ) (6) Substituting z as above, (6) transfors into relation (14). Tables I and II list soe nuerical values for the Wiener index of long TUVC 6, and short tubes stuvc 6,, resectively. TABLE I. Wiener index of long tubes, TUVC 6,, W W 3 3 507 4 4,176 3 8 5,11 4 8 10,64 3 16 3,136 4 16 6,336 5 5 6,685 6 6 16,704 5 10 3,560 6 1 81,16 5 15 89,685 6 18 3,488 5 0 190,560 6 4 474,64 7 7 36,183 8 8 70,656 7 14 175,784 8 16 343,040 7 1 483,455 8 4 943,104 7 8 1,06,44 8 3,001,90 Croat. Che. Acta 77 (1 ) 111 115 (004)
114 M. V. DIUDEA et al. TABLE II. Wiener index of short tubes, stuvc 6,, W W 9 9 17,449 10 10 16,000 9 8 99,07 10 8 134,400 9 7 74,745 10 6 73,90 9 6 54,16 10 5 50,880 9 5 37,33 10 4 3,30 9 4 3,616 10 3 18,080 9 5,83 10 8,000 METHOD (A). CASE OF»ZIG-ZAG«, TUHC 6 c,n TUBES We alied ethod (A) in the case of»zig-zag«, TUHC 6, tubes (Figure 3), as follows: and the total su u to : st c (,) = s (, s) c s1 (3) The total su fro v located at level = 1 to all vertices in TUHC 6 will be: s v (,) =st (,) +st c (,) (33) The Wiener index of a TUHC 6, is now: W(,) = 1 s v (,) + st (,) + s 1 (,z) d a () d b (,) (34) The last two negative ters in the above euation have ainly the sae reason as in the case of TUVs and account for the odulo(,3) and -diension, resectively. The first difference is of the following for: d a () =d 0 ()(1 od(,3))( od(,3)) / + d 1 ()( od(,3))(od(,3)) + d ()(1 od(,3))(od(,3)) / ( ) (35) where: d 0 () =4 + (trunc( / 3))( 3 ) (36) d 1 () = 4 + (trunc( / 3))( +1) (37) d () = 4 + (1 + trunc( / 3))( 1) (38) Evaluation of d a (), in (35) leads to: d a () =( / 3)(13 1) (39) Figure 3. A»zig-zag«olyhex lattice. The su fro v to all vertices on level = 1 is: s 1 (,z) = (7) For levels in the range 1 < k, the distance su is now: s 1 s (,s) = ( k) (8) k1 The total distance su u to level is given by: st (,) = + s (, s) s The distance su at level = is: 1 (9) s () = ( k) (30) k1 Calculate the su at levels > as: s c (,s) =s () +4(s ) (31) The second difference d b (,) in (34) is: d b (,) = ( )4 +( ) (40) Exansion of all the functions in (34) leads to the final forula for calculating W, in the case of long tubes : W TUHC6 (,) = 6 83 +4 6 3 + (41) which is identical to the forula reorted in a receding aer. 17 Forula for short tubes (case, TUHC 6 ), is as follows: W stuhc6 (,,) = where: st (,) + s 1 () ( / )( s 1 ()) d c (,) (4) d c (,) =kk ( 1) k (43) Croat. Che. Acta 77 (1 ) 111 115 (004)
WIENER INDEX OF ARMCHAIR POLYHEX NANOTUBES 115 Exansion of the above functions leads to the final forula for W, in the case of short tubes: W stuhc6 (,) = 6 3 +4 +6 4 (44) For =1,, the forula for sile cycles on vertices is recovered: CONCLUSIONS W(C P )= 3 (45) Forulas for calculating the su of all distances in»archair«olyhex nanotubes using two ethods are given. Method (A) was successfully alied in the case of»zig-zag«tubes. Acknowledgeent. This aer was suorted by the Roanian GRANT 003. REFERENCES 1. E. W. Wong, P. E. Sheehan, and C. M. Lieber, Science 77 (1997) 1971 1975.. B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76 (1996) 511 514. 3. W.-Zh. Liang, S. Yokojia, M.-F. Ng, G.-H. Chen, and G. He, J. A. Che. Soc. 13 (001) 9830 9836. 4. J. N. Colean, A. B. Dalton, S. Curran, A. Rubio, A. P. Davey, A. Drurry, B. McCarthy, B. Lahr, P. M. Ajayan, S. Roth, R. C. Barklie, and W. J. Blau, Adv. Mater. 1 (000) 13 16. 5. T. W. Odo, J.-L. Huang, Ph. Ki, and Ch. M. Lieber, J. Phys. Che. B, 104 (000) 794 809. 6. R. Saito, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Mater. Sci. Eng. B, 19 (1993) 185 191. 7. A. Hirsch, Angew. Che., Int. Ed. Engl. 41 (00) 1853 1859. 8. W. Han, S. Fan, Q. Li, and Y. Hu, Science 77 (1997) 187 189. 9. J. Sloan, J. Haer, M. Zwiefka-Sibley, and M. L. H. Green, Che. Coun. (1998) 347 348. 10. M. Wilson and P. A. Madden, J. A. Che. Soc. 13 (001) 101 10. 11. H. Wiener, J. A. Che. Soc. 69 (1947) 17 0. 1. M. V. Diudea and A. Graovac, MATCH Coun. Math. Cout. Che. 44 (001) 93 10 13. M. V. Diudea, I. Silaghi-Duitrescu, and B. Parv, MATCH Coun. Math. Cout. Che. 44 (001) 117 133. 14. M. V. Diudea and P. E. John, MATCH Coun. Math. Cout. Che. 44 (001) 103 116 15. M. V. Diudea, Bull. Che. Soc. Jn. 75 (00) 487 49 16. M. V. Diudea, MATCH Coun. Math. Cout. Che. 45 (00) 109 1. 17. P. E. John and M. V. Diudea, Croat. Che. Acta, 77 (004) 17 13 SA@ETAK Wienerov indeks za»archair«oliheksagonalne nanocijevi Mircea V. Diudea, Monica Stefu, Basil Pârv i Peter E. John Dana je forula za izra~unavanje Wienerova indeksa za»archair«oliheksagonalne nanocijevi. Ista je etoda riijenjena i na»zig-zag«oliheksagonalne nanocijevi. Croat. Che. Acta 77 (1 ) 111 115 (004)