Szeged Index of armchair polyhex Nanotube A. Iranmanesh * and Y. Pakravesh Department of Mathematice, Tarbiat Modares Universiry P. O. Box: 45-37, Tehran, Iran iranmana@modares.ac.ir Abstract Topological indices of nanotubes are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of nanotubes and their physico-chemical properties. The Szeged index is obtained as a bond additive quantity where bond contributions are given as the product of the number of atoms closer to each of the two end points of each bond. In this paper we find an exact expression for Szeged index of armchair polyhex nanotube ( TUAC6 [ p, k ] ). Keywords: Szeged index, TUAC 6 [ p, k ] Nanotube. Introduction A graph G consists of a set of vertices VG ( ) and a set of edges EG ( ). In chemical graphs, each vertex represented an atom of the molecule and covalent bonds between atoms are represented by edges between the corresponding vertices. This shape derived form a chemical compound is often called its molecular graph, and can be a path, a tree or in general a graph. A topological index is a single number, derived following a certain rule, which can be used to characterize the molecule. Usage of topological indices in biology and chemistry began in 947 when chemist Harold Wiener introduced Wiener index to demonstrate correlations between physico-chemical properties of organic compounds and the index of their molecular graphs. Wiener originally defined his index (W ) on trees and studied its use for correlation of physico chemical properties of alkenes, alcohols, amines and their analogous compounds. 3 A number of successful QSAR studies have been made based in the Wiener index and its decomposition forms. 4 In a series of papers, the Wiener index of some nanotubes is computed. 5- Another topological index was introduced by Gutman and called the Szeged index, abbreviated as Sz. Let e be an edge of a graph G connecting the vertices u and v. Define two sets N ( eg) and N ( eg) as N ( eg) = { x V( G) d( u, x) < d( v, x)} and N ( e G) = { x V( G) d( x, v) < d( x, u)}. The number of elements of N ( eg) and N ( eg) are denoted by n ( e G) and n ( e G) respectively. The szeged index of the graph G is defined as Sz ( G ) = Sz = n ( e G ) n ( ) e E( G) e G. The Szeged index is a * Corresponding Author
modification of Wiener index to cyclic molecules. The Szeged index was conceived by Gutman at the Attila jozsef university in Szeged. This index received considerable attention. It has attractive mathematical characteristics. In 3-7, another topological index of some nanotubes is computed. In this paper, we computed the Szeged index of TUAC 6 [ p, k ] nanotube. level level level 3 Two dimensional lattice of TUAC 6 [4,4] nanotube, p = 4, k = 4 Figure According to Figure, we denote the number of horizontal lines in one row by p and the number of levels by k. Throughout this paper, our notation is standard. The notation [ f ] is the greatest integer function..the Szeged index of TUAC 6 [ p, k ] nanotube In this section, the Szeged index of T = TUAC 6 [ p, k] nanotube is computed. Let e be an arbitrary edge of nanotube. Suppose n ( e G ) counts the vertices of G lying closer to one vertex than to other vertex and the meaning of n ( ) e G is analogues. For computing the Szeged index of T, we assume two cases: Case : p is even. Lemma. If e is a horizontal edge of T, then n ( e G ) n ( e G ) = p k Proof. Suppose that e is a horizontal edge of T, for example e= uv in Figure. In Figure, the region R has the vertices that belongs to N (e G)and the region R' has the vertices that belongs to N (e G). So we have n ( e G) = n ( e G) = pk, therefore n ( eg ) n ( e G ) = p k By the symmetry of. TUAC 6 [ p, k ] nanotube for every horizontal edges the above relations is hold..
R u v R e= uv is an horizontal edge in level m = 6. Figure For simplicity, we define k m a = and m b =. Lemma. Suppose p is even. If e is a oblique edge in level m( m k), then we have i) If m p and k m p then n ( e G ) = p(k+m-)+b(5-m+3b-p)+a(k-m-a-)+k-6m+. ( I ) ii) If m p and k m> p, then n ( eg ) = p(k-/p-)+b(5-p+3b-m)-4m+4. ( II ) iii) If m > p and k m p, then n ( e G ) = p(k-m+/p)+a(k-a-m-)+(k-m-). ( III ) iv) If m > p and k m> p, then n ( e G ) = p(k-m)+. ( IV ) Proof. Let e be a oblique edge of T, for example e= uv in Figure 3. In this figure, the region R has the vertices that belongs to N (e G)and the region R' has the vertices that belongs to N (e G). R v u R e= uv is an oblique edge in level m = 6 Figure 3 Number of vertices that is closer to u than to v is as follows: If m p and k m p, then a b n ( e G) p( k m ) (4 i) + (p 4 i) + ( k m a )(a+ ) + ( m b )(p 4b 4) = = + + i= i= p(k+m-)+b(5-m+3b-p)+a(k-m-a-)+k-6m+.
If m p and k m> p, then ( p ) n ( e G) p(k m p ) (4) i + (p 4) i + ( m b )(p 4b 4) = i= i= p(k-/p-)+b(5-p+3b-m)-4m+4. If m > p and k m p, then = + + a p n ( e G) p( k m ) (4 i) + (p 4 i) + ( k m a )(a+ ) + = i= i= p(k-m+/p)+a(k-a-m-)+(k-m-). And if m> p and k m> p, then = + + b ( p ) ( p ) n ( e G) = p(k m p+ ) + (4 i) + ( p 4 i) + = p(k-m)+. i= i= By the symmetry of TUAC 6 [ p, k ] nanotube for every oblique edge this relations is hold. Remark 3. According to Figure 3, let e be a oblique edge in level m( m k), then n ( e G) = pk n ( e G). Theorem 4. If p is even, then the Szeged index of TUAC 6 [ p, k ] nanotube is given as follows:. k is even i) If k p, then we have ii) If p < k p, then we have 3 3 5 4 3 Sz(T)=p (k -k -k+)+p (k -k)+p(-/6k +/3pk +/3k -/3k -/3k). Sz(T)=p (9/-3/3k)+p (4/3k -/3k-0/3)+p (k +/k +4/3k-4/3)+p (/3k -4k -/3k 5 4 3 3 3 4 iii) If k > p, then we have. k is odd. 5 4 3 6-4/3k+/5)+p(-/30k -7/k +7/3k -5/3k -4/5k)+3/5p. 4 5 3 4 3 ( ) p(/4k -/5k +/3k -/5k)+p (+/3k +/3k +/3k -4/3k+/5)+ 3 3 4 5 6 Sz T = p (-k +5/k -8/3k+8/3)+p (0k -6k-/3)+p (9/-3/3k)+3/5p. 5 4 3 3 3 i) If k p, then we have Sz(T)=p(-/6k +/3k -/3k -4/3k +5/6k+)+p (k -)+p (k -k -k+). ii) If p < k p, then we have 3 3 4 5 4 3 ( ) p (k +/k +3/3k+3/6)+p (+4/3k -/3k-7/3)+p(-/30k -/k +/3k +5/6k -3/0k-3/4) 4 3 5 6 Sz T = iii) If k > p, then we have +p (-/3k +5/3k -3k -3/3k+7/5)+9/p +3/5p. 3 3 4 6 5 4 Sz( T ) = p (-k +5/k -/3k+/6)+p (+0k -6k+/3)+3/5p +p(-/5k +/4k - 3 4 3 5 /3k -/k +8/5k+/4)+p (/3k +/3k +5/3k -8/3k+/5)+p (-3/3k+9/). Proof. At first, suppose A and B are the sets of all horizontal and oblique edges of T, respectively. Then we have
(*) Sz( T) = n ( e G) n ( e G) + n ( e G) n ( e G). e A e B The number of horizontal edges are pk. Thus we have The number of oblique edges are pk. So Let k be even. Now for k p, we have : 3 n( e G) n( e G) = p k. p = p k. e A k 5 4 3 n( e G) n( e G) = p.{ {( I)( pk ( I))}} = p{(-/k +/6k +/6k -/6k -/3k)+ e B m= 3 When p < k pwe have : p (k -k -/k+)+p(/k -k)}. k p p k e C m= m= k p m= p+ 5 3 4 3 3 3 n ( e G) n ( e G) = p.{ {( II )( pk ( II)) + ( I)( pk ( I )) + ( III )( pk ( III ))}} = p{(-/60k -5/6k +7/6k -/5k+p(-/3k +/6k -k -/3k+/5)-/3p k+p (+k -/4k +/3k-/3) And if k > p, then we have : 3 4 5 +p (7/3k -5/3) +p (9/4-3/6k)+3/0p }. k p p k e C m= m= k p m= p+ 5 4 3 4 3 3 n ( e G) n ( e G) = p.{ {( II)( pk ( II)) + ( IV)( pk ( IV)) + ( III)( pk ( III))}} = p{(-/0k +/8k +/3k -/5k)+p(+/6k +/3k +/3k -7/3k+/5)+p (-k +3/4k -4/3k+4/3) Suppose k is odd, in this case for k 3 4 5 +p (+5k -3k-/3)p (-3/6k+9/4)+3/0p }. p we have : 5 4 3 3 n( e G) n( e G) = p{-/k +/6k -/3k -/3k +/+5/k+p(k -)+p (/-/k-k +k )}. e C When p < k pwe have : 5 4 3 4 3 n( e G) n( e G) = p{-/60k -/4k +/6k +5/k -3/0k-3/8+p(-/3k +5/6k -3/k -3/6k+7/30) e C 3 3 4 5 +p (k -/4k +3/6k+3/)+p (7/3k -/3k-7/6)+p (-3/6k+9/4)+3/0p }. And if k > p, then we have : 5 3 4 3 n( e G) n( e G) = p{-3/40k -/k +9/0k+p(+/8k +/3k +5/4k -4/3k+/0) e C 3 3 4 5 p (-5/4k +k -/4k+/3) + p (9/4k -3k+9/)+p (-3/8k+/3)+93/40p }. So if p is even, then by using the above relations in (*) the result is hold. Case. p is odd. Lemma 5. If e is a oblique edge in level m( m k), then we have : i) If m p and k m p then n ( e G ) = p(k+m-)+b(5-m+3b-p)+a(k-m-a-)+k-6m+. ii) If m p and k m> p, then n ( eg ) = p(k-/p-)+b(5-p+3b-m)-4m+7/. iii) If m > p and k m p, then n ( e G ) = p(k-m+/p)-3/+a(k-a-m-)+(k-m).
iv) If m > p and k m> p, then n ( e G ) = p(k-m). Proof. The proof is similar to the proof of lemma. Theorem 6. If p is odd, then Szeged index of TUAC 6 [ p, k ] nanotube is given as follows:. k is even. i) If k p, then we have ii) If p < k p, then we have: 3 3 5 4 3 Sz(T)=p (k -k -k+)+p(-/6k +/3k +/3k -/3k -/3k)+p (k -k). 3 3 5 4 3 4 3 Sz(T)=p (k +/k -6/3k-4/6)+p(-/30k -7/k +/3k +/6k -7/5k-7/4)+p (-/3k +/3k + 4 5 6 iii) If k > p, then we have :. k is odd. k -0/3k-8/5)+p (4/3k -/3k+4/3)+p (9/-3/3k)+3/5p. Sz( T ) = p (-k +5/k -6/3k+7/6)+p(-/5k +/4k +/k -4/5k+/4)+p (/3k +/3k i) If k p, then we have 3 3 5 4 4 3 4 5 6 +8/3k -8/3k-6/5)+p (0k -6k+4)+p (9/-3/3k)+3/5p. 3 3 5 4 3 Sz(T)=p (k -k -k+)+p(-/6k +/3k -/3k -4/3k +5/6k+)+p (k -). ii) If p < k p, then we have: Sz(T)=p (+k -7/3k+/k -0/3)+p(-/30k -/k -4/3k -/3k +4/30k)+p (-/3k +5/3k 3 3 5 4 3 4 3 iii) If k > p, then we have : 4 5 6 +3k +/3k-8/5)+p (4/3k -/3k+7/3)+p (9/-3/3k)+3/5p. Sz( T ) = p (-k +5/k -9/3k-4/3)+p(-/5k +/4k -k +/5k)+p (/3k +/3k +/3k 3 3 5 4 3 4 3 4 5 6 -/3k-/5)+p (0k -6k+5)+p (9/-3/3k)+3/5p. Proof. The proof is similar to the proof of Theorem 4. Refrences. P. V. Khadikar, S. Singh,M. Jaiswal and M. Mandloi, Bioorganic & Medicinal. 4, 4795 (004).. H. Wiener, J. Am. Soc. 69, 7 (947). 3. P. V. Khadikar and S. Karmakar, Acta Chim. Slov. 49, 755 (00). 4. V. K. Agrawal, S. Bano, K. C. Mathur and P. V. Khadikar, Indian Acad. Sci., 37 (000). 5. H. Deng, MATCH Commun. Math Comput. Chem. 57, 393 (007). 6. H. Zhang, S. Xu and Y. Yang, MATCH Commun. Math Comput. Chem. 56, 53 (000). 7. M. Stefu and M. V. Diudea, MATCH Commun. Math Comput. Chem. 50, 33 (004). 8. M. V. Diudea, M. Stefu, B. Parv and P.E. John, Croat. Chem. Acad. 77, (004). 9. M. Randic, Acta Chim. Slov. 49, 483 (00). 0. P. E. John and M. V. Diudea, Croat. Chem. Acad. 77, 7 (004).. S. Yousefi and A. R. Ashrafi, MATCH Commun. Math Comput. Chem. 56, 69 (006).. I. Gutman, Graph Theory Notes, New York (994), p. 7. 3. A. Iranmanesh and B. Soleimani, MATCH Commun. Math Comput. Chem. 57, 5 (007). 4. Ali Reza Ashrafi and Amir Loghman, Nanotubes, J. Comput. Theor. Nanosci. 3, 378 (006). 5. A. Iranmanesh and O. Khormali, J. Comput. Theor. Nanosci (In press). 6. A. Iranmanesh and A. R. Ashrafi, J. Comput. Theor. Nanosci (In press).
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