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## Journal of the Chilean Chemical Society

*versión On-line* ISSN 0717-9707

### J. Chil. Chem. Soc. v.51 n.3 Concepción sep. 2006

#### http://dx.doi.org/10.4067/S0717-97072006000300008

J. Chil. Chem. Soc., 51, N°.3 (2006), p.968-970
Department of Mathematics, Faculty of Science, University of Kashan, Kashan, 87317-51167, Iran
The Padmakar–Ivan (PI) index of a graph G is defined as PI(G) = ∑[neu(e|G)+ n
A topological index is a real number related to a molecular graph. It must be a structural invariant, i.e., it must not depend on the labeling or the pictorial representation of a graph. Many topological indices have been defined and several of them have found applications as means to model chemical, pharmaceutical and other properties of molecules Here, we consider a new topological index, named the Padmakar-Ivan index, which is abbreviated as the PI index We now describe some notations which will be adhered to throughout. Benzenoid systems (graph representations of benzenoid hydrocarbons) are defined as finite connected plane graphs with no cut-vertices, in which all interior regions are mutually congruent regular hexagons. More details on this important class of molecular graphs can be found in the book of Gutman and Cyvin Let G be a simple molecular graph without directed or multiple edges and without loops, the vertex and edge-shapes of which are represented by V(G) and E(G), respectively. The graph G is said to be connected if for every pair of vertices x and y in V(G) there exists a path between x and y. In this paper we only consider connected graphs. If e is an edge of G, connecting the vertices u and v then we write e=uv. The number of vertices of G is denoted by n. The distance between a pair of vertices u and w of G is denoted by d(u,w). We now define the PI index of a graph G. To do this, suppose that e = uv and introduce the quantities neu(e|G) and n In a series of papers, Khadikar and coauthors2-17 defined and then computed the PI index of some chemical graphs. The present author20 computed the PI index of a zig-zag polyhex nanotube. In this paper we continue this study to prove an important result concerning the PI index and find an exact expression for the PI index of some other chemical graphs. Our notation is standard and mainly taken from the literature.
It is a well-known fact that the algebraic structure count of the linear phenylene with h six-membered rings is equal to h + 1. Gutman23 proved that the same expression is true if each four-membered ring in the phenylene is replaced by a linear array consisting of k, k = 4, 7, 10, ..., four-membered rings. Here we find the PI index of this graph, which is denoted by T, Figure 1.
Suppose N(e) = |E| - (neu(e|G) + n
If we put k = 0 in the last formula, then we obtain the PI index of polyacenes which has been computed before by Khadikar, Karmarkar and Varma
Vukievi and Trinajsti
We first consider a class of pericondensed benzenoid graphs consisting of two rows of hexagons of various lengths, Figure 2. Without loss of generality, we can assume that n ≥ m. It is easy to see that G(m,n) has exactly 5n + 3m + 2 edges. Suppose A and B are the set of all vertical and oblique edges. To compute the size of A, we note that there are two rows of vertical edges. In the first row, N(e) = n + 1 and in the second N(e) = m + 1. Thus ∑
It is easy to see that the collection P = {B1, B2, …, B2n} is a partition for B and if we define X = €
Finally, we consider a class of pericondensed benzenoid graphs consisting of three rows of hexagons of various lengths, Figure 3. To calculate the PI index of G(m,n,k), it is enough to consider four cases such that m ≤ n < k; m < n, k and n ≥ k; n < m, k; and n=m=k, see Figure 3. Two cases, where k ≤ n < m; k < n, m and n ≥ m are similar to the first two cases above. We have the following result:
Therefore ∑
The Wiener index W was the first topological index to be used in chemistry13. It was introduced in 1947 by Harold Wiener, as the path number for characterization of alkanes. In chemical language, the Wiener index is equal to the sum of all the shortest carbon-carbon bond paths in a molecule. In graph-theoretical language, the Wiener index is equal to the count of all the shortest distances in a graph.
Since every acyclic graph with n vertices has exactly m = n - 1 edges, the previous result states that in every acyclic graph G with m edges PI(G) = m(m-1). We now prove a weaker result about the relationship between the PI index and the number of edges in every connected graph. Before stating our result, we recall that the number of edges in a cycle T of a graph G is called the length of T.
Proof. Suppose e = uv is an edge of G. It is clear that neu(e|G) + n Case 1. Case 2. k is odd. Suppose that f = x Conversely, if G is a tree, then, according to Result 1, PI(G) = m(m-1), in which m = |E(G)|. Also, in every cycle G with odd length k, we have PI(G) = m(m-1), which completes the proof. It is easy to see that PI(
1. Todeschini, R.; Consonni, V. Handbook of Molecular Descriptors; Wiley: Weinheim, 2000. [ Links ] 2. Khadikar, P.V. Nat Acad Sci Lett-India, 2004, 23, 113. [ Links ] 3. Khadikar, P.V.; Karmarkar, S.; Agrawal, V.K. Nat Acad Sci Lett-India 2000, 23, 124. [ Links ] 4. Khadikar, P.V.; Karmarkar, S.; Agrawal, V.K. Nat Acad Sci Lett-India 2000, 23, 165. [ Links ] 5. Khadikar, P.V.; Karmarkar S.; Agrawal, V.K. J Chem Inf Comput Sci 2001, 41, 934. [ Links ] 6. Khadikar, P.V.; Kale, P.P.; Deshpande, N.V.; Karmarkar S.; Agrawal, V.K. J Math Chem 2001, 29, 143. [ Links ] 7. Khadikar, P.V.; Karmarkar S.; Varma, R.G. Acta Chim Slov 2002, 49, 755. [ Links ] 8. Khadikar, P.V.; Agrawal, V.K.; Karmarkar, S. Bioorg Med Chem 2002, 10, 3499. [ Links ] 9. Khadikar, P.V.; Karmarkar, S.; Singh, S.; Shrivastava, A. Bioorg Med Chem 2002, 10, 3163. [ Links ] 10. Khadikar, P.V.; Phadnis, A.; Shrivastava, A. Bioorg Med Chem 2002, 10, 1181. [ Links ] 11. Khadikar, P.V.; Mandloi, D.; Bajaj, A.V.; Joshi, S. Bioorg Med Chem Lett 2003, 13, 419. [ Links ] 12. Khadikar, P.V.; Joshi, S.; Bajaj, A.V.; Mandloi, D. Bioorg Med Chem Lett 2004, 14, 1187. [ Links ] 13. Jaiswal, M.; Khadikar, P.V.; Supuran, C.T. Bioorg Med Chem 2004, 12, 2477. [ Links ] 14. Jaiswal M.; Khadikar, P.V.; Bioorg Med Chem 2004, 12, 1731. [ Links ] 15. Khadikar, P.V.; Mandloi D.; Bajaj, A.V. Oxid Commun 2004, 27, 23. [ Links ] 16. Khadikar, P.V.; Mandloi D.; Bajaj, A.V. Oxid Commun 2004, 27, 29. [ Links ] 17. Khadikar, P.V.; Agrawal, V.K.; Karmarkar, S. Oxid Commun 2004, 27, 17. [ Links ] 18. H. Wiener, J Am Chem Soc 1947, 69, 17. [ Links ] 19. Gutman, I.; Cyvin, S.J. Introduction to the Theory of Benzenoid Hydrocarbons; Springer-Verlag: Berlin, 1989. [ Links ] 20. Ashrafi, A.R.; Loghman, A. MATCH Commun Math Comput Chem 2006, 55, 447. [ Links ] 21. Cameron, P.J. Combinatorics: Topics, Techniques, Algorithms; Cambridge University Press: Cambridge, 1994. [ Links ] 22. Trinajstic, N. Chemical Graph Theory, CRC Press: Boca Raton, FL. 1992. [ Links ] 23. Vukicevic, D.; Trinajstic, N. Bulletin of Chemist and Technologists of Macedonia 2004, 23, 113. [ Links ] 24. Gutman, I. J. Serb. Chem. Soc. 2003, 68, 391. [ Links ]
E-mail: ashrafi@kashanu.ac.ir |