Digest Journal of Nanomaterials and Biostructures Vol. 3, No.4, December 2008, p. 227 - 236 ON DISTANCE-BASED TOPOLOGICAL INDICES OF HC 5 C 7 [4p,8] NANOTUBES ALI REZA ASHRAFI • , HAMID SAATI AND MODJTABA GHORBANI Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-51167, I. R. Iran Let G be a connected graph, n u (e) is the number of vertices of G lying closer to u and n v (e) is the number of vertices of G lying closer to v. Then the Szeged index of G is defined as the sum of n u (e)n v (e), over edges of G.. The PI index of G is a Szeged-like topological index defined as the sum of [m u (e)+ m v (e)], where m u (e) is the number of edges of G lying closer to u than to v, m v (e) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, the PI and Szeged indices of a HC 5 C 7 [4p,8] nanotube are computed for the first time. (Received October 15, 2008, accepted October 22, 2008) Keywords: PI index, Szeged index, HC 5 C 7 [4p,8] nanotube 1. Introduction Carbon nanotubes are molecular-scale tubes of graphitic carbon with outstanding properties. They are among the stiffest and strongest fibres known, and have remarkable electronic properties and many other unique characteristics. For these reasons they have attracted huge academic and industrial interest, with thousands of papers on nanotubes being published every year. Commercial applications have been rather slow to develop, however, primarily because of the high production costs of the best quality nanotubes. A major part of the current research in mathematical chemistry, chemical graph theory and quantitative structure-activity-property relationship studies involves topological indices. 1 Topological indices (TIs) are numerical graph invariants that quantitatively characterize molecular structure. The problem of distances in graph continues to focus the attention of scientist both as theory and applications. In 1947, Harold Wiener has proposed his path number, as the total distance between all carbon atoms for correlating with the thermodynamic properties of alkanes. Numerous of its chemical applications were reported and its mathematical properties are well understood 2-5 . The Szeged index is another topological index which is introduced by Ivan Gutman. 6-8 To define the Szeged index of a graph G, we assume that e = uv is an edge connecting the vertices u and v. Suppose M eu (e|G) is the number of vertices of G lying closer to u and M ev (e|G) is the number of vertices of G lying closer to v. Edges equidistance from u and v are not taken into account. Then the Szeged index of the graph G is defined as Sz(G) = ∑ e=uv∈E(G) M eu (e|G)M ev (e|G). Khadikar and co-authors 9-13 defined a new topological index and named it Padmakar-Ivan index. They abbreviated this new topological index as PI. This newly proposed topological index does not coincide with the Wiener index for acyclic molecules. It is defined as PI(G) = ∑ e∈G [n eu (e|G)+ n ev (e|G)], where n eu (e|G) is the number of edges of G lying closer to u than to v and n ev (e|G) is the number of edges of G lying closer to v than to u. • Corresponding author. E-mail: ashrafi@kashanu..ac.ir