Symmetry and PI Index of C 60+12n Fullerenes F. KoorepazanMoftakhar and A. R. Ashrafi  Department of Nanocomputing, Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-51167, I. R. Iran Abstract A topological index is a numerical invariant of a chemical graph. In this paper, we apply the action of the symmetry group of a fullerenes molecule C60 + 12n on the set of its chemical bonds to compute the PI index of C60 + 12n. This is an efficient method that can be applied for other classes of fullerenes. Keywords: Fullerene, PI index, topological symmetry. 1. Introduction Throughout the paper, all graphs are finite, simple and connected. Let G be such a graph. We write V(G) and E(G) for the vertex and edge sets of G, respectively. A molecular graph is the graph in which the vertices are atoms and edges are chemical bonds of a given molecule. The degree of each vertex in such graphs is at most four. A topological index for a molecular graph G is a number invariant under each graph isomorphism from G into a graph H [1,2]. We now recall some algebraic definitions that will be used in the paper. Let G be a simple nvertex molecular graph and e = uv be an edge of G. The distance between vertices u and w of G is denoted by d(u,w). Define two quantities m u (e) and m v (e) as follows: m u (e) is defined as the number of edges lying closer to the vertex u than the vertex v, and m v (e) is defined analogously. The edges equidistant from both ends of the edge uv are not counted. The PI index of G is defined as PI(G) =  e=uv [m u (e) + m v (e)] [3,4]. A topological symmetry of a molecule M is an automorphism of its molecular symmetry. The topological symmetry of a molecular graph is not usually the same as its point group  Corresponding author (ashrafi@kashanu.ac.ir).