PI POLYNOMIAL OF ZIG-ZAG POLYHEX NANOTUBES

Let G be a graph with vertex and edge sets V(G) and E(G), respectively. As usual, the distance between the vertices u and v of G is denoted by d(u, v) and it is defined as the number of edges in a minimal path connecting the vertices u and v. A topological index is a real number related to a graph. It must be a structural invariant, i.e., it is fixed by any automorphism of the graph. There are several topological indices have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules. The Wiener index W is the first topological index to be used in chemistry. It was introduced in 1947 by Harold Wiener, as the path number for characterization of alkanes, [15]. In a graph theoretical language, the Wiener index is equal to the count of all shortest distances in a graph. For a survey in this topic we encourage the reader to consult [8,15]. Let G be a graph and f = uv an edge of G. nfu(f|G) denotes the number of edges lying closer to the vertex u than the vertex v, and nfv(f|G) is the number of edges lying closer to the vertex v than the vertex u. The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑[ nfu(f|G)+ nfv(f|G)] where summation goes over all edges of G see for details [7,9-11]. On can see that, for every f = uv ∈ E(G) we define PI(f) = nfu(f|G) + nfv(f|G) and N(f) = |E(G)| PI(f), Therefore ∑ ∈ − =