Computing Padmakar-Ivan Index of a TC4C8(R) Nanotorus

A graph G consists of a set of vertices V G and a set of edges E G . If the vertices u, v ∈ V G are connected by an edge e then we write e = uv. In chemical graphs, each vertex represents an atom of the molecule, and covalent bonds between atoms are represented by edges between the corresponding vertices. This shape derived from a chemical compound is often called its molecular graph, and can be a path, a tree, or in general a graph. The graph G is said to be connected if for every vertices x and y in V G there exists a path between x and y. The distance dG u v or d u v , between vertices u and v of a connected graph G is the number of edges in a minimum path from u to v. A topological index is a real number related to a molecular graph, which is a graph invariant. There are several topological indices already defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of the molecules. The Wiener index W is the first topological index proposed to be used in Chemistry. It was introduced in 1947 by Harold Wiener,1 as the path number for characterization of alkanes. It is defined as the sum of distances between all pairs of vertices in the graph under consideration. Here, we consider a new topological index, named Padmakar-Ivan index and abbreviated as PI index.2–4 To define the PI index of a connected graph G, we correspond to an edge e = uv of E G , two quantities neu(e G) and nev(e G) in which neu(e G) is the number of edges lying closer to the vertex u than the vertex v, and nev(e G) is the number of edges lying closer to the vertex v than the vertex u. Then the PI index of the graph G is defined as PI G =∑e=uv∈E G [neu(e G +nev(e G)].