A Numerical Method for Computing the Wiener Index of One-Heptagonal Carbon Nanocone

Abstract

In the past years, nanostructures involving carbon have been the focus of an intense research activity which is driven to a large extent by the quest for new materials with specific applications. One pentagonal carbon nanocones originally discovered by Ge and Sattler in 1994.1 These are constructed from a graphene sheet by removing a 60 wedge and joining the edges produces a cone with a single pentagonal defect at the apex. The inclusion of the heptagons in the hexagonal lattice leads to the appearance of negative curvature, Figure 1. The single sevenfold in the plain graphene lattice was theoretically studied but this situation, unfortunately, has not been observed in the experiment yet.2 We first describe some notations which will be kept throughout. Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V G and E G , respectively. A topological index of a graph G is a numeric quantity related to G. The oldest topological index is the Wiener index which introduced by Harold Wiener.3 The most important works on the geometric structures of nanotubes, nanotori and their topological indices were done by Diudea and his co-authors.4–9 One of the present authors (ARA) continued this program to calculate the Wiener index of some other nanostructures.10–15 In some research papers they computed the Wiener index of some nanotubes and nanotori. We encourage the reader to consult16–19 and references therein for background material as well as basic computational techniques. In this paper, we continue this program to compute the Wiener index of