PI and Szeged Indices of One-Pentagonal Carbon Nanocones

Abstract

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, Figure 1. Removing additional wedges introduces more such defects and reduces the opening angle. A cone with six pentagons has an opening angle of zero and is just a nanotube with one open end. We now recall some algebraic definitions that will be used in the paper. 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.2 Khadikar and co-authors3–7 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 neu e G +nev e G)], where neu e G) is the number of edges of G lying closer to u than to v and nev e G) is the number of edges of G lying closer to v than to u. Edges equidistant from both ends of the edge uv are not counted. The Szeged index is another topological index which is introduced by Ivan Gutman.8–10 To define the Szeged index of a graph G, we assume that e = uv is an edge connecting the vertices u and v. Suppose Meu e G) is the number of vertices of G lying closer to u and Mev e G) is the number of vertices of G lying closer to v. Then the Szeged index of the graph G is defined as Sz(G) = ∑ e=uv∈E G Meu e G)Mev e G). Notice that vertices equidistance from u and v are not taken into account.