The PI and Edge Szeged Indices of One-Heptagonal Carbon Nanocones

Abstract

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑(mu(e)+ mv(e)), where mu(e) is the number of edges of G lying closer to u than to v, mv(e) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. The edge Szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph G. In this paper, the PI and edge Szeged indices of one-heptagonal carbon nanocone CNC7(n) computed for the first time. The inclusion of the heptagons in the hexagonal lattice leads to the appearance of negative curvature, Fig. (1). The single sevenfold in the plane graphene lattice was theoretically studied in (2), but this situation, unfortunately, has not been observed in the experi- ment yet. The heptagons were observed in the nanotubes (3) and in the work (4) the magnetic properties of negatively curved structures were calculated. We now recall some algebraic definitions that will be used in the paper. Topological indices are graph invariants and are used for Quantitative Structure- Activity Relationship (QSAR) and Quanti- tative Structure-Property Relationship (QSPR) studies (5,6). Many topological indices have been defined and several of them have found applications as means to model physical, chemical, pharma- ceutical and other properties of molecules. Let G be a simple molecular graph without directed and multi- ple 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 nontrivial topological index is the Wiener index which was introduced by Harold Wiener, (7). John Platt was the only person who immedi- ately realized the importance of the Wiener's pioneering work and wrote papers analyzing and interpreting the physical meaning of the Wiener index. The name of topological index was introduced by Haruo Hosoya (8). We encourage the reader to consult (9-11) for historical background material as well as basic computational tech- niques.