Topological Index of Some Carbon Nanotubes and the Symmetry Group for Nanotubes and Unit Cells in Solids

Abstract

The Padmakar–Ivan (PI) index of a graph G is defined as $$ \mathrm{PI}(G)={\displaystyle \sum_{e\in G}\left[{n}_{eu}\left(e\left|G\right.\right)+{n}_{ev}\left(e\left|G\right.\right)\right]}, $$ where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u, and summation goes over all edges of G. Let G be a connected graph, n eu (e|G) be the number of vertices of G lying closer to u, and n ev (e|G) be the number of vertices of G lying closer to v. Then the Szeged index of G is defined as the sum of n eu (e|G)n ev (e|G) over the edges of G. The PI index of G is the Szeged-like topological index defined as the sum of [n eu (e|G) + n ev (e|G)], where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u, and summation goes over all edges of G. Different types of symmetry groups are commonly used in chemistry. Point groups are used for molecules, whereas, for solids, the 230 space groups are used. Neither of these types of symmetry groups are suitable for representing unit cells in solids, the symmetry of which is intermediate between that of point groups and space groups representing the symmetry of unit cells in an infinite lattice; the third type of symmetry group must be used. An algorithmic method of generating these symmetry groups is described. It could be demonstrated that these groups are appropriate by applying the conventional symmetry. This technique has been used in the case of the two-dimensional graphite lattice. Because the new method generates symmetry tables using only the topology of the system, the symmetry properties of graphs could also be readily derived. Finally, the relationship between these groups and the other two types of groups is established.