Molecular descriptors play a fundamental role in chemistry and pharmaceutical sciences since they can be correlated with a large number of physico-chemical properties of molecules. The most commonly used molecular descriptor is the Wiener index which is defined as the sum of distances between all the pairs of vertices in a molecular graph. The Graovac-Pisanski index, which is also called the modified Wiener index, considers the symmetries and the distances in molecular graphs. This index already has known correlations with the topological efficiency and the melting points for some molecules and nanostrucures. Carbon nanotubes are molecules made of carbon with a cylindrical structure possessing unusual valuable properties. Because of these properties carbon nanotubes are extremely valuable for nanotechnology and materials engineering. We focus on open-ended single-walled carbon nanotubes, which are also called tubulenes. In a mathematical model we can consider them as a subgraph of a hexagonal lattice embedded on a cylinder with some vertices being identified. In the present paper, we investigate the automorphisms and the orbits of armchair tubulenes and derive the closed formulas for their Graovac-Pisanski index.
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