# The Graovac-Pisanski Index of Armchair Nanotubes

## TL;DR

This paper derives closed-form formulas for the Graovac-Pisanski index, a symmetry-aware molecular graph invariant, specifically applied to armchair carbon nanotubes modeled as cylindrical hexagonal lattice subgraphs.

## Contribution

It provides the first explicit formulas for the Graovac-Pisanski index of armchair nanotubes, linking molecular symmetry with a modified Wiener index.

## Key findings

- Closed formulas for the Graovac-Pisanski index of armchair nanotubes
- Analysis of automorphisms and orbits in nanotube graphs
- Enhanced understanding of symmetry in molecular graph indices

## Abstract

The Graovac-Pisanski index, which is also called the modified Wiener index, considers the symmetries and the distances in molecular graphs. Carbon nanotubes are molecules made of carbon with a cylindrical structure possessing unusual valuable properties. 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 nanotubes and derive the closed formulas for their Graovac-Pisanski index.