# The Graovac-Pisanski Index of Zig-Zag Tubulenes and the Generalized Cut Method

## TL;DR

This paper generalizes the cut method to compute the Graovac-Pisanski index, a symmetry-aware variation of the Wiener index, for zig-zag tubulenes, providing new formulas and group structure insights.

## Contribution

It introduces a generalized cut method for calculating the Graovac-Pisanski index and analyzes the automorphism groups of zig-zag tubulenes.

## Key findings

- Automorphism group of zig-zag tubulenes can be isomorphic to a direct product of dihedral and cyclic groups.
- Closed formulas for the Graovac-Pisanski index of zig-zag tubulenes are derived.
- The generalized cut method extends previous calculations of the index.

## Abstract

The Graovac-Pisanski index, which is also called the modified Wiener index, was introduced in 1991 by A. Graovac and T. Pisanski. This variation of the classical Wiener index takes into account the symmetries of a graph. In 2016 M. Ghorbani and S. Klav\v{z}ar calculated this index by using the cut method, which we generalize in this paper. Moreover, we prove that in some cases the automorphism group of a zig-zag tubulene is isomorphic to the direct product of a dihedral group and a cyclic group. Finally, the closed formulas for the Graovac-Pisanski index of zig-zag tubulenes are calculated.