# On the Wiener complexity and the Wiener index of fullerene graphs

## TL;DR

This paper investigates the Wiener complexity and Wiener index of fullerene graphs, providing calculations for graphs up to 216 vertices, analyzing their structures, and deriving formulas for specific graph families.

## Contribution

It offers the first extensive calculations of Wiener complexity and index for fullerene graphs up to 216 vertices, and derives formulas for certain graph families.

## Key findings

- Calculated Wiener complexity and index for fullerene graphs up to 216 vertices.
- Identified structures with maximal Wiener complexity and Wiener index.
- Derived formulas for Wiener index of specific fullerene graph families.

## Abstract

Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex $v$ of a graph is the sum of distances from $v$ to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order $n \le 216$ are presented. Structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed and formulas for the Wiener index of several families of graphs are obtained.