# Sizes of Pentagonal Clusters in Fullerenes

## TL;DR

This paper investigates the possible distributions of pentagonal face clusters in fullerenes, determining which configurations can occur infinitely or finitely, and whether these clusters can be placed arbitrarily far apart.

## Contribution

It classifies all partitions of 12 into cluster sizes that can occur in fullerenes and analyzes their spatial arrangements and frequency.

## Key findings

- All partitions with max cluster size 5 or less occur infinitely with large distances.
- Nine partitions occur only finitely many times in fullerenes.
- Fifteen partitions do not occur in any fullerene.

## Abstract

Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes.