Some open mathematical problems on fullerenes

TL;DR

This paper discusses open mathematical problems related to fullerenes, including their random generation, properties of certain fullerene families, and introduces a new graph invariant that could aid in their enumeration.

Contribution

It presents new open questions, an infinite family of fullerenes resistant to certain operations, and a novel graph invariant for fullerene enumeration.

Findings

Identification of an infinite fullerene family resistant to Stone-Wales operations

Introduction of the character invariant for fullerenes

Numerical insights suggesting potential for linear enumeration

Abstract

Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets composed only of pentagons and hexagons. In this work, we outline a few of the many open questions about fullerenes, beginning with the problem of generating fullerenes randomly. We then introduce an infinite family of fullerenes on which the generalized Stone-Wales operation is inapplicable. Furthermore, we present numerical insights on a graph invariant, called \textit{character} of a fullerene, derived from its adjacency and degree matrices. This descriptor may lead to a new method for linear enumeration of all fullerenes.