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arXiv · 2410.19322

Some open mathematical problems on fullerenes

Artur Bille·Victor Buchstaber·Evgeny Spodarev
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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.

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Artur Bille, Victor Buchstaber, Evgeny Spodarev. 2024-10-25. Some open mathematical problems on fullerenes. https://arxiv.org/abs/2410.19322

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