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Some Open Mathematical Problems on Fullerenes
Artur Bille1, Victor Buchstaber2, Evgeny Spodarev1
1Ulm University, 89069 Ulm, Germany.
Journal of Chemical Information and Modeling
|March 10, 2025
Summary
This study explores fullerenes, carbon molecules mathematically modeled as graphs. Researchers introduce a new fullerene family and a graph invariant called character for potential linear enumeration.
Area of Science:
- Nanotechnology
- Computational Chemistry
- Graph Theory
Background:
- Fullerenes are carbon allotropes with unique cage-like structures.
- Their mathematical representation involves planar, 3-regular graphs with pentagonal and hexagonal faces.
- Understanding fullerene properties is crucial for materials science and chemistry.
Purpose of the Study:
- To address open questions in fullerene research, including random generation.
- To introduce a novel infinite family of fullerenes.
- To explore a graph invariant for fullerene characterization and enumeration.
Main Methods:
- Combinatorial modeling of fullerenes as graphs.
- Introduction of a specific fullerene family.
- Numerical analysis of graph invariants, specifically the 'character' derived from adjacency and degree matrices.
Main Results:
- An infinite family of fullerenes where the generalized Stone-Wales operation is inapplicable was identified.
- Numerical insights into the 'character' of fullerenes were presented.
- The 'character' shows potential as a descriptor for linear fullerene enumeration.
Conclusions:
- The study contributes to the theoretical understanding of fullerenes.
- The identified fullerene family presents unique structural properties.
- The 'character' invariant offers a promising avenue for efficient fullerene enumeration.
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