{"title":"Some Open Mathematical Problems on Fullerenes.","authors":"Artur Bille, Victor Buchstaber, Evgeny Spodarev","doi":"10.1021/acs.jcim.4c01997","DOIUrl":null,"url":null,"abstract":"<p><p>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 into a graph invariant, called the <i>character</i> of a fullerene, derived from its adjacency and degree matrices. As supported by numerical results, this descriptor may lead to a new method for linear enumeration of all fullerenes.</p>","PeriodicalId":44,"journal":{"name":"Journal of Chemical Information and Modeling ","volume":" ","pages":"2911-2923"},"PeriodicalIF":5.6000,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11938279/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Chemical Information and Modeling ","FirstCategoryId":"92","ListUrlMain":"https://doi.org/10.1021/acs.jcim.4c01997","RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/3/10 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"CHEMISTRY, MEDICINAL","Score":null,"Total":0}
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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 into a graph invariant, called the character of a fullerene, derived from its adjacency and degree matrices. As supported by numerical results, this descriptor may lead to a new method for linear enumeration of all fullerenes.
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The Journal of Chemical Information and Modeling publishes papers reporting new methodology and/or important applications in the fields of chemical informatics and molecular modeling. Specific topics include the representation and computer-based searching of chemical databases, molecular modeling, computer-aided molecular design of new materials, catalysts, or ligands, development of new computational methods or efficient algorithms for chemical software, and biopharmaceutical chemistry including analyses of biological activity and other issues related to drug discovery.
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