{"title":"Resonant hexagons in fullerene graphs","authors":"Jun Fujisawa","doi":"10.1007/s10910-024-01650-4","DOIUrl":null,"url":null,"abstract":"<div><p>A fullerene graph is a 3-connected plane cubic graph in which every face is pentagonal or hexagonal. A set of hexagons <span>\\(\\mathcal {H}\\)</span> of <i>G</i> is called a <i>resonant pattern</i> if there exists a perfect matching <i>M</i> of <i>G</i> such that exactly three edges of <i>H</i> is contained in <i>M</i> for each member <i>H</i> of <span>\\(\\mathcal {H}\\)</span>. In this paper we prove for any natural number <i>k</i> that almost all of the family of <i>k</i> disjoint hexagons are resonant patterns in sufficiently large fullerene graphs.</p></div>","PeriodicalId":648,"journal":{"name":"Journal of Mathematical Chemistry","volume":"62 9","pages":"2280 - 2288"},"PeriodicalIF":1.7000,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Chemistry","FirstCategoryId":"92","ListUrlMain":"https://link.springer.com/article/10.1007/s10910-024-01650-4","RegionNum":3,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
A fullerene graph is a 3-connected plane cubic graph in which every face is pentagonal or hexagonal. A set of hexagons \(\mathcal {H}\) of G is called a resonant pattern if there exists a perfect matching M of G such that exactly three edges of H is contained in M for each member H of \(\mathcal {H}\). In this paper we prove for any natural number k that almost all of the family of k disjoint hexagons are resonant patterns in sufficiently large fullerene graphs.
富勒烯图是一个三连平面立方图，其中每个面都是五边形或六边形。如果 G 中存在完美匹配的 M，使得 H 的每一个成员 H 都有三条边包含在 M 中，那么 G 的六边形集合 \(\mathcal {H}\) 就被称为共振图案。在本文中，我们证明了对于任意自然数 k，在足够大的富勒烯图中，几乎所有 k 个不相交的六边形族都是共振图案。
期刊介绍：
The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches.
Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.
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