Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96-145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website.

中文翻译：

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富勒烯的拓扑。

富勒烯是形成多面体笼的碳分子。它们的键结构恰好是只有五边形和六边形面的平面立方图。令人惊讶的是，富勒烯的许多化学性质可以从其图结构得到。自20世纪初期由Goldberg，Coxeter等人研究立方平面图和富勒烯图以来，已经有了丰富的数学运算，富勒烯的许多数学性质也找到了简单而优美的解决方案。然而，该领域中许多有趣的化学和数学问题仍未解决。在本文中，我们概述了过去二十年来富勒烯研究中拓扑和图论理论的最新发展，同时描述了已解决的问题和未解决的问题。WIRES Comput Mol Sci 2015，5：96-145。doi：10.1002 / wcms。1207利益冲突：作者没有为本文声明利益冲突。有关本文的更多资源，请访问WIREs网站。