We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C80 to C240. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Mobius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C80 with a chamfered dodecahedron structure, icosahedral C180, and C240 with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

中文翻译：

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边缘对称性的组合：二十面体巨富勒烯的手性和非手性边缘着色：C80、C180 和 C240 †

我们在 C80 到 C240 的二十面体巨富勒烯的边缘群的作用下开发了边缘对称性和边缘着色的组合。我们使用计算对称技术，这些技术采用 Sheehan 对 Pόlya 定理的修改和 Mobius 反演方法以及广义字符循环指数。这些技术用于生成边缘群对称性，包括诱导边缘排列，从而在边缘对称作用下为所有不可约表示生成巨型富勒烯的着色。我们主要考虑高对称二十面体富勒烯，例如具有倒角十二面体结构的 C80、二十面体 C180 和具有倒角截断二十面体几何形状的 C240。这些基于对称性的组合技术在有或没有约束的情况下列举了这种巨型富勒烯的非手性和手性边缘着色。我们的计算结果表明，巨富勒烯的边缘着色有几个等价类，其中大部分是手性的。通过枚举边缘着色的等价类，这些技术可以应用于超芳香性、六重奏多项式、共轭电路和共振能量的快速计算、手性测量等。