Parameterizing Fullerenes with Vertex Combinations

Abstract

A fullerene graph can be embedded in a piecewise linear 2-manifold with each non-hexagonal carbon ring corresponding to a cone vertex. Adjacent two or three such vertices can be combined as a cluster cut out from a parent cone round a parent vertex. The locations for the combined vertices are distributed on their parent cone with certain symmetries. With proper combinations, a fullerene can be simplified to a parent structure easily to be parameterized. And then, with additional parameters for the combined vertices in each cluster, the fullerene can also be parameterized. Such parameters may be different for fullerenes from type to type, but they are able to characterize the specific shape for each type.