Zooming in on fullerenes

Abstract

Carbon does not appear only as diamond and graphite. Fullerenes, forming a third family of allotropes of carbon (C), exist as large stable clusters of C atoms. A trivalent polyhedron P is a cubic graph which may be embedded on a convex 3-D surface of genus zero and a fullerene, Cn, is P with twelve pentagons, the remaining faces being hexagons. We introduce the concept of nut fullerenes, so called because their skeleton is a nut graph that implies one NBO with the charge contributed by the NBO electron being shared among all the C-centres. The charge distribution over all the framework of the molecule has strong chemical consequences. We study the substructures in fullerenes and other trivalent polyhedra, that determine the presence of a NBO. Together with the symmetry group of the graph, they shed new light on singular graphs and polyhedra in particular.