The discovery of the fullerene molecules and related forms of carbon, such as nanotubes, has generated an explosion of activity in chemistry, physics, and materials science, which is amply documented, for example, in and. In chemistry, the “classical” definition is that a fullerene is an all-carbon molecule in which the atoms are arranged as a map on a sphere made up entirely of 5-gons and 6-gons, which, therefore, necessarily includes exactly 12 5-gonal faces. We are concerned here with the following generalization: what fullerenes are possible if a fullerene is a finite 3-valent map with only 5- and 6-gonal faces embedded in any surface? This seemingly much larger concept leads only to three extensions to the class of spherical fullerenes. Embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective plane. In, the spectral properties of those fullerenes are examined. The usual spherical fullerenes have 12 5-gons, projective fullerenes 6, and toroidal and Klein bottle fullerenes none. Klein bottle and projective fullerenes are the antipodal quotients of centrally symmetric toroidal and spherical fullerenes, respectively. Extensions to infinite graphs (plane fullerenes, cylindrical fullerenes) are indicated. Detailed treatment of the concept of the extended fullerenes and their further generalization to higher dimensional manifolds are given in.

Classification of finite fullerenes

Define a 3-fullerene as a 3-valent map embedded on a surface and consisting of only 5-gonal and 6-gonal faces. Each such object has, say, v vertices, e edges, and f faces of which p5 are 5-gons and p 6 are 6-gons.