On octahedral fulleroids

The discovery of the first fullerene has raised an interest in the study of other candidates for modelling of carbon molecules. As a generalization of the fullerenes, fulleroids are defined as cubic convex polyhedra with all the faces of size five or greater. In this paper, we give necessary and sufficient condition for the existence of fulleroids with only pentagonal and image-gonal faces and with the symmetry group isomorphic to the full symmetry group of the regular octahedron. The existence is proved by finding infinite series of examples. The nonexistence is proved using symmetry invariants.