Symmetry extensions of Euler's theorem for polyhedral, toroidal and benzenoid molecules

Abstract

Euler's theorem is extended to the permutation symmetries of objects associated with vertices, edges and faces of general polyhedra imbedded in the sphere, torus or a surface of higher genus. An equivalent general theorem is derived for benzenoid hydrocarbons. Further relations are found for deltahedral and three-coordinate polyhedra imbedded in the various surfaces. Application of the new theorems to vibrational analysis and molecular electronic structure is sketched, with particular reference to bonding in possible toroidal analogues of the fullerenes.