SYMMETRY EXTENSIONS OF EULERS THEOREM FOR POLYHEDRAL, TOROIDAL AND BENZENOID MOLECULES

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