THEOREMS FOR CARBON CAGES

New theorems are established for cages (or polyhedra) with trivalent vertices. One theorem says that all such cages have at least three Kekule structures (or perfect matchings). Thence, resonance generally appears as a possibility. Another theorem says that for every even vertex count greater-than-or-equal-to 70 there is at least one cage of a "preferable" subclass, while for vertex count < 70 the sole preferable cage is that of the truncated icosahedron. Thence, the unique role of the buckminsterfullerene structure for C60 is mathematically indicated.