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 Kekulé structures (or perfect matchings). Thence, resonance generally appears as a possibility. Another theorem says that for every even vertex count >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.[/p]Work supported by the Welch Foundation of Houston, Texas.