Model semiempirical studies using quantum consistent force field/pi (QCFF/PI) arid density functional tight binding (DFTB) methods show the C-20 dodecahedral fullerene to have the lowest energy of all 7595 mathematically possible 20-vertex trivalent polyhedral cages by a substantial margin (estimated at 521 (QCFF/ PI) or 125 (DFTB) kJ mol(-1)). A topological invariant based on the distribution of face sizes is used to correlate the energies and predict the best polyhedral structure for C-22, which must be a nonfullerene cage. With 1 square, 10 pentagonal, and 2 hexagonal faces, this structure is verified to be more stable (by 259 (QCFF/PI), 60 (DFTB) kJ mol(-1)) than its nearest trivalent polyhedral rival.