The enumeration of all 60atom carbon cages associated to trivalent polyhedra with fiveand sixsided faces is addressed. This isomer problem is computationally solved to give 1790 cages, with a further resolution into subclasses of cages with differing numbers p of abutting pairs of pentagonal faces. The individual cages are generated, and then there are computed various graphtheoretic invariants, including Hckel MO energies, HOMOLUMO gaps, Kekul structure counts, and conjugatedcircuit counts. Associated properties as a function of p are reported and found to be in concert with earlier qualitative arguments. It is found that the most stable of these cages is the unqiue p = 0 Buckminsterfullerene structure. Copyright 1991 John Wiley & Sons, Inc.