The distance matrix of a chemical graph can be computed in quadratic time and from it can be obtained the distance level patterns (DLP); Wiener, Szeged, and Balaban indices; as well as the distance eigenvalues. Point-group symmetry places bounds on the numbers of distinct DLP and distance eigenvalues. Angular-momentum arguments rationalize the distance spectrum for near-spherical cages. Wiener and Balaban indices are inversely correlated and select fullerenes from general cubic polyhedra and isolated-pentagon from general fullerenes. In combination with hexagon-neighbor information, all three named indices select low-energy isolated-pentagon fullerenes at 84 and 100 atoms.