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 rationalise 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-neighbour information, all three named indices select low-energy isolated-pentagon fullerenes at 84 and 100 atoms.