The C-60 molecule and its fragments are studied using configuration interaction and variational Monte Carlo techniques, within the Hubbard model. Using benzene as a test case, we compare the results of the approximate calculations with exact calculations. The fragments of C-60 studied are pyracylene, fluoranthene, and corannulene. The energies, bond orders, spin-spin, and charge-correlation functions of these systems are obtained for various values of the Hubbard parameter, U. The analyses of bond orders and correlation functions of these individual molecules allow us to visualize pyracylene as a naphthalene unit with two ethylenic moieties and fluoranthene as weakly bridged benzene and naphthalene units. Corannulene is the largest fragment of C-60 that we have studied. The hexagon-hexagon (h-h) bond orders are slightly larger than those of the hexagon-pentagon (h-p) bonds, a feature also found in other fragments. We also find bonds between two coordinated carbon sites to be stronger than bonds involving three coordinated carbon sites. In C-60, the h-h bonds are stronger than in corannulene and the h-p bonds weaker than in corannulene for all correlation strengths. Introducing bond alternation in the buckyball enhances this difference.