  Ground states of Ising spin glasses on fully connected graphs are studied for
a broadly distributed bond family. In particular, bonds $J$ distributed
according to a Levy distribution P(J)\propto 1/|J|^{1+\alpha}, |J|>1, are
investigated for a range of powers \alpha. The results are compared with those
for the Sherrington-Kirkpatrick (SK) model, where bonds are Gaussian
distributed. In particular, we determine the variation of the ground state
energy densities with \alpha, their finite-size corrections, measure their
fluctuations, and analyze the local field distribution. We find that the
energies themselves at infinite system size attain universally the
Parisi-energy of the SK as long as the second moment of P(J) exists (\alpha>2),
and compare favorably with recent one-step replica symmetry breaking
predictions well below \alpha=2. At and just below \alpha=2, the simulations
deviate significantly from theoretical expectations. The finite-size
investigation reveals that the corrections exponent \omega decays from the SK
value \omega_{SK}=2/3 already well above \alpha=2, at which point it reaches a
minimum. This result is justified with a speculative calculation of a random
energy model with Levy bonds. The exponent \rho that describes the variations
of the ground state energy fluctuations with system size decays monotonically
from its SK value over the entire range of \alpha and apparently vanishes at
\alpha=1.
