  We study characteristics of the relativistic equation of state (EOS) for
collapse-driven supernovae, which is derived by relativistic nuclear many body
theory. Recently the relativistic EOS table has become available as a new
complete set of physical EOS for numerical simulations of supernova explosion.
We examine this EOS table by using general relativistic hydrodynamics of the
gravitational collapse and bounce of supernova cores. In order to study dense
matter in dynamical situation, we perform simplified calculations of core
collapse and bounce by following adiabatic collapse with the fixed electron
fraction for a series of progenitor models. This is intended to give us
``approximate models'' of prompt explosion. We investigate the profiles of
thermodynamical quantities and the compositions during collapse and bounce. We
also perform the calculations with the Lattimer-Swesty EOS to compare the
properties of dense matter. As a measure of the stiffness of the EOS, we
examine the explosion energy of the prompt explosion with electron capture
totally suppressed. We study the derivative of the thermodynamical quantities
obtained by the relativistic EOS to discuss the convective condition in
neutron-rich environment, which may be important in the delayed explosion.
