  Gravitational waves (GWs) generated by axisymmetric rotating collapse,
bounce, and early postbounce phases of a galactic core-collapse supernova will
be detectable by current-generation gravitational wave observatories. Since
these GWs are emitted from the quadrupole-deformed nuclear-density core, they
may encode information on the uncertain nuclear equation of state (EOS). We
examine the effects of the nuclear EOS on GWs from rotating core collapse and
carry out 1824 axisymmetric general-relativistic hydrodynamic simulations that
cover a parameter space of 98 different rotation profiles and 18 different EOS.
We show that the bounce GW signal is largely independent of the EOS and
sensitive primarily to the ratio of rotational to gravitational energy, and at
high rotation rates, to the degree of differential rotation. The GW frequency
of postbounce core oscillations shows stronger EOS dependence that can be
parameterized by the core's EOS-dependent dynamical frequency
$\sqrt{G\bar{\rho}_c}$. We find that the ratio of the peak frequency to the
dynamical frequency follows a universal trend that is obeyed by all EOS and
rotation profiles and that indicates that the nature of the core oscillations
changes when the rotation rate exceeds the dynamical frequency. We find that
differences in the treatments of low-density nonuniform nuclear matter, of the
transition from nonuniform to uniform nuclear matter, and in the description of
nuclear matter up to around twice saturation density can mildly affect the GW
signal. We find that approximations and uncertainties in electron capture rates
can lead to variations in the GW signal that are of comparable magnitude to
those due to different nuclear EOS. This emphasizes the need for reliable
nuclear electron capture rates and for self-consistent multi-dimensional
neutrino radiation-hydrodynamic simulations of rotating core collapse.
