  The density dependencies of various effective interaction strengths in the
relativistic mean field are studied and carefully compared for nuclear matter
and neutron stars. The influences of different density dependencies are
presented and discussed on mean field potentials, saturation properties for
nuclear matter, equations of state, maximum masses and corresponding radii for
neutron stars. Though the interaction strengths and the potentials given by
various interactions are quite different in nuclear matter, the differences of
saturation properties are subtle, except for NL2 and TM2, which are mainly used
for light nuclei, while the properties by various interactions for pure neutron
matter are quite different. To get an equation of state for neutron matter
without any ambiguity, it is necessary to constrain the effective interactions
either by microscopic many-body calculations for the neutron matter data or the
data of nuclei with extreme isospin. For neutron stars, the interaction with
large interaction strengths give strong potentials and large
Oppenheimer-Volkoff (OV) mass limits. The density-dependent interactions DD-ME1
and TW-99 favor a large neutron population due to their weak $\rho$-meson field
at high densities. The OV mass limits calculated from different equations of
state are 2.02 $\sim$ 2.81$ M_\odot$, and the corresponding radii are 10.78
$\sim$ 13.27 km. After the inclusion of the hyperons, the corresponding values
become 1.52 $\sim$ 2.06 $M_\odot$ and 10.24 $\sim$ 11.38 km.
