  Adopting the density dependent relativistic mean-field (RMF) and relativistic
Hartree-Fock (RHF) approaches, the properties of the nuclear fourth-order
symmetry energy $S_4$ are studied within the covariant density functional (CDF)
theory. It is found that the fourth-order symmetry energies are suppressed in
RHF at both saturation and supranuclear densities, where the extra contribution
from the Fock terms is demonstrated, specifically via the isoscalar
meson-nucleon coupling channels. The reservation of $S_4$ and higher-order
symmetry energies in the nuclear equation of state then affects essentially the
prediction of neutron star properties, which is illustrated in the quantities
such as the proton fraction, the core-crust transition density as well as the
fraction of crustal moment of inertia. Since the Fock terms enhance the density
dependence of the thermodynamical potential, the RHF calculations predict
systematically smaller values of density, proton fraction and pressure at the
core-crust transition boundary of neutron stars than density dependent RMF
ones. In addition, a linear anti-correlation between the core-crust transition
density $\rho_t$ and the density slope of symmetry energy $L$ is found which is
then utilized to constrain the core-crust transition density as
$\rho_t\thicksim[0.069, 0.098]~\rm{fm}^{-3}$ with the recent empirical
information on $L$. The study clarifies the important role of the fourth-order
symmetry energy in determining the properties of nuclear matter at extreme
isospin or density conditions.
