  A thorough understanding of properties of neutron stars requires both a
reliable knowledge of the equation of state (EOS) of super-dense nuclear matter
and the strong-field gravity theories simultaneously. To provide information
that may help break this EOS-gravity degeneracy, we investigate effects of
nuclear symmetry energy on the gravitational binding energy of neutron stars
within GR and the scalar-tensor subset of alternative gravity models. We focus
on effects of the slope $L$ of nuclear symmetry energy at saturation density
and the high-density behavior of nuclear symmetry energy. We find that the
variation of either the density slope $L$ or the high-density behavior of
nuclear symmetry energy leads to large changes in the binding energy of neutron
stars. The difference in predictions using the GR and the scalar-tensor theory
appears only for massive neutron stars, and even then is significantly smaller
than the difference resulting from variations in the symmetry energy.
