  We construct a class of nuclear equations of state based on a schematic
potential model, that originates from the work of Prakash et. al.
\cite{Prakash-88}, which reproduce the results of most microscopic
calculations. The equations of state are used as input for solving the
Tolman-Oppenheimer-Volkov equations for corresponding neutron stars. The
potential part contribution of the symmetry energy to the total energy is
parameterized in a generalized form both for low and high values of the baryon
density. Special attention is devoted to the construction of the symmetry
energy in order to reproduce the results of most microscopic calculations of
dense nuclear matter. The obtained nuclear equations of state are applied for
the systematic study of the global properties of a neutron star (masses, radii
and composition). The calculated masses and radii of the neutron stars are
plotted as a function of the potential part parameters of the symmetry energy.
A linear relation between these parameters, the radius and the maximum mass of
the neutron star is obtained. In addition, a linear relation between the radius
and the derivative of the symmetry energy near the saturation density is found.
We also address on the problem of the existence of correlation between the
pressure near the saturation density and the radius.
