  The parabolic approximation to the equation of state of the isospin
asymmetric nuclear matter (ANM) is widely used in the literature to make
predictions for the nuclear structure and the neutron star properties. Based on
the realistic M3Y-Paris and M3Y-Reid nucleon-nucleon interactions, we
investigate the effects of the higher-order symmetry energy on the proton
fraction in neutron stars and the location of the inner edge of their crusts
and their core-crust transition density and pressure, thermodynamically.
Analytical expressions for different-order symmetry energy coefficients of ANM
are derived using the realistic interactions mentioned above. It is found that
the higher-order terms of the symmetry energy coefficients up to its
eighth-order (E$_{sym8}$) contributes substantially to the proton fraction in
$\beta$ stable neutron star matter at different nuclear matter densities, the
core-crust transition density and pressure. Even by considering the symmetry
energy coefficients up to E$_{sym8}$, we obtain a significant change of about
40$\%$ in the transition pressure value from the one based on the exact
equation of state. Using equations of state based on both Paris and Reid
effective interactions which provide saturation incompressibility of symmetric
nuclear matter in the range of 220 MeV$\leq$K$_0$$\leq$270 MeV, we estimate the
ranges 0.090 fm$^{-3}$$\leq$$\rho_t$$\leq$0.095 fm$^{-3}$ and 0.49 MeV
fm$^{-3}$$\leq$P$_t$$\leq$ 0.59 MeV fm$^{-3}$ for the liquid core-solid crust
transition density and pressure, respectively. The corresponding range of the
proton fraction at this transition density range is found to be
0.029$\leq$x$_{p(t)}$$\leq$0.032.
