  The nuclear symmetry energy plays an important role in the description of the
properties of finite nuclei as well as neutron stars. Especially, for low
values of baryon density, the accurate description of the crust-core interface
strongly depends on the symmetry energy. Usually, the well known parabolic
approximation is employed for the definition of the symmetry energy without
avoiding some drawbacks. In the present work, a class of nuclear models,
suitable for the description of the inner and outer core of neutron stars, is
applied in studying the effect of higher orders of the expansion of the energy
on the location of the crust-core transition. The thermodynamical and dynamical
methods are used for the determination of the transition density $n_{\rm t}$
and pressure $P_{\rm t}$. The corresponding energy density functional is
applied for the study of some relevant properties of both non-rotating and slow
rotating neutron stars. We found that the larger the value of the slope
parameter $L$, the slower the convergence of the expansion. In addition, a
universal relation is presented between $n_{\rm t}$ and $L$, by employing the
full expression and dynamical approach. The crustal moment of inertia is very
sensitive on the location of the transition while the effects are moderated
concerning the critical angular velocity of the r-mode instability and minimum
mass configuration. The effect on the tidal deformability is less but not
negligible. In any case, the use of the parabolic approximation leads to the
overestimation of $n_{\rm t}$ and $P_{\rm t}$ and consequently, on inaccurate
predictions.
