  The stellar equilibrium and collapse, including mainly white dwarfs, neutron
stars and supper massive stars, is an interplay between general relativistic
effects and the equation of state of nuclear matter. In the present work, we
use the Chandrasekhar criterion of stellar instability by employing a large
number of realistic equations of state (EoS) of neutron star matter. We mainly
focus on the critical point of transition from stable to unstable
configuration. This point corresponds to the maximum neutron star mass
configuration. We calculate, in each case, the resulting compactness parameter,
$\beta=GM/c^2R$, and the corresponding effective adiabatic index, $\gamma_{\rm
cr}$. The role of the trial function $\xi(r)$ is presented and discussed in
details. We found that it holds a model-independent relation between
$\gamma_{\rm cr}$ and $\beta$. This statement is strongly supported by the
large number of EoS and it is also corroborated by using analytical solutions
of the Einstein's field equations. In addition, we present and discuss the
relation between the maximum rotation rate and the adiabatic index close to the
instability limit. Accurate observational measurements of the upper bound of
the neutron star mass and the corresponding radius, in connection with the
present predictions, may help to impose constraints on the high density part of
the neutron star equation of state.
