  The gravitational radiation has been proposed a long time before, as an
explanation for the observed relatively low spin frequencies of young neutron
stars and of accreting neutron stars in low-mass X-ray binaries as well. In the
present work we studied the effects of the neutron star equation of state on
the r-mode instability window of rotating neutron stars. Firstly, we employed a
set of analytical solution of the Tolman-Oppemheimer-Volkoff equations with
special emphasis on the Tolman VII solution. In particular, we tried to clarify
the effects of the bulk neutron star properties (mass, radius, density
distribution, crust size and elasticity) on the r-mode instability window. We
found that the critical angular velocity $\Omega_c$ depends mainly on the
neutron star radius. The effects of the gravitational mass and the mass
distribution are almost negligible. Secondly, we studied the effect of the
elasticity of the crust, via to the slippage factor $S$ and also the effect of
the nuclear equation of state, via the slope parameter $L$, on the instability
window. We found that the crust effects are more pronounced, compared to those
originated from the equation of state. Moreover, we proposed simple analytical
expressions which relate the macroscopic quantity $\Omega_c$ to the radius, the
parameter $L$ and the factor ${\cal S}$. We also investigated the possibility
to measure the radius of a neutron star and the factor ${\cal S}$ with the help
of accurate measures of $\Omega_c$ and the neutron star temperature. Finally,
we studied the effects of the mutual friction on the instability window and
discussed the results in comparison with previous similar studies.
