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\documentclass[12pt]{minimal}
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\begin{document}$r$\end{document}-mode instability window of rotating neutron stars. Firstly, we employed a set of analytical solution of the Tolman-Oppenheimer-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\documentclass[12pt]{minimal}
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\begin{document}$r$\end{document}-mode instability window. We found that the critical angular velocity Ωc\documentclass[12pt]{minimal}
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\begin{document}$\varOmega_{c}$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$S$\end{document} and also the effect of the nuclear equation of state, via the slope parameter L\documentclass[12pt]{minimal}
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\begin{document}$L$\end{document}, 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 Ωc\documentclass[12pt]{minimal}
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\begin{document}$\varOmega_{c}$\end{document} to the radius, the parameter L\documentclass[12pt]{minimal}
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\begin{document}$L$\end{document} and the factor S\documentclass[12pt]{minimal}
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\begin{document}$\mathcal{S}$\end{document}. We also investigated the possibility to measure the radius of a neutron star and the factor S\documentclass[12pt]{minimal}
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\begin{document}$\mathcal{S}$\end{document} with the help of accurate measures of Ωc\documentclass[12pt]{minimal}
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\begin{document}$\varOmega_{c}$\end{document} 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.
