  We study the $f$-mode frequencies and damping times of nonrotating neutron
stars (NS) in general relativity (GR) by solving the linearized perturbation
equations, with the aim to establish "universal" relations that depend only
weakly on the equations of state (EOS). Using a more comprehensive set of EOSs,
we re-examine some proposed linearizations that describe the $f$-mode
parameters in terms of mass and radius of the neutron star (NS), and we test a
more recent proposal for expressing the $f$-mode parameters as quadratic
functions of the effective compactness. Our extensive results for each equation
of state considered allow us to study the accuracy of each proposal. In
particular, we find that the damping time deviates quite considerably from the
proposed linearization. We introduce a new universal relation for the product
of the $f$-mode frequency and damping time as a function of the (ordinary)
compactness, which proved to be more accurate. The relations using the
effective compactness on the other hand also fit our data accurately. Our
results show that the maximum oscillation frequency depends strongly on the
EOS, such that the measurement of a high oscillation frequency would rule out
several EOSs. Lastly, we compare the exact mode frequencies to those obtained
in the Cowling approximation, and also to results obtained with a nonlinear
evolution code, validating the implementations of the different approaches.
