  A stability analysis of a spherically symmetric star in scalar-tensor
theories of gravity is given in terms of the frequencies of quasi-normal modes.
The scalar-tensor theories have a scalar field which is related to gravitation.
There is an arbitrary function, the so-called coupling function, which
determines the strength of the coupling between the gravitational scalar field
and matter. Instability is induced by the scalar field for some ranges of the
value of the first derivative of the coupling function. This instability leads
to significant discrepancies with the results of binary-pulsar-timing
experiments and hence, by the stability analysis, we can exclude the ranges of
the first derivative of the coupling function in which the instability sets in.
In this article, the constraint on the first derivative of the coupling
function from the stability of relativistic stars is found. Analysis in terms
of the quasi-normal mode frequencies accounts for the parameter dependence of
the wave form of the scalar gravitational waves emitted from the
Oppenheimer-Snyder collapse. The spontaneous scalarization is also discussed.
