  We consider the slowly rotating relativistic stars with a uniform angular
velocity in the scalar-tensor gravity, and examine the rotational effect around
such compact objects. For this purpose, we derive a 2nd order differential
equation describing the frame dragging in the scalar-tensor gravity and solve
it numerically. As a result, we find that the total angular momentum is
proportional to the angular velocity even in the scalar-tensor gravity. We also
show that one can observe the spontaneous scalarization in rotational effects
as well as the other stellar properties, if the cosmological value of scalar
field is zero. On the other hand, if the cosmological value of scalar field is
nonzero, the deviation from the general relativity can be seen in a wide range
of the coupling constant. Additionally, we find that the deviation from the
general relativity becomes larger with more massive stellar models, which is
independent of the cosmological value of scalar field. Thus, via precise
observations of astronomical phenomena associated with rotating relativistic
stars, one may be possible to probe not only the gravitational theory in the
strong-field regime, but also the existence of scalar field.
