  The observation of maximally rotating neutron stars (in comparison to
nonrotating ones) may provide more information on the behavior of nuclear
matter at high densities. We provide a theoretical treatment concerning the
effects of the upper bound of the sound speed in dense matter on the bulk
properties of maximally rotating (at mass-shedding limit) neutron stars. In
particular, we consider two upper bounds for the speed of sound, $v_s = c$ and
$v_s = c/\sqrt{3}$, and the one provided by the relativistic kinetic theory. We
investigate to what extent the possible predicted (from various theories and
conjectures) upper bounds on the speed of sound constrain the ones of various
key quantities, including the maximum mass and the corresponding radius,
Keplerian frequency, Kerr parameter and moment of inertia. We mainly focus on
the lower proposed limit, $v_{s}=c/\sqrt{3}$, and we explore in which mass
region a rotating neutron star collapses to a black hole. In any case, useful
relations of the mentioned bulk properties with the transition density are
derived and compared with the corresponding nonrotating cases. We concluded
that the proposed limit $v_{s}=c/\sqrt{3}$ leads to dramatic decrease on the
values of the maximum mass, Kerr parameter and moment of inertia preventing a
neutron star to reach values which derived with the consideration of realistic
equations of state or from other constraints. Possible measurements of the Kerr
parameter and moment of inertia would shed light on these issues and help to
reveal the speed of sound bound in dense matter.
