  Using the constraints from astrophysical observations and heavy-ion
experiments, we investigate the equation of state (EOS) of hybrid star matter
and the properties of quark-matter cores in hybrid stars. The quark matter
interactions in hybrid stars are described based on 3-flavor Nambu-Jona-Lasinio
model with various vector and vector-isovector coupling constants. In this
work, we find that the hybrid star matter EOS is more sensitive to the strength
of the vector interaction, and the EOS becomes stiffer with increasing vector
strength $R_V$. The vector-isovector interaction characterized by the coupling
constant $R_{IV}$ make main contribution to the hadron-quark mixed phase.
Meanwhile, we note that a step change of both the sound velocity and the
polytropic index $\gamma$ occurs in the hadron-quark phase transition, and it
is restored with the decrease of nucleon and lepton degrees of freedom in the
high density quark phase. Although the coupling constants increase the hybrid
star maximum mass up to $2.08M_{\odot}$, they also decrease the mass and radius
of the quark core and the mixed core. With different quark coupling constants,
we also find that the maximum mass and radius of the quark matter core in a
stable hybrid star can reach $0.80M_{\odot}$ and 6.95 km, which are close to
half of the maximum mass and radius of the complete star. However, properties
of quark matter have no effect on the $M = 1.4M_{\odot}$ hybrid star as a
result of no quark matter inner core, which can also be confirmed by the
criterion of the polytropic index, and thus our results also indicate that the
quark interactions have no effect on the tidal deformability $\Lambda_{1.4}$ of
hybrid stars.
