10 Tolman-Oppenheimer-Volkoff (TOV) equation [73, 74] where m$_{g}$ is the gravitational mass inside the radius r of the hybrid star given by the lepton fraction decreases while the proton fraction in- creases when the hadron-quark phase transition occurs. Furthermore, only Λ hyperons (no other hyperons) ap- pear in the mixed phase in our model. The fraction of Λ hyperons is, however, sensitive to the value of the bag constant B , and with a smaller value of B its fraction becomes smaller. with ϵ ( r ) being the energy density. Above equations are solved by starting from a central energy density ϵ ( r = 0) ≡ ϵ$_{c}$ and integrating outward until the pressure on the surface of the hybrid star defined by r = R vanishes, i.e, P ( R ) = 0. This gives the radius R , and the total gravitational mass of a hybrid star is then given by M = m$_{g}$ ( R ). In our calculations, the hybrid star is divided into three parts from the center to the surface: the liquid core, the inner crust and the outer crust as in our previous work [12, 13]. In the liquid core it is assumed to be the hypernuclear matter or that with the hadron-quark phase transition, and the resulting EOSs shown in pre- vious subsections are used. In the inner crust, a param- eterized EOS of P = a + bϵ 4 / 3 is used as in the previous treatment [12, 13]. The outer crust usually consists of heavy nuclei and the electron gas, where we use the BPS EOS [75]. The transition density ρ$_{t}$ between the liquid core and the inner crust has been consistently determined in our previous work [12, 13], and for the density which distinguishes the edge of inner crust and outer crust, we take it to be ρ$_{out}$ = 2 . 46 × 10 − 4 fm − $^{3}$. The parameters a and b can then be determined by the pressures and energy densities at ρ$_{t}$ and ρ$_{out}$ . For the EOSs in the presence of the hadron-quark phase transition, they are shown in Fig. 10 with B 1 / 4 = 180 MeV and 170 MeV. Our results show that the EOSs of the mixed phase and the quark phase are both softened in comparison with a pure hadron phase. The difference between the results from x = 0 and x = − 1 is not large except for the different starting density of the hadron- quark phase transition. Since the EOS is more sensitive to the value of bag constant than to the value of x when the quark degrees of freedom are introduced, the bag constant B is thus the main parameter in determining the EOS of dense matter as reported in other work [22]. The energy density at the end of the hadron-quark phase transition is about 1 . 2 GeV/fm 3 for B 1 / 4 = 180 MeV and about 1 . 6 GeV/fm 3 for B 1 / 4 = 170 MeV, with the former closer to the value of about 1 GeV/fm 3 obtained from the lattice QCD calculation [71] and extracted from heavy-ion collision experiments [72]. Compared with the equation of state constrained by the collective flow data in heavy-ion collisions [70], our results from both values of bag constants satisfy the empirical constraint. In Fig. 11, we show the mass-radius (M-R) and mass- central density (M- ρ$_{c}$ ) relations of hybrid stars from the EOS of the hypernuclear matter without the hadron- quark phase transition as obtained above based on the MDI-Hyp interaction. Results using both attractive and repulsive ΣN interactions are shown, and the results from a pure nucleonic approach without hyperons are also shown for comparison. The maximum mass obtained with hyperons in hybrid stars is seen to be smaller as the EOS of the hypernuclear matter is softer than that of the nuclear matter. The EOSs at moderate densi- ties play an important role in determining the maximum mass, as the M- ρ$_{c}$ relation shows that the neutron star mass can not increase even the central density reaches a very high value. The mass is larger when a repulsive ΣN interaction is used as a result of the smaller number of degrees of freedom and stiffer EOS, as discussed in the previous subsection. For smaller values of ρ$_{c}$ a stiffer symmetry energy gives a larger mass as the nuclear mat- ter dominates this density range, and the results are thus the same as those in our previous work [12, 13]. With larger ρ$_{c}$ , the effect due to the nuclear symmetry energy is small, and the mass from x = 0 is similar to that from x = − 1 because of the opposite effects from the symme- try pressure and the fraction of hyperons on the equation In this subsection, we use the MDI-Hyp interaction to study the properties of static hybrid stars with spher- ically symmetric mass distributions. In particular, we calculate the mass-radius relation of a hybrid stars using C. Hybrid stars FIG. 10: (Color online) Same as Fig. 9 for the equation of state. 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