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ter from the MDI interaction with x = 0 and x = − 1 and of hypernuclear matter from the MDI-Hyp-A and MDI- Hyp-R interactions. It is seen that the EOS is softened when hyperons are present as compared to the EOSs of pure nuclear matter. We note that the EOS at mod- erate densities plays an important role in determining the maximum mass of neutron stars. Without hyperons, the EOS at moderate densities is stiffer for smaller value of x , while it becomes stiffer for larger value of x when hyperons are included, especially for a repulsive ΣN in- teraction. As shown in Fig. 6, a stiffer symmetry energy leads to a larger number of hyperons, which results in a softer EOS as a result of the lower pressure due to the presence of more degrees of freedom, and the effect of hy- perons on the EOS is smaller for a soft symmetry energy, as fewer hyperons are then present in the hypernuclear matter. One the other hand, the symmetry energy con- tribution to the pressure of the hypernuclear matter is larger for x = − 1 than x = 0. As a result, the EOS of hypernuclear matter shows similar stiffness for differ- ent symmetry energy parameters. Furthermore, the EOS from the MDI-Hyp-R interaction is stiffer than that from the MDI-Hyp-A interaction, as Σ particles do not appear in the former case as shown in Fig. 6. Our results thus show that the hypernuclear matter has a stiffer (softer) EOS at lower densities but a softer (stiffer) EOS at mod- erate densities with a stiffer (softer) nuclear symmetry energy. We note that the results from both x = 0 and x = − 1 are consistent with the constraints on the nu- clear equation of state obtained from the analysis of the collective flow data in heavy-ion collisions [70]. 

At higher densities in the core of a neutron star, a tran- sition from hadron matter to quark matter is expected to occur. Here we only consider the hadron-quark phase transition for the repulsive ΣN interaction as it is more consistent with the latest empirical information [56–60]. In Fig. 9, we display the particle fractions of each species in the presence of a hadron-quark phase transition, with the hadron phase from the MDI-Hyp-R interaction with x = 0 and x = − 1 and the quark phase from the MIT bag model with bag constants B 1 / 4 = 180 MeV and 170 MeV. The particle fractions are weighted by the baryon number and corresponding phase fraction. It is seen that for B 1 / 4 = 180 MeV, the phase transition begins at the density 0 . 35 fm − 3 and ends at the density 1 . 38 fm − 3 for x = 0, and begins at the density 0 . 26 fm − 3 and ends at the density 1 . 38 fm − 3 for x = − 1, while for B 1 / 4 = 170 MeV it begins at the density 0 . 21 fm − 3 and ends at the density 1 . 05 fm − 3 for x = 0, and begins at the den- sity 0 . 18 fm − 3 and ends at the density 1 . 05 fm − 3 for x = − 1. The hadron-quark phase transition thus hap- pens at lower baryon number density for a stiffer symme- try energy and for a smaller value of B , while the density at which the hadron-quark phase transition ends seems to depend only on the value of B but not much on the value of the symmetry energy parameter x . With a smaller value of the bag constant, the hadron-quark phase tran- sition both begins and ends earlier. It is also seen that d and s quarks occupy a larger fraction than u quarks in the mixed phase because of their negative charges, so 

## B. The hadron-quark phase transition 

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