  Using in-medium hadron properties according to the Brown-Rho scaling due to
the chiral symmetry restoration at high densities and considering naturalness
of the coupling constants, we have newly constructed several relativistic
mean-field Lagrangians with chiral limits. The model parameters are adjusted
such that the symmetric part of the resulting equation of state at supra-normal
densities is consistent with that required by the collective flow data from
high energy heavy-ion reactions, while the resulting density dependence of the
symmetry energy at sub-saturation densities agrees with that extracted from the
recent isospin diffusion data from intermediate energy heavy-ion reactions. The
resulting equations of state have the special feature of being soft at
intermediate densities but stiff at high densities naturally. With these
constrained equations of state, it is found that the radius of a 1.4$M_\odot$
canonical neutron star is in the range of 11.9 km$\leq$R$\leq$13.1 km, and the
maximum neutron star mass is around 2.0$M_\odot$ close to the recent
observations.
