  Using the relativistic mean-field model with nonlinear couplings between the
isoscalar and isovector mesons, we study the properties of isospin-asymmetric
nuclear matter. Not only the vector mixing,
$\omega_{\mu}\omega^{\mu}\mathbf{\rho}_{\nu}\mathbf{\rho}^{\nu}$, but also the
quartic interaction due to the scalar mesons, $\sigma^{2}\mathbf{\delta}^{2}$,
is taken into account to investigate the density dependence of nuclear symmetry
energy, $E_{\rm sym}$, and the neutron-star properties. It is found that the
$\delta$ meson increases $E_{\rm sym}$ at high densities, whereas the
$\sigma$-$\delta$ mixing makes $E_{\rm sym}$ soft above the saturation density.
Furthermore, the $\delta$ meson and its mixing have a large influence on the
radius and tidal deformability of a neutron star. In particular, the
$\sigma$-$\delta$ mixing reduces the neutron-star radius, and, thus, the
present calculation can simultaneously reproduce the dimensionless tidal
deformabilities of a canonical $1.4M_{\odot}$ neutron star observed from the
binary neutron star merger, GW170817, and from the compact binary coalescence,
GW190814.
