  We study the extended Bose-Hubbard model on a two-dimensional honeycomb
lattice by using large scale quantum Monte Carlo simulations. We present the
ground state phase diagrams for both the hard-core case and the soft-core case.
For the hard-core case, the transition between $\rho=1/2$ solid and the
superfluid is first order and the supersolid state is unstable towards phase
separation. For the soft-core case, due to the presence of the multiple
occupation, a stable particle induced supersolid (SS-p) phase emerges when
$1/2<\rho<1$. The transition from the solid at $\rho=1/2$ to the SS-p is second
order with the superfluid density scaling as $ \rho_{s} \sim \rho-1/2 $. The
SS-p has the same diagonal order as the solid at $ \rho=1/2 $. As the chemical
potential increasing further, the SS-p will turn into a solid where two bosons
occupying each site of a sublattice through a first order transition. We also
calculate the critical exponents of the transition between $\rho=1/2$ solid and
superfluid at the Heisenberg point for the hard core case. We find the
dynamical critical exponent $z=0.15$, which is smaller than results obtained on
smaller lattices. This indicates that $ z $ approaches zero in the
thermodynamic limit, so the transition is also first order even at the
Heisenberg point.
