  We studied some phases and phase transitions in an extended boson Hubbard
model slightly away from half filling on bipartite lattices such as honeycomb
and square lattice. We find that in the insulating side, different kinds of
supersolids are generic stable states slightly away from half filling. We
propose a new kind of supersolid: valence bond supersolid. We show that the
quantum phase transitions from solids to supersolids driven by a chemical
potential are in the same universality class as that from a Mott insulator to a
superfluid, therefore have exact exponents $ z=2, \nu=1/2, \eta=0 $ (with
logarithmic corrections). Comparisons with previous quantum Monte-Carlo (QMC)
simulations on some microscopic models on a square lattice are made.
Implications on possible future QMC simulations are given.
