  We consider a tight-binding model with the nearest neighbour hopping
integrals on the honeycomb lattice in a magnetic field. Assuming one of the
three hopping integrals, which we denote t_a, can take a different value from
the two others, we study quantum phase structures controlled by the anisotropy
of the honeycomb lattice.For weak and strong t_a regions, respectively, the
Hall conductances are calculated algebraically by using the Diophantine
equation. Except for a few specific gaps, we completely determine the Hall
conductances in these two regions including those for subband gaps. In a weak
magnetic field, it is found that the weak t_a region shows the unconventional
quantization of the Hall conductance, \sigma_{xy}=-(e^2/h)(2n+1), (n=0,\pm
1,\pm 2,...), near the half-filling, while the strong t_a region shows only the
conventional one, \sigma_{xy}=-(e^2/h)n,(n=0,1,2,...). From topological nature
of the Hall conductance, the existence of gap closing points and quantum phase
transitions in the intermediate t_a region are concluded. We also study
numerically the quantum phase structure in detail, and find that even when
t_a=1, namely in graphene case, the system is in the weak t_a phase except when
the Fermi energy is located near the van Hove singularity or the lower and
upper edges of the spectrum.
