  Inspired by a recent discovery of a peculiar integer quantum Hall effect
(QHE) in graphene, we study
  QHE on a honeycomb lattice in terms of the topological quantum number, with
two-fold interests: First, how the zero-mass Dirac QHE around the center of the
tight-binding band crosses over to the ordinary finite-mass fermion QHE around
the band edges. Second, how the bulk QHE is related with the edge QHE for the
entire spectrum including Dirac and ordinary behaviors. We find the following:
  (i) The zero-mass Dirac QHE persists up to the van Hove singularities, at
which the ordinary fermion behavior abruptly takes over. Here a technique
developed in the lattice gauge theory enabled us to calculate the behavior of
the topological number over the entire spectrum. This result indicates a
robustness of the topological quantum number, and should be observable if the
chemical potential can be varied over a wide range in graphene. (ii) To see if
the honeycomb lattice is singular in producing the anomalous QHE, we have
systematically surveyed over square-honeycomb-$\pi$-flux lattices, which is
scanned by introducing a diagonal transfer $t'$. We find that the massless
Dirac QHE forms a critical line, that is, the presence of Dirac cones in the
Brillouin zone is preserved by the inclusion of $t'$ and the Dirac region sits
side by side with ordinary one persists all through the transformation. (iii)
We have compared the bulk QHE number obtained by an adiabatic continuity of the
Chern number across transformation and numerically obtained edge QHE numbers
calculated from the whole energy spectra for sample with edges, which shows
that the bulk QHE number coincides, as in ordinary lattices, with the edge QHE
number throughout the lattice transformation.
