  We discuss topological aspects of electronic properties of graphene,
including edge effects, with the tight-binding model on a honeycomb lattice and
its extensions to show the following: (i) Appearance of the pairn of massless
Dirac dispersions, which is the origin of anomalous properties including a
peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is
rather generic for a class of two-dimensional lattices that interpolate between
square and $\pi$-flux lattices. Persistence of the peculiar QHE is interpreted
as a topological stability. (ii) While we have the massless Dirac dispersion
only around E=0, the anomalous QHE associated with the Dirac cone unexpectedly
persists for a wide range of the chemical potential. The range is bounded by
van Hove singularities, at which we predict a transition to the ordinary
fermion behavior acompanied by huge jumps in the QHE with a sign change. (iii)
For edges we establish a coincidence between the quantum Hall effect in the
bulk and the quantum Hall effect for the edge states, which is a manifestation
of the topological bulk-edge correspondence. We have also explicitly shown that
the E=0 edge states in honeycomb in zero magnetic field persist in magnetic
field.
