| We discuss several topological aspects of graphene with and without boundaries by tight binding models on a honeycomb lattice supplemented by the adiabatic principle. Appearance of gapless semiconductors as massless Dirac fermions is not accidental and it has a topological stability to form a critical region in the parameter space. The quantum Hall effects of the models are discussed topologically and the bulk-edge correspondence is justified explicitly. Characteristic zero energy boundary states at the Zigzag edges still exist under a magnetic field. Also varying the chemical potential away from the zero energy, we predict a sign change of the Hall conductance at the van Hove energies associated with change of its effective theory from the Dirac fermions' to the free electrons' or holes'. |
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