This paper presents an unconventional integer quantum Hall effect (IQHE) in graphene, where the quantized Hall conductivity is given by σ_xy = -(2e²/h)(2n + 1), with n = 0, 1, ..., which differs from the conventional IQHE observed in other materials. This unique quantization arises from the quantum anomaly of the n = 0 Landau level (LL), which has half the degeneracy of higher LLs and does not depend on the magnetic field. This effect was experimentally observed in ultrathin graphite films. Graphene, a single layer of carbon, exhibits remarkable properties such as high electrical conductivity and mechanical strength. Its quasiparticle excitations are described by a 2+1 dimensional Dirac theory, leading to a unique relativistic-like electronic system. The quantum Hall effect in graphene is studied using the Kubo formalism and the Landau level quantization. The Hall conductivity is derived as σ_xy = -2e²/h(2n + 1), which is distinct from the conventional IQHE where σ_xy = -νe²/h with ν being an integer. The unconventional quantization is due to the special nature of the n = 0 LL in Dirac theory, which has half the degeneracy of higher LLs and is not affected by the magnetic field. This leads to a different filling factor and Hall conductivity behavior compared to conventional semiconductors. The paper also discusses the quantum magnetic oscillations in graphene, showing that the minima of the diagonal conductivity occur at fillings ν_B = 2n + 1, indicating the possible positions of the plateaux in the IQHE. The study highlights the unique properties of graphene, including its 2+1 dimensional Dirac theory and the quantum anomaly of the n = 0 LL. These properties lead to an unconventional IQHE with a different quantization pattern, which has been experimentally confirmed in ultrathin graphite films. The results demonstrate that the theoretical description of graphene is based on a 2+1 dimensional Dirac theory, where the lowest LL has half the degeneracy of higher LLs.This paper presents an unconventional integer quantum Hall effect (IQHE) in graphene, where the quantized Hall conductivity is given by σ_xy = -(2e²/h)(2n + 1), with n = 0, 1, ..., which differs from the conventional IQHE observed in other materials. This unique quantization arises from the quantum anomaly of the n = 0 Landau level (LL), which has half the degeneracy of higher LLs and does not depend on the magnetic field. This effect was experimentally observed in ultrathin graphite films. Graphene, a single layer of carbon, exhibits remarkable properties such as high electrical conductivity and mechanical strength. Its quasiparticle excitations are described by a 2+1 dimensional Dirac theory, leading to a unique relativistic-like electronic system. The quantum Hall effect in graphene is studied using the Kubo formalism and the Landau level quantization. The Hall conductivity is derived as σ_xy = -2e²/h(2n + 1), which is distinct from the conventional IQHE where σ_xy = -νe²/h with ν being an integer. The unconventional quantization is due to the special nature of the n = 0 LL in Dirac theory, which has half the degeneracy of higher LLs and is not affected by the magnetic field. This leads to a different filling factor and Hall conductivity behavior compared to conventional semiconductors. The paper also discusses the quantum magnetic oscillations in graphene, showing that the minima of the diagonal conductivity occur at fillings ν_B = 2n + 1, indicating the possible positions of the plateaux in the IQHE. The study highlights the unique properties of graphene, including its 2+1 dimensional Dirac theory and the quantum anomaly of the n = 0 LL. These properties lead to an unconventional IQHE with a different quantization pattern, which has been experimentally confirmed in ultrathin graphite films. The results demonstrate that the theoretical description of graphene is based on a 2+1 dimensional Dirac theory, where the lowest LL has half the degeneracy of higher LLs.