This paper presents a detailed analysis of the magneto-optical conductivity in graphene, focusing on the behavior of the optical spectral weight and its redistribution as a function of frequency and chemical potential. The study is based on the Dirac nature of graphene's quasiparticles and the unusual quantum Hall effect observed in the material. The optical conductivity is calculated using the Kubo formula and the Green's function approach, considering both diagonal and Hall components of the conductivity. The results are derived for various values of the magnetic field and chemical potential, and the behavior of the conductivity is analyzed in the context of Landau level quantization. The optical conductivity is found to exhibit characteristic behavior due to the Dirac nature of the quasiparticles, with the lowest Landau level being shared equally by electrons and holes. The spectral weight of the optical conductivity is analyzed under different absorption peaks, and the redistribution of this weight as the chemical potential varies is discussed. The results are also presented for different magnetic fields and chemical potentials, showing how the conductivity changes with these parameters. The paper provides analytic expressions for the optical conductivity, including the real and imaginary parts of the diagonal and Hall conductivities. These expressions are derived under different assumptions about the scattering rate and the dependence of the conductivity on the Landau level index. The results are validated against numerical calculations and compared with experimental data, showing good agreement. The study also considers the low-field limit of the magneto-optical Lorentzian model and demonstrates how the conductivity behaves in this regime. The results are used to understand the behavior of the optical conductivity in graphene, including the effects of the excitonic gap and the influence of the magnetic field on the spectral weight. The paper concludes with a discussion of the implications of the results for the understanding of the optical properties of graphene, highlighting the importance of the Dirac nature of the quasiparticles and the role of the magnetic field in determining the optical conductivity. The results are also relevant for the development of experimental techniques to measure the optical properties of graphene.This paper presents a detailed analysis of the magneto-optical conductivity in graphene, focusing on the behavior of the optical spectral weight and its redistribution as a function of frequency and chemical potential. The study is based on the Dirac nature of graphene's quasiparticles and the unusual quantum Hall effect observed in the material. The optical conductivity is calculated using the Kubo formula and the Green's function approach, considering both diagonal and Hall components of the conductivity. The results are derived for various values of the magnetic field and chemical potential, and the behavior of the conductivity is analyzed in the context of Landau level quantization. The optical conductivity is found to exhibit characteristic behavior due to the Dirac nature of the quasiparticles, with the lowest Landau level being shared equally by electrons and holes. The spectral weight of the optical conductivity is analyzed under different absorption peaks, and the redistribution of this weight as the chemical potential varies is discussed. The results are also presented for different magnetic fields and chemical potentials, showing how the conductivity changes with these parameters. The paper provides analytic expressions for the optical conductivity, including the real and imaginary parts of the diagonal and Hall conductivities. These expressions are derived under different assumptions about the scattering rate and the dependence of the conductivity on the Landau level index. The results are validated against numerical calculations and compared with experimental data, showing good agreement. The study also considers the low-field limit of the magneto-optical Lorentzian model and demonstrates how the conductivity behaves in this regime. The results are used to understand the behavior of the optical conductivity in graphene, including the effects of the excitonic gap and the influence of the magnetic field on the spectral weight. The paper concludes with a discussion of the implications of the results for the understanding of the optical properties of graphene, highlighting the importance of the Dirac nature of the quasiparticles and the role of the magnetic field in determining the optical conductivity. The results are also relevant for the development of experimental techniques to measure the optical properties of graphene.