We carry out an explicit calculation of the vacuum polarization tensor for an effective low-energy model of monolayer graphene in the presence of a weak magnetic field of intensity $B$ perpendicularly aligned to the membrane. By expanding the quasiparticle propagator in the Schwinger proper time representation up to order $(eB)^2$, where $e$ is the unit charge, we find an explicitly transverse tensor, consistent with gauge invariance. Furthermore, assuming that graphene is radiated with monochromatic light of frequency $omega$ along the external field direction, from the modified Maxwells equations we derive the intensity of transmitted light and the angle of polarization rotation in terms of the longitudinal ($sigma_{xx}$) and transverse ($sigma_{xy}$) conductivities. Corrections to these quantities, both calculated and measured, are of order $(eB)^2/omega^4$. Our findings generalize and complement previously known results reported in literature regarding the light absorption problem in graphene from the experimental and theoretical points of view, with and without external magnetic fields.
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