  We proposed a minimal model to describe the Floquet band structure of
two-dimensional materials with light-induced resonant inter-band transition. We
applied it to graphene to study the band features caused by the light
irradiation. Linearly polarized light induces pseudo gaps (gaps are functions
of wavevector), and circularly polarized light causes real gaps on the
quasi-energy spectrum. If the polarization of light is linear and along the
longitudinal direction of zigzag ribbons, flat edge bands appear in the pseudo
gaps, and if is in the lateral direction of armchair ribbons, curved edge bands
can be found. For the circularly polarized cases, edge bands arise and
intersect in the gaps of both types of ribbons. The edge bands induced by the
circularly polarized light are helical and those by linearly polarized light
are topologically trivial ones. The Chern number of the Floquet band, which
reflects the number of pairs of helical edge bands in graphene ribbons, can be
reduced into the winding number at resonance.
