The corrugation pattern and electronic structure of graphene on MgO (111), which is a lattice-mismatched substrate, were investigated using the ab initio method. On Mg-terminated octo-polar reconstructed MgO (111), graphene was physisorbed and shaped into a stripe-like corrugation pattern. Due to weak charge transfer at the interface, the graphene was of the n-type doped, and its low effective mass was preserved. On the other hand, graphene on O-terminated octo-polar reconstructed MgO (111) showed a larger corrugation triggered by periodic oxygen-carbon chemisorption. In addition, a band-gap opening, which was induced by the periodic chemisorption, of 0.294 eV was observed. The results indicate that a corrugation structure induced by the lattice-mismatched substrate is highly effective for band-gap engineering treatment.