Enhanced interactions of light with graphene on the surface of a lossless dielectric magnetic mirror (DMM) are studied theoretically and experimentally in the visible range, where the DMM is composed of truncated dielectric photonic crystals (PCs). The absorption of graphene on the DMM was enhanced by about 4-fold for the spectral range within the forbidden gap of PCs over a wide range of incidence angles for both transverse electric and transverse magnetic polarizations compared with that of free-standing graphene. Moreover, the enhanced local electric field on the DMM surface led to much better detection efficiencies of the photocurrent, Raman spectroscopy and enhanced third-harmonic generation of graphene. These results offer a new way to achieve an enhanced interaction of light with graphene and develop new compact graphene-based devices.