We theoretically study photon transmission and mechanical ground state cooling in a two-dimensional optomechanical system that is formed of a suspended graphene sheet on an one-dimensional optomechanical crystal. When the frequencies of graphene resonator and nanobeam resonator(phononic mode of optomechanical crystal) are approximately the same, the $Lambda$-type degenerate four-level system of two-dimensional optomechanics shows two-color optomechanically-induced transparency , and the transparency window could be switched among probe signals absorption, transparency, and amplification. According to our calculations, the graphene resonator could also effectively assist the ground state cooling of large damping nanobeam resonator in two-dimensional optomechanics.
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