In this paper, we design and analyze MC2G (Matrix Completion with 2 Graphs), an algorithm that performs matrix completion in the presence of social and item similarity graphs. MC2G runs in quasilinear time and is parameter free. It is based on spectral clustering and local refinement steps. The expected number of sampled entries required for MC2G to succeed (i.e., recover the clusters in the graphs and complete the matrix) matches an information-theoretic lower bound up to a constant factor for a wide range of parameters. We show via extensive experiments on both synthetic and real datasets that MC2G outperforms other state-of-the-art matrix completion algorithms that leverage graph side information.