We present the first ab initio and molecular dynamics study of Na adsorption and diffusion on ideal graphene that considers Na-Na interaction and dispersion forces. From density functional theory (DFT) calculations using the generalized gradient approximation (GGA), the binding energy (vs. the vacuum reference state) of -0.75 eV is higher than the cohesive energy of Na metal (E-coh(DFT) = -1.07 eV). The binding energy approaches the Na metal cohesive energy (E-coh(DFT-D) = -1.21 eV) when dispersion correction is included (DFT-D), with E-b = -1.14 eV. Both DFT and DFT-D predict that the increase of Na concentration on graphene results in formation of Na complexes. This is evidenced by smaller Bader charge on Na atoms of Na dimer, 0.55e (0.48e for DFT) compared to 0.86e (for both DFT and DFT-D) for the single atom adsorption as well as by the formation of a Na Na bond identified by analysis of the electron density. These results suggest that ideal graphene is not a promising anode material for Na-ion batteries. Analysis of diffusion pathways for a Na dimer shows that the dimer remains stable during the diffusion, and computed migration barriers are significantly lower for the dimer than that for the single atom diffusion. This indicates that Na-Na interaction should be taken into account during the analysis of Na transport on graphene. Finally, we show that the typical defects (vacancy and divacancy) induce significant strengthening of the Na-C interaction. In particular, the largest change to the interaction is computed for vacancy-defected graphene, where the found lowest binding energy (vs. the metal reference state) is about 1.15 eV (1.21 eV for DFT) lower than that for ideal graphene. (C) 2015 Elsevier B.V. All rights reserved.