We study energetics, electronic and magnetic structures, and magnetic anisotropy barriers of a monolayer of single-molecule magnets (SMM), [Mn 12 O 12 (COOR) 16 ](H 2 O) 4 (abbreviated as Mn 12 , with R=H, CH 3 , C 6 H 5 and CHCl 2 ), on a graphene surface using spin-polarized density-functional theory with generalized gradient corrections and the inclusion of van der Waals interactions. We find that Mn 12 molecules with ligands -H, -CH 3 , and -C 6 H 5 are physically adsorbed on graphene through weak van der Waals interactions, and a much stronger ionic interaction occurs using a -CHCl 2 ligand. The strength of bonding is closely related to the charge transfer between the molecule and the graphene sheet and can be manipulated by strain in the graphene; specifically, tension enhances n-doping of graphene and compression encourages p-doping. The magnetic anisotropy barrier is computed by including the spin-orbit interaction within density-functional theory. The barriers for the Mn 12 molecules with ligands -H, -CH 3 and -C 6 H 5 on graphene surfaces remain unchanged (within 1 K) from those of isolated molecules because of their weak interaction, and a much larger reduction (10 K) is observed when using the -CHCl 2 ligand on graphene due to a substantial structural deformation as a consequence of the much stronger interaction. Neither strain in graphene nor charge transfer affects the magnetic anisotropy barrier significantly. Finally, we discuss the effect of strong correlation in the high spin state of a Mn 12 SMM and the consequence in SMM-surface adsorption.