To identify families of stable planar anchor groups for use in single molecule electronics, we report detailed results for the binding energies of two families of anthracene and pyrene derivatives adsorbed onto graphene. We find that all the selected derivatives functionalized with either electron donating or electron accepting substituents bind more strongly to graphene than the parent non-functionalized anthracene or pyrene. The binding energy is sensitive to the detailed atomic alignment of substituent groups over the graphene substrate leading to larger than expected binding energies for –OH and –CN derivatives. Furthermore, the ordering of the binding energies within the anthracene and pyrene series does not simply follow the electron affinities of the substituents. Energy barriers to rotation or displacement on the graphene surface are much lower than binding energies for adsorption and therefore at room temperature, although the molecules are bound to the graphene, they are almost free to move along the graphene surface. Binding energies can be increased by incorporating electrically inert side chains and are sensitive to the conformation of such chains.