The interaction of polycyclic aromatic hydrocarbon molecules with hydrogen-terminated graphene is studied using density functional theory with empirical dispersion correction. The effective potential energy surfaces for the interaction of benzene, C(6)H(6), naphthalene, C(10)H(8), coronene, C(24)H(12), and ovalene, C(32)H(14), with hydrogen-terminated graphene are calculated as functions of the molecular displacement along the substrate. The potential energy surfaces are also described analytically using the lowest harmonics of the Fourier expansion. It is shown that inclusion of the dispersive interaction, which is the most important contribution to the binding of these weakly bound systems, does not change the shape of the interaction energy surfaces or the value of the barriers to the motion of polycyclic aromatic hydrocarbon molecules on graphene. The potential energy surfaces are used in the estimation of the friction forces acting on the molecules along the direction of motion. These results underpin the modelling, using density functional theory, of electromechanical devices based on the relative vibrations of graphene layers and telescoping carbon nanotubes.