We explore the spatial variations of the unoccupied electronic states of graphene epitaxially grown on Ru(0001) and observed three unexpected features: the first graphene image state is split in energy; unlike all other image states, the split state does not follow the local work function modulation, and a new interfacial state at +3 eV appears on some areas of the surface. First-principles calculations explain the observations and permit us to conclude that the system behaves as a self-organized periodic array of quantum dots.