Using low-temperature scanning tunneling spectroscopy, we map the local density of states (LDOS) of graphene quantum dots supported on Ir(111). Due to a band gap in the projected Ir band structure around the graphene K point, the electronic properties of the QDs are dominantly graphene-like. Indeed, we compare the results favorably with tight binding calculations on the honeycomb lattice based on parameters derived from density functional theory. We find that the interaction with the substrate near the edge of the island gradually opens a gap in the Dirac cone, which implies soft-wall confinement. Interestingly, this confinement results in highly symmetric wave functions. Further influences of the substrate are given by the known moir{e} potential and a 10% penetration of an Ir surface resonance
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