We performed scanning tunneling microscopy/spectroscopy (STM/S) on the edges of monolayer graphene islands grown on Ir(111). Constant current STM images reveal that the atomic corrugation at the graphene edges correlates with the moiré pattern of graphene on Ir(111). The graphene islands terminate with a zigzag edge and moiré-periodic kinks in the regions of the on-top stacked carbon rings. Graphene edges near an Ir(111) monatomic step also show the formation of periodic kinks. STS indicates a significant spatial dependence of the differential conductance dI/dV near graphene edges. We observe a considerably reduced differential conductance at the graphene island edges. We tentatively ascribe these observations to the electronic interaction of graphene with the Ir(111) substrate, which affects the differential conductance differently near graphene edges as compared to the inner part of a graphene island.