The chemical and physical properties of nanoclusters largely depend on their sizes and shapes. This is partly due to finite size effects influencing the local electronic structure of the nanocluster atoms which are located on the nanofacets and on their edges. Here we present a thorough study on graphene-supported Rh nanocluster assemblies and their geometry-dependent electronic structure obtained by combining high-energy resolution core level photoelectron spectroscopy, scanning tunneling microscopy, and density functional theory. We demonstrate the possibility to finely control the morphology and the degree of structural order of Rh clusters grown in register with the template surface of graphene/Ir(111). By comparing measured and calculated core electron binding energies, we identify edge, facet, and bulk atoms of the nanoclusters. We describe how small interatomic distance changes occur while varying the nanocluster size, substantially modifying the properties of surface atoms. The properties of under-coordinated Rh atoms are discussed in view of their importance in heterogeneous catalysis and magnetism.