Motivated by the experimentally observed high mobility of gold atoms on graphene and their tendency to form nanometer-sized clusters, we present a density functional theory study of the ground state structures of small gold clusters on graphene, their mobility and clustering. Our detailed analysis of the electronic structures identifies the opportunity to form strong gold-gold bonds and the graphene-mediated interaction of the pre-adsorbed fragments as the driving forces behind gold's tendency to aggregate on graphene. While clusters containing up to three gold atoms have one unambiguous ground state structure, both gas phase isomers of a cluster with four gold atoms can be found on graphene. In the gas phase the diamond-shaped Au(4)(D) cluster is the ground state structure, whereas the Y-shaped Au(4)(Y) becomes the actual ground state when adsorbed on graphene. As we show, both clusters can be produced on graphene by two distinct clustering processes. We also studied in detail the stepwise formation of a gold dimer out of two pre-adsorbed adatoms, as well as the formation of Au(3). All reactions are exothermic and no further activation barriers, apart from the diffusion barriers, were found. The diffusion barriers of all studied clusters range from 4 to 36 meV only, and are substantially exceeded by the adsorption energies of - 0.1 to - 0.59 eV. This explains the high mobility of Au(1-4) on graphene along the C-C bonds.