Quadrature amplitude modulation (QAM)-like signal sets are considered in this paper as coset constellations placed on regular graphs on surfaces known as flat tori. Such signal sets can be related to spherical, block, and trellis codes and may be viewed as geometrically uniform (GU) in the graph metric in a sense that extends the concept introduced by Forney [13]. Homogeneous signal sets of any order can then be labeled by a cyclic group, induced by translations on the Euclidean plane. We construct classes of perfect codes on square graphs including Lee spaces, and on hexagonal and triangular graphs, all on flat tori. Extension of this approach to higher dimensions is also considered. Index Terms-Codes on graphs, coset codes, flat torus, geometrically uniform (GU) codes, perfect codes, spherical codes.501023632377