Learning the Graphical Nature of Symmetries

The paper constructs a dataset of 131,406 Cayley graphs covering all groups of order at most 767 (except order 512), recording algebraic labels and graph statistics. The census contributes new sequences to the OEIS and identifies empirical regularities including conjectures on square clustering and spectral eigengaps. Comparing classical models, an MLP, and graph neural networks, the authors find that engineered graph statistics are highly informative while GNNs recover substantial structural signal directly from the graphs.