Learning the Graphical Nature of Symmetries

2026-07-13

AI Summary

This study investigates how to effectively learn the algebraic structure and symmetry properties of finite groups from their Cayley graphs. We construct a large-scale dataset comprising 131,406 Cayley graphs, covering all finite groups of order up to 767 (excluding 512), thereby establishing for the first time a systematic correspondence between graph structure and group-theoretic properties. Using graph neural networks (GIN, GCN), multilayer perceptrons, and handcrafted graph statistics—such as cycle counts, distance distributions, and spectral features—we predict various group properties. Our results reveal that handcrafted features are highly informative, and GNNs effectively recover structural signals of the underlying groups, significantly outperforming conventional models on specific tasks. The work demonstrates the capacity of graph representations to encode group symmetries, proposes several empirically verifiable patterns, and contributes multiple new integer sequences to the OEIS.

Abstract

Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.

Problem

• Cayley graphs
• finite groups
• group properties
• graph representations
• symmetries

Innovation

• Cayley graphs
• graph neural networks
• finite groups
• spectral statistics
• group-theoretic properties