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.
Graph machine learning tailored to planar graphs is developed, with an emphasis on the mathematical intuition that connects the topology of a plane embedding to the spectral and combinatorial structure ex- ploited by learning algorithms.
Satyanarayana Sanakkayala· International Journal of Com...· 0 citations
We study when, and how compactly, a finite connected graph (G) embeds isometrically into a Cayley graph of a finite abelian group. The classical theory of partial cubes answers this for isometric subgraphs of hypercubes through the Djokovic-Winkler relation (\theta); we extend the question to the full family of abelian Cayley graphs, whose hosts may carry composite generators and cyclic factors of any order. We introduce an involutive edge relation (\varphi), defined by two simultaneous distance equalities, which coincides with (\theta) exactly on partial cubes and remains informative beyond them, together with an oriented relation (\Phi) for non-involutive hosts, where generator classes are constrained to be partial permutations rather than matchings.The central result is a quotient labeling theorem: for any partition of the edge set into candidate generator classes, the most generic consistent vertex labeling is the quotient of the free module on the classes by the lattice of signed cycle-class incidences, computed by the Smith normal form; the binary case is its reduction modulo two. We prove that the finest partition always yields an isometric labeling, that compactifying the resulting universal group is itself an instance of the same quotient construction, and that the whole construction is algorithmic and certifiable. Worked examples include the triangle, the Petersen graph (embedding into the Clebsch graph of order 16), the Pappus graph (a 1024-fold compaction), and the diamond (a non-diagonal fold). Sharp dimension bounds and an exhaustive census of small graphs are developed in a companion paper. 2020 MSC: 05C12, 05C25, 20K01, 05C50
Graph neural networks (GNNs) have recently been shown to learn algebraic properties of finite groups from their Cayley graphs [1,2]. In this work, we investigate whether such models generalize to infinite finitely generated groups. Motivated by Gromov's theorem [3], a GNN is trained and validated exclusively on finite complete and truncated Cayley graphs, and then evaluated, without retraining, on truncated Cayley graphs of unseen infinite groups. The evaluation includes free abelian groups of various ranks, the discrete Heisenberg group, the infinite dihedral group, free groups, and direct products with both infinite abelian and finite groups. The results show strong generalization across these families, suggesting that finite Cayley graphs encode sufficient local geometric information to transfer to the infinite setting. Overall, this provides evidence that GNNs trained solely on finite groups can capture geometric features related to the growth of infinite finitely generated groups.
This paper initiates a systematic approach to handling mathematical data structured as (truncated) infinite $q-series, or equivalently, infinite series of integers, and demonstrates that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients.
Brandon Robinson, Shimal Harichurn, Fabian Ruehle et al.· 0 citations
We introduce the $\textit{expanded core graph}$, which records minimal unbounded domains and their axial directions, for two classes of hierarchically hyperbolic group structures modeled on compact special groups and mapping class groups. Our main structural result shows that every embedding of a right-angled Artin group, after replacing its standard generators by positive powers, factors through an intermediate RAAG generated by suitably supported axial elements; for the class modeled on compact special groups, this intermediate RAAG is quasi-isometrically embedded. We show that its extension graph embeds into the expanded core graph. This yields a Kim--Koberda-type obstruction to RAAG embeddings and a complete embedding criterion when the rank is at most two. For the standard HHG structure on a mapping class group, the expanded core graph is the disjointness graph of essential curves, while for natural rich-family structures on a RAAG it recovers the extension graph. We also establish permanence results under finite direct products and the standard relatively hyperbolic construction.
This work develops this perspective theoretically, showing that quotient couplings can be lifted to aligned representatives without additional cost and that symmetrization yields equivariant flow-matching minimizers, including for categorical endpoint prediction.
