# CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)

arXiv: https://arxiv.org/abs/2509.19162 · HTML: https://mlsanity.com/paper/2509.19162/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 124 references that are in the map.
- First posted on arXiv 23 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
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- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [Cayley Graph Propagation](https://mlsanity.com/paper/2410.03424/) (2024)
- [TransPath: Learning Heuristics For Grid-Based Pathfinding via Transformers](https://mlsanity.com/paper/2212.11730/) (2022)
- [Solving the Rubik's Cube Optimally is NP-complete](https://mlsanity.com/paper/1706.06708/) (2017)
- [Machine learning detects terminal singularities](https://mlsanity.com/paper/2310.20458/) (2023)
- [What makes math problems hard for reinforcement learning: a case study](https://mlsanity.com/paper/2408.15332/) (2024)
- [Advancing Geometry with AI: Multi-agent Generation of Polytopes](https://mlsanity.com/paper/2502.05199/) (2025)
- [Puzzle Solving without Search or Human Knowledge: An Unnatural Language Approach](https://mlsanity.com/paper/2109.02797/) (2021)
- [Learning Graph Search Heuristics](https://mlsanity.com/paper/2212.03978/) (2022)
- [CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs](https://mlsanity.com/paper/2502.18663/) (2025)
- [Contrastive Representations for Temporal Reasoning](https://mlsanity.com/paper/2508.13113/) (2025)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [Self-Supervision is All You Need for Solving Rubik's Cube](https://mlsanity.com/paper/2106.03157/) (2021)
- [Towards Learning Foundation Models for Heuristic Functions to Solve Pathfinding Problems](https://mlsanity.com/paper/2406.02598/) (2024)
- [A Machine Learning Approach That Beats Large Rubik's Cubes](https://mlsanity.com/paper/2502.13266/) (2025)
- [CubeTR: Learning to Solve The Rubiks Cube Using Transformers](https://mlsanity.com/paper/2111.06036/) (2021)
- [Probabilistic estimates of the diameters of the Rubik's Cube groups](https://mlsanity.com/paper/2404.07337/) (2024)

## Built on by (most cited first, 1 of 1)

- [Learning the Graphical Nature of Symmetries](https://mlsanity.com/paper/2607.12026/) (2026)

## Abstract (arXiv)

This is the third paper of the CayleyPy project applying artificial intelligence to problems in group theory. We announce the first public release of CayleyPy, an open source Python library for computations with Cayley and Schreier graphs. Compared with systems such as GAP and Sage, CayleyPy handles much larger graphs and performs several orders of magnitude faster. Using CayleyPy we obtained about 200 new conjectures on Cayley and Schreier graphs, focused on diameters and growth. For many Cayley graphs of symmetric groups Sn we observe quasi polynomial diameter formulas: a small set of quadratic or linear polynomials indexed by n mod s. We conjecture that this is a general phenomenon, giving efficient diameter computation despite the problem being NP hard. We propose a refinement of the Babai type conjecture on diameters of Sn: n^2/2 + 4n upper bounds in the undirected case, compared to previous O(n^2) bounds. We also provide explicit generator families, related to involutions in a square with whiskers pattern, conjectured to maximize the diameter; search confirms this for all n up to 15. We further conjecture an answer to a question posed by V M Glushkov in 1968 on directed Cayley graphs generated by a cyclic shift and a transposition. For nilpotent groups we conjecture an improvement of J S Ellenberg's results on upper unitriangular matrices over Z/pZ, showing linear dependence of diameter on p. Some conjectures are LLM friendly, naturally stated as sorting problems verifiable by algorithms or Python code. To benchmark path finding we created more than 10 Kaggle datasets. CayleyPy works with arbitrary permutation or matrix groups and includes over 100 predefined generators. Our growth computation code outperforms GAP and Sage up to 1000 times in speed and size.