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

A. Chervov, D. Fedoriaka, E. Konstantinova, A. Naumov, I. Kiselev, A. Sheveleva, I. Koltsov, S. Lytkin, A. Smolensky, A. Soibelman, F. Levkovich-Maslyuk, R. Grimov, D. Volovich, A. Isakov, A. Kostin, M. Litvinov, N. Vilkin-Krom, A. Bidzhiev, A. Krasnyi, M. Evseev, E. Geraseva, L. Grunwald, S. Galkin, E. Koldunov, S. Diner, A. Chevychelov, E. Kudasheva, A. Sychev, A. Kravchenko, Z. Kogan, A. Natyrova, L. Shishina, L. Cheldieva, V. Zamkovoy, D. Kovalenko, O. Papulov, S. Kudashev, D. Shiltsov, R. Turtayev, O. Nikitina, D. Mamayeva, S. Nikolenko, M. Obozov, A. Titarenko, A. Dolgorukova, A. Aparnev, O. Debeaupuis, S. Alami C., H. Isambert

Abstract

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. Moreover. 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.

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

Abstract

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. Moreover. 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.

Paper Structure

This paper contains 36 sections, 9 equations, 31 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Context. Cayley graphs as a testbed for RL methods
  3. Problem statements on Cayley graphs from the mathematical perspective
  4. Main Contributions
  5. Background and related works
  6. Reminder on Cayley graphs, diameters, growth
  7. Related works
  8. Results Summary
  9. Brief features of CayleyPy
  10. Main features
  11. Examples and Tutorials
  12. Benchmarks for growth computations
  13. Benchmarks for AI path-finding
  14. Diameter quasi-polynomiality conjecture
  15. Table of results: hundreds of new conjectures on Cayley graphs
  16. ...and 21 more sections

Figures (31)

  • Figure 1: Defining a Cayley graph with given permutations (one-line notation) and computations of the growth.
  • Figure 2: Defining a Cayley graph from the predefined collection ("LRX" generators), computing its adjacency matrix, visualization.
  • Figure 3: Defining a Schreier coset graph by specifying the initial vector ("central state" -- the vector whose stabilizer defines a subgroup $H$ for the factor set $G/H$), computing the adjacency matrix, visualization.
  • Figure 4: Diameters of all possible Cayley graphs for $S_7$ generated by two permutations with/without their inverses.
  • Figure 5: Three generators of $S_{12}$ represented graphically, showing the pattern "square with whiskers". Three involutions, so the edges are undirected.
  • ...and 26 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Definition 3
  • Conjecture 5
  • Definition 4
  • Definition 5
  • ...and 10 more