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A census of Cayley graphs

Rhys J. Evans, Primož Potočnik

TL;DR

This work develops a structured census framework for enumerating small-order Cayley graphs by combining sub-(a,b)-valent group construction with (a,b)-valent Cayley-set generation, followed by isomorphism reduction. It employs dual strategies—library filtering and group extensions via cohomology—to build comprehensive lists up to a given order, and two generation schemes (orderly and ordered) to cover Aut(G)-orbits of Cayley sets. The authors produce complete datasets of all 3-valent graphs up to 5,000 vertices and all 4-valent graphs up to 1,025 vertices, totaling nearly 1.94 million cubic and 11.36 million quartic graphs, and analyze their structural properties (GRRs, arc-transitivity, girth, bipartiteness) to pose open questions about asymptotic behavior. The work provides reproducible algorithms and datasets that support further investigations into the geometry and symmetry of vertex-transitive graphs and their Cayley representations.

Abstract

Given positive integers $k$ and $n$, we present methods to construct all groups of order at most $n$ that contain a Cayley set of size $k$, and to enumerate the Cayley sets of order $k$ in a given group, up to the action of the automorphism group. We use these methods to generate complete lists of pairwise nonisomorphic 3-valent Cayley graphs with at most 5000 vertices and 4-valent Cayley graphs with at most 1025 vertices.

A census of Cayley graphs

TL;DR

This work develops a structured census framework for enumerating small-order Cayley graphs by combining sub-(a,b)-valent group construction with (a,b)-valent Cayley-set generation, followed by isomorphism reduction. It employs dual strategies—library filtering and group extensions via cohomology—to build comprehensive lists up to a given order, and two generation schemes (orderly and ordered) to cover Aut(G)-orbits of Cayley sets. The authors produce complete datasets of all 3-valent graphs up to 5,000 vertices and all 4-valent graphs up to 1,025 vertices, totaling nearly 1.94 million cubic and 11.36 million quartic graphs, and analyze their structural properties (GRRs, arc-transitivity, girth, bipartiteness) to pose open questions about asymptotic behavior. The work provides reproducible algorithms and datasets that support further investigations into the geometry and symmetry of vertex-transitive graphs and their Cayley representations.

Abstract

Given positive integers and , we present methods to construct all groups of order at most that contain a Cayley set of size , and to enumerate the Cayley sets of order in a given group, up to the action of the automorphism group. We use these methods to generate complete lists of pairwise nonisomorphic 3-valent Cayley graphs with at most 5000 vertices and 4-valent Cayley graphs with at most 1025 vertices.

Paper Structure

This paper contains 18 sections, 3 theorems, 3 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Outline of the computational approach
  4. Constructing a list of sub-$(a,b)$-valent groups
  5. Quotients by minimal normal subgroups
  6. Using libraries of small groups
  7. Group extensions
  8. The inductive step.
  9. Computing cohomology groups.
  10. Further optimisations.
  11. Constructing a list of $(a,b)$-valent Cayley sets
  12. Orderly generation
  13. Ordered generation
  14. Constructing the Cayley graphs
  15. Statistical observations
  16. ...and 3 more sections

Key Result

Lemma 2.3

Let $G$ be a group and $N\unlhd G$. Then;

Figures (7)

  • Figure 1: Cumulative number of cubic (L) and quartic (R) Cayley graphs of each order.
  • Figure 2: Cumulative proportion of cubic (L) and quartic (R) Cayley graphs that are GRRs.
  • Figure 3: Cumulative proportion of quartic Cayley graphs that are GRRs for odd and even orders.
  • Figure 4: Cumulative proportion of cubic (L) and quartic (R) Cayley graphs that are arc-transitive.
  • Figure 5: The proportion $a(n,s)$ for $s\in\lbrace1,2,3,4,5\rbrace$.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Definition 2.1: for Cayley sets
  • Definition 2.2: for groups
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof