A census of Cayley graphs

TL;DR

This paper develops methods to classify and enumerate Cayley graphs based on group order and Cayley set size, providing comprehensive lists of certain valency Cayley graphs up to specified sizes.

Contribution

It introduces algorithms for constructing all groups with specific Cayley set properties and enumerates Cayley sets up to automorphism, enabling complete classification of certain Cayley graphs.

Findings

Generated complete lists of 3-valent Cayley graphs up to 5000 vertices.

Generated complete lists of 4-valent Cayley graphs up to 1025 vertices.

Provided methods for classifying groups with Cayley sets of given sizes.

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.