Association schemes on triples over few vertices

TL;DR

This paper classifies association schemes on triples (ASTs) for small vertex counts, providing enumeration algorithms and identifying unique structures for three to five vertices.

Contribution

It introduces an algorithm to enumerate ASTs invariant under group actions and identifies unique ASTs for small vertex sets, advancing the understanding of higher-dimensional association schemes.

Findings

Unique AST over three vertices identified

Symmetric ASTs over four and five vertices classified

Unique circulant AST over five vertices found

Abstract

In this paper, we obtain classification results for higher-dimensional analogues of classical association schemes called association schemes on triples (ASTs). We present an algorithm that enumerates all ASTs on a fixed number of vertices whose nontrivial relations are invariant under the action of some group. Applying this algorithm to three, four, and five vertices along with appropriate group actions yields the unique AST over three vertices, the unique symmetric ASTs over four or five vertices, the unique AST over four vertices with two nontrivial relations, and the unique nontrivial circulant AST over five vertices.