On uniform Higmanian association schemes

TL;DR

This paper characterizes when a Higmanian association scheme with two nontrivial parabolics is uniform and provides examples of such schemes, advancing understanding of their structure.

Contribution

It establishes a necessary and sufficient condition for uniformity in Higmanian association schemes with specific properties and supplies concrete examples.

Findings

Derived a criterion for uniformity in Higmanian schemes with two nontrivial parabolics.

Provided explicit examples of uniform Higmanian Cayley schemes.

Abstract

An imprimitive symmetric indecomposable association scheme of rank $5$ is said to be Higmanian. In the present paper, we prove a necessary and sufficient condition for a Higmanian association scheme with two nontrivial parabolics to be uniform. We also provide examples of uniform Higmanian Cayley schemes.