On a family of divisible design digraphs

TL;DR

This paper constructs a family of highly symmetric Cayley digraphs over Heisenberg groups, revealing their indistinguishability by the Weisfeiler-Leman algorithm and establishing their Weisfeiler-Leman dimension as 3.

Contribution

It introduces a new family of divisible design Cayley digraphs with specific symmetry properties and analyzes their Weisfeiler-Leman algorithm behavior.

Findings

Digraphs are pairwise nonisomorphic and normal arc-transitive.

Neighborhood designs are isomorphic over the Heisenberg group.

Digraphs are not distinguished by the Weisfeiler-Leman algorithm, with WL dimension 3.

Abstract

For every odd prime power $q$, a family of pairwise nonisomorphic normal arc-transitive divisible design Cayley digraphs with isomorphic neighborhood designs over a Heisenberg group of order $q^3$ is constructed. It is proved that these digraphs are not distinguished by the Weisfeiler-Leman algorithm and have the Weisfeiler-Leman dimension $3$.