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On a family of divisible design digraphs

Mikhail Muzychuk, Grigory Ryabov

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 $3$.

On a family of divisible design digraphs

Abstract

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

Paper Structure

This paper contains 7 sections, 23 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coherent configurations
  4. $S$-rings
  5. WL-closure and WL-dimension
  6. An $S$-ring over a Heisenberg group
  7. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $q$ be an odd prime power. There exist $I\subseteq \mathbb{F}_q$ and a family $\Gamma_i$, $i\in I$, of pairwise nonisomorphic normal arc-transitive divisible design Cayley digraphs with parameters $(q^3,q^2,0,q,q^2,q)$ over $H_3(q)$ such that: for all $i,j\in I$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 3.1
  • Lemma 3.2
  • ...and 25 more