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Divisible design graphs from Higmanian association schemes

Grigory Ryabov

TL;DR

This work characterizes when unions of basis relations from a Higmanian, rank‑5, imprimitive symmetric scheme yield divisible design graphs (DDGs), linking DDGs to fusions of Higmanian schemes. The authors derive necessary and sufficient conditions in terms of intersection numbers and parabolic structure, and show that several known DDG families arise as fusions (class $ ext{K}_1$ and $ ext{K}_2$). They provide concrete constructions: (i) Cayley DDGs from Higmanian S-rings built from explicit groups, (ii) DDGs from weighing-matrix–based Higmanian schemes, and (iii) a new infinite family of DDGs on generalized dihedral groups built from divisible difference sets in abelian groups. The results deepen the connection between association schemes, S-rings, and divisible difference sets, and yield numerous new infinite families of DDGs with explicit parameters, expanding the catalog of known DDGs and suggesting further avenues for fusion-based design constructions.

Abstract

An imprimitive symmetric indecomposable association scheme of rank 5 is said to be Higmanian. A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a symmetric divisible design. We establish conditions which guarantee that a union of some basis relations of a Higmanian association scheme is an edge set of a divisible design graph. Further, we show that several known families of divisible design graphs can be obtained as fusions of Higmanian association schemes. Finally, using our approach we construct new infinite families of divisible design graphs.

Divisible design graphs from Higmanian association schemes

TL;DR

This work characterizes when unions of basis relations from a Higmanian, rank‑5, imprimitive symmetric scheme yield divisible design graphs (DDGs), linking DDGs to fusions of Higmanian schemes. The authors derive necessary and sufficient conditions in terms of intersection numbers and parabolic structure, and show that several known DDG families arise as fusions (class and ). They provide concrete constructions: (i) Cayley DDGs from Higmanian S-rings built from explicit groups, (ii) DDGs from weighing-matrix–based Higmanian schemes, and (iii) a new infinite family of DDGs on generalized dihedral groups built from divisible difference sets in abelian groups. The results deepen the connection between association schemes, S-rings, and divisible difference sets, and yield numerous new infinite families of DDGs with explicit parameters, expanding the catalog of known DDGs and suggesting further avenues for fusion-based design constructions.

Abstract

An imprimitive symmetric indecomposable association scheme of rank 5 is said to be Higmanian. A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a symmetric divisible design. We establish conditions which guarantee that a union of some basis relations of a Higmanian association scheme is an edge set of a divisible design graph. Further, we show that several known families of divisible design graphs can be obtained as fusions of Higmanian association schemes. Finally, using our approach we construct new infinite families of divisible design graphs.
Paper Structure (11 sections, 24 theorems, 90 equations)

This paper contains 11 sections, 24 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schemes
  4. Adjacency algebras
  5. $S$-rings
  6. Divisible difference sets
  7. Proof of Theorem \ref{['main']}
  8. DDGs from $\mathcal{K}_1$
  9. DDGs from $\mathcal{K}_2$
  10. New family of DDGs
  11. Further discussion

Key Result

Theorem 1.1

Let $\mathcal{X}=(V,S)$ be a Higmanian scheme and $\{s_0,\ldots,s_4\}$ a standard ordering of $S$. The graph $\Gamma=(V,E)$, where $E=s_2\cup s_i$ for some $i\in \{3,4\}$, is a divisible design graph if and only if $c_{33}^3=c_{33}^4$ and one of the following equalities holds: The canonical partition of $\Gamma$ is the partition into the classes of $e_1$$($$e_0$, resp.$)$ whenever Eq. cond1$($Eq.

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 30 more