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Every nonsymmetric $4$-class association scheme can be generated by a digraph

Yuefeng Yang

TL;DR

This work addresses when the Bose-Mesner algebra of a commutative $d$-class association scheme can be generated by a digraph, extending prior results for the 3-class case to schemes with exactly one pair of nonsymmetric relations. The authors leverage eigenvalue techniques, the Bannai–Muzychuk fusion criterion, and fission/fusion arguments, distinguishing skew-symmetric versus non-skew-symmetric (nonsymmetric) structures and using amorphic/ERD99 classifications to derive precise generation criteria. They prove that for commutative schemes with a single nonsymmetric relation pair, the digraph $(X,R_d)$ generates the scheme when the symmetrization is generated by $(X,R_d\cup R_{d+1})$, yielding the key result that every nonsymmetric $4$-class scheme is digraph-generated under these conditions. The findings connect to strongly regular graphs via symmetric 2-class cases and provide a structured pathway to construct digraphs realizing the Bose–Mesner algebras of these schemes, with implications for explicit combinatorial realizations and further study of amorphic schemes.

Abstract

A (di)graph $Γ$ generates a commutative association scheme $\mathfrak{X}$ if and only if the adjacency matrix of $Γ$ generates the Bose-Mesner algebra of $\mathfrak{X}$. In [17, Theorem 1.1], Monzillo and Penjić proved that, except for amorphic symmetric association schemes, every $3$-class association scheme can be generated by the adjacency matrix of a (di)graph. In this paper, we characterize when a commutative association scheme with exactly one pair of nonsymmetric relations can be generated by a digraph under certain assumptions. As an application, we show that each nonsymmetric $4$-class association scheme can be generated by a digraph.

Every nonsymmetric $4$-class association scheme can be generated by a digraph

TL;DR

This work addresses when the Bose-Mesner algebra of a commutative -class association scheme can be generated by a digraph, extending prior results for the 3-class case to schemes with exactly one pair of nonsymmetric relations. The authors leverage eigenvalue techniques, the Bannai–Muzychuk fusion criterion, and fission/fusion arguments, distinguishing skew-symmetric versus non-skew-symmetric (nonsymmetric) structures and using amorphic/ERD99 classifications to derive precise generation criteria. They prove that for commutative schemes with a single nonsymmetric relation pair, the digraph generates the scheme when the symmetrization is generated by , yielding the key result that every nonsymmetric -class scheme is digraph-generated under these conditions. The findings connect to strongly regular graphs via symmetric 2-class cases and provide a structured pathway to construct digraphs realizing the Bose–Mesner algebras of these schemes, with implications for explicit combinatorial realizations and further study of amorphic schemes.

Abstract

A (di)graph generates a commutative association scheme if and only if the adjacency matrix of generates the Bose-Mesner algebra of . In [17, Theorem 1.1], Monzillo and Penjić proved that, except for amorphic symmetric association schemes, every -class association scheme can be generated by the adjacency matrix of a (di)graph. In this paper, we characterize when a commutative association scheme with exactly one pair of nonsymmetric relations can be generated by a digraph under certain assumptions. As an application, we show that each nonsymmetric -class association scheme can be generated by a digraph.
Paper Structure (8 sections, 29 theorems, 31 equations)

This paper contains 8 sections, 29 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic results
  4. Symmetric 2-class association schemes: strongly regular graphs
  5. Proofs of Theorems \ref{['one pair']} and \ref{['main-general']}
  6. Proof of Theorem \ref{['main']}
  7. $\mathfrak{X}$ is skew-symmetric
  8. $\mathfrak{X}$ is not skew-symmetric

Key Result

Theorem 1.2

Let $\mathfrak{X}=(X,\{R_i\}_{i=0}^{d+1})$ be a commutative association scheme with $R_d^{\top}=R_{d+1}$ and $R_i^{\top}=R_i$ for $0\leq i\leq d-1$, and $\tilde{\mathfrak{X}}$ be its symmetrization. Suppose that the graph $(X,R_d\cup R_{d+1})$ generates $\tilde{\mathfrak{X}}$. Then the following hol

Theorems & Definitions (45)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 35 more