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

TL;DR

This paper proves that every nonsymmetric 4-class association scheme can be generated by a digraph, extending previous results on association schemes with fewer classes and specific symmetry conditions.

Contribution

It characterizes when a commutative association scheme with one pair of nonsymmetric relations can be generated by a digraph, and shows all nonsymmetric 4-class schemes can be generated.

Findings

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

Provides conditions for generating schemes with one pair of nonsymmetric relations.

Extends previous results on association schemes with fewer classes.

Abstract

A (di)graph $\Gamma$ generates a commutative association scheme $\mathfrak{X}$ if and only if the adjacency matrix of $\Gamma$ generates the Bose-Mesner algebra of $\mathfrak{X}$. In [17, Theorem 1.1], Monzillo and Penji\'{c} 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.