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

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.