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

A (di)graph Γ generates a commutative association scheme $X$ if and only if the adjacency matrix of Γ generates the Bose-Mesner algebra of $X$. In [18, 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.