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 [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 an 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 the certain assumptions. As an application, we show that each nonsymmetric $4$-class association scheme can be generated by a digraph.