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

A (di) graph generates a commutative association scheme X if and only if the adjacency matrix of 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.