  Let ${\cal M}$ denote the Bose--Mesner algebra of a commutative $d$-class
association scheme ${\mathfrak X}$ (not necessarily symmetric), and $\Gamma$
denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under
the assumption that $A$ belongs to ${\cal M}$, we describe the combinatorial
structure of $\Gamma$. Moreover, we provide an algebraic-combinatorial
characterization of $\Gamma$ when $A$ generates ${\cal M}$.
  Among else, we show that, if ${\mathfrak X}$ is a commutative $3$-class
association scheme that is not an amorphic symmetric scheme, then we can always
find a (directed) graph $\Gamma$ such that the adjacency matrix $A$ of $\Gamma$
generates the Bose--Mesner algebra ${\cal M}$ of ${\mathfrak X}$.
