On commutative association schemes and associated (directed) graphs

Let $\mathcal{M}$ denote the Bose–Mesner algebra of a commutative $d$-class association scheme $\mathfrak{X}$ (not necessarily symmetric), and $\mathrm{\Gamma }$ denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under the assumption that $A$ belongs to $\mathcal{M}$, we describe the combinatorial structure of $\mathrm{\Gamma }$. Moreover, we provide an algebraic-combinatorial characterization of $\mathrm{\Gamma }$ when $A$ generates $\mathcal{M}$. 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 $\mathrm{\Gamma }$ such that the adjacency matrix $A$ of $\mathrm{\Gamma }$ generates the Bose–Mesner algebra $\mathcal{M}$ of $\mathfrak{X}$.