  Let $W$ be a Coxeter group whose proper parabolic subgroups are finite.
According to Theorem~1.12 of [1], if the module of a finite $W$-digraph
$\Gamma$ is isomorphic to the module of a $W$-graph over $Q$, then $\Gamma$ is
acyclic. We extend this result to Coxeter groups with finite dihedral parabolic
subgroups and $W$-graphs over arbitrary fields $F$ of $C$. Also, an example is
provided showing the converse of this theorem is false. That is, there is an
example of a finite, acyclic $W$-digraph whose module does not afford a
$W$-graph.
