  We study the family of \emph{derangement action digraphs}, which are a
subfamily of the group action graphs introduced in [Fred Annexstein, Marc
Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel
architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any
non-empty set $X$ and a non-empty subset $S$ of $\Der(X)$, the set of
derangments of $X$, we define the derangement action digraph
$\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to
$y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and
digraphs, derangement action digraphs may be useful to model networks as the
same routing and communication scheme can be implemented at each vertex. We
determine necessary and sufficient conditions on $S$ under which
$\rm\overrightarrow{DA}(X;S)$ may be viewed as a simple graph of valency $|S|$,
and we call such graphs derangement action graphs. Also we investigate the
structural and symmetry properties of these digraphs and graphs. Several open
problems are posed and many examples are given.
