Derangement action digraphs and graphs

A derangement of a set [Formula presented] is a fixed-point-free permutation of [Formula presented]. Derangement action digraphs are closely related to group action digraphs introduced by Annexstein, Baumslag and Rosenberg in 1990. For a non-empty set [Formula presented] and a non-empty subset [Formula presented] of derangements of [Formula presented], the derangement action digraph [Formula presented] has vertex set [Formula presented], and an arc from [Formula presented] to [Formula presented] if and only if [Formula presented] is the image of [Formula presented] under the action of some element of [Formula presented], so by definition it is a simple digraph. In common with Cayley graphs and Cayley digraphs, derangement action digraphs may be useful to model networks since the same routing and communication schemes can be implemented at each vertex. We prove that the family of derangement action digraphs contains all Cayley digraphs, all finite vertex-transitive simple graphs, and all finite regular simple graphs of even valency. We determine necessary and sufficient conditions on [Formula presented] under which [Formula presented] may be viewed as a simple undirected graph of valency [Formula presented]. We investigate structural and symmetry properties of these digraphs and graphs, pose several open problems, and give many examples.