Deza digraphs

We consider a directed version of Deza graphs. A digraph is said to be a Deza digraph if it is regular and the number of common out-neighbors of any two distinct vertices takes on at most two values. We introduce some constructions and develop some basic theory. In particular, constructions from Cayley digraphs, graph lexicograph products and graph directed products are described. Moreover, we characterize a family of normal Deza digraphs, and list all cyclic strictly Deza digraphs with at most six vertices. Finally, connections to weakly distance-regular digraphs are discussed, and all strictly Deza digraphs which are also weakly distance-regular digraphs with at most ten vertices are listed.