Normally Regular Digraphs.

A Normally Regular Digraph with parameters (v,k,lambda,mu) is a directed graph on v vertices with the following properties

every vertex has out-degree k
every pair of adjacent vertices have lambda common out-neighbours
every pair of non-adjacent vertices have mu common out-neighbours
the graph has no directed 2-cycles (i.e. the graph is "oriented")

A v x v matrix A is the adjacency matrix of a Normally Regular Digraph if and only if

A is a {0,1} matrix
A+A^T is a {0,1} matrix
AA^T= k I+ lambda (A+A^T) + mu (J-I-A-A^T)

A matrix satisfying these properties is normal, i.e., properties of out-neighbours in the definition holds for in-neighbours as well.

The basic results about these graphs are contained in a report from 1994: On Normally Regular Digraphs .
It is my intention that this report should be updated soon.

Some results about the particular case mu=0 are contained in Isomorphic switching in tournaments .
This paper was published in: Congressus Numerantium 104 (1994), 217-222.

Some results of a computer search for Normally Regular Digraphs are contained the report: Search for directed strongly regular graphs .