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 .
A new and updated version has now been published in
Leif K. Jørgensen, Normally Regular Digraphs,
The Electronic Journal of Combinatorics
Volume 22, Issue 4 (2015) > Paper #P4.21

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

In the following recently published paper we study relations between normally regular digraphs and other combinatorial structures:
Leif Kjær Jørgensen, Gareth A. Jones, Mikhail H. Klin and Sung Y. Song,
Normally Regular Digraphs, Association Schemes and Related Combinatorial Structures,
Séminaire Lotharingien de Combinatoire, B71c (2014), 39 pp.

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