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 outdegree k

every pair of adjacent vertices have lambda common outneighbours

every pair of nonadjacent vertices have mu common outneighbours

the graph has no directed 2cycles (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 (JIAA^{T})
A matrix satisfying these properties is normal, i.e., properties
of outneighbours in the definition holds for inneighbours 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), 217222
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 .