Normally Regular Digraphs

A normally regular digraph with parameters (v,k,λ,μ) is a directed graph with adjacency matrix A satisfying the equation AA^T=kI+λ(A+A^T) +μ(J-I-A-A^T).  I.e., the number of common out-neighbours of vertices x and y is k if x=y, μ if x and y are non-adjacent, λ if x and y are adjacent in one direction and 2λ-μ if they are adjacent in both directions.
The adjacency matrix of a normally regular graph is normal, and all eigenvalues other than k are on one circle in the complex plane. This property characterizes normally regular digraphs.
We consider constructions and structural characterizations and we also consider connections to association schemes, symmetric 2-designs, generalized difference sets, and partition of a projective plane into subplanes.