Normally Regular Digraphs

A normally regular digraph with parameters (v, k, λ, μ) is a directed graph on v vertices whose adjacency matrix A satisfies the equation AAt = kI + λ(A + At) + μ(J − I − A − At). This means that every vertex has out-degree k, a pair of nonadjacent vertices have μ common out-neighbours, a pair of vertices connected by an edge in one direction have λ common out-neighbours and a pair of vertices connected by edges in both directions have 2λ− μ common out-neighbours. We often assume that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regular digraph is normal. A connected k-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than k are on one circle in the complex plane. We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs are Cayley graphs of abelian groups and the construction is then based on a generalization of difference sets. We also show connections to other combinatorial objects: strongly regular graphs, symmetric 2-designs and association schemes.