Normally Regular Digraphs

A normally regular digraph with parameters $(v,k,\lambda ,\mu )$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$\mathrm{.}Thismeansthateveryvertexhasout-degree$k$,apairofnon-adjacentverticeshave$\mu$ common out-neighbours, a pair of vertices connected by an edge in one direction have $\lambda$ common out-neighbours and a pair of vertices connected by edges in both directions have $2\lambda -\mu$ 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.