Normally Regular Digraphs

TL;DR

This paper introduces normally regular digraphs, explores their algebraic properties, provides non-existence results, structural characterizations, and constructions, often using Cayley graphs and difference sets, and connects them to other combinatorial objects.

Contribution

It defines normally regular digraphs, proves their adjacency matrices are normal, characterizes their eigenvalues, and offers new constructions and non-existence results.

Findings

Adjacency matrix of a normally regular digraph is normal.

Eigenvalues other than k lie on a single circle in the complex plane.

Many such graphs are Cayley graphs of abelian groups.

Abstract

A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $\mu$ 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.