An extension of regular colouring of graphs to digraphs, networks and hypergraphs *

Everett and Borgatti (1991) have suggested an alternative formulisation of Regular Equivalence (White and Reitz 1983) using similar ideas to classical graph colouring. This fo~ulisation provides a simple representation of regular equivalence without the need to resort to complicated algebraic constructions. In this paper we shall extend the ideas in Everett and Borgatti’s paper to digraphs and networks in keeping with White and Reitz’s original usage of regular equivalence. In addition, using the new terminology, we examine how regular equivalence can be extended to edges and hypergraphs. We first introduce the idea of regular colouring for graphs. (Note that Everett and Borgatti (1991) call this role-colouring.) Let C(V, E) be an undirected graph (self-loops are permissible) with vertex set V and edge let E. The neighbourhood N(u) of a vertex u E Y is the set of all verticles adjacent to U. So that N(U) = (x:(u, X) EE).