A note on regular colorings of two mode data

In a recent paper, Everett and Borgatti (1992bl generalized their earlier (1991) definition of regular (or role) coloring of graphs. They extended their previous results to demonstrate that colorings can be used to find regular partitionings in digraphs, networks and hypergraphs. The purpose of this note is to suggest an alternative to their hypergraph representation one that requires only familiar graph theoretic concepts and that lends itself to clear visual displays. In their earlier paper, Everett and Borgatti (1991) defined a graph G = (V, E) in which the vertices u E v are social actors and the edges e E E refer to a symmetric social relation linking pairs of actors. The neighborhood of a vertex N(u) is defined as the set of all points that are adjacent to U. Each vertex u E I/ is assigned a color C(u). Then any subset of vertices S c P can be characterized by its spectrum C(S), the set of all the colors assigned to its members. The assignment of colors to vertices partitions the actors into equivalence sets. Everett and Borgatti showed that these equivalence sets are regular (White and Reitz 1983) when, for all U, w E V,