On D. G. Higman's note on regular 3-graphs

Abstract

We introduce the notion of a t-graph and prove that regular 3-graphs are equivalent to cyclic antipodal 3-fold covers of a complete graph. This generalizes the equivalence of regular two-graphs and Taylor graphs. As a consequence, an equivalence between cyclic antipodal distance regular graphs of diameter 3 and certain rank 6 commutative association schemes is proved. New examples of regular 3-graphs are presented.

On t-fold covers of coherent configurations

“…We refer to two weights as switching equivalent if one is obtained from the other by some sequence of switches, and observe that δω is invariant under switching. This is one of a number of natural generalizations of the regular two-graph ( [9,16,22,23,24,25]). Since a 2-cochain is equivalent to a weight on the edges of a complete graph, the notion of regularity can be extended to weights on CAs.…”

“…It is not necessarily distance regular. This case encompasses the regular two-graphs (t = 2), and the regular 3-graphs (t = 3) of Higman [9] and Kalmanovich [16].…”