On small world non-Sunada twins and cellular Voronoi diagrams

Abstract

Special infinite families of regular graphs of unbounded degree and of bounded diameter (small world graphs) are considered. Two families of small world graphs \(G_i\) and \(H_i\) form a family of non-Sunada twins if \(G_i\) and \(H_i\) are isospectral of bounded diameter but groups \(Aut(G_i)\) and \(Aut(H_i)\) are nonisomorphic.

We say that a family of non-Sunada twins is unbalanced if each \(G_i\) is edge-transitive but each \(H_i\) is edge-intransitive. If all \(G_i\) and \(H_i\) are edge-transitive we have a balanced family of small world non-Sunada twins. We say that a family of non-Sunada twins is strongly unbalanced if each \(G_i\) is edge-transitive but each \(H_i\) is edge-intransitive.

We use term edge disbalanced for the family of non-Sunada twins such that all graphs \(G_i\) and \(H_i\) are edge-intransitive. We present explicit constructions of the above defined families. Two new families of distance-regular—but not distance-transitive—graphs will be introduced.