### 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.

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PDFDOI: http://dx.doi.org/10.12958/adm1343

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