Routing in a Class of Cayley Graphs of Semidirect Products of Finite Groups

Abstract

Recently, Draper initiated the study of interconnection networks based on Cayley graphs of semidirect products of two cyclic groups called supertoroids. Interest in this class of graphs stems from their relatively smaller diameter compared to toroids of the same size. The Borel graphs introduced by Arden and Tang are a family of Cayley graphs based on a special class of matrix groups. In this paper, we describe a deterministic, distributed routing scheme for supertoroids. While we do not have a proof of correctness of our scheme, experimental evidence leads to a natural conjecture that our scheme is a shortest path routing algorithm. By proving the similarities among supertoroids, Borel graphs, and metacyclic graphs, this routing scheme is then extended to Borel graphs.