Schur rings : some theory and applications to graphical regular representations

Abstract

A Schur ring is a specific type of group ring constructed via the partitioning of a group while the Graphical Regular Representation (GRR) of a group 􀀀 is a graph which has an automorphism group isomorphic to 􀀀 which acts regularly on it. In this dissertation we will be looking at some background theory related to these two concepts and use it to produce an original theorem which allows us to quickly produce trivalent GRRs for all dihedral groups Dp where p is a prime number greater than 5. The background theory we will be considering is mostly work done by other authors over recent years although some new proofs by the author are included, especially in situations where we will be looking at special cases of established work and so prefer a proof specific. to the special cases being considered. The original theorem which we will be building up to is the following: If p is a prime number greater than 5 and 3r 􀀀 2s = t mod p then Cay(Dp; fabr; abs; abtg) is a GRR of Dp. The use of Schur rings will be central to proving this theorem. We will also see a few examples of GRRs produced using this theorem as well as construct a table of connecting sets which admit GRRs for all dihedral groups Dp with p prime and greater than a certain value.