Using Schur Rings to Produce GRRs for Dihedral Groups

Abstract

In this paper we shall be looking at several results relating Schur rings to sufficient conditions for a graph to be a graphical regular representation (GRR) of a finite group, and then applying these specifically in the case of certain subfamilies of dihedral groups. Numerical methods are given for constructing trivalent GRRs for these dihedral groups very quickly.

Figures (4)

• Figure 1: The basic Cayley graphs associated with the Schur ring in Example \ref{['exm:basiccayley']}
• Figure 2: The colour graph for the Schur ring in Example \ref{['exm:basiccayley']}
• Figure 3: $Cay(D_{11}, \{ab, ab^3, ab^4\})$, a GRR of $D_{11}$.
• Figure 4: $Cay(D_{13}, \{ab, ab^3, ab^4\})$, a GRR of $D_{13}$.

Theorems & Definitions (25)

• Definition 1.1.1: Graphical Regular Representations
• Definition 1.1.2: Cayley graphs
• Definition 1.2.1: Group Rings
• Definition 1.2.2: Schur Rings
• Example 1.2.1
• Definition 1.2.3: Structure Constants
• Example 1.2.2
• Definition 1.3.1: Basic Cayley Graphs
• Definition 1.5.1: Automorphisms of Schur Rings
• Theorem 1.5.1