On Minimal Distance-Regular Cayley Graphs of Generalized Dihedral Groups

Štefko Miklavič,Primož Šparl

doi:10.37236/9755

Abstract

Let $G$ denote a finite generalized dihedral group with identity $1$ and let $S$ denote an inverse-closed subset of $G \setminus \{1\}$, which generates $G$ and for which there exists $s \in S$, such that $\langle S \setminus \{s,s^{-1}\} \rangle \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\mathrm{Cay}(G;S)$ for such pairs of $G$ and $S$.

Highlights

Two of the most extensively studied phenomena in graph theory are regularity and symmetry properties of graphs
While it is usually quite easy to see that a certain degree of symmetry that a graph possesses implies certain degree of regularity, the opposite question seems to be much harder to handle
The graph K6,6−6K2 can be obtained as the Cayley graph of the dihedral group GD( a ), where |a| = 6, with respect to the connection set S = {a±1, t, ta2, ta4}

Summary

Introduction

Two of the most extensively studied phenomena in graph theory are regularity and symmetry properties of graphs. In [12] the authors obtained a complete classification of distance-regular graphs, which are Cayley over an abelian group and the corresponding connection set S is minimal. In this paper the work from [12] is extended by classifying distance-regular graphs, which are Cayley over a generalized dihedral group, and for which the corresponding connection set S is minimal.

Preliminaries

The graphs from the main theorem

Distance-regular Cartesian products