Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On arc-transitive inner-automorphic Cayley graphs on dihedral groups

Jun-Jie Huang, Jin-Hua Xie

Abstract

A Cayley graph $\Cay(G,S)$ is said to be inner-automorphic if $S$ is a union of conjugacy classes of a group $G$, and arc-transitive if its full automorphism group acts transitively on the set of arcs. In this paper, we characterize four well-known families of arc-transitive graphs that arise as connected inner-automorphic Cayley graphs on dihedral groups, and we provide a necessary condition for other connected arc-transitive Cayley graphs on dihedral groups to be inner-automorphic. We further construct an infinite family of examples satisfying this condition, thereby demonstrating the existence of such graphs. Finally, we complete the classification of all 2-distance-transitive connected inner-automorphic Cayley graphs on dihedral groups.

On arc-transitive inner-automorphic Cayley graphs on dihedral groups

Abstract

A Cayley graph is said to be inner-automorphic if is a union of conjugacy classes of a group , and arc-transitive if its full automorphism group acts transitively on the set of arcs. In this paper, we characterize four well-known families of arc-transitive graphs that arise as connected inner-automorphic Cayley graphs on dihedral groups, and we provide a necessary condition for other connected arc-transitive Cayley graphs on dihedral groups to be inner-automorphic. We further construct an infinite family of examples satisfying this condition, thereby demonstrating the existence of such graphs. Finally, we complete the classification of all 2-distance-transitive connected inner-automorphic Cayley graphs on dihedral groups.

Paper Structure

This paper contains 4 sections, 19 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['Thm:dihedrant']}
  4. Construction of arc-transitive inner-automorphic Cayley graphs

Key Result

Theorem 1.1

Let $\mathrm{\Gamma}=\mathrm{Cay}(G,S)$ be a connected arc-transitive inner-automorphic Cayley graph, where $G=\langle a,b\mid a^n=b^2=1,a^b=a^{-1}\rangle\cong\mathrm{D}_{2n}$ with $n\geq 2$. Then one of the following holds: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • ...and 11 more