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The classification of two-distance transitive dihedrants

Jun-Jie Huang, Yan-Quan Feng, Jin-Xin Zhou, Fu-Gang Yin

Abstract

A vertex transitive graph $Γ$ is said to be $2$-distance transitive if for each vertex $u$, the group of automorphisms of $Γ$ fixing the vertex $u$ acts transitively on the set of vertices at distance $1$ and $2$ from $u$, while $Γ$ is said to be $2$-arc transitive if its automorphism group is transitive on the set of $2$-arcs. Then $2$-arc transitive graphs are $2$-distance transitive. The classification of $2$-arc transitive Cayley graphs on dihedral groups was given by Du, Malnič and Marušič in [Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349--1372]. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order $2n$ is either $2$-arc transitive, or isomorphic to the complete multipartite graph $K_{m[b]}$ for some $m\geq3$ and $b\geq2$ with $mb=2n$.

The classification of two-distance transitive dihedrants

Abstract

A vertex transitive graph is said to be -distance transitive if for each vertex , the group of automorphisms of fixing the vertex acts transitively on the set of vertices at distance and from , while is said to be -arc transitive if its automorphism group is transitive on the set of -arcs. Then -arc transitive graphs are -distance transitive. The classification of -arc transitive Cayley graphs on dihedral groups was given by Du, Malnič and Marušič in [Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349--1372]. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order is either -arc transitive, or isomorphic to the complete multipartite graph for some and with .
Paper Structure (6 sections, 19 theorems, 9 equations, 2 tables)

This paper contains 6 sections, 19 theorems, 9 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group theoretical results
  4. Graph theoretical results
  5. $2$-distance transitive covers of graphs
  6. Proof of Theorem \ref{['Thm-1']}

Key Result

Theorem 1.1

Let $\mathrm{\Gamma}$ be a connected Cayley graph on the dihedral group $\mathrm{D}_{2n}$ of order $2n$ with $n\geq2$. Then $\mathrm{\Gamma}$ is $2$-distance transitive if and only if either $\mathrm{\Gamma}$ is a $2$-arc transitive graph, or $\mathrm{\Gamma}\cong\mathsf{K}_{m[b]}$ for some integers

Theorems & Definitions (20)

  • Conjecture
  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8: DGLP12
  • ...and 10 more