Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A family of symmetric graphs in relation to 2-point-transitive linear spaces

Teng Fang, Sanming Zhou, Shenglin Zhou

Abstract

A graph $Γ$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $Γ$, where an arc is an ordered pair of adjacent vertices. Let $Γ$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition ${\cal B}$, and let ${\cal D}(Γ, {\cal B})$ be the incidence structure with point set ${\cal B}$ and blocks $\{B\} \cup Γ_{\cal B}(α)$, for $B \in {\cal B}$ and $α\in B$, where $Γ_{\cal B}(α)$ is the set of blocks of ${\cal B}$ containing at least one neighbour of $α$ in $Γ$. In this paper we classify all $G$-symmetric graphs $Γ$ such that $Γ_{\cal B}(α) \ne Γ_{\cal B}(β)$ for distinct $α, β\in B$, the quotient graph of $Γ$ with respect to ${\cal B}$ is a complete graph, and ${\cal D}(Γ, {\cal B})$ is isomorphic to the complement of a $(G, 2)$-point-transitive linear space.

A family of symmetric graphs in relation to 2-point-transitive linear spaces

Abstract

A graph is -symmetric if it admits as a group of automorphisms acting transitively on the set of arcs of , where an arc is an ordered pair of adjacent vertices. Let be a -symmetric graph such that its vertex set admits a nontrivial -invariant partition , and let be the incidence structure with point set and blocks , for and , where is the set of blocks of containing at least one neighbour of in . In this paper we classify all -symmetric graphs such that for distinct , the quotient graph of with respect to is a complete graph, and is isomorphic to the complement of a -point-transitive linear space.
Paper Structure (12 sections, 5 theorems, 22 equations, 1 table)

This paper contains 12 sections, 5 theorems, 22 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminology
  4. Flag graphs
  5. Necessary conditions
  6. Proof of Theorem \ref{['thm:main theorem']}: Part (a)
  7. Doubly point-transitive proper linear spaces
  8. Almost simple case
  9. Affine case
  10. Proof of Theorem \ref{['thm:main theorem']}: Part (b)
  11. Almost simple case
  12. Affine case

Key Result

Theorem 1.2

Let $\Gamma$ be a $G$-symmetric graph such that $V(\Gamma)$ admits a nontrivial $G$-invariant partition $\mathcal{B}$, where $G \le {\rm Aut}(\Gamma)$. Suppose that $\Gamma_{\mathcal{B}}$ is a complete graph, $\mathcal{D}^{*}(B)$ contains no repeated blocks, and $\mathcal{D}(\Gamma, \mathcal{B})$ is

Theorems & Definitions (7)

  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Definition 4.1