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2-Geodesic-transitive graphs of order twice a prime power

Jiangmin Pan, Cixuan Wu, Yingnan Zhang, Hanlin Zou

Abstract

In this paper, we study 2-geodesic-transitive graphs of order twice an odd prime power. Classifications of corresponding basic graphs and such graphs with almost simple automorphism groups are given, and a reduction theorem for general case is obtained. Certain new 2-geodesic-transitive graphs are found, and a Magma code regarding 2-geodesic-transitive graphs is provided.

2-Geodesic-transitive graphs of order twice a prime power

Abstract

In this paper, we study 2-geodesic-transitive graphs of order twice an odd prime power. Classifications of corresponding basic graphs and such graphs with almost simple automorphism groups are given, and a reduction theorem for general case is obtained. Certain new 2-geodesic-transitive graphs are found, and a Magma code regarding 2-geodesic-transitive graphs is provided.

Paper Structure

This paper contains 7 sections, 13 theorems, 5 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Examples of $2$-geodesic-transitive graphs
  4. Background results
  5. Basic $2$-geodesic-transitive graphs of order $2p^n$
  6. Proofs of Theorems \ref{['Thm']} and \ref{['Thm-2']}
  7. Magma code regarding $2$-geodesic-transitive graphs

Key Result

Theorem 1.1

Let ${\it \Gamma}$ be a connected $(G,2)$-geodesic-transitive graph of order $2p^n$ with $p$ an odd prime and $n$ a positive integer. If $G$ is quasiprimitive or biquasiprimitive on $V{\it \Gamma}$, then either $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Proposition 2.7
  • ...and 6 more