Pentavalent symmetric graphs of order twice a prime power

TL;DR

This paper classifies connected pentavalent symmetric graphs of order twice a prime power, identifying basic graphs and their automorphism groups, and explores their structure as covers of a dipole graph.

Contribution

It provides a complete classification of basic connected pentavalent symmetric graphs of order 2p^n and determines their automorphism groups and covering structures.

Findings

Basic graphs are either of order 6, 16, 250, or belong to three infinite Cayley graph families.

Automorphism groups of these graphs are explicitly computed.

Connected pentavalent symmetric graphs of order 2p^2 are classified.

Abstract

A connected symmetric graph of prime valency is {\em basic} if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let $p$ be a prime and $n$ a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order $2p^n$, and it is shown that a connected pentavalent symmetric graph of order $2p^n$ is basic if and only if it is either a graph of order $6$, $16$, $250$, or a graph of three infinite families of Cayley graphs on generalized dihedral groups -- one family has order $2p$ with $p=5$ or $5 \mid (p-1)$, one family has order $2p^2$ with $5 \mid (p\pm 1)$, and the other family has order $2p^4$. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order $2p^n$ have been done. It is shown that basic graphs of connected pentavalent symmetric graphs of order $2p^n$ are symmetric elementary abelian covers of the dipole $\Dip_5$, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric $\mz_p^n$-covers of the dipole $\Dip_5$ are classified. As a byproduct, connected pentavalent symmetric graphs of order $2p^2$ are classified.