Pentavalent symmetric graphs of order twice a prime power

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.