Pentavalent symmetric graphs of order twice a prime power

A connected symmetric graph of prime valency is basic   if its automorphism group contains no nontrivial normal subgroup having more than two orbits. Let pp be a prime and nn a positive integer. In this paper, we investigate properties of connected pentavalent symmetric graphs of order 2pn2pn, and it is shown that a connected pentavalent symmetric graph of order 2pn2pn is basic if and only if it is either a graph of order 66, 1616, 250250, or a graph of three infinite families of Cayley graphs on generalized dihedral groups—one family has order 2p2p with p=5p=5 or 5∣(p−1)5∣(p−1), one family has order 2p22p2 with 5∣(p±1)5∣(p±1), and the other family has order 2p42p4. Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2pn2pn have been done.It is shown that basic graphs of connected pentavalent symmetric graphs of order 2pn2pn are symmetric elementary abelian covers of the dipole Dip5Dip5, and with covering techniques, uniqueness and automorphism groups of these basic graphs are determined. Moreover, symmetric Zpn-covers of the dipole Dip5Dip5 are classified. As a byproduct, connected pentavalent symmetric graphs of order 2p22p2 are classified.