A connected symmetric graph of prime valency is 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 2 p n , and it is shown that a connected pentavalent symmetric graph of order 2 p 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 2 p with p = 5 or 5 ź ( p - 1 ) , one family has order 2 p 2 with 5 ź ( p ź 1 ) , and the other family has order 2 p 4 . Furthermore, the automorphism groups of these basic graphs are computed. Similar works on cubic and tetravalent symmetric graphs of order 2 p n have been done.It is shown that basic graphs of connected pentavalent symmetric graphs of order 2 p 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 Z p n -covers of the dipole Dip 5 are classified. As a byproduct, connected pentavalent symmetric graphs of order 2 p 2 are classified.
['Yan-Quan Feng', 'Jin-Xin Zhou', 'Yan-Tao Li']
Pentavalent symmetric graphs of order twice a prime power