Cubic symmetric graphs of order twice an odd prime-power

Abstract

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2p(n) for an odd prime p, we show that if p not equal 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p not equal 3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s-1)-regular subgroup for each 1 <= s <= 5. As an application, we show that every connected cubic symmetric graph of order 2p(n) is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p(2) for each 1 <= s <= 5 and each prime p, as a continuation of the authors&apos; classification of 1-regular cubic graphs of order 2p(2). The same classification of those of order 2p is also done.