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 2pn for an odd prime p, we show that if p a‰? 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p a‰?3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s a?’ 1)-regular subgroup for each 1 a‰| s a‰| 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1a‰| sa‰| 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.
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