Cubic symmetric graphs of order twice an odd prime-power

Abstract

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 2 p n for an odd prime p , we show that if p ≠ 5, 7 then every Sylow p -subgroup of the full automorphism group Aut( X ) of X is normal, and if p ≠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 2 p n is a Cayley graph if p > 5 and we classify the s -regular cubic graphs of order 2 p 2 for each 1≦ s≦ 5 and each prime p . as a continuation of the authors' classification of 1-regular cubic graphs of order 2 p 2 . The same classification of those of order 2 p is also done.