Classification of cubic symmetric tetracirculants and pentacirculants

A graph X is said to be an m -Cayley graph on a group G ( | G | ≠ 1 ) if its automorphism group contains a semiregular subgroup isomorphic to G having m orbits on the vertex set of X . If G is cyclic and m = 1 , 2 , 3 , 4 , or 5 then X is said to be a circulant, a bicirculant, a tricirculant, a tetracirculant, or a pentacirculant, respectively. h is said to be symmetric if its automorphism group acts transitively on the set of its arcs. All cubic symmetric circulants, bicirculants and tricirculants are known, and in this paper we give complete classifications of cubic symmetric tetracirculants and pentacirculants. In particular, it is shown that there are infinitely many connected cubic symmetric tetracirculants whereas there are only two connected cubic symmetric pentacirculants.