Metacirculants were introduced by Alspach and Parsons in 1982 and have been a rich source of various topics since then, including the Hamiltonian path problem in metacirculants. A metacirculant has a vertex-transitive metacyclic subgroup of automorphisms, and a long-standing interesting question in the area is if the converse statement is true, namely, whether a graph with a vertex-transitive metacyclic automorphism group is a metacirculant. We shall answer this question in the negative, and then present a classification of cubic metacirculants.
We study s-arc-transitive graphs with s 2, and give a characterisation of the actions of vertex-transitive normal subgroups. An interesting consequence of this characterisation states that each non-bipartite 3-arc-transitive graph is a normal cover of a 2-arc-transitive graph admitting a simple group, or a locally primitive graph admitting a simple group with a soluble vertex stabiliser.
We prove that the automorphism group of a self-complementary metacirculant is either soluble or has A 5 as the only insoluble composition factor, extending a result of Li and Praeger which says the automorphism group of a self-complementary circulant is soluble. The proof involves a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. To the best of our knowledge, these are the first examples of self-complementary graphs with this property.
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