Weak metacirculants of odd prime power order

Metacirculants are a basic and well-studied family of vertex-transitive graphs, and weak metacirculants are generalizations of them. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. This paper is devoted to the study of weak metacirculants with odd prime power order. We first prove that a weak metacirculant of odd prime power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. We then prove that for any odd prime p and integer l ≥ 4, there exist weak metacirculants of order p which are Cayley graphs but not Cayley graphs of any metacyclic group; this answers a question in [C. H. Li, S. J. Song and D. J. Wang, A characterization of metacirculants, J. Combin. Theory Ser. A 120 (2013) 39–48]. We construct such graphs explicitly by introducing a construction which is a generalization of generalized Petersen graphs. Finally, we determine all smallest possible metacirculants of odd prime power order which are Cayley graphs but not Cayley graphs of any metacyclic group.