Weak metacirculants of odd prime power order

TL;DR

This paper investigates weak metacirculant graphs of odd prime power order, characterizing when they are metacirculants and constructing examples that are Cayley graphs but not of metacyclic groups, answering an open question.

Contribution

It provides a characterization of weak metacirculants of odd prime power order and constructs explicit examples that resolve a previously open problem.

Findings

A weak metacirculant of odd prime power order is a metacirculant iff it has a vertex-transitive split metacyclic automorphism group.

Existence of weak metacirculants of order p^l that are Cayley graphs but not of any metacyclic group for l ≥ 4.

Identification of the smallest such metacirculants that are Cayley graphs but not of metacyclic groups.

Abstract

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 $\ell\geq 4$, there exist weak metacirculants of order $p^\ell$ which are Cayley graphs but not Cayley graphs of any metacyclic group; this answers a question in Li et al. (2013). 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.