Metacirculants are a basic and well-studied family of vertex-transitivegraphs, and weak metacirculants are generalizations of them. A graph is calleda weak metacirculant if it has a vertex-transitive metacyclic automorphismgroup. This paper is devoted to the study of weak metacirculants with odd primepower order. We first prove that a weak metacirculant of odd prime power orderis a metacirculant if and only if it has a vertex-transitive split metacyclicautomorphism group. We then prove that for any odd prime $p$ and integer$\ell\geq 4$, there exist weak metacirculants of order $p^\ell$ which areCayley graphs but not Cayley graphs of any metacyclic group; this answers aquestion in Li et al. (2013). We construct such graphs explicitly byintroducing a construction which is a generalization of generalized Petersengraphs. Finally, we determine all smallest possible metacirculants of odd primepower order which are Cayley graphs but not Cayley graphs of any metacyclicgroup.
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机译：元循环量是顶点-传递图的基础知识和研究充分的族，而弱元循环量是它们的概括。如果图具有顶点可传递的元环自同构群，则该图称为弱元循环。本文致力于研究具有奇数幂次阶的弱元循环体。我们首先证明，当且仅当它具有顶点可传递的分裂亚环自同构群时，奇质数次幂阶的弱亚循环才是亚循环。然后我们证明，对于任何奇数素数$ p $和integer $ \ ell \ geq 4 $，都存在阶次为$ p ^ \ ell $的弱元循环体，它们是Cayley图而不是任何元环群的Cayley图；这回答了李等人的质疑。 （2013）。我们通过引入一种构造来显式构造此类图，该构造是广义Petersengraph的概括。最后，我们确定奇数次幂阶的所有最小可能的元环，它们是Cayley图，而不是任何元环群的Cayley图。
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