Metacirculants and split weak metacirculants

TL;DR

This paper explores the properties and existence conditions of split weak metacirculants, a generalization of metacirculants, providing new criteria and constructions, especially for graphs of composite order, and addressing an open question.

Contribution

It establishes new sufficient and necessary conditions for the existence of split weak metacirculants that are not metacirculants, including for composite orders, and constructs infinite examples.

Findings

Provided a sufficient condition for split weak metacirculants of order m not being metacirculants.

Derived a necessary and sufficient condition for split weak metacirculants of order n (product of two prime powers) not being metacirculants.

Constructed infinitely many split weak metacirculant graphs that are not metacirculants.

Abstract

Metacirculants are a rich resource of many families of interesting graphs, and weak metacirculants are generalizations of them. A graph is called a {\em split weak metacirculant} if it has a vertex-transitive split metacyclic automorphism group. In two recent papers, it is shown that a graph of prime power order is a metacirculant if and only if it is a split weak metacirculant. Let $m$ is a positive integer. In this paper, we first give a sufficient condition for the existence of split weak metacirculants of order $m$ which are not metacirculants. This is then used to give a sufficient and necessary condition for the existence of split weak metacirculants of order $n$ which are not metacirculants, where $n$ is a product of two prime-powers. As byproducts, we construct infinitely many split weak metacirculant graphs which are not metacirculant graphs, and answer an open question reported in the literature.