Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order

TL;DR

This paper studies bipartite bi-Cayley graphs over non-abelian metacyclic p-groups, proving normality of the group in automorphisms and classifying certain symmetric graphs with small valency.

Contribution

It establishes the normality of the group in the automorphism group and classifies half-arc-transitive bipartite bi-Cayley graphs over these groups with small valency.

Findings

G is normal in Aut(Γ) when G is a Sylow p-subgroup.

Classifies half-arc-transitive bipartite bi-Cayley graphs with valency less than 2p.

No semisymmetric or arc-transitive bipartite bi-Cayley graphs with valency less than p.

Abstract

A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(\Gamma)$ of $\Gamma$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(\Gamma)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $p$.