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

A graph $\mathrm{\Gamma }$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\mathrm{\Gamma }$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\mathrm{\Gamma }$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{\Gamma })$ of $\mathrm{\Gamma }$ when $G$ is a Sylow $p$-subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{\Gamma })$. 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 $p$.