Abstract
A graph is a bi-Cayley graph over a group if the group acts semiregularly on the vertex set of the graph with two orbits. Let G be a non-abelian metacyclic p -group for an odd prime p . In this paper, we prove that if G is a Sylow p -subgroup in the full automorphism group Aut(Γ) of a graph Γ , then G is normal in Aut(Γ) . As an application, we classify the half-arc-transitive bipartite bi-Cayley graphs over G of valency less than 2 p , while the case for valency 4 was given by Zhang and Zhou in 2019. It is further shown that there are no semisymmetric or arc-transitive bipartite bi-Cayley graphs over G of valency less than p .
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