Generalizing Praeger--Xu graphs

We uniformly study directed Praeger--Xu graphs and Praeger's generalized family PX(Γ,r,s), without requiring that s r and $Γ$ is an undirected graph. Assuming that Γ×K₂ is connected, we determine the full automorphism group of PX(Γ,r,s) for every r and s, prove connectivity, characterize vertex-transitivity, and give an equivalent criterion for t-arc-transitivity. We also study when PX(Γ,r,s) is a Cayley graph. We also give two applications. First, we give a new family of twin-free connected arc-transitive digraphs of valency $4$ whose vertex stabilizers have exponential growth. We also give a connected regular digraph that is not vertex-transitive but whose direct product with a directed cycle is Cayley.