Generalizing Praeger--Xu graphs

We uniformly study directed Praeger--Xu graphs and Praeger's generalized family $\overrightarrow{\operatorname{PX}}(Γ,r,s)$, without requiring that $s\mid r$ and $Γ$ is an undirected graph. Assuming that $Γ\times\overrightarrow{K}_2$ is connected, we determine the full automorphism group of $\overrightarrow{\operatorname{PX}}(Γ,r,s)$ for every $r$ and $s$, prove connectivity, characterize vertex-transitivity, and give an equivalent criterion for $t$-arc-transitivity. We also study when $\overrightarrow{\operatorname{PX}}(Γ,r,s)$ 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.