Finite $s$-Geodesic-Transitive Digraphs - Jin, Wei (The Electronic Journal of Combinatorics, 2024)
This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s≥ 2$. We first give a global analysis by providing a reduction result. Let $Γ$ be such a digraph and let $N$ be a normal subgroup of $G$ maximal with respect to having at least $3$ orbits. Then the quotient digraph $Γ_N$ is $(G/N,s')$-geodesic-transitive where $s'=\min\{s,diam(Γ_N)\}$, $G/N$ is either quasiprimitive or bi quasiprimitive on $V(Γ_N)$, and $Γ_N$ is either directed or an undirected complete graph. Moreover, it is further shown that if $Γ $ is not $(G,2)$-arc-transitive, then $G/N$ is quasiprimitive on $V(Γ_N)$.On the other hand, we also consider the case that the normal subgroup $N$ of $G$ has one orbit on the vertex set. We show that if $N$ is regular on $V(Γ)$, then $Γ$ is a circuit, and particularly each $(G,s)$-geodesic-transitive normal Cayley digraph with $s≥ 2$, is a circuit.Finally, we investigate $(G,2)$-geodesic-transitive digraphs with either valency at most 5 or diameter at most 2. Let $Γ$ be a $(G,2)$-geodesic-transitive digraph. It is proved that: if $Γ$ has valency at most $5$, then $Γ$ is $(G,2)$-arc-transitive; if $Γ$ has diameter $2$, then $Γ$ is a balanced incomplete block design with the Hadamard parameters.