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’={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.