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