VERTEX-PRIMITIVE <i>s</i>-ARC-TRANSITIVE DIGRAPHS OF ALMOST SIMPLE GROUPS

The investigation of s-arc-transitivity can be dated back to 1947.Tutte [7] studied cubic graphs and showed that a cubic graph can be at most 5-arc-transitive.A more general result for s-arc-transitivity of graphs was obtained by Weiss [8] and it turns out that finite undirected graphs of valency at least 3 that are not cycles can be at most 7-arc-transitive.In stark contrast with the situation in undirected graphs, Praeger [6] showed that for each s and d, there are infinitely many finite s-arc-transitive digraphs of valency d that are not (s + 1)-arc-transitive.However, once we add the condition of primitivity, the situation is quite different.Given the lack of evidence of the existence of vertex-primitive 2-arc-transitive digraphs, Praeger [6] asked if there exists any vertex-primitive 2-arc-transitive digraph.This question was answered in [2,4] by constructing infinite families of G-vertex-primitive (G, 2)-arc-transitive digraphs such that G has AS and SD type, respectively.In [4], Giudici and Xia then asked for the upper bound on s for a G-vertex-primitive (G, s)-arc-transitive digraph that is not a directed cycle.A reasonable conjecture is that s 2. At the same time, Giudici and Xia [4] showed that to answer that question, it suffices for us to consider the case when G is almost simple.Various attempts have been made to analyse the s-arc-transitivity of different almost simple groups.For instance, Giudici et al. [3] showed that s 2 when the socle of G is a projective special linear group, Pan et al. [5] proved that s 2 when the socle of G is an alternating group except for one subcase and Chen et al. [1] addressed the case when the socle of G is a Suzuki group or a small Ree group, when it turns out that the upper bound on s is 1.The result from [1] is part of Chapter 4.In this thesis, we investigate the upper bound on s for G-vertex-primitive (G, s)-arc-transitive digraphs for almost simple groups G with Soc(G) = PSp 2n (q) ,