The investigation of s-arc-transitivity can be dated back to 1947. Tutte [Reference Tutte7] 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 [Reference Weiss8] 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 [Reference Praeger6] 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 [Reference Praeger6] asked if there exists any vertex-primitive
$2$
-arc-transitive digraph. This question was answered in [Reference Giudici, Li and Xia2, Reference Giudici and Xia4] by constructing infinite families of G-vertex-primitive
$(G,2)$
-arc-transitive digraphs such that G has AS and SD type, respectively. In [Reference Giudici and Xia4], 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
$s\leqslant 2$
. At the same time, Giudici and Xia [Reference Giudici and Xia4] 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. [Reference Giudici, Li and Xia3] showed that
$s\leqslant 2$
when the socle of G is a projective special linear group, Pan et al. [Reference Pan, Wu and Yin5] proved that
$s\leqslant 2$
when the socle of G is an alternating group except for one subcase and Chen et al. [Reference Chen, Giudici and Praeger1] 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 [Reference Chen, Giudici and Praeger1] 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
$\mathrm {Soc}(G)=\mathrm {PSp}_{2n}(q)'$
,
$\mathrm {PSU}_{n}(q)$
(for certain cases),
$\mathrm {Sz}(q)$
,
$\mathrm {Ree}(q)$
,
${}^{2}\mathrm {F}_{4}(q)$
,
${}^{3}\mathrm {D}_{4}(q)$
and
$\mathrm {G}_{2}(q)$
. It turns out that such an upper bound is
$s\leqslant 2$
for all the groups mentioned above, giving some evidence to the conjecture that
$s\leqslant 2$
.
