48 research outputs found
Bounding the size of a vertex-stabiliser in a finite vertex-transitive graph
In this paper we discuss a method for bounding the size of the stabiliser of
a vertex in a -vertex-transitive graph . In the main result the
group is quasiprimitive or biquasiprimitive on the vertices of ,
and we obtain a genuine reduction to the case where is a nonabelian simple
group.
Using normal quotient techniques developed by the first author, the main
theorem applies to general -vertex-transitive graphs which are -locally
primitive (respectively, -locally quasiprimitive), that is, the stabiliser
of a vertex acts primitively (respectively
quasiprimitively) on the set of vertices adjacent to . We discuss how
our results may be used to investigate conjectures by Richard Weiss (in 1978)
and the first author (in 1998) that the order of is bounded above by
some function depending only on the valency of , when is
-locally primitive or -locally quasiprimitive, respectively
Locally -distance transitive graphs
We give a unified approach to analysing, for each positive integer , a
class of finite connected graphs that contains all the distance transitive
graphs as well as the locally -arc transitive graphs of diameter at least
. A graph is in the class if it is connected and if, for each vertex ,
the subgroup of automorphisms fixing acts transitively on the set of
vertices at distance from , for each from 1 to . We prove that
this class is closed under forming normal quotients. Several graphs in the
class are designated as degenerate, and a nondegenerate graph in the class is
called basic if all its nontrivial normal quotients are degenerate. We prove
that, for , a nondegenerate, nonbasic graph in the class is either a
complete multipartite graph, or a normal cover of a basic graph. We prove
further that, apart from the complete bipartite graphs, each basic graph admits
a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a
biquasiprimitive action. These results invite detailed additional analysis of
the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types
Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups
In this paper we study -arc-transitive graphs where the
permutation group induced by the stabiliser of the
vertex on the neighbourhood satisfies the two conditions given
in the introduction. We show that for such a -arc-transitive graph ,
if is an arc of , then the subgroup of
fixing pointwise and is a -group for some prime .
Next we prove that every -locally primitive (respectively quasiprimitive,
semiprimitive) graph satisfies our two local hypotheses. Thus this provides a
new Thompson-Wielandt-like theorem for a very large class of arc-transitive
graphs.
Furthermore, we give various families of -arc-transitive graphs where our
two local conditions do not apply and where has arbitrarily
large composition factors