104 research outputs found
On finite edge-primitive and edge-quasiprimitive graphs
Many famous graphs are edge-primitive, for example, the Heawood graph, the
Tutte--Coxeter graph and the Higman--Sims graph. In this paper we
systematically analyse edge-primitive and edge-quasiprimitive graphs via the
O'Nan--Scott Theorem to determine the possible edge and vertex actions of such
graphs. Many interesting examples are given and we also determine all
-edge-primitive graphs for an almost simple group with socle .Comment: 30 pages To appear in Journal of Combinatorial Theory Series
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
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
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