63 research outputs found
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
Basic and degenerate pregeometries
We study pairs , where is a 'Buekenhout-Tits'
pregeometry with all rank 2 truncations connected, and is transitive on the set of elements of each type. The family of such
pairs is closed under forming quotients with respect to -invariant
type-refining partitions of the element set of . We identify the
'basic' pairs (those that admit no non-degenerate quotients), and show, by
studying quotients and direct decompositions, that the study of basic
pregeometries reduces to examining those where the group is faithful and
primitive on the set of elements of each type. We also study the special case
of normal quotients, where we take quotients with respect to the orbits of a
normal subgroup of . There is a similar reduction for normal-basic
pregeometries to those where is faithful and quasiprimitive on the set of
elements of each type
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
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
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