996 research outputs found
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
Enumerating planar locally finite Cayley graphs
We characterize the set of planar locally finite Cayley graphs, and give a
finite representation of these graphs by a special kind of finite state
automata called labeling schemes. As a result, we are able to enumerate and
describe all planar locally finite Cayley graphs of a given degree. This
analysis allows us to solve the problem of decision of the locally finite
planarity for a word-problem-decidable presentation.
Keywords: vertex-transitive, Cayley graph, planar graph, tiling, labeling
schemeComment: 19 pages, 6 PostScript figures, 12 embedded PsTricks figures. An
additional file (~ 438ko.) containing the figures in appendix might be found
at http://www.labri.fr/Perso/~renault/research/pages.ps.g
The vertex-transitive TLF-planar graphs
We consider the class of the topologically locally finite (in short TLF)
planar vertex-transitive graphs, a class containing in particular all the
one-ended planar Cayley graphs and the normal transitive tilings. We
characterize these graphs with a finite local representation and a special kind
of finite state automaton named labeling scheme. As a result, we are able to
enumerate and describe all TLF-planar vertex-transitive graphs of any given
degree. Also, we are able decide to whether any TLF-planar transitive graph is
Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures
Submitted to Discrete Mathematics The appendix is accessible at
http://www.labri.fr/~renault/research/research.htm
Abelian Carter subgroups in finite permutation groups
We show that a finite permutation group containing a regular abelian
self-normalizing subgroup is soluble.Comment: 6 page
Graphs, permutations and topological groups
Various connections between the theory of permutation groups and the theory
of topological groups are described. These connections are applied in
permutation group theory and in the structure theory of topological groups.
The first draft of these notes was written for lectures at the conference
Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has
been corrected
Characterizing a vertex-transitive graph by a large ball
It is well-known that a complete Riemannian manifold M which is locally
isometric to a symmetric space is covered by a symmetric space. Here we prove
that a discrete version of this property (called local to global rigidity)
holds for a large class of vertex-transitive graphs, including Cayley graphs of
torsion-free lattices in simple Lie groups, and Cayley graph of torsion-free
virtually nilpotent groups. By contrast, we exhibit various examples of Cayley
graphs of finitely presented groups (e.g. SL(4,Z)) which fail to have this
property, answering a question of Benjamini, Ellis, and Georgakopoulos.
Answering a question of Cornulier, we also construct a continuum of non
pairwise isometric large-scale simply connected locally finite
vertex-transitive graphs. This question was motivated by the fact that
large-scale simply connected Cayley graphs are precisely Cayley graphs of
finitely presented groups and therefore have countably many isometric classes.Comment: v1: 38 pages. With an Appendix by Jean-Claude Sikorav v2: 48 pages.
Several improvements in the presentation. To appear in Journal of Topolog
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