616 research outputs found
Infinite primitive and distance transitive directed graphs of finite out-valency
We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph
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
Subdegree growth rates of infinite primitive permutation groups
A transitive group of permutations of a set is primitive if the
only -invariant equivalence relations on are the trivial and
universal relations.
If , then the orbits of the stabiliser on
are called the -suborbits of ; when acts transitively
the cardinalities of these -suborbits are the subdegrees of .
If acts primitively on an infinite set , and all the suborbits of
are finite, Adeleke and Neumann asked if, after enumerating the subdegrees
of as a non-decreasing sequence , the subdegree
growth rates of infinite primitive groups that act distance-transitively on
locally finite distance-transitive graphs are extremal, and conjecture there
might exist a number which perhaps depends upon , perhaps only on ,
such that .
In this paper it is shown that such an enumeration is not desirable, as there
exist infinite primitive permutation groups possessing no infinite subdegree,
in which two distinct subdegrees are each equal to the cardinality of
infinitely many suborbits. The examples used to show this provide several novel
methods for constructing infinite primitive graphs.
A revised enumeration method is then proposed, and it is shown that, under
this, Adeleke and Neumann's question may be answered, at least for groups
exhibiting suitable rates of growth.Comment: 41 page
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
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Constructing continuum many countable, primitive, unbalanced digraphs
AbstractWe construct continuum many non-isomorphic countable digraphs which are highly arc transitive, have finite out-valency and infinite in-valency, and whose automorphism groups are primitive
Strongly regular edge-transitive graphs
In this paper, we examine the structure of vertex- and edge-transitive
strongly regular graphs, using normal quotient reduction. We show that the
irreducible graphs in this family have quasiprimitive automorphism groups, and
prove (using the Classification of Finite Simple Groups) that no graph in this
family has a holomorphic simple automorphism group. We also find some
constraints on the parameters of the graphs in this family that reduce to
complete graphs.Comment: 23 page
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