897 research outputs found
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
Distance-regular graph with large a1 or c2
In this paper, we study distance-regular graphs that have a pair of
distinct vertices, say x and y, such that the number of common neighbors of x
and y is about half the valency of . We show that if the diameter is at
least three, then such a graph, besides a finite number of exceptions, is a
Taylor graph, bipartite with diameter three or a line graph.Comment: We submited this manuscript to JCT
Line graphs and -geodesic transitivity
For a graph , a positive integer and a subgroup G\leq
\Aut(\Gamma), we prove that is transitive on the set of -arcs of
if and only if has girth at least and is
transitive on the set of -geodesics of its line graph. As applications,
we first prove that the only non-complete locally cyclic -geodesic
transitive graphs are the complete multipartite graph and the
icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and
girth 3, and determine which of them are geodesic transitive
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
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