897 research outputs found

    Subdegree growth rates of infinite primitive permutation groups

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    A transitive group GG of permutations of a set Ω\Omega is primitive if the only GG-invariant equivalence relations on Ω\Omega are the trivial and universal relations. If α∈Ω\alpha \in \Omega, then the orbits of the stabiliser GαG_\alpha on Ω\Omega are called the α\alpha-suborbits of GG; when GG acts transitively the cardinalities of these α\alpha-suborbits are the subdegrees of GG. If GG acts primitively on an infinite set Ω\Omega, and all the suborbits of GG are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of GG as a non-decreasing sequence 1=m0≤m1≤...1 = m_0 \leq m_1 \leq ..., 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 cc which perhaps depends upon GG, perhaps only on mm, such that mr≤c(m−2)r−1m_r \leq c(m-2)^{r-1}. 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

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    In this paper, we study distance-regular graphs Γ\Gamma 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 Γ\Gamma. 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 22-geodesic transitivity

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    For a graph Γ\Gamma, a positive integer ss and a subgroup G\leq \Aut(\Gamma), we prove that GG is transitive on the set of ss-arcs of Γ\Gamma if and only if Γ\Gamma has girth at least 2(s−1)2(s-1) and GG is transitive on the set of (s−1)(s-1)-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic 22-geodesic transitive graphs are the complete multipartite graph K3[2]K_{3[2]} 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

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    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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