4,317 research outputs found
Bernoulli and self-destructive percolation on non-amenable graphs
In this note we study some properties of infinite percolation clusters on
non-amenable graphs. In particular, we study the percolative properties of the
complement of infinite percolation clusters. An approach based on
mass-transport is adapted to show that for a large class of non-amenable
graphs, the graph obtained by removing each site contained in an infinite
percolation cluster has critical percolation threshold which can be arbitrarily
close to the critical threshold for the original graph, almost surely, as p
approaches p_c. Closely related is the self-destructive percolation process,
introduced by J. van den Berg and R. Brouwer, for which we prove that an
infinite cluster emerges for any small reinforcement.Comment: 7 page
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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