6,710 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
All reducts of the random graph are model-complete
We study locally closed transformation monoids which contain the automorphism
group of the random graph. We show that such a transformation monoid is locally
generated by the permutations in the monoid, or contains a constant operation,
or contains an operation that maps the random graph injectively to an induced
subgraph which is a clique or an independent set. As a corollary, our
techniques yield a new proof of Simon Thomas' classification of the five closed
supergroups of the automorphism group of the random graph; our proof uses
different Ramsey-theoretic tools than the one given by Thomas, and is perhaps
more straightforward. Since the monoids under consideration are endomorphism
monoids of relational structures definable in the random graph, we are able to
draw several model-theoretic corollaries: One consequence of our result is that
all structures with a first-order definition in the random graph are
model-complete. Moreover, we obtain a classification of these structures up to
existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed
by the more recent article 1003.4030. Length 14 pages
Hitting time results for Maker-Breaker games
We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property \mP. We focus on three natural
properties for Maker's graph, namely being -vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the -vertex connectivity game
exactly at the time the random graph process first reaches minimum degree ;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree ; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree . The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
The isoperimetric constant of the random graph process
The isoperimetric constant of a graph on vertices, , is the
minimum of , taken over all nonempty subsets
of size at most , where denotes the set of
edges with precisely one end in . A random graph process on vertices,
, is a sequence of graphs, where
is the edgeless graph on vertices, and
is the result of adding an edge to ,
uniformly distributed over all the missing edges. We show that in almost every
graph process equals the minimal degree of
as long as the minimal degree is . Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically , the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value
The integrated density of states of the random graph Laplacian
We analyse the density of states of the random graph Laplacian in the
percolating regime. A symmetry argument and knowledge of the density of states
in the nonpercolating regime allows us to isolate the density of states of the
percolating cluster (DSPC) alone, thereby eliminating trivially localised
states due to finite subgraphs. We derive a nonlinear integral equation for the
integrated DSPC and solve it with a population dynamics algorithm. We discuss
the possible existence of a mobility edge and give strong evidence for the
existence of discrete eigenvalues in the whole range of the spectrum.Comment: 4 pages, 1 figure. Supplementary material available at
http://www.theorie.physik.uni-goettingen.de/~aspel/data/spectrum_supplement.pd
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