102 research outputs found
High-Precision Entropy Values for Spanning Trees in Lattices
Shrock and Wu have given numerical values for the exponential growth rate of
the number of spanning trees in Euclidean lattices. We give a new technique for
numerical evaluation that gives much more precise values, together with
rigorous bounds on the accuracy. In particular, the new values resolve one of
their questions.Comment: 7 pages. Revision mentions alternative approach. Title changed
slightly. 2nd revision corrects first displayed equatio
The densest subgraph problem in sparse random graphs
We determine the asymptotic behavior of the maximum subgraph density of large
random graphs with a prescribed degree sequence. The result applies in
particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of
Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in
extending the notion of balanced loads from finite graphs to their local weak
limits, using unimodularity. This is a new illustration of the objective method
described by Aldous and Steele [In Probability on Discrete Structures (2004)
1-72 Springer].Comment: Published at http://dx.doi.org/10.1214/14-AAP1091 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A formula for the number of spanning trees in circulant graphs with non-fixed generators and discrete tori
We consider the number of spanning trees in circulant graphs of
vertices with generators depending linearly on . The matrix tree theorem
gives a closed formula of factors, while we derive a formula of
factors. Using the same trick, we also derive a formula for the
number of spanning trees in discrete tori. Moreover, the spanning tree entropy
of circulant graphs with fixed and non-fixed generators is compared.Comment: 8 pages, 2 figure
Uniqueness of maximal entropy measure on essential spanning forests
An essential spanning forest of an infinite graph is a spanning forest of
in which all trees have infinitely many vertices. Let be an
increasing sequence of finite connected subgraphs of for which . Pemantle's arguments imply that the uniform measures on spanning trees
of converge weakly to an -invariant measure
on essential spanning forests of . We show that if is a
connected, amenable graph and acts
quasitransitively on , then is the unique -invariant measure
on essential spanning forests of for which the specific entropy is maximal.
This result originated with Burton and Pemantle, who gave a short but incorrect
proof in the case . Lyons discovered the error and
asked about the more general statement that we prove.Comment: Published at http://dx.doi.org/10.1214/009117905000000765 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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