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Identities and Inequalities for Tree Entropy

Published online by Cambridge University Press:  15 December 2009

RUSSELL LYONS
RUSSELL LYONS*
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405-5701, USA (e-mail: rdlyons@indiana.eduhttp://mypage.iu.edu/~rdlyons/)
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Abstract

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede–Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lück's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 2 , March 2010 , pp. 303 - 313
Copyright
Copyright © Cambridge University Press 2009

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