Identities and Inequalities for Tree Entropy

Details Identities and Inequalities for Tree Entropy Identities and Inequalities for Tree Entropy

2007-12-18

arxiv.org/abs/0712.3035v2

Combin. Probab. Comput. 19, no. 2 (2010), 303-313.

Mathematics

Combinatorics

12 pages; revision contains more background

Scientific paper

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 Lueck's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.

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