# Spanning tree generating functions and Mahler measures

@article{Guttmann2012SpanningTG,
title={Spanning tree generating functions and Mahler measures},
author={Anthony J. Guttmann and Mathew Rogers},
journal={arXiv: Mathematical Physics},
year={2012}
}
• Published 12 July 2012
• Mathematics
• arXiv: Mathematical Physics
We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form…
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The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees NST and establish inequalities relating the numbers of spanning trees
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For n-vertex, d-dimensional lattices Λ with d ≥ 2, the number of spanning trees NST(Λ) grows asymptotically as exp(nzΛ) in the thermodynamic limit. We present an exact closed-form result for the
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