Geometric bounds for spanning tree entropy of planar lattice graphs

TL;DR

This paper establishes new geometric bounds for the spanning tree entropy of planar lattice graphs, linking graph theory with hyperbolic geometry, and provides practical estimates for these bounds.

Contribution

It proves numerous cases of conjectured bounds for spanning tree entropy using hyperbolic geometric structures, a novel approach in this context.

Findings

Bounds are easy to compute for many graphs

Bounds provide excellent numerical estimates

Infinitely many cases of conjectures are proven

Abstract

We prove infinitely many cases of conjectured sharp upper and lower bounds for the spanning tree entropy of any planar lattice graph. These bounds come from volumes of associated hyperbolic alternating links, right-angled hyperbolic polyhedra and hyperbolic regular ideal bipyramids. For many planar lattice graphs, we show these bounds are easy to compute and provide excellent numerical estimates for the spanning tree entropy.