The diameter of uniform spanning trees in high dimensions

TL;DR

This paper demonstrates that in high-dimensional graphs, the diameter of a uniformly chosen spanning tree generally scales with the square root of the number of vertices, revealing a universal behavior across various graph classes.

Contribution

It establishes a universal asymptotic growth rate for the diameter of uniform spanning trees in high-dimensional graphs, including expanders and tori, under certain conditions.

Findings

Diameter grows like Θ(√n) in high-dimensional graphs

Results apply to expanders, tori, hypercubes, and their perturbations

Universal behavior across multiple graph classes in high dimensions

Abstract

We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $\Theta(\sqrt{n})$. In particular this result applies to expanders, finite tori $\mathbb{Z}_m^d$ of dimension $d \geq 5$, the hypercube $\{0,1\}^m$, and small perturbations thereof.