The diameter of the uniform spanning tree of dense graphs

TL;DR

This paper proves that in dense graphs with high minimal degree, the typical diameter of a uniform spanning tree is about the square root of the number of vertices, linking graph expansion properties to spectral gap.

Contribution

It establishes a new relationship between the diameter of uniform spanning trees and the graph's expansion and spectral properties in dense graphs.

Findings

Spanning tree diameter is typically of order √n in dense graphs.

Cheeger constant and spectral gap are comparable in such graphs.

Provides insights into the structure of uniform spanning trees in dense graphs.

Abstract

We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on $n$ vertices with minimal degree linear in $n$ is typically of order $\sqrt{n}$. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.