Anticoncentration of random spanning trees in graphs with large minimum degree

TL;DR

This paper proves that graphs with large minimum degree have exponentially many non-isomorphic spanning trees and that a uniformly random spanning tree is unlikely to be isomorphic to any fixed tree, confirming a conjecture.

Contribution

It establishes tight bounds on the number of non-isomorphic spanning trees and demonstrates anticoncentration for random spanning trees in graphs with large minimum degree.

Findings

Graphs with minimum degree d have at least n^{Ω(d)} non-isomorphic spanning trees.

Probability that a random spanning tree is isomorphic to any fixed tree is at most n^{-Ω(d)}.

Results confirm a conjecture of Lee about anticoncentration in such graphs.

Abstract

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider graphs of large minimum degree. We show that every connected graph $G$ with $n$ vertices and minimum degree $d$ has at least $n^{\Omega(d)}$ non-isomorphic spanning trees. This is tight up to the constant factor in the exponent. In fact, we prove the following anticoncentration result: if $\mathcal{T}$ is a uniformly random spanning tree of $G$, then for every tree $T$, the probability that $\mathcal{T}$ is isomorphic to $T$ is at most $n^{-\Omega(d)}$. This proves a conjecture of Lee in a strong form.