Anticoncentration of random spanning trees in graphs with large minimum degree

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.