Anticoncentration of random spanning trees in almost regular graphs

TL;DR

This paper demonstrates that large, almost regular graphs contain exponentially many non-isomorphic spanning trees, and shows that the probability of a specific tree appearing as a random spanning tree is exponentially small.

Contribution

It establishes exponential abundance of non-isomorphic spanning trees in almost regular graphs and introduces a novel graph-theoretic balls-into-bins model for analysis.

Findings

Large almost regular graphs have exponentially many spanning trees.

Probability of a specific tree as a random spanning tree is exponentially small.

Introduces a new graph-theoretic balls-into-bins model.

Abstract

The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with sufficiently large degree already contains exponentially many non-isomorphic spanning trees. Indeed, we prove a stronger statement: for every fixed $n$-vertex tree $T$, $$ \Pr\bigl[\mathcal{T} \simeq_{\mathrm{iso}} T\bigr] = e^{-\Omega(n)}, $$ where $\mathcal{T}$ is a uniformly random spanning tree of a connected $n$-vertex almost regular graph with sufficiently large degree. To prove this, we introduce a graph-theoretic variant of the classical balls--into--bins model, which may be of independent interest.