On the number of distinct spanning trees in pseudorandom graphs

TL;DR

This paper extends Otter's classical result by showing that pseudorandom graphs with certain degree and eigenvalue conditions contain exponentially many distinct unlabelled spanning trees, close to the maximum in complete graphs.

Contribution

It proves that pseudorandom graphs with high degree and spectral gap have nearly as many spanning trees as complete graphs, generalizing Otter's result.

Findings

Pseudorandom graphs have at least (-)^n spanning trees.

The number of spanning trees is exponential in the number of vertices.

The result holds for graphs with sufficiently large degree and spectral gap.

Abstract

A celebrated result of Otter says the number of distinct unlabelled spanning trees in $K_n$ is $\alpha^n$ up to subexponential factors for an absolute constant $\alpha>0$. In this note, we prove that for every $0<\varepsilon<\alpha$, there are constants $C$ and $d_0$ such that every $(n,d,\lambda)$-graph with $d\geq d_0$ and $d/\lambda \geq C$ has at least $(\alpha-\varepsilon)^n$ distinct unlabelled spanning trees.