Sharp estimates for spanning trees

TL;DR

This paper establishes a sharp upper bound on the number of spanning trees in a graph based on vertex degrees, improving previous estimates and extending to weighted graphs.

Contribution

It provides a new, tight bound for spanning trees in simple and weighted graphs, refining earlier results by Alon and Kostochka.

Findings

The maximum number of spanning trees is bounded by (1/n^2) times the product of (d(v)+1) over all vertices.

The bound is tight for complete graphs.

An analogous bound applies to weighted spanning tree enumeration.

Abstract

We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight (for complete graphs), and improves earlier estimates of Alon from 1990 and Kostochka from 1995 by a factor of about $1/n$ (for dense graphs). We additionally show that an analogous bound holds for the weighted spanning tree enumerator of a (nonnegatively) weighted graph as well.