On the number of spanning trees in bipartite graphs

TL;DR

This paper investigates Ehrenborg's conjecture on the number of spanning trees in bipartite graphs, proving it for one-side regular graphs and Ferrers graphs, thereby advancing understanding of graph spanning tree counts.

Contribution

The paper proves Ehrenborg's conjecture for one-side regular bipartite graphs and provides a new proof for Ferrers graphs, extending known cases.

Findings

Ehrenborg's conjecture holds for one-side regular bipartite graphs.

A new proof confirms the equality for Ferrers graphs.

The results expand the classes of bipartite graphs where the conjecture is verified.

Abstract

In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph the number of spanning trees does not exceed the product of the degrees of the vertices divided by the product of the sizes of the graph components. We show that the conjecture is true for a one-side regular graph (that is a graph for which all degrees of the vertices of at least one of the components are equal). We also present a new proof of the fact that the equality holds for Ferrers graphs.