The number of trees in distance-hereditary graphs and their friends

TL;DR

This paper investigates the number of spanning trees in distance-hereditary graphs and related classes, providing unified proofs, generalizations, and exploring conjectures about bipartite graphs with fixed degree sequences.

Contribution

It offers new proofs and generalizations for counting spanning trees in distance-hereditary graphs and establishes the equivalence of Ehrenborg's conjecture with its polynomial form.

Findings

Unified proofs for spanning tree counts in distance-hereditary graphs

Generalizations of previous results from the 2020s

Equivalence of Ehrenborg's conjecture and its polynomial form

Abstract

Counting the number of spanning trees in specific classes of graphs has attracted increasing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained in the 2020s. The main method is to study the behavior of the vertex (degree) enumerator of a distance-hereditary graph under the operations of copying vertices. Ehrenborg conjecture says that a Ferrer--Young graph maximizes the number of spanning trees among bipartite graphs with the same degree sequence. The second result of this paper is the equivalence of the Ehrenborg conjecture and its polynomial form.