Around the number of trees in distance-hereditary graphs

Counting the number of spanning trees in specific classes of graphs has attracted growing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained during the 2020s. Our main approach is to study the behavior of the vertex (degree) enumerator polynomial of a distance-hereditary graph under certain graph-theoretical operations. The first result provides a factorization formula applicable to graphs admitting a cut whose edges form a complete bipartite subgraph. One of the central open problems in this area is Ehrenborg's conjecture, which asserts that a Ferrers–Young graph maximizes the number of spanning trees among all bipartite graphs with the same degree sequence. The second main result of this paper shows the equivalence between Ehrenborg's conjecture and its polynomial version.