On stability of spanning tree degree enumerators

TL;DR

This paper characterizes when the spanning tree degree enumerator polynomial of a connected graph is real stable, establishing a precise equivalence with the graph being distance-hereditary.

Contribution

It provides a complete characterization of graphs for which the spanning tree degree enumerator polynomial is real stable, linking graph structure to polynomial stability.

Findings

The polynomial is real stable if and only if the graph is distance-hereditary.

Distance-hereditary graphs are exactly those with stable spanning tree degree enumerators.

The result bridges graph theory and polynomial stability concepts.

Abstract

We show that the spanning tree degree enumerator polynomial of a connected graph $G$ is a real stable polynomial if and only if $G$ is distance-hereditary.