Generalized degree polynomials of trees

TL;DR

This paper explores the generalized degree polynomial of trees, showing it encodes various structural properties and is determined by the chromatic symmetric function, thus linking combinatorial invariants.

Contribution

It demonstrates that the generalized degree polynomial and its generalizations can recover multiple tree invariants from the chromatic symmetric function, revealing new connections.

Findings

Double-degree sequence of trees can be recovered from the polynomial.

Leaf adjacency sequence is determined by the polynomial.

Generalized polynomial for vertex tuples is also determined by the chromatic symmetric function.

Abstract

The generalized degree polynomial $\mathbf{G}_T(x,y,z)$ of a tree $T$ is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that $\mathbf{G}_T$ is determined linearly by the chromatic symmetric function $\mathbf{X}_T$, introduced by Stanley. We present several classes of information about $T$ that can be recovered from $\mathbf{G}_T$ and hence also from $\mathbf{X}_T$. Examples of such information include the double-degree sequence of $T$, which enumerates edges of $T$ by the pair of degrees of their endpoints, and the leaf adjacency sequence of $T$, which enumerates vertices of $T$ by degree and number of adjacent leaves. We also discuss a further generalization of $\mathbf{G}_T$ that enumerates tuples of vertex sets and show that this is also determined by $\mathbf{X}_T$.