A class of trees determined by their chromatic symmetric functions

TL;DR

This paper investigates whether trees are uniquely identified by their chromatic symmetric functions, proving it for a specific class of trees and providing evidence supporting a broader conjecture.

Contribution

It proves that trees with exactly two vertices of degree at least 3 are determined by their chromatic symmetric functions and shows that the generalized degree sequence of subtrees is also determined.

Findings

Trees with two high-degree vertices are uniquely identified by their chromatic symmetric functions.

The generalized degree sequence of subtrees is determined by the chromatic symmetric function for any tree.

Provides evidence supporting Crew's conjecture.

Abstract

Stanley introduced the concept of chromatic symmetric functions of graphs which extends and refines the notion of chromatic polynomials of graphs, and asked whether trees are determined up to isomorphism by their chromatic symmetric functions. Using the technique of differentiation with respect to power-sum symmetric functions, we give a positive answer to Stanley's question for the class of trees with exactly two vertices of degree at least 3. In addition, we prove that for any tree $T$, the generalized degree sequence for subtrees of $T$ is determined by the chromatic symmetric function of $T$, providing evidence to a conjecture of Crew.