The Chromatic Symmetric Function for Unicyclic Graphs

TL;DR

This paper investigates the chromatic symmetric function of unicyclic graphs, demonstrating how key structural properties can often be inferred from it, and develops methods for extracting these features.

Contribution

It extends known results from trees to unicyclic graphs, providing explicit formulas and methods for analyzing their chromatic symmetric functions.

Findings

Structural properties like number of leaves and cycle size can often be recovered from the CSF.

Non-isomorphic unicyclic graphs with the same CSF are rare for graphs with up to 17 vertices.

Explicit formulas for star-expansions of certain graph classes are provided.

Abstract

Motivated by the question of which structural properties of a graph can be recovered from the chromatic symmetric function (CSF), we study the CSF of connected unicyclic graphs. While it is known that there can be non-isomorphic unicyclic graphs with the same CSF, we find experimentally that such examples are rare for graphs with up to 17 vertices. In fact, in many cases we can recover data such as the number of leaves, number of internal edges, cycle size, and number of attached non-trivial trees, by extending known results for trees to unicyclic graphs. These results are obtained by analyzing the CSF of a connected unicyclic graph in the $\textit{star-basis}$ using the deletion-near-contraction (DNC) relation developed by Aliste-Prieto, Orellana and Zamora, and computing the "leading" partition, its coefficient, as well as coefficients indexed by hook partitions. We also give explicit formulas for star-expansions of several classes of graphs, developing methods for extracting coefficients using structural properties of the graph.