Chromatic Symmetric Functions of Trees and Unicycles

Given any simple graph, there is a corresponding symmetric function called the chromatic symmetric function (CSF). Introduced by Richard Stanley in 1995, the CSF of a graph G = (V(G), E(G)) is defined as follows:

where the sum is over all proper colorings κ of G. A proper coloring is a labeling of a graph such that no two adjacent vertices have the same label. In Geoffrey Scott's senior thesis, published in 2008, several open problems in graph theory were presented. In our poster, we investigate these open problems and generalize some of his results. Our goal is to find necessary conditions for any two graphs that will ensure that they have the same CSF. We first consider two special types of graphs: trees and unicycles and write a program in SAGE to compare the CSF for any simple graph, and thus, compile a library of graphs with a small number of vertices alongside their CSFs.

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