A Linear Algebraic Method on the Chromatic Symmetric Function

Abstract

The Stanley-Stembridge conjecture is a longstanding conjecture that has evaded proof for nearly 30 years. Concerned with the e-basis expansions of the chromatic symmetric functions of unit-interval graphs, this conjecture has served as a significant motivator of research in algebraic graph theory in recent years. We summarize a great deal of the existing work done in favor of this conjecture, giving an overview of the various techniques that have previously been used in the study of this problem. Moreover, we develop a novel technique using methods from linear algebra and use it to obtain an e-basis expansion of graphs known as single clique-blowups of paths. Using this same method, we use this result to prove the e-positivity of double clique-blowups of paths, a previously unknown result.