The Chromatic Symmetric Function of a Graph Centred at a Vertex

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function YG in noncommuting variables in this quotient algebra, which defines yG : v, the chromatic symmetric function of a graph G centred at a vertex v. We then apply our methods to yG : v and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical e-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge. In our study of yG : v, we also describe methods of constructing new e-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of yG : v to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.