  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 $Y_G$ in noncommuting variables in this quotient
algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph
$G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ 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 $y_{G : v}$, 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 $y_{G
: v}$ to acyclic orientations of graphs, including a noncommutative refinement
of Stanley's sink theorem.
