  This work is inspired by Shareshian and Wachs's exquisite formula for the
chromatic symmetric function of paths. We develop a composition method to
unearth neat noncommutative analogs of chromatic symmetric functions. A
symmetric function is $e$-positive if and only if it has a $\Lambda$-positive
noncommutative analog. We bring to light short and sweet $\Lambda$-positive
noncommutative analogs for the chromatic symmetric functions of tadpoles and
barbells. Using these elegant formulas and the composition method, we discover
a new family of $e$-positive graphs and call it hat graphs, which are the
unicyclic graphs obtained by adding an edge to a path. We also obtain a compact
ribbon Schur analog for cycles.
