  We use a Dyck path model for unit-interval graphs to study the chromatic
quasisymmetric functions introduced by Shareshian and Wachs, as well as
vertical strip --- in particular, unicellular LLT polynomials.
  We show that there are parallel phenomena regarding $e$-positivity of these
two families of polynomials. In particular, we give several examples where the
LLT polynomials behave like a "mirror image" of the chromatic quasisymmetric
counterpart.
  The Dyck path model is also extended to circular arc digraphs to obtain
larger families of polynomials. This circular extensions of LLT polynomials has
not been studied before. A lot of the combinatorics regarding unit interval
graphs carries over to this more general setting, and we prove several
statements regarding the $e$-coefficients of chromatic quasisymmetric functions
and LLT polynomials.
  In particular, we believe that certain $e$-positivity conjectures hold in all
these families above. Furthermore, we study vertical-strip LLT polynomials, for
which there is no natural chromatic quasisymmetric counterpart. These
polynomials are essentially modified Hall--Littlewood polynomials, and are
therefore of special interest.
  In this more general framework, we are able to give a natural combinatorial
interpretation for the $e$-coefficients for the line graph and the cycle graph,
in both the chromatic and the LLT setting.
