Melting lollipop chromatic quasisymmetric functions and Schur expansion of unicellular LLT polynomials

In this work, we generalize and utilize the linear relations of LLT polynomials introduced by Lee \cite{Lee}. By using the fact that the chromatic quasisymmetric functions and the unicellular LLT polynomials are related via plethystic substitution and thus they satisfy the same linear relations, we can apply the linear relations to both sets of functions. As a result, in the chromatic quasisymmetric function side, we find a class of $e$-positive graphs, called \emph{melting lollipop graphs}, and explicitly prove the $e$-unimodality. In the unicellular LLT side, we obtain Schur expansion formulas for LLT polynomials corresponding to certain set of graphs, namely, complete graphs, path graphs, lollipop graphs and melting lollipop graphs.