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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.

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