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

## TL;DR

This paper explores the relationships between chromatic quasisymmetric functions and unicellular LLT polynomials, introducing melting lollipop graphs and deriving Schur expansions for specific graph classes, advancing understanding of their algebraic properties.

## Contribution

It generalizes linear relations of LLT polynomials, introduces melting lollipop graphs, and provides explicit Schur expansion formulas for certain graph-related LLT polynomials.

## Key findings

- Identified a class of e-positive melting lollipop graphs
- Proved e-unimodality for melting lollipop graphs
- Derived Schur expansion formulas for LLT polynomials of specific graphs

## Abstract

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.