Chromatic quasisymmetric functions of the path graph

TL;DR

This paper investigates the symmetry properties of chromatic quasisymmetric functions (CQFs) for path and star graphs, revealing that symmetry occurs only under specific labelings for paths and is absent for star graphs.

Contribution

It demonstrates that CQFs of path graphs are generally not symmetric unless labeled naturally, and shows star graphs have nonsymmetric CQFs for all labelings, advancing understanding of graph labeling effects.

Findings

CQFs of path graphs are symmetric only with natural labelings.

Star graphs have nonsymmetric CQFs for all labelings.

Symmetry of CQFs depends critically on graph labeling.

Abstract

We show that the chromatic quasisymmetric function (CQF) of a labeled path graph on $n$ vertices is not symmetric unless the labeling is the natural labeling $1, 2, ..., n$ or its reverse $n, ..., 2, 1$. We also show that the star graph $K_{1, n-1}$ with $n\geq 3$ has a nonsymmetric CQF for all labelings.