Chromatic quasisymmetric functions of the path graph

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 $1,2,\mathrm{.}\mathrm{.}\mathrm{.},n$ or its reverse $n,\mathrm{.}\mathrm{.}\mathrm{.},2,1$. We also show that the star graph $K_{1,n-1}$ with $n\ge 3$ has a nonsymmetric CQF for all labelings.