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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_762_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$1, 2,\ldots , n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> or its reverse <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_762_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$n,\ldots , 2, 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We also show that the star graph <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_762_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$K_{1, n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="26_2025_762_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> has a nonsymmetric CQF for all labelings.

Chromatic Quasisymmetric Functions of the Path Graph

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2025年5月26日

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Farid Aliniaeifard,
Shamil Asgarli,
Maria Esipova,
Ethan Shelburne,
Stephanie van Willigenburg,
Tamsen Whitehead McGinley

摘要

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