Sink polynomials of line graphs and weighted matching polynomials
Abstract
Chromatic quasisymmetric functions and unicellular LLT polynomials give symmetric-function interpretations of graph orientations counted by sinks. For natural unit interval graphs, we prove that the acyclic sink polynomial is real-rooted for every nonnegative ascent weight. Equivalently, the elementary coefficients of the chromatic quasisymmetric function, grouped by partition length, form a real-rooted polynomial. The Carlsson--Mellit relation expresses this polynomial as a plethystic specialization of the corresponding LLT polynomial, and the Hessenberg correspondence gives an interpretation in cohomology. More generally, the acyclic sink polynomial is real-rooted for chordal claw-free graphs, and the all-orientations sink polynomial is real-rooted for every claw-free graph. We also study a real-rooted polynomial counting joint minima in local edge orders. It agrees with the acyclic sink polynomial of the line graph exactly for forests. An exact formula for unicyclic roots gives a line-graph counterexample to acyclic sink real-rootedness.
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BibTeX
@misc{alexandersson2026sink,
title = {{Sink polynomials of line graphs and weighted matching polynomials}},
author = {Per Alexandersson and Leonardo Saud Maia Leite},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41502},
eprint = {2610.41502},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41502},
}
RIS
TY - UNPB
TI - Sink polynomials of line graphs and weighted matching polynomials
AU - Per Alexandersson
AU - Leonardo Saud Maia Leite
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41502
UR - https://arxiv.org/abs/2610.41502
AB - Chromatic quasisymmetric functions and unicellular LLT polynomials give symmetric-function interpretations of graph orientations counted by sinks. For natural unit interval graphs, we prove that the acyclic sink polynomial is real-rooted for every nonnegative ascent weight. Equivalently, the elementary coefficients of the chromatic quasisymmetric function, grouped by partition length, form a real-rooted polynomial. The Carlsson--Mellit relation expresses this polynomial as a plethystic specialization of the corresponding LLT polynomial, and the Hessenberg correspondence gives an interpretation in cohomology. More generally, the acyclic sink polynomial is real-rooted for chordal claw-free graphs, and the all-orientations sink polynomial is real-rooted for every claw-free graph. We also study a real-rooted polynomial counting joint minima in local edge orders. It agrees with the acyclic sink polynomial of the line graph exactly for forests. An exact formula for unicyclic roots gives a line-graph counterexample to acyclic sink real-rootedness.
ER -