Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 29 min ago
PreprintCC BY 4.0arXiv

Unimodality of Independence Polynomials for Sufficiently Large Forests

  • Ethan X. Fang
  • Junwei Lu
  • Eran Nevo
  • Yuan Yao
  • Hailun Zheng
Open at sourcePDFDOICite

Abstract

We prove that the independence sequence of every sufficiently large forest is unimodal. The result follows from establishing log-concavity on a central interval of the sequence, along with monotonicity of the initial and final segments. The main analytic step is a central limit theorem for the size of a random independent set sampled with the hard-core model, uniform over all forests and over an interval of positive fugacities.

Related

  1. Unimodality of Forest Independence Polynomials
    Wei Li, Kevin Vallier, Tong Zhang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids
    Matthew H. Y. Xie, Zuo-Ru Zhang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Proof of Almkvist's conjecture on the unimodality of partition polynomials
    Jianxi Mao, Wenle Shi, Bao-xuan Zhu
    arXivPreprintCC0Full textCombinatorics & discrete mathematicsarXiv
  4. Strict unimodality without log-concavity for partial Petrial polynomials
    Ruiqing Feng, Qi Yan
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Unbounded log-concavity breaks in independence polynomials of spherically symmetric trees
    César Bautista-Ramos
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{fang2026unimodality,
  title = {{Unimodality of Independence Polynomials for Sufficiently Large Forests}},
  author = {Ethan X. Fang and Junwei Lu and Eran Nevo and Yuan Yao and Hailun Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20961},
  eprint = {2609.20961},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20961},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Unimodality of Independence Polynomials for Sufficiently Large Forests
AU  - Ethan X. Fang
AU  - Junwei Lu
AU  - Eran Nevo
AU  - Yuan Yao
AU  - Hailun Zheng
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20961
UR  - https://arxiv.org/abs/2609.20961
AB  - We prove that the independence sequence of every sufficiently large forest is unimodal. The result follows from establishing log-concavity on a central interval of the sequence, along with monotonicity of the initial and final segments. The main analytic step is a central limit theorem for the size of a random independent set sampled with the hard-core model, uniform over all forests and over an interval of positive fugacities.
ER  -