Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 19 min ago
PreprintCC BY 4.0arXiv

Independence polynomials and the weak Lefschetz property for tadpole graphs

  • Tran Quang Hoa
  • Nguyen Duy Phuoc
  • Tran Nguyen Thanh Son
Open at sourcePDFDOICite

Abstract

Let Tm,n be the tadpole graph obtained by joining a cycle Cm to a path Pn by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fixed vertex. Over a field of characteristic zero, we also give a complete classification of the pairs (m,n) for which the Artinian algebra defined by the edge ideal of Tm,n together with the squares of all variables has the weak Lefschetz property.

Related

  1. Lorentzian polynomials and log-concavity of the independence polynomials of graphs
    Lily Li Liu, Hongyue Tang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Strong Lefschetz property of the Kirchhoff algebra of the complete tripartite graph K1,1,n
    Yohji Akama
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  3. Weak and strong Lefschetz properties for vertex cover Artinian algebras associated to graphs
    Trung Chau, Tran Quang Hoa, Dang Thi Kieu Uyen
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. Unimodality of Independence Polynomials for Sufficiently Large Forests
    Ethan X. Fang, Junwei Lu, Eran Nevo +2
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsProbabilityarXiv
  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{hoa2026independence,
  title = {{Independence polynomials and the weak Lefschetz property for tadpole graphs}},
  author = {Tran Quang Hoa and Nguyen Duy Phuoc and Tran Nguyen Thanh Son},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13888},
  eprint = {2609.13888},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13888},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Independence polynomials and the weak Lefschetz property for tadpole graphs
AU  - Tran Quang Hoa
AU  - Nguyen Duy Phuoc
AU  - Tran Nguyen Thanh Son
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13888
UR  - https://arxiv.org/abs/2609.13888
AB  - Let T_m,n be the tadpole graph obtained by joining a cycle C_m to a path P_n by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fixed vertex. Over a field of characteristic zero, we also give a complete classification of the pairs (m,n) for which the Artinian algebra defined by the edge ideal of T_m,n together with the squares of all variables has the weak Lefschetz property.
ER  -