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Lorentzian polynomials and log-concavity of the independence polynomials of graphs

  • Lily Li Liu
  • Hongyue Tang
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Abstract

In this paper, we first construct two graphs ℱ(l,m,t,s) and 𝒢4(l,m,t,s). Then we introduce the graph ℱn(l,m,t,s) and the operator E𝒢4(l,m,t,s), where ℱn(l,m,t,s) is defined by identifying the vertex c of n copies of ℱ(l,m,t,s), and E𝒢4(l,m,t,s) is defined by replacing each edge of G with 𝒢4(l,m,t,s), for any simple finite undirected graph G. By using the theory of Lorentzian polynomials, we prove that the independence polynomials of the graphs ℱn(l,m,t,s) and the image graphs of E𝒢4(l,m,t,s) are log-concave, respectively. As applications, our results not only make progress on the conjecture of Alavi, Malde, Schwenk and Erdős, but also generalize the results of Bendjeddou and Hardiman.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026lorentzian,
  title = {{Lorentzian polynomials and log-concavity of the independence polynomials of graphs}},
  author = {Lily Li Liu and Hongyue Tang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37553},
  eprint = {2609.37553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37553},
}
RIS
TY  - UNPB
TI  - Lorentzian polynomials and log-concavity of the independence polynomials of graphs
AU  - Lily Li Liu
AU  - Hongyue Tang
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37553
UR  - https://arxiv.org/abs/2609.37553
AB  - In this paper, we first construct two graphs ℱ(l,m,t,s) and 𝒢_4(l,m,t,s). Then we introduce the graph ℱ_n(l,m,t,s) and the operator E_𝒢_4(l,m,t,s), where ℱ_n(l,m,t,s) is defined by identifying the vertex c of n copies of ℱ(l,m,t,s), and E_𝒢_4(l,m,t,s) is defined by replacing each edge of G with 𝒢_4(l,m,t,s), for any simple finite undirected graph G. By using the theory of Lorentzian polynomials, we prove that the independence polynomials of the graphs ℱ_n(l,m,t,s) and the image graphs of E_𝒢_4(l,m,t,s) are log-concave, respectively. As applications, our results not only make progress on the conjecture of Alavi, Malde, Schwenk and Erdős, but also generalize the results of Bendjeddou and Hardiman.
ER  -