• DOI:10.48550/arXiv.2211.00464
• Corpus ID: 253244560

On the zeroes of hypergraph independence polynomials

@article{Galvin2022OnTZ,
  title={On the zeroes of hypergraph independence polynomials},
  author={David J. Galvin and Gweneth McKinley and Will Perkins and Michail Sarantis and Prasad Tetali},
  journal={ArXiv},
  year={2022},
  volume={abs/2211.00464}
}

• David J. Galvin, Gweneth McKinley, +2 authors P. Tetali
• Published 1 November 2022
• Mathematics
• ArXiv

We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree ∆ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree ∆ established by Shearer (of radius ∼ 1 / ( e ∆)). Up to logarithmic factors in ∆ this is optimal, even for hypergraphs with all edge-sizes strictly greater than 2. We conjecture that for k ≥ 3, k -uniform linear hypergraphs have a much larger zero-free disk of radius Ω(∆ − 1 k − 1 ). We establish this in the case of linear hypertrees.