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Hypergraph independence polynomials with a zero close to the origin

Part of: Equilibrium statistical mechanics Graph theory

Published online by Cambridge University Press:  22 January 2025

Shengtong Zhang Open the ORCID record for Shengtong Zhang [Opens in a new window]
Shengtong Zhang*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA, USA
*
Email: stzh1555@stanford.edu
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Abstract

For each uniformity $k \geq 3$, we construct $k$ uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a zero $\lambda$ with $\left \vert \lambda \right \vert = O\left (\frac {\log \Delta }{\Delta }\right )$. This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.

Keywords

Independence polynomialhypergraphzero-free region

MSC classification

Primary: 05C31: Graph polynomials
Secondary: 05C69: Dominating sets, independent sets, cliques 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 34 , Issue 4 , July 2025 , pp. 486 - 490
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

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