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<p>We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|z+1/3|=1/3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> <mo stretchy="false">|</mo> <mo>=</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin, <i>Trees with non log-concave independent set sequences</i>, arXiv:2502.10654v1, 2025].</p>

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Linear Recurrences for Non-Log-Concave Independence Polynomials of Trees

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  • 出版模式：

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  • 发表日期：

    2026年6月22日

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  • César Bautista-Ramos,
  • Carlos Guillén-Galván,
  • Paulino Gómez-Salgado

摘要

We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle \(|z+1/3|=1/3\) | z + 1 / 3 | = 1 / 3 in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv:2502.10654v1, 2025].