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arXiv · 2603.14204

Linear recurrences for non-log-concave independence polynomials of trees

César Bautista-Ramos·Carlos Guillén-Galván·Paulino Gómez-Salgado
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Abstract

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 $|z+1/3|=1/3$ 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].

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César Bautista-Ramos, Carlos Guillén-Galván, Paulino Gómez-Salgado. 2026-03-15. Linear recurrences for non-log-concave independence polynomials of trees. https://arxiv.org/abs/2603.14204

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