Linear recurrences for non-log-concave independence polynomials of trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917344348995584 |
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| author | Bautista-Ramos, César Guillén-Galván, Carlos Gómez-Salgado, Paulino |
| author_facet | Bautista-Ramos, César Guillén-Galván, Carlos Gómez-Salgado, Paulino |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_14204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear recurrences for non-log-concave independence polynomials of trees Bautista-Ramos, César Guillén-Galván, Carlos Gómez-Salgado, Paulino Combinatorics 05C69, 05C31, 05A20 (Primary) 30C15 (Secondary) 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]. |
| title | Linear recurrences for non-log-concave independence polynomials of trees |
| topic | Combinatorics 05C69, 05C31, 05A20 (Primary) 30C15 (Secondary) |
| url | https://arxiv.org/abs/2603.14204 |