Lorentzian polynomials and the independence sequences of graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917986457092096 |
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| author | Bendjeddou, Amire Hardiman, Leonard |
| author_facet | Bendjeddou, Amire Hardiman, Leonard |
| contents | We study the multivariate independence polynomials of graphs and the log-concavity of the coefficients of their univariate restrictions. Let $R_{W_4}$ be the operator defined on simple and undirected graphs which replaces each edge with a caterpillar of size $4$. We prove that all graphs in the image of $R_{W_4}$ are what we call pre-Lorentzian, that is, their multivariate independence polynomial becomes Lorentzian after appropriate manipulations. In particular, as pre-Lorentzian graphs have log-concave (and therefore unimodal) independence sequences, our result makes progress on a conjecture of Alavi, Malde, Schwenk and Erdős which asks if the independence sequence of trees or forests is unimodal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_00511 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lorentzian polynomials and the independence sequences of graphs Bendjeddou, Amire Hardiman, Leonard Combinatorics 05C69 (Primary) 52B40 (Secondary) We study the multivariate independence polynomials of graphs and the log-concavity of the coefficients of their univariate restrictions. Let $R_{W_4}$ be the operator defined on simple and undirected graphs which replaces each edge with a caterpillar of size $4$. We prove that all graphs in the image of $R_{W_4}$ are what we call pre-Lorentzian, that is, their multivariate independence polynomial becomes Lorentzian after appropriate manipulations. In particular, as pre-Lorentzian graphs have log-concave (and therefore unimodal) independence sequences, our result makes progress on a conjecture of Alavi, Malde, Schwenk and Erdős which asks if the independence sequence of trees or forests is unimodal. |
| title | Lorentzian polynomials and the independence sequences of graphs |
| topic | Combinatorics 05C69 (Primary) 52B40 (Secondary) |
| url | https://arxiv.org/abs/2405.00511 |