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arXiv · 2405.00511

Lorentzian polynomials and the independence sequences of graphs

Amire Bendjeddou·Leonard Hardiman
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Abstract

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 Erd\H{o}s which asks if the independence sequence of trees or forests is unimodal.

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Amire Bendjeddou, Leonard Hardiman. 2024-05-01. Lorentzian polynomials and the independence sequences of graphs. https://doi.org/10.1112/blms.70031

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