Lorentzian polynomials and the independence sequences of graphs
Amire Bendjeddou, Leonard Hardiman
TL;DR
This work develops a Lorentzian-polynomial framework for independence polynomials by introducing coloured independence polynomials and a glueing calculus. It defines pre-Lorentzian graphs via homogenisation and a $(xy)^k$-scaling, and proves pre-Lorentzian-ness is preserved under glueing across free vertices. The main result shows that applying the edge-replacement operator $R_{W_4}$ to any graph yields graphs whose independence sequences are log-concave, achieved by representing these graphs as glueings of leafier-star gadgets $\mathcal{L}_n$ and proving each gadget is pre-Lorentzian. This yields progress towards unimodality conjectures for trees/forests and provides a robust method to obtain log-concavity via Lorentzian theory, with potential extensions to broader graph families.
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 Erdős which asks if the independence sequence of trees or forests is unimodal.