Lorentzian polynomials and the independence sequences of graphs

TL;DR

This paper introduces the concept of pre-Lorentzian graphs, showing that certain graph transformations produce Lorentzian independence polynomials, advancing understanding of log-concavity and unimodality in graph independence sequences.

Contribution

It proves that applying a specific edge-replacement operator yields pre-Lorentzian graphs with Lorentzian independence polynomials, supporting a conjecture about unimodality in trees and forests.

Findings

Graphs in the image of the operator are pre-Lorentzian.

Pre-Lorentzian graphs have log-concave independence sequences.

Progress on the conjecture about unimodality of independence sequences in trees.

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.