Multiple breaks of log-concavity in the independence polynomials of trees

TL;DR

This paper constructs infinite families of trees with independence polynomials that violate log-concavity at multiple points, answering a question about the behavior of these polynomials in trees.

Contribution

It introduces new examples of trees with independence polynomials that break log-concavity at multiple indices, expanding understanding of polynomial properties in graph theory.

Findings

Infinite families of trees with multiple log-concavity violations

Counterexamples to log-concavity in independence polynomials

Resolution of a question posed by D. Galvin

Abstract

We construct infinite families of trees whose independence polynomials violate log-concavity at an arbitrary number of indices. This affirmatively answers a question of D. Galvin.