Unimodality of independence polynomials of two family of trees

TL;DR

This paper proves that two infinite families of trees, which have independence polynomials that are not log-concave, still have unimodal independence polynomials, confirming a longstanding conjecture for these cases.

Contribution

It demonstrates that the independence polynomials of the two newly constructed families of trees are unimodal, despite not being log-concave.

Findings

The independence polynomials of the families $T_{3,m,n}$ and $T_{3,m,n}^*$ are unimodal.

These families provide counterexamples to log-concavity but support unimodality.

The results confirm the unimodality conjecture for these specific infinite families.

Abstract

In 1987, Alavi, Malde, Schwenk and Erd\H{o}s conjectured that the independence polynomials of trees are unimodal. Subsequently, many researchers proposed strengthening this conjecture to log-concavity. In 2023, Kadrawi, Levit, Yosef, and Mizrachi discovered independence polynomials of trees of order 26 that are not log-concave, which led them to construct two infinite families of such polynomials, denoted by $T_{3,m,n}$ and $T_{3,m,n}^*$. In this paper, we show that these two infinite families also satisfy the unimodal conjecture raised by Alavi, Malde, Schwenk, and Erd\H{o}s.