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

TL;DR

This work proves that independence polynomials of trees can have multiple log-concavity breaks by constructing the families $TG_{m,t}$ and showing that for each fixed $m$ and sufficiently large $t$, the polynomial $I(TG_{m,t})$ breaks log-concavity at $m$ indices. The method centers on the reflected polynomial $R I(TG_{m,t})$, whose coefficients satisfy $c_k(m,t) \in \Theta\bigl(2^{(k+\lfloor k/2\rfloor) t}\bigr)$, enabling precise identification of leading-term contributions that cause violations. The main results include explicit recurrence-derived formulas for $I(TG_{m,t})$ and a clear asymptotic framework, plus concrete examples such as $TG_{4,6}$ and $TG_{5,6}$ demonstrating multiple breaks. Overall, the paper answers Galvin's question by showing that log-concavity can fail in multiple, controllable locations for trees, enriching the understanding of how independence polynomials can deviate from log-concavity in combinatorial settings.

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.

Figures (1)

• Figure 1: The tree $TG_{2,5}$, whose independence polynomial has degree 37 and breaks log-concavity at indices 34 and 36. For $TG_{m,t}$, the parameter $m$ equals the number of log-concavity violations for sufficiently large $t$.

Theorems & Definitions (12)

• Definition 1
• Lemma 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Lemma 2
• proof
• Lemma 3
• proof