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Linear recurrences for non-log-concave independence polynomials of trees

César Bautista-Ramos, Carlos Guillén-Galván, Paulino Gómez-Salgado

Abstract

We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle $|z+1/3|=1/3$ in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv:2502.10654v1, 2025].

Linear recurrences for non-log-concave independence polynomials of trees

Abstract

We identify a structural pattern in the construction of known infinite families of trees whose independence polynomials are not log-concave. Using this pattern and properties of polynomial ring ideals, we derive linear recurrences for these polynomials. As a consequence, we prove that the set of non-isolated limit points of their zeros lies on the circle in the complex plane. Building on these recurrences, we also exhibit infinite families of trees whose independence polynomials break log-concavity at one, two, and three consecutive indices, as well as finite families that break log-concavity at four and five consecutive indices. Our approach suggests that arbitrarily many consecutive breaks may be achievable, offering further insight into a question posed by Galvin [D. Galvin, Trees with non log-concave independent set sequences, arXiv:2502.10654v1, 2025].
Paper Structure (10 sections, 18 theorems, 54 equations, 4 figures, 1 table)

This paper contains 10 sections, 18 theorems, 54 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Patterns and their linear recurrences
  3. Pattern graphs
  4. Linear recurrences for the pattern graphs
  5. Polynomial rings, recursive patterns, and their linear recurrences
  6. Applications
  7. Non-isolated limit points of zeros
  8. Trees breaking log-concavity at one index
  9. Trees breaking log-concavity at two indices
  10. Log-concavity broken at three or more indices

Key Result

Theorem 2.2

Let $G^v$ and $H^w$ be simple rooted graphs. For all $k \geq 2$, This is a linear recurrence with characteristic polynomial

Figures (4)

  • Figure 1: Trees with non-log-concave independence polynomials, for $k\geq 4$.
  • Figure 2: The scale graph $Z_k(H^w)$ (left) and the pattern graph $(G^v:H^w)_k$ (right), with base graph $G$ rooted at $v$ and pendant graph $H$ rooted at $w$.
  • Figure 3: Calculation of the independence polynomial of $(G^v:H^w)_k$ by applying Eq. \ref{['eq:Aro']} at the root $w$ (filled circle) of a copy of the pendant graph $H$.
  • Figure 4: The pattern graph $(G^v:H^w)_k^{(m)}$. Eq. \ref{['eq:Aro']} is applied to the filled vertex $v$.

Theorems & Definitions (42)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 32 more