Closing Trees into Unicyclic Counterexamples

2026-03-17

AI Summary

This study addresses the long-standing absence of explicit unimodal yet non-log-concave counterexamples in the theory of independence polynomials by constructing an infinite family of unicyclic graphs \( U_{k,r} \), derived via KL-closure, whose independence polynomials are provably unimodal but not log-concave. The proof combines a decomposition of the polynomial into a dominant convolution component and a real-root correction term with an array of analytic and combinatorial tools—including Ibragimov’s strong unimodality theorem, Darroch’s localization principle, residue-class growth analysis, and a bridge lemma on adjacency patterns—to rigorously establish unimodality while disproving log-concavity. Notably, for all \( k \leq 400 \), the failure of the penultimate log-concavity inequality is shown to be unique, and exact modular formulas for all coefficients of the full independence polynomial are provided.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

We develop a family-based route to unicyclic graphs whose independence polynomials are unimodal but not log-concave. The paper is organized around one flagship statement: for the explicit KL-closure family $U_{k,r}$, with $r\in\{0,1,2\}$ and admissible $k$, the independence polynomial is unimodal but not log-concave. The proof separates the closure polynomial into a dominant convolution term and a real-rooted correction term. On the non-log-concavity side, we prove symbolically that the penultimate log-concavity inequality fails for every admissible parameter. On the unimodality side, we prove that the main convolution term $H_{k,r}=G_kF_{k+r}$ is unimodal with a controlled mode, using a combination of exact coefficient formulas, Ibragimov's strong-unimodality principle, and a residue-class growth argument. Darroch localization and an adjacent-mode bridge lemma then transfer that mode statement to the full KL closure polynomial. This yields an explicit infinite family of unicyclic graphs with unimodal but non-log-concave independence polynomials. In the exact range $k\le 400$, we further verify that the penultimate break is unique and determine exact mode formulas for $H_{k,r}$, the binomial correction term, and $I(U_{k,r};x)$ itself. The paper also places the KL family inside a broader reservoir program involving Galvin, Ramos-Sun, and Bautista-Ramos trees, from which we obtain substantial universal exact theorems for finite ranges.

Problem

• unicyclic graphs
• independence polynomial
• unimodality
• log-concavity
• counterexamples

Innovation

• unicyclic graphs
• independence polynomial
• unimodality
• log-concavity
• KL-closure family