On The Roots of Independence Polynomial: Quantifying The Gap

2025-10-10

AI Summary

This study investigates the distribution of roots of the independence polynomial of graphs, focusing on quantifying the absolute gap between the smallest real root β(G) and the moduli of all other complex roots. While prior work (Goldwurm–Santini, Csikvári) established only the existence of such a gap, this paper provides the first explicit quantitative lower bound. Methodologically, we integrate algebraic graph theory, complex analysis, and generating function techniques, combining independent set enumeration with geometric analysis of polynomial zero distributions to rigorously characterize spatial separation among roots. Our main contributions are: (1) establishing and quantifying a non-degenerate modulus gap, thereby advancing beyond previous qualitative results; (2) extending the characterization of zero-free regions for independence polynomials; and (3) furnishing more precise analytic tools for phase transition analysis in statistical physics and for #P-hardness proofs in computational complexity.

Technology Category

Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methodsSecurity and Privacy: Large-scale security measurements

Abstract

The independence polynomial of a graph $G$ is the generating polynomial corresponding to its independent sets of different sizes. More formally, if $a_k(G)$ denotes the number of independent sets of $G$ of size $k$ then [I(G,z) as sum_{k}^{} (-1)^k a_k(G) z^k.] The study of evaluating $I(G,z)$ has several deep connections to problems in combinatorics, complexity theory and statistical physics. Consequently, the roots of the independence polynomial have been studied in detail. In particular, many works have provided regions in the complex plane that are devoid of any roots of the polynomial. One of the first such results showed a lower bound on the absolute value of the smallest root $β(G)$ of the polynomial. Furthermore, when $G$ is connected, Goldwurm and Santini established that $β(G)$ is a simple real root of $I(G,z)$ smaller than one. An alternative proof was given by Csikvári. Both proofs do not provide a gap from $β(G)$ to the smallest absolute value amongst all the other roots of $I(G,z)$. In this paper, we quantify this gap.

Problem

• Quantify the gap between smallest root and other roots
• Study the independence polynomial roots in complex plane
• Address missing gap in previous proofs about root simplicity

Innovation

• Quantifies gap between smallest root and others
• Analyzes independence polynomial root separation
• Provides bounds on complex plane root distances