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On The Roots of Independence Polynomial: Quantifying The Gap

Om Prakash, Vikram Sharma

TL;DR

This work addresses the problem of quantifying the gap between the smallest real root $\beta(G)$ of the independence polynomial $I(G,z)$ and the other roots. It develops a two-pronged approach: (i) local univalence of $I(G,z)$ near $\beta(G)$ to isolate the smallest root, and (ii) a majorant-function framework that bounds $|f_u(z)|$ on the circle $|z|=\beta(G)$ and yields explicit exclusion discs for all other roots. The main result shows a concrete isolation radius $D\left(0,\beta(G)+\left(\dfrac{\beta(G)}{n}\right)^{O(\mathrm{dia}(G))}\right)$ containing only $\beta(G)$, together with explicit gap bounds for families such as $P_n$, $C_n$, and $K_{n\times n}$. The paper thus provides the first quantitative root separation for independence polynomials, built from a combination of univalence arguments, majorant bounds, and detailed analysis of the associated $f_u$-maps, with potential algorithmic implications for graph families with large gaps.

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.

On The Roots of Independence Polynomial: Quantifying The Gap

TL;DR

This work addresses the problem of quantifying the gap between the smallest real root of the independence polynomial and the other roots. It develops a two-pronged approach: (i) local univalence of near to isolate the smallest root, and (ii) a majorant-function framework that bounds on the circle and yields explicit exclusion discs for all other roots. The main result shows a concrete isolation radius containing only , together with explicit gap bounds for families such as , , and . The paper thus provides the first quantitative root separation for independence polynomials, built from a combination of univalence arguments, majorant bounds, and detailed analysis of the associated -maps, with potential algorithmic implications for graph families with large gaps.

Abstract

The independence polynomial of a graph is the generating polynomial corresponding to its independent sets of different sizes. More formally, if denotes the number of independent sets of of size then The study of evaluating 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 of the polynomial. Furthermore, when is connected, Goldwurm and Santini established that is a simple real root of smaller than one. An alternative proof was given by Csikvári. Both proofs do not provide a gap from to the smallest absolute value amongst all the other roots of . In this paper, we quantify this gap.

Paper Structure

This paper contains 8 sections, 12 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Univalence of $I(G,z)$ around $\beta(G)$
  4. Gap around every point on the circle with radius $\beta(G)$
  5. Upper bound on the Majorant Function near the origin
  6. Some Explicit Lower Bounds On The Gap
  7. Concluding Remarks
  8. Basic Results

Key Result

Theorem 1.1

Let $G$ be a connected graph on $n$ vertices. Then the disc centered at the origin contains only the smallest root $\beta(G)$ of $I(G,z)$.

Figures (1)

  • Figure 1: The absolute value function is not always monotone. In (a) we have the plot of $\left\lvert z/(1-z/(1-z)^{2})\right\rvert$ in red color, corresponding to $S_3$ as given in (\ref{['eq:fuex']}), where $z\textcolor{red}{\mathop{\rm :=}} \beta e^{i t}$, and $\beta\sim 0.318$ is the point where the function takes the value one. The plot in blue color shows the corresponding majorizing function. (b) Shows the derivative with respect to $t$ of the absolute value in $[0, \pi]$ and (c) shows the same graph zoomed in to highlight an additional root of the derivative besides $0$ and $\pi$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 6 more