Parity-Dependent Real-Rootedness in Independence Polynomials of Generalized Petersen Graphs

TL;DR

This paper examines the zeros of the independence polynomial $I(\mathrm{GP}(n,k),x)$ for Generalized Petersen graphs ${GP(n,k)}$ and uncovers a sharp parity-driven pattern: odd $k$ yields complex-conjugate root curves while even $k$ yields strictly real, negative roots. Using an exact transfer-matrix method parameterized by $k$, the authors compute $I(\mathrm{GP}(n,k),x)$ for $n\le 30$ and $k\in\{1,2,3,4\}$, and perform precise root-finding with verification to $10^{-10}$. They conjecture that $I(\mathrm{GP}(n,k),x)$ is real-rooted if and only if $k$ is even, linking the algebraic root structure to inner-edge bipartiteness and the zero-free regions relevant to the hard-core lattice gas model. If proven, this parity conjecture would identify a new infinite family of real-rooted independence polynomials beyond claw-free graphs and sharpen our understanding of when cubic graphs exhibit real-rootedness and log-concavity in their coefficient sequences.

Abstract

We investigate the distribution of zeros of the independence polynomial ${\rm I}(G, x)$ for the family of Generalized Petersen graphs ${\rm GP}(n, k)$ in the complex plane. While the independence numbers and coefficients of these graphs have been studied, the global behavior of their roots remains largely unexplored. Using an exact transfer matrix algorithm parameterized by $k$, we compute ${\rm I}({\rm GP}(n,k), x)$ for $n$ up to $30$ and $k \in \{1, 2, 3, 4\}$. Our numerical analysis reveals a striking parity-based dichotomy: for odd $k$, the roots exhibit complex conjugate structures accumulating on closed curves, whereas for even $k$, the roots appear to be strictly real and negative. Motivated by this evidence, we conjecture that ${\rm I}({\rm GP}(n,k), x)$ is real-rooted, and hence log-concave, if and only if $k$ is even. This phenomenon connects algebraic properties of ${\rm GP}(n,k)$ to questions about zero-free regions and limiting behavior in the hard-core lattice gas model.

Figures (1)

• Figure 1: Root distributions for $\mathop{\mathrm{GP}}\nolimits(n,k)$. Even values of $k$ produce exclusively real negative roots, while odd values of $k$ generate complex conjugate curves.

Theorems & Definitions (2)

• Definition 2.1: watkins1969theorem
• Conjecture 4.1: Parity Conjecture