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

TL;DR

This paper explores the roots of independence polynomials of Generalized Petersen graphs, revealing a parity-based pattern where roots are complex for odd k and real for even k, leading to a conjecture about their real-rootedness.

Contribution

It introduces an exact transfer matrix algorithm for computing independence polynomials of GP(n,k) and uncovers a parity-dependent root distribution pattern, proposing a new conjecture.

Findings

Roots form complex conjugate structures for odd k

Roots are strictly real and negative for even k

Conjecture: independence polynomials are real-rooted iff k is even

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.