The stability of independence polynomials of complete bipartite graphs

TL;DR

This paper investigates the stability of independence polynomials of complete bipartite graphs, proving stability for certain classes and identifying conditions under which stability does not hold.

Contribution

It establishes stability for $K_{2,n}$ and $K_{3,n}$, and characterizes stability for $K_{m,m+k}$ and $K_{m, rac{p}{q}m}$ graphs.

Findings

$K_{2,n}$ and $K_{3,n}$ are stable.

Stability of $K_{m,m+k}$ for large $m$.

Non-stability of $K_{m, rac{p}{q}m}$ for certain ratios.

Abstract

The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z \in \mathbb{C} : \mathrm{Re}(z) \leq 0\}$, and the graph itself is also referred to as stable. Brown and Cameron (Electron. J. Combin. 25(1) (2018) \#P1.46) proved that the complete bipartite graph $K_{1,n}$ is stable and posed the question: \textbf{Are all complete bipartite graphs stable?} We answer this question by establishing the following results: \begin{itemize} \item The complete bipartite graphs $K_{2,n}$ and $K_{3,n}$ are stable. \item For any integer $k\geq0$, there exists an integer $N(k)\in \mathbb{N}$ such that $K_{m,m+k}$ is stable for all $m>N(k)$. \item For any rational $\ell> 1$, there exists an integer $N(\ell) \in \mathbb{N}$ such that whenever $m >N(\ell)$ and $\ell \cdot m$ is an integer, $K_{m, \ell \cdot m}$ is \textbf{not} stable. \end{itemize}