The stability of independence polynomials of complete bipartite graphs

The independence polynomial of a graph is termed {\it stable} if all its roots are located in the left half-plane $\{z\in \mathbb{C}:\mathrm{R}\mathrm{e}(z)\le 0\}$, 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: $\mathbf{\text{Are}}\mathbf{\text{all}}\mathbf{\text{complete}}\mathbf{\text{bipartite}}\mathbf{\text{graphs}}\mathbf{\text{stable?}}$ 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 $k\ge 0$, 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 $\mathrm{\ell }>1$, there exists an integer $N(\mathrm{\ell })\in \mathbb{N}$ such that whenever $m>N(\mathrm{\ell })$ and $\mathrm{\ell }\cdot m$ is an integer, $K_{m,\mathrm{\ell }\cdot m}$ is $\mathbf{\text{not}}$ stable. \end{itemize}