Unimodality conjecture for independence polynomials of zero-divisor graphs of integer rings

Let $n$ be a positive integer and let $\varGamma(\mathbb{Z}_n)$ denote the zero-divisor graph of the ring $\mathbb{Z}_n$. Its independence polynomial is

$$I\bigl(\varGamma(\mathbb{Z}_n),x\bigr)=\sum_{k=0}^{\alpha(\varGamma(\mathbb{Z}_n))}s_k\bigl(\varGamma(\mathbb{Z}_n)\bigr)x^k.$$

Zero-divisor graph unimodality conjecture. The independence polynomial $I\bigl(\varGamma(\mathbb{Z}_n),x\bigr)$ is unimodal. The paper presents this as a conjecture following computational experiments and results for the families $n\in\{p,p^2,p^3,pq,p^2q\}$; the supplied text does not establish a resolution for arbitrary $n$.