Independent domination polynomial of comaximal graphs of commutative rings

The comaximal graph $\Gamma(R)$ of a commutative ring $R$ is a simple graph with vertex set $R$ and two distinct vertices $a$ and $b$ of $\Gamma(R)$ are adjacent if and only if $aR+bR=R$, where $aR$ is the ideal generated by $a$ in $R$. In this article, the independent domination polynomial $D_{i}(\Gamma(\mathbb{Z}_{n}),x)$ of $\Gamma(\mathbb{Z}_{n})$ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(\Gamma(\mathbb{Z}_{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $I(\Gamma(\mathbb{Z}_{n}),x )$ of $\Gamma(\mathbb{Z}_{n})$ for special values of $n$ and provide a general result associated with it. The bounds for the zero of the polynomial $I(\Gamma(\mathbb{Z}_{n}),x )$ are established, and their log-concave and unimodal properties are examined.