Independent domination polynomial of comaximal graphs of commutative rings

The comaximal graph $\mathrm{\Gamma }(R)$ of a commutative ring $R$ is a simple graph with vertex set $R$ and two distinct vertices $a$ and $b$ of $\mathrm{\Gamma }(R)$ 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 $D_i(\mathrm{\Gamma }({\mathbb{Z}}_n),x)$ of $\mathrm{\Gamma }({\mathbb{Z}}_n)$ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_i(\mathrm{\Gamma }({\mathbb{Z}}_n),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $I(\mathrm{\Gamma }({\mathbb{Z}}_n),x)$ of $\mathrm{\Gamma }({\mathbb{Z}}_n)$ for special values of $n$ and provide a general result associated with it. The bounds for the zero of the polynomial $I(\mathrm{\Gamma }({\mathbb{Z}}_n),x)$ are established, and their log-concave and unimodal properties are examined.