Independent domination polynomial of comaximal graphs of commutative rings

Abstract

The comaximal graph $ Γ(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ Γ(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}(Γ(\mathbb{Z}_{n}),x) $ of $ Γ(\mathbb{Z}_{n}) $ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(Γ(\mathbb{Z}_{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $ I(Γ(\mathbb{Z}_{n}),x ) $ of $ Γ(\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(Γ(\mathbb{Z}_{n}),x ) $ are established, and their log-concave and unimodal properties are examined.