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.

Figures (4)

• Figure 1: Comaximal graphs of $\mathbb{Z}_{5}$ and $\mathbb{Z}_{15}$
• Figure 2: Comaximal graphs of $\mathbb{Z}_{30}$
• Figure 3: Comaximal graphs of $\mathbb{Z}_{16}$ and $\mathbb{Z}_{36}$
• Figure 4: Representation of zeros of $I(\Gamma(\mathbb{Z}_{7\cdot 11}),x)$ and $I(\Gamma(\mathbb{Z}_{2^{5}}),x)$ on plane.

Theorems & Definitions (10)

• Lemma 2.1: afkhamibanerjeeMatrices
• Theorem 2.2
• Corollary 2.3
• Proposition 2.4
• Corollary 2.5
• Proposition 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 3.1
• Theorem 3.2