Independent domination polynomial of comaximal graphs of commutative rings

TL;DR

This paper investigates the independent domination polynomial and independence polynomial of comaximal graphs of integers modulo n, exploring their properties such as unimodality, log-concavity, zeros, and bounds.

Contribution

It introduces new results on the properties and zeros of these polynomials for comaximal graphs of Z_n, including specific cases and general bounds.

Findings

Independent domination polynomial exhibits unimodal and log-concave properties for certain n.

Zeros of the independence polynomial are characterized and bounded.

Explicit formulas for the independence polynomial are provided for special n.

Abstract

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.