Independent domination polynomial of comaximal graphs of commutative rings

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.

arXiv comment22 pages, 7 figues, Accepted for publication in the journal "Algebra Colloquium" on April 16, 2025, and is about to appear online in the second issue of 2026