Independent domination polynomial of comaximal graphs of commutative rings
This work addresses a niche problem in algebraic graph theory for researchers studying combinatorial properties of ring structures, and it is incremental as it extends existing graph-theoretic concepts to specific ring cases.
The paper tackles the problem of analyzing the independent domination polynomial and independence polynomial of comaximal graphs for commutative rings, specifically for rings of integers modulo n, and presents results on their unimodal, log-concave properties and zero bounds for certain values of n.
The comaximal graph $ Î(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ Î(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}(Î(\mathbb{Z}_{n}),x) $ of $ Î(\mathbb{Z}_{n}) $ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(Î(\mathbb{Z}_{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $ I(Î(\mathbb{Z}_{n}),x ) $ of $ Î(\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(Î(\mathbb{Z}_{n}),x ) $ are established, and their log-concave and unimodal properties are examined.