Domination polynomial of co-maximal graphs of integer modulo ring

TL;DR

This paper studies the domination polynomial of co-maximal graphs over integer modulo rings, providing explicit formulas, structural insights, and analyzing the roots to reveal properties like unimodality and log-concavity.

Contribution

It derives explicit formulas for specific cases, connects the domination polynomial to subgraph structures, and analyzes the roots to establish bounds and properties.

Findings

Polynomials are unimodal and log-concave for certain n.

Explicit formulas are provided for n = p^{n_1} and n = p^{n_1}q^{n_2}.

Bounds for domination roots are established using Eneström–Kakeya theorem.

Abstract

We investigate the domination polynomial of the co-maximal graph $\Gamma(\mathbb{Z}_n)$ related to the ring of integers modulo $n$. Explicit formulas are derived for \( n = p^{n_1} \) and \( n = p^{n_1}q^{n_2} \), demonstrating that the resulting polynomials exhibit unimodality and log-concavity. For general $n$, we present structural expressions that connect $D(\Gamma(\mathbb{Z}_n),x)$ to appropriate induced subgraphs. Finally, we examine domination roots and establish bounds for their moduli using the Enestr\"om--Kakeya theorem.