A note on the Independent domination polynomial of zero divisor graph of rings

TL;DR

This paper investigates the independent domination polynomial of zero divisor graphs of certain rings, revealing non-unimodality, real zeros, and log-concavity, while correcting previous misconceptions and extending understanding of their algebraic properties.

Contribution

It corrects prior claims about unimodality, proves log-concavity, and analyzes the zero distribution of these polynomials for specific ring structures.

Findings

Independent domination polynomials are not unimodal for certain rings.

These polynomials have only real zeros, contradicting earlier assumptions.

The polynomials are proven to be log-concave despite having complex zeros.

Abstract

In this note we consider the independent domination polynomial problem along with their unimodal and log-concave properties which were earlier studied by G\"ursoy, \"Ulker and G\"ursoy (Soft Comp. 2022). We show that the independent domination polynomial of zero divisor graphs of $\mathbb{Z}_{n}$ for $n\in \{ pq, p^{2}q, pqr, p^{\alpha}\}$ where $p,q,r$ are primes with $2<p<q<r$ are not unimodal thereby contradicting the main result of G\"ursoy, \"Ulker and G\"ursoy \cite{gursoy}. Besides the authors show that the zero of the independent domination polynomial of these graphs have only real zero and used concept of Newton's inequalities to establish the log-concave property for the afore said polynomials. We show that these polynomials have complex zeros and the technique of Newton's inequalities are not applicable. Finally, by definition of log-concave, we prove that these polynomials are log-concave and fix the flaws in Theorem 10 of G\"ursoy, \"Ulker and G\"ursoy \cite{gursoy}.