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

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}.