A note on the Independent domination polynomial of zero divisor graph of rings
Bilal Ahmad Rather
TL;DR
The paper re-examines the independent domination polynomial $D_i(G,x)$ of the zero-divisor graph $\Gamma(\mathbb{Z}_n)$ for $n$ in $\{pq,p^2q,pqr,p^\alpha\}$ and questions the universality of unimodality and log-concavity claimed in prior work. It constructs explicit counterexamples with complex zeros (e.g., for $\mathbb{Z}_{pq}$, $D_i(G,x)=x^2+x^4$ has zeros $0$ and $\pm i$) and provides a corrected classification that several cases are not unimodal, while log-concavity may still hold in some instances, independent of Newton's inequalities. The authors show Newton's inequalities are inapplicable when zeros are nonreal and fix flaws in Theorem 10 by detailing the precise conditions under which unimodality or log-concavity fails or holds. The work highlights the delicate behavior of zeros in the complex plane and outlines directions to study other graph polynomials on zero-divisor graphs and to locate zeros more precisely.
Abstract
In this note we consider the independent domination polynomial problem along with their unimodal and log-concave properties which were earlier studied by Gürsoy, Ülker and Gürsoy (Soft Comp. 2022). We show that the independent domination polynomial of zero divisor graphs of $\mathbb{Z}_{n}$ for $n\in \{ pq, p^{2}q, pqr, p^α\}$ where $p,q,r$ are primes with $2<p<q<r$ are not unimodal thereby contradicting the main result of Gürsoy, Ülker and Gürsoy \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 Gürsoy, Ülker and Gürsoy \cite{gursoy}.