This article examines the independent domination polynomial of graphs, which encodes the number of independent dominating sets of all possible sizes, a problem classified as NP-hard in computational complexity. We analyze this topic for zero divisor graphs of commutative ring \(\mathbb {Z}_{n},\) and examine its complex zeros, previously examined by Gürsoy et al. (Soft Comput 26(15):6989–6997, 2022). We illustrate that the independent domination polynomial of zero divisor graphs of the ring \(\mathbb {Z}_{n}\) for \(n\in \{p^{2}q, pqr\},\) possesses complex zeros, which we identify in the plane, thereby refine their existing results, where \(2<p<q<r\) are prime numbers. Additionally, we present the independent domination polynomial of zero divisor graphs of ring \(\mathbb {Z}_{(pq)^{2}}\) , examine its log-concave and unimodal characteristics, and investigate its zeros. Finally, we demonstrate the applicability of independent dominating sets in the virtual communication backbone of wireless ad-hoc or sensor networks, as well as the single-error-correcting code in coding theory.
