Whole Domination Parameter in Zero-Divisor Graphs with Applications to Minimal Biomarker Identification

Abstract

Domination theory plays a significant role in the structural analysis of graphs and their applications to real-world systems. In this study, we introduce and investigate whole domination parameter in zero-divisor graphs associated with commutative rings. We establish characterization for whole dominating sets, and identify algebraic conditions under which such sets exist or fail to exist. Also whole domination number of zero-divisor graphs corresponding to rings that are product of finite fields and rings such as  are determined. Furthermore, the concept is applied to a biological framework, where vertices represent metabolic biomarkers and edges denote pathological interactions. Through this mapping, whole domination provides a mathematical approach for identifying minimal sets of biomarkers that can monitor or predict metabolic dysfunction. The study bridges algebraic graph theory and biomedical modeling, offering both a new theoretical perspective and a potential diagnostic interpretation.