Domination-Degree-Based Topological Indices of Zero-Divisor Graphs

Source abstract

We examined the distribution of domination behavior in the zero-divisor graphs associated with [Formula: see text] for all [Formula: see text]. For every non-zero zero-divisor [Formula: see text], we define its domination-degree [Formula: see text] as the number of minimal dominating sets in which [Formula: see text] appears. Incorporating these values into Zagreb-type constructions yields the following three descriptors: [Formula: see text] These indices capture the concentration of domination-degrees across vertices, the interaction of dominating contributions along adjacency relations, and the structural imbalance arising from their distribution within the graph, respectively. Our contributions are twofold. First, we provide a complete catalogue of [Formula: see text], trio [Formula: see text], and the total number of minimal dominating sets for every [Formula: see text], organized by the factorization pattern. Second, because enumerating minimal dominating sets is notoriously expensive, we present a deterministic algorithm tailored to [Formula: see text] that uses Constraint Programming and Boolean Satisfiability (CP-SAT) to compute the number of minimal dominating sets and thereby reliably compute [Formula: see text] within this range. We close with the application sketches in fault-tolerant sensor placement and emergency facility placement illustrating how these indices translate algebraic constraints into actionable design guidance.