Strong domination over zero-divisor graphs Γ(ℤn)

This paper investigates the concept of strong domination within zero-divisor graphs (ZDGs), aiming to compute their strong domination numbers. A zero-divisor graph (ZDG), denoted by [Formula: see text], is constructed by taking the nonzero zero-divisors of a commutative ring [Formula: see text] with unity as its vertices. An edge connects two distinct vertices ([Formula: see text] and [Formula: see text]) if their product [Formula: see text] is zero. This specific definition, excluding the zero element, was a redefinition and simplification by Anderson and Livingston in 1999 from Istvan Beck’s original concept in 1988, allowing for a clearer study of the properties of zero-divisors. The broader concept of domination in graphs has a notable history, initially defined by Berge [Theory of Graphs and its Applications (Methuen, London, 1962)] in 1962 and later popularized by Cockayne and Hedetniemi [Towards a theory of domination in graphs, Networks 7 (1977) 247–261] in 1977, and remains a vibrant and rapidly developing area in graph theory. A strong dominating set [Formula: see text] is a subset of ZDG vertices such that for every vertex [Formula: see text] not in [Formula: see text], there exists a vertex [Formula: see text] in [Formula: see text] where [Formula: see text] and [Formula: see text] are connected by an edge (their product is zero), and crucially, the degree of [Formula: see text] is less than or equal to the degree of [Formula: see text]. The strong domination number [Formula: see text] is then defined as the minimum possible number of vertices in such a strong dominating set(SDS). The results related to strong domination number of [Formula: see text] are stated and proved. Aside from the theoretical findings, we also formulate a Python algorithm that can calculate the strong domination number of [Formula: see text].