Fault-Tolerant Edge Metric Dimension of Zero-Divisor Graphs of Commutative Rings

Abstract

In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring S with unity. Consider a graph Γ(S) with vertex set V(Γ) = Z*(S), and two vertices in Γ(S) are adjacent if and only if their product is zero. This graph Γ(S) is known as zero-divisor graph of S. Zero-divisor graphs provide a powerful bridge between abstract algebra and graph theory. The zero-divisor graphs for finite commutative rings and their minimum fault-tolerant edge-resolving sets are studied in this article. Through analytical and constructive techniques, we highlight how the algebraic properties of the ring influence the edge metric structure of its associated graph. In addition to this, the existence of a connected graph G having a resolving set of cardinality of 2n + 2 from a star graph K1,2n, is studied.

Keywords

• zero-divisor graph
• resolving set
• metric dimension
• edge metric dimension
• fault-tolerant edge metric dimension