Resolving Properties of Graphs Associated to Commutative Rings

Abstract

Let 𝑆 be a finite commutative unital ring having some non-zero elements 𝑥 and 𝑦 such that 𝑥𝑦=0. The elements of 𝑆 possesses such property are called the zero-divisors, the set of all these elements is denoted by 𝑍(𝑆). We can associate a graph to 𝑆 by means of a zero-divisor set 𝑍(𝑆) denoted by 𝜁(𝑆) (called the zero-divisor graph) to study the algebraic properties of the ring 𝑆. In this research work, we aim to produce some general bounds for edge and mixed version of metric dimension regarding some zero-divisor graphs of some families of rings. Then we prove general result for the upper and lower bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of 𝜁(𝑆). Finally, we discuss some relationships between girth, diameter and mixed metric dimension of zero-divisor graphs.