On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings
Hafiz Muahmmad Afzal Siddiqui, Ammar Mujahid, Muhammad Ahsan Binyamin, Muhammad Faisal Nadeem
Source abstract
Given a finite commutative unital ring S having some non-zero elements x , y such that x . y = 0 , the elements of S that possess such property are called the zero divisors, denoted by Z S . We can associate a graph to S with the help of 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 the edge version of metric dimension regarding zero-divisor graphs of S . To do so, we will discuss the zero-divisor graphs for the ring of integers ℤ m modulo m , some quotient polynomial rings, and the ring of Gaussian integers ℤ m i modulo m . Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ζ S . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.
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