On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

Hafiz Muahmmad Afzal Siddiqui, Ammar Mujahid, Muhammad Ahsan Binyamin and Muhammad Faisal Nadeem

Mathematical Problems in Engineering, 2021, vol. 2021, 1-7

Abstract:

Given a finite commutative unital ring having some non-zero elements such that , the elements of that possess such property are called the zero divisors, denoted by . We can associate a graph to with the help of zero-divisor set , denoted by (called the zero-divisor graph), to study the algebraic properties of the ring . In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of . To do so, we will discuss the zero-divisor graphs for the ring of integers modulo , some quotient polynomial rings, and the ring of Gaussian integers modulo . 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 . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlmpe:5826722

DOI: 10.1155/2021/5826722

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