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

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Listed:
  • Hafiz Muahmmad Afzal Siddiqui
  • Ammar Mujahid
  • Muhammad Ahsan Binyamin
  • Muhammad Faisal Nadeem

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.

Suggested Citation

  • Hafiz Muahmmad Afzal Siddiqui & Ammar Mujahid & Muhammad Ahsan Binyamin & Muhammad Faisal Nadeem, 2021. "On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-7, December.
    • Handle: RePEc:hin:jnlmpe:5826722
    DOI: 10.1155/2021/5826722
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