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A graph-theoretic approach to ring analysis: Dominant metric dimensions in zero-divisor graphs
Nasir Ali1, Hafiz Muhammad Afzal Siddiqui1, Muhammad Bilal Riaz2,3
1Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan.
Heliyon
|May 30, 2024
Summary
This study explores dominant metric dimensions in zero divisor graphs (ZD-graphs) of commutative rings. Researchers established bounds for these dimensions, revealing structural insights into different ring types.
Area of Science:
- Abstract Algebra
- Graph Theory
- Commutative Ring Theory
Background:
- Zero divisor graphs (ZD-graphs) visually represent algebraic structures of rings.
- Understanding metric dimensions in ZD-graphs is crucial for analyzing ring properties.
Purpose of the Study:
- To investigate and establish general bounds for the dominant metric dimension (Ddim) of ZD-graphs.
- To analyze the structural properties of ZD-graphs for specific commutative rings.
Main Methods:
- Construction of ZD-graphs for finite commutative rings with unity.
- Examination of ZD-graphs for specific rings: Gaussian integers modulo n, integers modulo n, and quotient polynomial rings.
- Derivation of bounds for Ddim based on graph properties.
Main Results:
- Established general bounds for the dominant metric dimension of ZD-graphs.
- Identified structural similarities and differences among rings with identical metric dimensions.
- Presented a general result linking Ddim to maximum degree, girth, clique number, and diameter.
Conclusions:
- The study provides a framework for analyzing commutative rings via their ZD-graphs.
- Findings contribute to understanding the relationship between algebraic properties and graph-theoretic invariants.
- Results advance theoretical knowledge in abstract algebra and graph theory.
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