On Multiplicative Regular Graphs of Commutative Rings
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Abstract
Background:
The zero-divisor graph is central in algebraic graph theory. Let be a commutative ring with identity. In this work we define the multiplicative regular graph as a generalization that captures the behavior of regular elements, those satisfying or.
Materials and Methods:
The graph is defined using ring-theoretic conditions and graph-theoretic representations, Two distinct none-zero elements and are adjacent if and only if . A Python-based algorithm is used to construct and visualize for finite commutative rings.
Results:
We prove that if and only if , and when is a regular ring. The study includes analysis of key graph invariants such as connectivity, diameter, girth, and regularity.
Conclusion:
The graph offers a generalization of existing graphs of commutative rings. By emphasizing regular elements, this model uncovers new structural relationships within the ring and explores ring-theoretic properties through graph-theoretic methods.
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