On Multiplicative Regular Graphs of Commutative Rings

الملخص

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.