Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on $\mathbb{Z}_n$

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Document Type : Original paper

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Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia

Abstract

Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent element of } R\}$. In this study, we examine the structure of clean graphs over $\mathbb{Z}_{n}$ derived from their $Cl_2$ graphs and investigate their relationship with the structure of their idempotent graphs. Furthermore, we obtain an equivalence between the isomorphism of two clean graphs and the isomorphism of their corresponding idempotent graphs over an Artinian ring.

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References

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Communications in Combinatorics and Optimization

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Available Online from 08 May 2026

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