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Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on $\mathbb{Z}_n$ | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 18 اردیبهشت 1405 اصل مقاله (475.68 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31307.2811 | ||
| نویسندگان | ||
| Felicia Servina Djuang؛ Indah Emilia Wijayanti* ؛ Yeni Susanti | ||
| Department of Mathematics, Universitas Gadjah Mada, Yogyakarta, Indonesia | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Clean graph؛ idempotent graph؛ isomorphism graph؛ local ring؛ Artinian ring | ||
| مراجع | ||
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