Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on Z_n

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 $C{l}_2(R)$ 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 $C{l}_2$ graphs and investigate their relationship with the structure of their idempotent graphs.