Isomorphism of Clean Graphs over Zₙ and Structural Insight into M₂(Zₚ)

Let R be a finite ring with identity. The clean graph $Cl(R)$ of a ring R is a graph whose vertices are pairs $(e, u)$, 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 Cl₂(R) is the induced subgraph of $Cl(R)$ induced by the set \(e, u): e is a nonzero idempotent and u is a unit of R\. In this study, we present properties that arise from the isomorphism of two clean graphs and conditions under which two clean graphs over direct product rings are isomorphic. We also examine the structure of the clean graph over the ring M₂(Zₚ) through their Cl₂ graph.