Clean Graphs and Idempotent Graphs over Finite Rings: An Approach Based on Z_n
Felicia Servina Djuang, Indah Emilia Wijayanti, Yeni Susanti
TL;DR
The paper addresses how clean graphs $Cl_2(R)$ over finite rings relate to idempotent graphs $I(R)$, focusing on $R=\mathbb{Z}_n$. It corrects a previously reported degree formula for $Cl_2(R)$ and establishes the general isomorphism $Cl_2(R) \cong Shu^{|U'(R)|}_{|U(R)|}(I(R))$, linking clean graphs to the underlying idempotent structure via unit groups. It then derives explicit structures for $Cl_2(\mathbb{Z}_{p^n})$, $Cl_2(\mathbb{Z}_{p^n q^m})$, and $Cl_2(\mathbb{Z}_{p_1^{n_1}p_2^{n_2}p_3^{n_3}})$ as Shuriken graphs with parameters determined by unit groups, and extends these ideas to four primes and beyond. Finally, it presents a general decomposition result: $Cl_2(\mathbb{Z}_n) \cong Shu^{2^{k-1}}_m(I(\mathbb{Z}_n))$ when $2\mid n$ and $4\nmid n$ (with variants for higher 2-powers), offering a unified graph-theoretic framework for clean graphs over $\mathbb{Z}_n$ in terms of $I(\mathbb{Z}_n)$.
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.