On the Clean Graph of a Ring

TL;DR

This work studies the induced clean graph $Cl_2(R)$ of a ring with identity, focusing on the Wiener index $W(Cl_2(R))$ and matchings. By leveraging the decomposition $R\cong R_1\times\cdots\times R_n$ into local rings and the unit-idempotent structure, it derives a closed-form Wiener index in terms of $n$, $|U(R)|$, and $|U'(R)|$, namely $W(Cl_2(R)) = \frac{1}{2} \left[ |U(R)|^2 (2\cdot 2^{2n} - 5\cdot 2^{n} + 5) - |U(R)|(2^{2n} - 2^{n} + 3) + |U'(R)| 2^{n} \right]$, and specializes to $R=\mathbb{Z}_n$ with explicit formulas and the example $W(Cl_2(\mathbb{Z}_{15}))=332$.$ It proves that $Cl_2(R)$ possesses a perfect matching when $|U(R)|$ is even and determines the matching number in the odd-unit case, linking algebraic properties to graph-theoretic parameters. The results contribute tools for analyzing algebraic graphs and suggest a product-graph viewpoint that could unify idempotent and unit-structure graphs.

Abstract

Let $R$ be a ring (not necessarily a commutative ring) with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph with vertices in the form of an ordered pair $(e,u)$, where $e$ is an idempotent and $u$ is a unit of ring $R$, respectively. Two distinct vertices $(e,u)$ and $(f,v)$ are adjacent in $Cl(R)$ if and only if $ef=fe=0$ or $uv=vu=1$. In this study, we considered the induced subgraph $Cl_2(R)$ of $Cl(R)$. We determined the Wiener index of $Cl_2(R)$, and we showed $Cl_2(R)$ has a perfect matching. In addition, we determined the matching number of $Cl_2(R)$ if $|U(R)|$ is not even.

Figures (3)

• Figure 1: Partition of $V(\mathbf{R})$
• Figure 2: $Cl_2{(\mathbb{Z}_{pq})}$
• Figure 3: Perfect matching in $\mathbf{Cl_2(R)}$

Theorems & Definitions (12)

• Proposition 2.1: Hewitt
• Theorem 2.1: Habibi
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Proposition 3.1
• proof
• Corollary
• Example 3.1