On the Clean Graph of a Ring

TL;DR

This paper investigates the structure of the induced subgraph of the clean graph of a ring, determining its Wiener index, perfect matchings, and matching number under certain conditions.

Contribution

It introduces the analysis of the induced subgraph $Cl_2(R)$ of the clean graph of a ring and computes key graph invariants like Wiener index and matching properties.

Findings

Determined the Wiener index of $Cl_2(R)$.

Proved $Cl_2(R)$ has a perfect matching.

Calculated the matching number when $|U(R)|$ is odd.

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.