On the Clean Graph of a Ring

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 $C{l}_2(R)$ of $Cl(R)$. We determined the Wiener index of $C{l}_2(R)$, and we showed $C{l}_2(R)$ has a perfect matching. In addition, we determined the matching number of $C{l}_2(R)$ if $\mathrm{\mid }U(R)\mathrm{\mid }$ is not even.