Sombor index of clean graphs

Let $G=(V,E)$ be a graph with the vertex set $V(G)$ and edge set $E(G)$. The Sombor index of $G$, $SO(G)$, is defined as ${\sum }_{uv\in E(G)}\sqrt{deg(u{)}^2+deg(v{)}^2}$, where $deg(u)$ is the degree of vertex $u$ in $V(G)$. The clean graph of a ring R, denoted by $Cl(R)$, is a graph with vertex set $\{(e,u):e\in Id(R),u\in U(R)\}$ and two distinct vertices $(e,u)$ and$(f,v)$ are adjacent if and only if $ef=0$ or $uv=1$ ($Id(R)$ and $U(R)$ are the sets of idempotents and unit elements of R, respectively). The induced subgraph on $\{(e,u):e\in I{d}^{\ast }(R),u\in U(R)\}$ is denoted by $C{l}_2(R)$. In this paper, $SO(Cl2({\mathbb{Z}}_n))$, for different values of the positive integer $n$, is investigated.