Roman domination number of zero-divisor graphs over commutative rings

For a graph $G=(V,E)$, a Roman dominating function is a map $f:V\to \{0,1,2\}$ satisfies the property that if $f(v)=0$, then $v$ must have adjacent to at least one vertex $u$ such that $f(u)=2$. The weight of a Roman dominating function $f$ is the value $f(V)={\mathrm{\Sigma }}_{u\in V}f(u)$, and the minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$, denoted by ${\gamma }_R(G)$. The main focus of this paper is to study the Roman domination number of zero-divisor graph $\mathrm{\Gamma }(R)$ and find the bounds of the Roman domination number of $T(\mathrm{\Gamma }(R))$.