Roman domination number of zero-divisor graphs over commutative rings

For a graph $G= (V, E)$, a Roman dominating function is a map $f : V \rightarrow \{0, 1, 2\}$ 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 $f(V)= \Sigma_{u \in V} f(u)$, 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 $\Gamma(R)$ and find the bounds of the Roman domination number of $T(\Gamma(R))$.