on the roman domination polynomials

‎‎a roman dominating function (rdf) on a graph $g$ is a function $ f:v(g)to {0,1,2}$  satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. the weight of an rdf $f$ is the sum of the weights of the vertices under $f$. the roman domination number, $gamma_r(g)$ of $g$ is the minimum weight of an rdf in $g$. the roman domination polynomial of a graph $g$ of order $n$ is the polynomial $rd(g,x)=sum_{i=gamma_r(g)}^{2n} d_r(g,i) x^{i}$, where $d_r(g,i)$ is the number of rdfs of $g$ with weight $i$. in this paper we prove properties of roman domination polynomials and determine $rd(g,x)$ in several classes of graphs $g$ by new approaches. we also present bounds on the number of all roman domination polynomials in a graph.