Domination polynomials of cubic graphs of order 10

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=sum_{i=gamma(G)}^n d(G,i) x^i, where d(G,i) is the number of dominating sets of G of size i, and gamma(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.