DOMINATION POLYNOMIALS OF THE JEWEL GRAPH AND ITS COMPLEMENT

Let $G=(V(G),E(G))$ be a simple graph. The Jewel graph $J_n$ is a graph with vertex set $V(J_n)=\{u,v,x,y,u_i : 1\leq i\leq n\}$ and edge set $E(J_n)=\{ux,uy,xy,xv,yv,uu_i,vu_i :1\leq i\leq n\}$. The domination polynomial of a graph $G$ of order $n$ 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 cardinality $i$. In this paper, we present various domination polynomials of the Jewel graph $J_n$. Also we determine the same results for the complement of the Jewel graph.