Let $G$ be a simple graph of order $n$. The connected domination polynomial of $G$ is the polynomial $D_{c}(G, x)=\sum_{i=\gamma_{c}(G)}^{|V(G)|}d_{c}(G, i)x^i$, where $d_{c}(G, i)$ is the number of connected dominating sets of $G$ of size $i$ and $\gamma_{c}(G)$ is the connected domination number of $G$. In this paper we study $D_{c}(G, x)$ of any graph. We classify many families of graphs by studying their connected domination polynomial.
