On the number of connected edge cover sets in a graph

Let $G=(V,E)$ be a simple graph of order $n$ and size $m$. A connected edge cover set of a graph is a subset $S$ of edges such that every vertex of the graph is incident to at least one edge of $S$ and the subgraph induced by $S$ is connected. We initiate the study of the number of the connected edge cover sets of a graph $G$ with cardinality $i$, $e_{c}(G,i)$ and consider the generating function for $e_{c}(G,i)$ which is called the connected edge cover polynomial of $G$. After obtaining some results for this polynomial, we investigate this polynomial for some certain graphs.