Counting specific subgraphs
Your problem is known as counting the number of connected spanning subgraphs, and is pretty hard even for restricted classes of graphs. See this question on cstheory.

The number of connected spanning subgraphs of a graph $G$ equals $T_G(1,2)$, where $T_G$ is the Tutte polynomial of $G$ (this is mentioned in one of the answers to the linked question). Hence this paper of Björklund et al., which shows how to compute said polynomial in vertex-exponential time (i.e., $\exp O(n)$), gives us hope, since in your case $n$ is pretty small ($n \leq 15$).