Given a finite simple graph $G$ with $n$ vertices, for each 0-1 colouring $\alpha \in \mathbb{Z}_2^n$ of its vertices consider the subgraph $G_\alpha$, whose vertices are the $1$-coloured vertices in $\alpha$, created as follows: if two vertices with labels $1$ are adjacent, then add the edge between them. Is there a way of computing $$\Theta(G) = \sum_{k = 1}^n (-1)^k \sum_{\substack{\alpha \in 2^n\\|\alpha|= k}} \# G_\alpha,$$ where $|\alpha|$ is the sum of the components of $\alpha$ (so the number of vertices with label $1$) and $\#G_\alpha$ is the number of connected components of $G_\alpha$.

As an example, for complete graphs we have $\Theta(K_m)= -1$.

Are the values of $\Theta$ known at least on simple families of graphs, such as the cycle graphs or the $1$-skeletons of the $m$-dimensional cubes?