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?



number of components of subgraphs
Something to start with.


We have $$\Theta(G)=\sum\_{\alpha} (-1)^{\alpha}\#G\_\alpha=\sum\_{\alpha}(-1)^\alpha\sum\_U \mathbf{1}(U\,\text{is a component of}\,G\_{\alpha}),$$
where $U$ runs over all induced connected subgraphs of $G$. Changing the order of summation, we get
$$
\Theta(G)=\sum\_U \sum\_{\alpha:\, U\,\text{is a component of}\,G\_{\alpha}} (-1)^{\alpha}.
$$
Look at the inner sum. Let $\tilde{U}$ denote the set of vertices $v\in V\setminus U$ (here $V=$set of vertices of $G$), which have a neighbour in $U$, and $U^\*$ denote $V\setminus (U\cup \tilde{U})$. Then $G\_\alpha$ must contain $U$, should not have vertices in $\tilde{U}$ and may contain arbitrary subset of $U^\*$. It yields that the inner sum has $2^{|U^\*|}$ summands, and that it vanishes unless $U^\*=\emptyset$. In the latter case, the inner sum equals $(-1)^{|U|}$. Such $U$ for which $U^\*=\emptyset$ (that is, every vertex either belongs to $U$ or has a neighbour in $U$) are called dominating. Thus
$$
\Theta(G)=\sum\_{U\,\text{is dominating and connected}} (-1)^{|U|}.
$$
If $G=C\_n$ is a cycle of length $n$, then $U$ should be either the whole cycle (1 variant), or a path of length $n-1$ ($n$ variants) or a path of length $n-2$ ($n$ variants), so we get $\Theta(C\_n)=(-1)^n$.


