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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?

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  • $\begingroup$ did you try findstat.org/StatisticFinder?Domain=Cc0020 yet? $\endgroup$ Nov 11, 2021 at 14:17
  • $\begingroup$ no, wasn't aware of this webpage, will check it out, but at a first glance it doesn't seem to cover topics specifically related to this problem $\endgroup$ Nov 11, 2021 at 14:21
  • $\begingroup$ what I meant is, if you plug in the first few numbers, and you are lucky, you will find that someone has studied your statistic already. $\endgroup$ Nov 11, 2021 at 17:21

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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 $\Theta(C_n)=(-1)^n$.

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  • $\begingroup$ this is a great answer, thanks a lot! $\endgroup$ Nov 15, 2021 at 11:23

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