$G_1, G_2$.

For a function $f: \bigcup_s \Gamma_{n,s} \to \Complex$ and a graph $G \in
\Gamma_{n,k}$ introduce the notation $$\label{Eq:SumSubgr}
\SumSub(f;G) {\stackrel{\mbox{\scriptsize def}}{=}}\sum_{H \subseteq G} f(H).$$ For a set of graphs $\mathfrak B \subset \Gamma_{n,k}$ denote $$\begin{aligned}
&U(\mathfrak B) {\stackrel{\mbox{\scriptsize def}}{=}}\sum_{G \in \mathfrak B} G \in \Graph_{n,k}, \\
&X(\mathfrak B) {\stackrel{\mbox{\scriptsize def}}{=}}\sum_{G \in \mathfrak B} (-1)^{\beta_0(G)} G \in
\Graph_{n,k};
 \end{aligned}$$ $\beta_0(G)$ here means the $0$-th Betti number of $G$, i.e. the number of its connected components (in the topological sense).

\[Df:Minors\] The element $$\det_{n,k}^I {\stackrel{\mbox{\scriptsize def}}{=}}\frac{(-1)^k}{k!} X(\SSC_{n,k}^I) \in
\Graph_{n,k}$$ is called a [*universal diagonal $I$-minor*]{} of degree $k$; in particular, $\det_{n,k}^{\varnothing}$ is called a [*universal determinant*]{} of degree $k$.

The element $$\det_{n,k}^{i/j} {\stackrel{\mbox{\scriptsize def}}{=}}\frac{(-1)^k}{k!} X(\{G \in \Graph_{n,k}
\mid ([ij])*G \in \SSC_{n,k+1}^{\varnothing}\})$$ is called a [*universal (codimension $1$) $(i,j)$-minor*]{} of degree $k$.

\[Ex:Det\] Example \[Ex:SSC\] implies that if $I = \{i_1 {, \dots ,} i_s\}$ and $k < n-s$ then $\det_{n,k}^I = 0$.

Let $k = n$ and $I = {\varnothing}$. By Example \[Ex:SSC\] the graphs $G \in
\SSC_{n,n}^{\varnothing}$ are in one-to-one correspondence with permutations $\sigma$ of $\{1 {, \dots ,} n\}$. It is easy to see that $(-1)^{\beta_0(G)}$ is equal to $(-1)^n$ if $\sigma$ is even and to $-(-1)^n$ if 