The diagonal minor of the Laplace matrix obtained by deletion of the rows and columns numbered $i_1 {, \dots ,} i_s$ is equal to $\frac{(-1)^{n-s}}{(n-s)!}\langle W \mid U(\AC_{n,n-s}^I)\rangle$, that is, to $(-1)^{n-s}$ times the sum of monomials $w_{a_1 b_1} \dots w_{a_{n-s}
b_{n-s}}$ such that the graph $([a_1 b_1] {, \dots ,} [a_{n-s} b_{n-s}])$ is a $s$-component forest where every component contains exactly one vertex $i_\ell$ for some $\ell = 1 {, \dots ,} s$, and all the edges of the component are directed towards $i_\ell$.

and

\[Cr:Codim1\] The minor of the Laplace matrix obtained by deletion of its $i$-th row and its $j$-th column is equal to $(-1)^{n-1}$ times the sum of monomials $w_{a_1 b_1} \dots w_{a_{n-1} b_{n-1}}$ such that the graph $([a_1 b_1]
{, \dots ,} [a_{n-1} b_{n-1}])$ is a tree with all the edges directed towards the vertex $i$.

Corollaries \[Cr:Diag\] and \[Cr:Codim1\] are particular cases of the celebrated matrix-tree theorem first discovered by G.Kirchhoff [@Kirch] in 1847 (for symmetric matrices and diagonal minors of codimension $1$) and proved in its present form by W.Tutte [@TutteMTT].

Consider now the following functions on $\Gamma_{n,k}$: $$\begin{aligned}
\sigma(G) &= \begin{cases}
(-1)^{\beta_1(G)}, &G \in \Gamma_{n,k} \text{ is strongly semiconnected},\\
0 &\text{otherwise},
 \end{cases}\\
