or, if $q > 0$, to $\SSC_{n,k-q}^{\{i\}}$. Vice versa, if $q > 0$ and $\hat G
\in \SSC_{n,k-q}^{[i:0]} \cup \SSC_{n,k-q}^{\{i\}}$ then $G \in
\SSC_{n,k}^{[i:q]}$. Deletion of a loop does not break a graph, so $\beta_0(G) = \beta_0(\hat G)$.

If $G \in \SSC_{n,k}^{[i:q]}$ then there are $\binom{k}{q}$ ways to assign numbers to the loops of $G$ attached to $i$. Since $\langle W \mid
G\rangle$ does not depend on the edge numbering, one has for $q > 0$ $$\langle W \mid X(\SSC_{n,k}^{[i:q]})\rangle = \binom{k}{q} w_{ii}^q
\langle W \mid X(\SSC_{n,k-q}^{[i:0]}) + X(\SSC_{n,k-q}^{\{i\}})\rangle,$$ so that $$\begin{aligned}
\langle W \mid \det_{n,k}^{\varnothing}\rangle &= \frac{1}{k!}\sum_{q=0}^k
\langle W \mid X(\SSC_{n,k}^{[i:q]})\rangle\nonumber \\
&= \frac{1}{k!} \langle W \mid X(\SSC_{n,k}^{[i:0]})\rangle + \sum_{q=1}^k
\frac{w_{ii}^q}{q!(k-q)!} \langle W \mid X(\SSC_{n,k-q}^{[i:0]}) +
X(\SSC_{n,k-q}^{\{i\}})\rangle\nonumber \\
&= \sum_{q=0}^k \frac{w_{ii}^q}{q!(k-q)!} \langle W \mid
X(\SSC_{n,k-q}^{[i:0]}) + X(\SSC_{n,k-q}^{\{i\}})\rangle - \langle W \mid
\det_{n,k}^{\{i\}}\rangle.\label{Eq:Develop}
 \end{aligned}$$ The expressions $\langle W \mid X(\SSC_{n,k-q}^{[i:0]}) +
X(\SSC_{n,k-q}^{\{i\}})\rangle$ and $\langle W \mid
\det_{n,k}^{\{i\}}\rangle$ do not contain $w_{ii}$. So, applying the operator $\frac{\partial^m}{\partial w_{ii}^m}$ to equation and using the equation again with $k-m$ in place of $k$ one gets .

Theorems \[Th:Direct\] and \[Th:Direct\]’ are equivalent. {#SSec:Equiv}
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Denote by $E(G)$ the set of edges of the graph $G \in \Gamma_{n,k}$. The functions $\alpha$ and $\sigma$ do 