\text{and\hspace{3.5cm}}\\
\alpha(G) &=  \begin{cases}
(-1)^k, &G \in \Gamma_{n,k} \text{ is acyclic},\\
0 &\text{otherwise}.
 \end{cases}
 \end{aligned}$$ Theorem \[Th:Diag\] follows from the two equivalent statements (see Section \[Sec:Proofs\] for details):

\[Th:Direct\] $\SumSub(\alpha;G) = (-1)^k \sigma(G)$ for $G \in \Gamma_{n,k}$.

$\SumSub(\sigma;G) = (-1)^k\alpha(G)$ for $G \in \Gamma_{n,k}$.

Here $\SumSub$ is summation over subgraphs, as defined by . These theorems are essentially [@Bernardi Proposition 6.16]. We will nevertheless give their proofs in Section \[Sec:Proofs\] thus answering a request for a direct proof expressed in [@Bernardi] (the original proof in [@Bernardi] is a specialization of a much more general identity).

A digression: undirected graphs and the universal Potts partition function
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Denote by $\Upsilon_{n,k}$ the set of all [*undirected*]{} graphs with $n$ vertices numbered $1 {, \dots ,} n$ and $k$ edges numbered $1 {, \dots ,} k$. Denote by $\lmod \,\cdot\,\rmod: \Gamma_{n,k} \to \Upsilon_{n,k}$ the “forgetful” map replacing every edge by its undirected version; the edge numbering is preserved. By $\Undir_{n,k}$ denote a vector space spanned by $\Upsilon_{n,k}$; then $\lmod \,\cdot\,\rmod$ is extended to the linear map $\Graph_{n,k} \to \Undir_{n,k}$. The notion of a subgraph and the notation $\SumSub$ (see ) for undirected graphs are similar to those for $\Graph_{n,k}$. One can also define 