School of Economics.

Proofs {#Sec:Proofs}
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We start with proving Proposition \[Pp:DetProp\] (Section \[SSec:PrDetProp\]), to continue with Theorems \[Th:Direct\] and \[Th:Direct\]’ (Sections \[SSec:Equiv\] and \[SSec:PrDirect\]). Theorem \[Th:Diag\] will then follow from Theorem \[Th:Direct\]’ (Section \[SSec:DiagFromDirect\]), and Theorem \[Th:Codim1\], from Theorem \[Th:Diag\] and assertion \[It:RowCol\] of Proposition \[Pp:DetProp\] (Section \[SSec:PrCodim1\]).

For two vertices $a, b \in G \in \Gamma_{n,k}$ we will write $a \succeq b$ if $G$ contains a directed path starting at $a$ and finishing at $b$; also $a \succeq a$ for any $a$ by definition.

Proof of Proposition \[Pp:DetProp\] {#SSec:PrDetProp}
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Let $G$ be a strongly semiconnected graph, and $[ij] \in G$ be its edge carrying number $1$. Since $G$ is strongly semiconnected, $j \succeq i$ in $G \setminus ([ij])$, and therefore $\beta_0(G \setminus ([ij])) =
\beta_0(G)$. Then $G$ enters the left-hand side and the $(i,j)$-th term of the sum at the right-hand side of with the same coefficient.

Denote by $\SSC_{n,k}^{[i:q]}$ the set of all graphs $G \in
\SSC_{n,k}^{\varnothing}$ having $q$ loops ($0 \le q \le k$) attached to vertex $i$. The graph $\hat G$ obtained from $G$ by deletion of all these loops (with the relevant renumbering of the remaining edges) belongs either to $\SSC_{n,k-q}^{[i:0]} \subset \SSC_{n,k-q}^{\varnothing}$ 