Background: An electrical network is modeled by a complex. Branch current distributions $\mathbf I\in C_1$ are represented by $1$-chains; branch voltage drop distributions $\mathbf V\in C^1$ are $1$-cochains; $\dim C_1=\dim C^1=b$, the number of branches. We define $Z_1=\ker\operatorname{\partial}$ (where $\operatorname{\partial}:C_1\to C_0$ is the boundary operator) as the subspace of cycles; $\dim Z_1=m$, the number of independent meshes in the network. Branch current distributions that satisfy Kirchhoff’s current law, $\operatorname{\partial}\mathbf I=0$ are thus elements of $Z_1$.
We establish a basis for $Z_1$ that consists of linearly independent meshes by selecting a maximal tree $T$ for the network. For each branch $\beta\notin T$, there is a unique mesh $\mathbf M_\beta$ that is obtained by adjoining $\beta$ to the maximal tree $T$. Remember that a mesh has coefficients in $\{-1, 0, +1\}$. The orientation of $\mathbf M_\beta$ is chosen so that the coefficient of $\beta$ is $+1$. Then, we define the space $H_1$ as $Z_1$ considered as a vector space in its own right instead of as a subspace of $C_1$. The map $\sigma:H_1\to C_1$ identifies this space with $Z_1$, e.g., if the mesh $\mathbf M\in H_1$ consists of the branch currents $\alpha+\beta-\gamma\in C_1$, then $\sigma(\mathbf M)=\alpha+\beta-\gamma$. The matrix of $\sigma$ relative to this basis has these meshes as its columns. Note that there are $m$ such meshes for a given tree $T$, hence also $m$ non-tree branches.
We are given the vector of branch current sources $\mathbf K\in C_1$, voltage sources $\mathbf W\in C^1$ and the map $Z:C_1\to C^1$ that relates branch currents to branch voltages. The matrix of $Z$ relative to the standard bases is the diagonal matrix of branch impedances, which for a resistive network are resistances—positive reals. We must then find $\mathbf I\in C_1$ and $\mathbf V\in C^1$ such that $$ \mathbf V-\mathbf W = Z(\mathbf I-\mathbf K) $$ and Kirchhoff’s laws $$ \operatorname{\partial}\mathbf I=0 \\ \mathbf V=-\operatorname{d}\mathbf\Phi $$ are satisfied. (I haven’t defined the zero-cochain $\mathbf\Phi$ or the coboundary operator $\operatorname{d}$ here, but I don’t think they’re directly relevant to my problem.)
One way to solve a resistive network is due to Weyl: We can use $Z$ to define a scalar product $\langle\mathbf I,\mathbf I'\rangle_Z = \sum_\alpha z_\alpha I_\alpha I'_\alpha$. If $\pi:C_1\to Z_1$ is the orthogonal projection operator relative to this scalar product, then $\mathbf I=\pi(\mathbf K-Z^{-1}\mathbf W)$ and $\mathbf V=Z(1-\pi)(\mathbf K-Z^{-1}\mathbf W)$ are the solution. Comparing this to the solution obtained via the mesh-current method provides an explicit expression for this operator in terms of $Z$ and $\sigma$:$$ \pi = \sigma(\sigma^*Z\sigma)^{-1}\sigma^*Z. \tag{*} $$ Kirchhoff’s construction of this orthogonal projection uses a weighted average of a set of (possibly) non-orthogonal projections as follows. Each maximal tree $T$ has an associated projection $\rho_T: C_1\to Z_1$ defined by: $$ \rho_T(\beta) = \begin{cases} 0, &\text{if }\beta\in T \\ \mathbf M_\beta, &\text{if }\beta\notin T, \end{cases} $$ where $\mathbf M_\beta$ is the mesh associated with the non-tree branch $\beta$. $\sum _T\lambda_T\rho_T$, with $0\le\lambda_T\le1$ and $\sum_T\lambda_T=1$ and summed over all maximal trees, is also a projection of $C_1$ onto $Z_1$. By taking $$ \lambda_T = \frac{Q_T}R \\ Q_T = \prod_{\beta\notin T}z_\beta \\ R = \sum_TQ_T, $$ the resulting projection is orthogonal relative to $\langle\cdot,\cdot\rangle_Z$.
The Problem: From solving various networks as exercises, it looks to me like $R=\det(\sigma^*Z\sigma)$, the Gram determinant for the chosen basis of $Z_1$ relative to $\langle\cdot,\cdot\rangle_Z$. Since the entries of the matrices for $\sigma$ and $\sigma^*$ are all integers, the only source of non-integer factors in $(*)$ is $Z$. Similarly, $Z$, and hence $\frac1R$ is the only source of non-integer factors in Kirchhoff’s sum. Setting all resistances to integer values makes it obvious that $R\; | \det(\sigma^*Z\sigma)$ or $\det(\sigma^*Z\sigma)\;|\;R$, but I haven’t been able to come up with a proof of equality. I tried expanding the determinant by cofactors and comparing to the the coefficients in Kirchhoff’s sum, but couldn’t get anywhere with that. Perhaps if I had a better idea of how those coefficients were determined, that might do the trick. Hints and suggestions are welcome.
Update: The main diagonal elements of $\sigma^*Z\sigma$ are $\sum_\beta z_\beta[\beta\in\mathbf M_\alpha]$ ($[\cdot]$ are Iverson brackets), i.e., the total resistances in each basis mesh, while the off-diagonal elements are $\pm\sum_\gamma z_\gamma[\gamma\in\mathbf M_\alpha][\gamma\in\mathbf M_\beta]$, i.e., the total common resistances of each pair of basis meshes, negative if the meshes have opposite orientation. Using linear combinations of rows/columns of this matrix to isolate the non-tree resistances looks promising.