All-Minors Matrix-Tree Theory for Superport Networks: Completed Quotient-Incidence Determinants and Conductance-Weighted Subdivision Extensions
An electrical network can be summarized at its boundary by a response matrix: prescribed boundary voltages determine boundary currents. A superport network adds a constraint by grouping boundary terminals into superports, requiring the total current in each group to be zero and making voltage differences inside the groups the natural coordinates. Earlier work determined forest formulas for a single response entry and for the determinant of the whole response matrix. The missing case was an arbitrary subdeterminant, or minor: one needs to know not only which spanning forests contribute, but also the sign carried by each forest. This paper supplies that sign rule. After choosing one reference vertex in each superport, the response is \[ L=\left(D^{T}K^{-1}D\right)^{-1}, \] with \(K\) the grounded weighted Laplacian and \(D\) recording the selected voltage differences. Contracting the components of a physical spanning forest \(F\) produces a much smaller quotient port graph \(H_F\). Its reduced incidence matrix \[ B_F=Q_FD \] has columns only of the forms \[ 0,\qquad \pm e_a,\qquad e_a-e_b. \] Hence every square incidence minor is exactly \(0\) or \(\pm1\). For a \(k\)-set of response coordinates \(I\), append to \(B_F\) the selector rows \(E_I^{T}\) and define the completed quotient-incidence determinant \[ \widehat{\chi}_F(I) = \det \begin{pmatrix} B_F\\ E_I^{T} \end{pmatrix}. \] For coordinate sets \(I,J\) of the same size, the arbitrary response minor is a weighted spanning-forest sum whose coefficient is simply \[ \widehat{\chi}_F(I)\widehat{\chi}_F(J). \] Thus the Jacobi complementary-minor factors used in the derivation disappear from the final theorem. Direct block-triangular reduction gives \[ \widehat{\chi}_F(I)\in\{0,\pm1\}, \] with nonvanishing occurring exactly when the complementary quotient edges \(N\setminus I\) form a spanning tree of \(H_F\).