All-Minors Matrix-Tree Theory for Superport Networks: Completed Quotient-Incidence Determinants and Conductance-Weighted Subdivision Extensions
Abstract
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=(DTK−1D)−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 HF. Its reduced incidence matrix BF=QFD has columns only of the forms 0, ± ea, ea−eb. Hence every square incidence minor is exactly 0 or ±1. For a k-set of response coordinates I, append to BF the selector rows EIT and define the completed quotient-incidence determinant χ̂F(I) = det BF EIT . For coordinate sets I,J of the same size, the arbitrary response minor is a weighted spanning-forest sum whose coefficient is simply χ̂F(I)χ̂F(J). Thus the Jacobi complementary-minor factors used in the derivation disappear from the final theorem. Direct block-triangular reduction gives χ̂F(I)∈{0,±1}, with nonvanishing occurring exactly when the complementary quotient edges N∖ I form a spanning tree of HF.
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BibTeX
@misc{newton2026allminors,
title = {{All-Minors Matrix-Tree Theory for Superport Networks: Completed Quotient-Incidence Determinants and Conductance-Weighted Subdivision Extensions}},
author = {Tony Newton},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01672},
eprint = {2609.01672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01672},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - All-Minors Matrix-Tree Theory for Superport Networks: Completed Quotient-Incidence Determinants and Conductance-Weighted Subdivision Extensions
AU - Tony Newton
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01672
UR - https://arxiv.org/abs/2609.01672
AB - 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=(D^TK^−1D)^−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, ± e_a, e_a−e_b. Hence every square incidence minor is exactly 0 or ±1. 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 χ̂_F(I) = det B_F E_I^T . For coordinate sets I,J of the same size, the arbitrary response minor is a weighted spanning-forest sum whose coefficient is simply χ̂_F(I)χ̂_F(J). Thus the Jacobi complementary-minor factors used in the derivation disappear from the final theorem. Direct block-triangular reduction gives χ̂_F(I)∈{0,±1}, with nonvanishing occurring exactly when the complementary quotient edges N∖ I form a spanning tree of H_F.
ER -