Superport networks. arXiv:2309.14039
Preprint, arXiv:2309.14039 [math.CO] (2023).
Summary: We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff’s matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon-D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.
MSC:
05C82 | Small world graphs, complex networks (graph-theoretic aspects) |
05C22 | Signed and weighted graphs |
94C05 | Analytic circuit theory |
31C20 | Discrete potential theory |
35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |
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