We study multiport networks, which have boundary conditions different from the usual 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. In electrical engineering, such networks are more common than ordinary electrical networks. 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 Kenyon and Wilson, determinantal identities, and combinatorial bijections. We prove our theorems in the generality of superport networks, which generalize both ordinary networks and multiport ones. This is joint work with Svetlana Shirokovskikh and Mikhail Skopenkov.
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