Superport networks

TL;DR

This paper introduces superport networks, a generalization of multiport and ordinary electrical networks, and extends Kirchhoff's matrix-tree theorem to analyze their boundary conditions and response matrices.

Contribution

It generalizes Kirchhoff's matrix-tree theorem to superport networks, incorporating boundary conditions with paired vertices and signed forest contributions.

Findings

Derived a generalized matrix-tree theorem for superport networks

Connected superport networks to response matrix minors using combinatorial methods

Established the mathematical foundation for analyzing complex boundary conditions in electrical networks

Abstract

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.