Matroids arising from electrical networks

TL;DR

This paper introduces Dirichlet matroids, a new class derived from electrical networks, and explores their properties, dualities, and polynomial bounds, connecting matroid theory with network analysis and algebraic geometry.

Contribution

It presents the concept of Dirichlet matroids, characterizes their Bergman fans, and establishes new results on response matrix zeros and polynomial bounds.

Findings

Dirichlet matroids generalize graphic matroids from electrical networks.

Bergman fans of Dirichlet matroids are subfans of graphic Bergman fans.

Response matrix trace zeros and poles follow an interlacing pattern.

Abstract

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main theorems. First, we exhibit a matroid quotient involving geometric duals of networks embedded in surfaces with boundary. Second, we characterize the Bergman fans of Dirichlet matroids as subfans of graphic Bergman fans. Third, we prove an interlacing result on the real zeros and poles of the trace of the response matrix. And fourth, we bound the coefficients of the precoloring polynomial of a network by the coefficients of the associated chromatic polynomial.