Matroids arising from electrical networks

This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main results. First, we exhibit a matroid quotient formed by the dual of a network embedded in a surface with boundary and the dual of the associated Dirichlet matroid. This generalizes an analogous result for graphic matroids of cellularly embedded graphs. Second, we characterize the Bergman fans of Dirichlet matroids as explicit subfans of graphic Bergman fans. In doing so, we generalize the connection between Bergman fans of complete graphs and phylogenetic trees. Third, we use the half-plane property of Dirichlet matroids to 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 chromatic polynomial of the underlying graph.Comment: 27 pages, 14 figure