  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.
