Electrical networks and frame matroids

Bob Lutz

Published 2018-09-26Version 1

This paper defines Dirichlet matroids, a class of matroids arising from electrical networks and almost-balanced biased graphs. We prove four main results. First we use properties of the electrical response matrix to show that every Dirichlet matroid has the half-plane property. Second we bound the coefficients of the precoloring polynomial in terms of the chromatic polynomial. Third we prove a simple characterization of 3-connected Dirichlet matroids. And fourth we prove a circular network analog of the duality theorem for planar graphic matroids.