Electrical Networks, Hyperplane Arrangements and Matroids

Abstract

This thesis introduces a class of hyperplane arrangements, called Dirichlet arrangements, arising from electrical networks with Dirichlet boundary conditions. Dirichlet arrangements encode harmonic functions on electrical networks and generalize graphic arrangements, a fundamental class of hyperplane arrangements arising from finite graphs.

The first part of the thesis studies the main combinatorial properties of Dirichlet arrangements in detail. We characterize these in ways that directly generalize well-known results on graphic arrangements. Particular attention is paid to the matroids underlying Dirichlet arrangements, called Dirichlet matroids. We prove a number of results concerning Dirichlet matroids, including some on the half-plane property, Bergman fans, and duals of circular electrical networks. These results are applied to related objects and problems, including response matrices of electrical networks, order polytopes of finite posets, and graph coloring problems.

The latter part of the thesis studies two specific problems. First, we show that a given Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This answers an open question in the special case of Dirichlet arrangements. Second, we establish a relationship between structural rigidity of graphs and topological complexity of complements of hyperplane arrangements. The notion of topological complexity originates from the motion planning problem in topological robotics.