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

Infinite graphic matroids Part I

An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable

Broken circuit complexes and hyperplane arrangements

We study Stanley-Reisner ideals of broken circuits complexes and characterize those ones admitting a linear resolution or being complete intersections. These results will then be used to characterize arrangements whose Orlik-Terao ideal has the same properties. As an application, we improve a result of Wilf on upper bounds for the coefficients of the chromatic polynomial of a maximal planar graph. We also show that for an ordered matroid with disjoint minimal broken circuits, the supersolvability of the matroid is equivalent to the Koszulness of its Orlik-Solomon algebra.Comment: 21 page

Reconfiguration of basis pairs in regular matroids

In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange distance of two matroid basis sequences, a problem that appears in several areas of computer science and mathematics. In 1980, White proposed a conjecture for the characterization of two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gr\"obner bases. In this work, we verify White's conjecture for basis sequences of length two in regular matroids, a problem that was formulated as a separate question by Farber, Richter, and Shan and Andres, Hochst\"attler, and Merkel. Most of previous work on White's conjecture has not considered the question from an algorithmic perspective. We study the problem from an optimization point of view: our proof implies a polynomial algorithm for determining a sequence of symmetric exchanges that transforms a basis pair into another, thus providing the first polynomial upper bound on the exchange distance of basis pairs in regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on the serial symmetric exchange property of matroids for the regular case.Comment: 28 pages, 6 figure