122 research outputs found
Rank three matroids are Rayleigh
A Rayleigh matroid is one which satisfies a set of inequalities analogous to
the Rayleigh monotonicity property of linear resistive electrical networks. We
show that every matroid of rank three satisfies these inequalities.Comment: 11 pages, 3 figures, 3 table
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
Correlation bounds for fields and matroids
Let be a finite connected graph, and let be a spanning tree of
chosen uniformly at random. The work of Kirchhoff on electrical networks can be
used to show that the events and are negatively
correlated for any distinct edges and . What can be said for such
events when the underlying matroid is not necessarily graphic? We use Hodge
theory for matroids to bound the correlation between the events ,
where is a randomly chosen basis of a matroid. As an application, we prove
Mason's conjecture that the number of -element independent sets of a matroid
forms an ultra-log-concave sequence in .Comment: 16 pages. Supersedes arXiv:1804.0307
Matroids on Eight Elements with the Half-plane Property and Related Concepts
We classify all matroids with at most 8 elements that have the half-plane
property, and we provide a list of some matroids on 9 elements that have, and
that do not have the half-plane property. Furthermore, we prove that several
classes of matroids and polynomials that are motivated by the theory of
semidefinite programming are closed under taking minors and under passing to
faces of the Newton polytope.Comment: Test results on the half-plane property of matroids on 9 elements are
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