The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice Z of sets and a function r on Z to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to Z. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids.Comment: 15 pages, 1 figure. The new version addresses earlier work by Julie Sims that the authors learned of after submitting the first versio

Isotropical Linear Spaces and Valuated Delta-Matroids

The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D

Lattice path matroids: enumerative aspects and Tutte polynomials

Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the beta invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure

On Local Equivalence, Surface Code States and Matroids

Recently, Ji et al disproved the LU-LC conjecture and showed that the local unitary and local Clifford equivalence classes of the stabilizer states are not always the same. Despite the fact this settles the LU-LC conjecture, a sufficient condition for stabilizer states that violate the LU-LC conjecture is missing. In this paper, we investigate further the properties of stabilizer states with respect to local equivalence. Our first result shows that there exist infinitely many stabilizer states which violate the LU-LC conjecture. In particular, we show that for all numbers of qubits $n\geq 28$, there exist distance two stabilizer states which are counterexamples to the LU-LC conjecture. We prove that for all odd $n\geq 195$, there exist stabilizer states with distance greater than two which are LU equivalent but not LC equivalent. Two important classes of stabilizer states that are of great interest in quantum computation are the cluster states and stabilizer states of the surface codes. To date, the status of these states with respect to the LU-LC conjecture was not studied. We show that, under some minimal restrictions, both these classes of states preclude any counterexamples. In this context, we also show that the associated surface codes do not have any encoded non-Clifford transversal gates. We characterize the CSS surface code states in terms of a class of minor closed binary matroids. In addition to making connection with an important open problem in binary matroid theory, this characterization does in some cases provide an efficient test for CSS states that are not counterexamples.Comment: LaTeX, 13 pages; Revised introduction, minor changes and corrections mainly in section V

Compatible systems of representatives

AbstractThe main result of this paper can be quickly described as follows. Let G be a bipartite graph and assume that for any vertex v of G a strongly base orderable matroid is given on the set of edges adjacent with v. Call a subgraph of G a system of representatives of G if the edge neighborhood of each vertex of this subgraph is independent in the corresponding matroid. Two systems of representatives we call compatible if they have no common edge. We give a necessary and sufficient condition for G to have k pairwise compatible systems of representatives with at least d edges. Unfortunately, this condition is not sufficient if we deal with arbitrary matroids. Furthermore, we establish a listing variant of the Edmonds' covering theorem for strongly base orderable matroids

Some characteristics of matroids through rough sets

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to matroids through defining a family of sets which are constructed from the upper approximation operator with respect to an equivalence relation. First, we prove the family of sets satisfies the support set axioms of matroids, and then we obtain a matroid. We say the matroids induced by the equivalence relation and a type of matroid, namely support matroid, is induced. Second, through rough sets, some characteristics of matroids such as independent sets, support sets, bases, hyperplanes and closed sets are investigated.Comment: 13 page

The Contributions of Dominic Welsh to Matroid Theory

Dominic Welsh began writing papers in matroid theory nearly forty years ago. Since then, he has made numerous important contributions to the subject. This chapter reviews Dominic Welsh\u27s work in and influence on the development of matroid theory