On algebras and matroids associated to undirected graphs (original) (raw)

A Commutative Algebra for Oriented Matroids

Discrete & Computational Geometry, 2002

Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K-algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred: • Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A? We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik-Terao, which is the main tool of our proofs.

A note on mixed graphs and matroids

A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.

Matroids in terms of Cayley graphs and some related results1

The point we try to get across is that the generalization of the coun- terparts of the matroid theory in Cayley graphs since the matroid theory frequently simplify the graphs and so Cayley graphs. We will show that, for a Cayley graph G, the cutset matroid M ( G) is the dual of the circuit matroid M( G). We will also deduce that if G is an abstract-dual of a Cayley graph , then M( G) is isomorphic to (M( G)) . 2000 Mathematics Subject Classication : 05C20; 05C25; 05C60.

On incidence algebras and directed graphs

International Journal of Mathematics and Mathematical Sciences, 2002

The incidence algebraI(X,ℝ)of a locally finite poset(X,≤)has been defined and studied by Spiegel and…

Matroids, Delta-matroids and Embedded Graphs

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We illustrate how the connections between embedded graphs and delta-matroids can be exploited by using these connections to find an excluded minor characterization and a rough structure theorem for the class of delta-matroids that arise as twists of matroids. Also, we show that several polynomials of embedded graphs, including the Bollobás-Riordan, Penrose, and transition polynomials, are in fact delta-matroidal. . Ralf Rueckriemen was financed by the DFG through grant RU 1731/1-1. 1 8.1. The Penrose polynomial 36 8.2. The Penrose and characteristic polynomials 38 8.3. The transition polynomial 40 References 43 1. Overview

Line-Closed Matroids, Quadratic Algebras, and Formal Arrangments

Advances in Applied Mathematics, 2002

Let G be a matroid on ground set A. The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra E on A by the ideal I generated by circuit boundaries. The quadratic closure A(G) of A(G) is the quotient of E by the ideal generated by the degree-two component of I. We introduce the notion of nbb set in G, determined by a linear order on A, and show that the corresponding monomials are linearly independent in the quadratic closure A(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. These results generalize to the degree r closure of A(G).

On a class of algebras associated to directed graphs

2005

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over noncommutative algebras. We also construct a basis for our algebras associated to layered graphs.

Line-closed matroids, quadratic algebras, and formal arrangements

2000

Let G be a matroid on ground set . The Orlik-Solomon algebra A(G) is the quotient of the exterior algebra on by the ideal generated by circuit boundaries. The quadratic closure A̅(G) of A(G) is the quotient of by the ideal generated by the degree-two component of . We introduce the notion of set in G, determined by a linear order on , and show that the corresponding monomials are linearly independent in the quadratic closure A̅(G). As a consequence, A(G) is a quadratic algebra only if G is line-closed. An example of S. Yuzvinsky proves the converse false. These results generalize to the degree r closure of (G). The motivation for studying line-closed matroids grew out of the study of formal arrangements. This is a geometric condition necessary for to be free and for the complement M of to be a K(π,1) space. Formality of is also necessary for A(G) to be a quadratic algebra. We clarify the relationship between formality, line-closure, and other matroidal conditions related to formalit...