Spatial Graphs and Oriented Matroids: the Trefoil

Energy of Graphs, Matroids and Fibonacci Numbers

The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G. In this article we consider the problem whether generalized Fibonacci constants $varphi_n$ $(ngeq 2)$ can be the energy of graphs. We show that $varphi_n$ cannot be the energy of graphs. Also we prove that all natural powers of $varphi_{2n}$ cannot be the energy of a matroid.

Fachbereich 3 Mathematik Oriented Matroids Oriented Matroids

Oriented Matroids

The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations. Apparently totally different mathematical objects such as point and vector configurations, arrangements of hyperplanes, convex polytopes, directed graphs, and linear programs find a common generalization in the language of oriented matroids. The...

On tope graphs of complexes of oriented matroids

We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, anoth...

A characterization of cocircuit graphs of uniform oriented matroids

The cocircuit graph of an oriented matroid is the 1-skeleton of the cellular decomposition induced by the Topological Representation Theorem due to Folkman and Lawrence (1978) [J. Folkman, J. Lawrence, Oriented matroids, J. Combin. Theory Ser. B 25 (1978) 199–236]. In this paper we exhibit a characterization of such graphs (for the uniform case) via their natural embedding into Qn—the 1-skeleto...