Hypergraph Laplace Operators for Chemical Reaction Networks

We generalize the normalized combinatorial Laplace operator for graphs by defining two Laplace operators for hypergraphs that can be useful in the study of chemical reaction networks. We also investigate some properties of their spectra.Comment: 23 pages, 7 figure

On the spectrum of the normalized graph Laplacian

The spectrum of the normalized graph Laplacian yields a very comprehensive set of invariants of a graph. In order to understand the information contained in those invariants better, we systematically investigate the behavior of this spectrum under local and global operations like motif doubling, graph joining or splitting. The eigenvalue 1 plays a particular role, and we therefore emphasize those constructions that change its multiplicity in a controlled manner, like the iterated duplication of nodes.Comment: 9 pages, no figure

Topology and curvature of metric spaces

We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Buesemann, our concept also applies to metric spaces that might be discrete. The natural comparison spaces that emerge from our discussion are no longer Euclidean spaces, but rather tripod spaces. These tripod spaces include the hyperconvex spaces which have trivial Cech homology. This suggests a link of our geometrical method to the topological method of persistent homology in topological data analysis. We also investigate the geometry of general tripod spaces.Comment: 21 pages, no figure