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

Spectra of Normalized Laplace Operators for Graphs and Hypergraphs

In this thesis, we bring forward the study of the spectral properties of graphs and we extend this theory for chemical hypergraphs, a new class of hypergraphs that model chemical reaction networks

Spectral classes of hypergraphs

The notions of spectral measures and spectral classes, which are well known for graphs, are generalized and investigated for oriented hypergraphs

Maximal colourings for graphs

We consider two different notions of graph colouring, namely, the $t$-periodic colouring for vertices that has been introduced in 1974 by Bondy and Simonovits, and the periodic colouring for oriented edges that has been recently introduced in the context of spectral theory of non-backtracking operators. For each of these two colourings, we introduce the corresponding colouring number which is given by maximising the possible number of colours. We first investigate these two new colouring numbers individually, and we then show that there is a deep relationship between them

The Aleph of Borges and the Paradise of Cantor

The mathematician Georg Cantor, the writer Jorge Luis Borges, and the protagonist of Borges\u27 short story The Aleph, Carlos Argentino Daneri, are seen here as three pieces of a single puzzle. We put these pieces together and we look at the surprising figure that we obtain

Spectral Theory of Laplace Operators on Oriented Hypergraphs

Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities that bound the eigenvalues are introduced. In particular, the Cheeger constant is generalized and it is shown that the classical Cheeger bounds can be generalized for some classes of hypergraphs; it is shown that a geometric quantity used to study zonotopes bounds the largest eigenvalue from below, and that the notion of coloring number can be generalized and used for proving a Hoffman-like bound. Finally, the spectrum of the unnormalized Laplacian for Cartesian products of hypergraphs is discussed.Comment: 39 page

Cheeger-like inequalities for the largest eigenvalue of the graph Laplace Operator

We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.Comment: 17 pages, 1 figur

Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limit

We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected components, counted according to isotopy type, converges in probability to a deterministic measure. More generally, we also prove similar convergence results for the counting measure of types of components of each $k$-skeleton of a random geometric complex. As a consequence, in the case of the $1$-skeleton (i.e. for random geometric graphs) we show that the empirical spectral measure associated to the normalized Laplace operator converges to a deterministic measure