51 research outputs found
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
-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 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
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 -skeleton of a random geometric complex. As a
consequence, in the case of the -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
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
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