37,332 research outputs found
On spectra of weighted graphs of order ≤5
Producción CientíficaThe problem of characterizing the real spectra of weighted graphs is only solved
for weighted graphs of order n ≤ 4. We overview these known results, that come
from the context of nonnegative matrices, and give a new method to rule out many
unresolved spectra of size 5.Ministerio de Economía, Industria y Competitividad ( grant MTM2015-365764-C3-1-P)Universidad de Valladolid (GIR TAMCO
Functional control of network dynamics using designed Laplacian spectra
Complex real-world phenomena across a wide range of scales, from aviation and
internet traffic to signal propagation in electronic and gene regulatory
circuits, can be efficiently described through dynamic network models. In many
such systems, the spectrum of the underlying graph Laplacian plays a key role
in controlling the matter or information flow. Spectral graph theory has
traditionally prioritized unweighted networks. Here, we introduce a
complementary framework, providing a mathematically rigorous weighted graph
construction that exactly realizes any desired spectrum. We illustrate the
broad applicability of this approach by showing how designer spectra can be
used to control the dynamics of various archetypal physical systems.
Specifically, we demonstrate that a strategically placed gap induces chimera
states in Kuramoto-type oscillator networks, completely suppresses pattern
formation in a generic Swift-Hohenberg model, and leads to persistent
localization in a discrete Gross-Pitaevskii quantum network. Our approach can
be generalized to design continuous band gaps through periodic extensions of
finite networks.Comment: 9 pages, 5 figure
Spectral preorder and perturbations of discrete weighted graphs
In this article, we introduce a geometric and a spectral preorder relation on
the class of weighted graphs with a magnetic potential. The first preorder is
expressed through the existence of a graph homomorphism respecting the magnetic
potential and fulfilling certain inequalities for the weights. The second
preorder refers to the spectrum of the associated Laplacian of the magnetic
weighted graph. These relations give a quantitative control of the effect of
elementary and composite perturbations of the graph (deleting edges,
contracting vertices, etc.) on the spectrum of the corresponding Laplacians,
generalising interlacing of eigenvalues.
We give several applications of the preorders: we show how to classify graphs
according to these preorders and we prove the stability of certain eigenvalues
in graphs with a maximal d-clique. Moreover, we show the monotonicity of the
eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic
Cheeger constants with respect to the geometric preorder. Finally, we prove a
refined procedure to detect spectral gaps in the spectrum of an infinite
covering graph.Comment: 26 pages; 8 figure
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