do you know what is spectra in graph theory?

Answer 1

Spectral graph theory and its applications

Spectral graph theory—the study of the eigenvectors and eigenvalues of matrices associated
with graphs—is a large field with many beautiful results. Most of the work in this area has
been descriptive, determining how combinatorial features of a graph are revealed by its spectra.
On the other hand, many applications require us to control the spectra of a graph. One of
the main goals of the course will be to use descriptive results to improve efforts at control of
spectra. We will not restrict ourselves to graphs, and will generally consider how combinatorial
features of matrices affect their spectra. We will consider application areas including quantum
computation, error-correcting codes, graph partitioning and preconditioning. I hope that some
in the class will help me develop spectral hypergraph theory, an area in which I have many
conjectures and few theorems.
The exact list of topics to be covered will be dictated by student interest. Possible topics
include:
1. Spectral graph drawing
Relations to the work of Tutte, and Colin de Verdiere embeddings of graphs.
2. Using spectra to test graph isomorphism.
3. Spectra of special graphs
(a) Strongly regular graphs.
(b) Cayley graphs, and connections to group representation theory.
(c) Path graphs, and discretizations of the continuous Laplacian.
4. The second eigenvalue of a graph
(a) Cheeger’s inequality, with three proofs.
(b) Spectral graph partitioning.
(c) The second eigenvalue of planar graphs.
(d) The diameter of graphs, with applications to concentration of measure.
(e) Expansion in graphs.
(f) Pseudo-random phenomena in graphs.
5. Eigenvalues and eigenvectors of random graphs
(a) Continuous distributions, extreme eigenvalues.
(b) Concentration of eigenvalues.
(c) Discrete distributions, many conjectures.
(d) Spectral Partitioning of semi-random graphs.
(e) Finding Cliques hidden in random graphs.
(f) Coloring random 3-colorable graphs.
(g) Free probability, simplified.
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6. Preconditioning matrices.
(a) Preconditioning with spanning trees.
(b) Sparsifying Laplacian matrices.
(c) Preconditioning Laplacian matrices.
(d) Preconditioning matrices from hypergraphs weighted by elasticity.
7. Quantum computation.
(a) Quantum computation in one lecture.
(b) Discrete proof of the adiabatic theorem.
(c) Analysis of the adiabatic algorithm in special cases.
8. Constructing error-correcting codes from expanding graphs.
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Answer 2

the spectra are the coefficients that belongs to the eigenvectors

Av = 12Ae, then 12 is part of the spectrum of A

Answer 3

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Answer 4

I think it's the spectrum of the associated adjacency matrix or the Laplacian matrix. For applications, you can see the answer above.

Answer 5

no idea. you are cute, understanding only the SEXY part of my question in spanish, thanx though, merry chistmas, and happy new year.