Bounds on sums of graph eigenvalues · smallest eigenvalues, and some other spectral properties,...
Transcript of Bounds on sums of graph eigenvalues · smallest eigenvalues, and some other spectral properties,...
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Bounds on sums of graph eigenvalues
Copyright 2013 by Evans M. Harrell II.
Evans Harrell Georgia Tech
www.math.gatech.edu/~harrell GT math physics seminar
1 February, 2013
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Abstract
I'll discuss two methods for finding bounds on sums of graph eigenvalues (variously for the Laplacian, the renormalized Laplacian, or the adjacency matrix). One of these relies on a Chebyshev-type estimate of the statistics of a subsample of an ordered sequence, and the other is an adaptation of a variational argument used by P. Kröger for Neumann Laplacians. Some of the inequalities are sharp in suitable senses.
This is ongoing work with J. Stubbe of ÉPFL.
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The essential message of this seminar
• It is well known that the largest and smallest eigenvalues, and some other spectral properties, such as determinants, satisfy simple inequalities and provide information about the structure of a graph.
• It will be shown that statistical properties of spectra (means, variance of samples) also satisfy simple inequalities and provide information about the structure of a graph.
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Why focus on sums of eigenvalues? λ0 + …+ λk-1
• These are ground-state energies, in a weakly interacting approximation for matter obeying Fermi statistics.
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• These are ground-state energies, in a weakly interacting approximation for matter obeying Fermi statistics.
• (And consequently:) Estimates of sums are helpful for establishing stability of matter.
Why focus on sums of eigenvalues? λ0 + …+ λk-1
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• These are ground-state energies, in a weakly interacting approximation for matter obeying Fermi statistics.
• (And consequently:) Estimates of sums are helpful for establishing stability of matter.
• They appear in “semiclassical” theorems related to Weyl limits and phase space, which have taken on a life of their own.
Why focus on sums of eigenvalues? λ0 + …+ λk-1
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• These are ground-state energies, in a weakly interacting approximation for matter obeying Fermi statistics.
• (And consequently:) Estimates of sums are helpful for establishing stability of matter.
• They appear in “semiclassical” theorems related to Weyl limits and phase space, which have taken on a life of their own
• Berezin-Li-Yau • Lieb-Thirring What would the
semiclassical limit mean
for a graph?
Why focus on sums of eigenvalues? λ0 + …+ λk-1
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A nanotutorial on graph spectra
• A graph on n vertices is in 1-1 correspondence with an an n by n adjacency matrix A, with aij = 1 when i and j are connected, otherwise 0.
Generic assumptions: connected, not directed, finite, at most one edge between vertices, no self-connection...
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A nanotutorial on graph spectra
• A graph on n vertices is in 1-1 correspondence with an an n by n adjacency matrix A, with aij = 1 when i and j are connected, otherwise 0.
• How is the structure of the graph reflected in the spectrum of A?
• What sequences of numbers might be spectra of A?
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A nanotutorial on graph spectra
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A nanotutorial on graph spectra
• The graph Laplacian is a matrix that compares values of a function at a vertex with the average of its values at the neighbors.
H := -Δ := Deg – A, where Deg := diag(dv), dv := # neighbors of v.
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A nanotutorial on graph spectra
• The graph Laplacian is a matrix that compares values of a function at a vertex with the average of its values at the neighbors.
H := -Δ := Deg – A, where Deg := diag(dv), dv := # neighbors of v.
• How is the structure of the graph reflected in the spectrum of -Δ?
• What sequences of numbers might be spectra of -Δ?
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A nanotutorial on graph spectra
• There is also a normalized graph Laplacian, favored by Fan Chung
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A nanotutorial on graph spectra
• There is also a normalized graph Laplacian, favored by Fan Chung.
• The spectra of the three operators are trivially related if the graph is regular (all degrees equal), but otherwise not.
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The most basic spectral facts
• The spectrum of A allows one to count “spanning subgraphs.”
• It easily determines whether the graph has 2 colors. “bipartite”
• The max eigenvalue is ≤ the max degree. • There is an interlacing theorem when an
edge is added.
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The most basic spectral facts
• H ≥ 0 and H 1 = 0 1. (Like Neumann) • Taking unions of disjoint edge sets,
• This implies a relation between the spectra of a graph and of its edge complement, and various useful simple inequalities.
• The spectrum determines the number of spanning trees (classic thm of Kirchhoff)
• There is an interlacing theorem when an edge is added
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The most basic spectral facts
• For none of the operators on graphs is it known which precise sets of eigenvalues are feasible spectra.
• Examples of nonequivalent isospectral graphs are known (and not too tricky)
• Eigenfunctions can sometimes be supported on small subsets
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Notation for the eigenvalues is not standardized!
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Two good philosophies for understanding spectra
Make variational estimates.
Exploit algebraic properties of the operator.
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Variational bounds on graph spectra
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Variational bounds on graph spectra
• By adapting Pawel Kröger’s variational argument for the Neumann counterpart to Berezin-Li-Yau, using the basis for the complete graph in place of exp(i x•z), and averaging over edges instead of integrating over z, we get upper bounds on in terms of L and the degrees, and corresponding lower bounds on sums from L+1 to n-1.
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An abstract version of Kröger’s inequality
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Proof of Kröger’s inequality
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Kröger’s inequality is used in an odd way
Although it looks as though we want an upper bound on λk, Kröger instead arranged that the left side be ≥ 0, and focused on the implication that
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Kröger’s inequality is used in an odd way
Although it looks as though we want an upper bound on λk, Kröger instead arranged that the left side be ≥ 0, and focused on the implication that
We want the “bigger” average to simplify the Fourier coefficients, while choosing the “smaller” average sufficiently large to arrange that ≥ 0.
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How to use Kröger’s lemma to get sharp results for graphs? (A deep question)
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M =
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Meanwhile, on the left we require that
Giving
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Variational bounds on graph spectra
Another way to apply the Kröger Lemma to graphs is to let M be the set of pairs of vertices. The reason is that the complete graph has a superbasis of nontrivial eigenfunctions consisting of functions equal to 1 on one vertex, -1 on a second, and 0 everywhere else. Let these functions be hz, where z is a vertex pair.
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Variational bounds on graph spectra
Two facts are easily seen:
1.
2.
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Variational bounds on graph spectra
It follows from Kröger’s lemma that
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Variational bounds on graph spectra
• Extensions to renormalized Laplacian
The coefficient of λk works out to be 4|M0|− k + 1, where |M0| is defined asthe number of pairs of vertices in M0. It follows that if 4|M0| ≥ k − 1, then
k−1�
j=1
λj ≤1
n
�
M0
(du + dv + 2Auv).
✷
Next we apply the same ideas to the renormalized Laplacian:
Corollary 8 Let G be any finite graph on n vertices, and let M0 be any set ofp pairs of vertices {u, v} with
�M0
du + dv ≥ 2(k− 1)E. Then the eigenvaluesof the renormalized Laplacian CG satisfy
k−1�
j=1
cj ≤1
2E
�
M0
(2 + du + dv).
Proof. We use Lemma 5, again choosing H as the orthogonal complement of1, and taking M as the set of all pairs {u, v}. For each such pair, this timewe define the vector fv,w :=
√dvδu,� −
√duδv,�. As before we calculate the
quantities on the right side of (2.14), beginning with
�Hfv,w, fv,w� =�HGDeg−1/2
fv,w, Deg−1/2fv,w
�=
�
edges(k�)
�(Deg−1/2
fu,v)k − (Deg−1/2fu,v)�
�2
=
�
dv
du+
�du
dv
2
+ (du − 1)
�dv
du
�
+ (dv − 1)
�du
dv
�
= 2 + du + dv.
It follows thatA0 =
�
M0
(2 + du + dv).
For the term a, recall that for j > 0, the eigenvectors φj are orthogonal toφ0 = Deg1/21. Therefore
�
all pairs
| �fv,w, φj� |2 =1
2
�
u,v
|d1/2v φj,u − d
1/2u φj,v|
2 =1
2
�
u,v
�dv|φj,u|
2 + du|φj,v|2�
= 2E .
The coefficient of λk on the left side of (2.14) works out to be�
M0du + dv −
2(k − 1)E . It follows that if this quantity is nonnegative, then
k−1�
j=1
cj ≤1
2E
�
M0
(2 + du + dv),
yielding the claim. ✷
The analogous result for the adjacency matrix reads as follows
9
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Variational bounds on graph spectra
• Variant for adjacency matrix?
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Variational bounds on graph spectra
• Variant for adjacency matrix? • Caution! Since the spectrum
is not a fortiori positive, to get a useful inequality the coefficient of λk in the Kröger lemma should be 0, not just ≥ 0. That turns out to be possible to arrange, but we are debugging our arithmetic.!
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Variational bounds on graph spectra
• Other inequalities arise from min-max and good choices of trial functions.
• For example, Fiedler showed in 1973 that for the graph Laplacian
(0 = λ0 < λ1 ≤ ... ≤ λn-1 ≤ n)
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Variational bounds on graph spectra
Alternative for λ1 + λ2:
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Variational bounds on graph spectra
• where the degrees are in decreasing order. Optimal for the complete and star graphs
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Variational bounds on graph spectra
• Generalization of Fiedler:
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Variational bounds on graph spectra
• Use eigenvectors of
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Variational bounds on graph spectra
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A deeper look at the statistics of spectra
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1. Inequalities involving means and standard deviations of ordered sequences. References: Hardy-Littlewood-Pólya, Mitrinovic.
Пафнутий Львович Чебышёв
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The counting function, N(z) := #(λk ≤ z) Integrals of the counting function,
known as Riesz means
Chandrasekharan and Minakshisundaram, 1952; Safarov, Laptev, Weidl, ...
Riesz means
p p
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If the sequence happens to be the spectrum of a self-adjoint matrix, then
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How can a general identity give information about graphs?
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How can a general identity give information about graphs?
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How can a general identity give information about graphs?
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An analogue of Lieb-Thirring
Consider the operator s Deg – A, which interpolates between -A and H as s goes from 0 to 1. Then (writing D for Deg)
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An analogue of Lieb-Thirring
When integrated,
i.e.,
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THE END