Spectral theory on combinatorial and quantum graphs
Copyright 2016 by Evans M. Harrell II.
Evans Harrell Georgia Tech
www.math.gatech.edu/~harrell
القيروان
Topic 1: The Ubiquitous Laplacian
November, 2016
Atlanta
Spectral theory on combinatorial and quantum graphs
القيروان
Atlanta
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Just what is a Laplacian, and why are Laplacians ubiquitous?
I. The simplest and most symmetric second-order differential.
ª Linear partial differential equations can be converted to normal forms with a change of variables. The leading term is:
where we may assume that A is symmetric. By diagonalizing A and enforcing invariance under symmetries, we find that A is a multiple of the identity.
What about a surface or manifold? What about a surface or manifold?
The essence of a manifold is that it looks locally like Euclidean space. In fact, ifyou single out a given point, you can find coordinates, called Fermi coordinates, inwhich the metric tensor at that spot becomes the identity, just like for Euclideanspace. Of course, it does this only momentarily. Think, for example of the spherewith spherical coordinates ✓,� for which
x = rsin✓ cos �
y = rsin✓ sin�
x = r cos ✓
As we know, fixing r = 1, the arc length and Laplacian look like:
ds2 = d✓2 + sin2 ✓d�2
� =1
sin ✓
@
@✓sin ✓
@
@✓+
@2
@�2,
but on the equator, where sin ✓ = 1 and its derivative is zero, we obtain thefamiliar Laplacian as the unweighted sum of the second derivatives with respectto an orthogonal coordinate system.
One could develop the notion of the Laplace-Beltrami operator by using Fermicoordinates and then calculating the complications that arrive as soon as onemoves avay from the special point, but there is an easier way to proceed, whichis to think about the weak form of the Laplace operator, which is the quadraticform on a domain or manifold ⌦ defined by
(f, g)!Zrf ·rg dV ol
or, by polarization,
f ! E(f) :=Z
|rf |2 dV ol. (1.1)
Again, among all quadratic expressions in the first derivatives of f , the integrand in(1.3) is uniquely defined, up to a constant multiple, by respecting the symmetriesof Euclidean space. The infamously complicated form of the Laplace-Beltramioperator in terms of a metric tensor,
�LB
f :=X
ij
1pg@
i
gij
pg@
i
f, (1.2)
where g := det(gij
), is simply what you obtain if you introduce local coordinatesand integrate (1.3) by parts:
Z|rf |2 dV ol =
Zf(��
LB
f) dVol + possible boundary contribs. (1.3)
Verification Exercise. Verify (1.2), (1.3).
2
The weak, or quadratic form
ª If you ask what the Laplace-Beltrami operator looks like in general coordinates, and you work hard enough, from this you find
The weak, or quadratic form
ª However, there is a simpler way, when you define the Laplacian by its weak form,
By partial integration, if allowed, the latter term contains the Laplacian but the weak form requires less regularity.
The weak, or quadratic form
It suffices to consider
by polarization:
The weak, or quadratic form
ª Verification exercise: Check by partial integration that
The usual Laplace-Beltrami operator on, for example a closed manifold (no boundary), is defined by using the Friedrichs extension from a suitable dense set of test functions f. Cf. the lectures by Hatem Najar.
II. The generator of the simplest probabilistic process.
II. The generator of the simplest probabilistic process.
ª Verification exercise: Check that P satisfies the heat equation and that provides the general solution for the initial-value problem for the heat equation on Euclidean space.
III. The Laplacian measures how a function differs from its averages.
ª A basic question of analysis: How does a quantity compare with its average value?
ª Subharmonic functions: f(x) always less than its average over balls centered at x. Superharmonic refers to the opposite inequality. A harmonic function is both sub- and superharmonic.
III. The Laplacian measures how a function differs from its averages.
III. The Laplacian measures how a function differs from its averages.
III. The Laplacian measures how a function differs from its averages.
ª In words: The Laplacian of a function f at x measures the rate at which nearby averages of f increase as you move away from x.
ª This point of view makes no reference to differentiation!
ª While we won’t develop the subject here, this gives one a way to define Laplacians on abstract measure spaces.
Spectral theory on combinatorial and quantum graphs
Copyright 2016 by Evans M. Harrell II.
Evans Harrell Georgia Tech
www.math.gatech.edu/~harrell
Topic 2: The Ubiquitous Notion of a Graph
القيروان
Atlanta
Just what is a graph, and why are graphs ubiquitous?
Combinatorial graphs
ª Sets of n “vertices” and m edges, with m ≤ n(n-1). (Or n(n-1)/2 if we don’t “orient” the edges).
ª There is a vertex space isomorphic to Cn and an edge space isomorphic to Cm.
ª A matrix can be used to efficiently describe which edges connect to which vertices.
Uses of combinatorial graphs
ª Electrical networks ª Social networks (communication,
internet). ª Discrete approximations of physical
problems modeled by PDEs or dynamical systems.
ª Biochemical pathways. ª Molecular structure
From Constance Harrell et al., Psychoneuroendocrinology 62 (2015) 252–264.
Uses of combinatorial graphs
ª A “graph of knowledge”
1/12/2015 128.61.105.123/wikigraph/tree.html
http://128.61.105.123/wikigraph/tree.html 1/1
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Wikigraph – a project of P. Laban
“Standard” combinatorial graphs
ª Finite. m, n < ∞. ª Connected. ª Undirected. ª Loop-free. ª Unweighted.
Matrices describing graphs
ª The n×n adjacency matrix specifies which vertices are connected to which other vertices.
ª The n×m incidence matrix specifies which vertices attach to a given edge.
ª The m×n discrete gradient specifies how oriented edges attach to the vertices.
Complete graph
Ring and star
Tree
Bipartite
A classic problem in graph theory is to determine how many labels, or “colors,” are necessary for the vertices, so that no two vertices with the same label are connected. A bipartite graph is one where only two labels are necessary.
Prof. H’s favorite example
Prof. H’s favorite example
Incidence matrix:
Prof. H’s favorite example
The diagonal is the set of “degrees” (or valences, i.e., the number of neighbors.
Prof. H’s favorite example
= Deg + A
Prof. H’s favorite example
Gradient: (An arbitrary orientation has been put on the edges)
Prof. H’s favorite example
= Deg - A
Prof. H’s favorite example
Gradient: (Both orientations are allowed on the edges.)
Prof. H’s favorite example
(exactly twice the previous caculation, 2 (Deg – A). )
The Laplacian on a graph
ª The operator d*d is what we will define (up to a sign) as the graph Laplacian,
- Δ = d*d . The quadratic form of this is:
ª The operator d*d is what we will define (up to a sign) as the graph Laplacian,
- Δ = d*d . Notice that - Δ = Deg - A ª This means that at a vertex v, [- Δ f](v) = Deg(v)(f(v) - <f>w~v).
The Laplacian on a graph
ª The renormalized graph Laplacian of Fan Chung is defined as
Deg-1/2 (- Δ) Deg-1/2 ª This is related by a similarity
transformation to Deg-1 [- Δ f](v) = f(v) - <f>w~v.
The Laplacian on a graph
ª In the next lecture we will discuss quantum graphs, where the edges have the metric structure of intervals.
The Laplacian on a graph
