LMS/EPSRC Short Course
Analysis on Graphs and its Applications
Gregynog Hall, University of Wales, 10-15 January 2007
Quantum Graphs
Part I. Basic Structures
Peter Kuchment
kuchment@math.tamu.edu
Department of Mathematics
Texas A& M University
College Station, TX, USA
1. An outline of the three lectures
• What is a quantum graph?
interesting?
Why are quantum graphs
• Self-adjoint vertex conditions.
• Spectral theory. Usual and unusual spectral features.
• Special problems: justification of the model, gaps, slowing
down waves, etc.
• Inverse problems.
• Generalizations: “leaky graphs”, “quantum buildings”,etc.
• The outer space.
2. Books, surveys and collections of articles
(MANY more references at the end of the file)
Books and collections of articles
(a) G. Berkolaiko et al (Editors), Quantum Graphs and Their
Applications, Contemp. Math., v. 415, Amer. Math.
Soc. 2006.
(b) V. A. Kozlov, V. G. Mazýa, and A. B. Movchan, Asymptotic Analysis of Fields in Multi-Structures, Oxford Sci.
Publ. 1999.
(c) P. Kuchment (Editor), Waves in Periodic and Random
Media, Contemp. Math. v. 339, Amer. Math. Soc.
2004.
(d) P. Kuchment (Editor), Quantum graphs and their applications, a special issue of Waves in Random media,
14(2004), no.1.
(e) F. Mehmeti et al (Editors), Partial Differential Equations
on Multistructures, CRC 2001
(f) Yu.V. Pokornyi et al, Differential equations on geometrical
graphs. (in Russian) Moscow: Fizmatlit, 2004.
Surveys and introductory articles
(a) P. Exner and P. Seba, Electrons in semiconductor microstructures: a challenge to operator theorists, in Proc.
Workshop “Schrödinger Operators, Standard and Nonstandard” (Dubna 1988), World Sci., Singapore 1989; pp.
79–100.
(b) V. Kostrykin and R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys. A 32(1999), 595-630.
(c) T. Kottos and U. Smilansky, Periodic orbit theory and
spectral statistics for quantum graphs, Ann. Phys. 274
(1999), 76–124.
(d) P. Kuchment, The Mathematics of Photonic Crystals, Ch.
7 in Mathematical Modeling in Optical Science, Bao, G.,
Cowsar, L. and Masters, W.(Editors), 207–272, Philadelphia: SIAM, 2001.
(e) P. Kuchment, Graph models of wave propagation in thin
structures, Waves in Random Media 12(2002), no. 4,
R1-R24.
(f) P. Kuchment, Differential and pseudo-differential operators on graphs as models of mesoscopic systems, in
“Analysis and Applications”, H. Begehr et al (Editors),
Kluwer 2003, 7–30.
(g) P. Kuchment, Quantum graphs I. Some basic structures,
Waves in Random media, 14(2004), S107–S128.
(h) P. Kuchment, On some spectral problems of mathematical physics, in Partial Diff. Equat. and Inverse Problems,
C. Conca et al(Editors), Contemp. Math. v. 362, AMS
2004
(i) Yu. V. Pokornyi and V. L. Pryadiev, Some problems of
the qualitative SturmLiouville theory on a spatial network,
Russian Math. Surveys 59 (2004), no. 3, 515–552.
Books on spectral graph theory
(a) F. Chung, Spectral Graph Theory, Amer. Math. Soc.,
Providence R.I., 1997.
(b) Y. Colin de Verdière, Spectres De Graphes, Societe Mathematique De France, 1998
(c) D. Cvetkovic, M. Doob, and H. Sachs, Spectra of Graphs,
Acad. Press., NY 1979.
(d) D. Cvetkovic, M. Doob, I. Gutman, A. Targasev, Recent
Results in the Theory of Graph Spectra, Ann. Disc. Math.
36, North Holland, 1988.
3. What is a metric graph?
Locally finite graph Γ = (V, E). Graphs will be assumed to
be connected.
Metric graph - graph equipped with a length 0 < le ≤ ∞
assigned to each edge e.
Example: Γ embedded into Rn, le - the “true” length of e.
We consider here Γ as a singular manifold.
Metric graphs are also sometimes (in combinatorial and geometric group theory) called R-graphs.
4. What is a quantum graph?
Introduce coordinate xe (or just x) on e with values in [0, le].
Measure dx on Γ, integration, and differentiation along edges
are defined, as well as some functional spaces.
L 2
2
L (e).
E.g., space L (Γ) =
e∈E
One can define Sobolev space H 1(Γ) as follows: include functions u that satisfy:


u ∈ H 1(e) for each edge e ∈ E,

 P
kuk2
1 (e) < ∞ (needed for infinite graphs only),
H

e∈E


u is continuous at each v ∈ V, and thus on the whole Γ.
Higher order Sobolev spaces are not uniquely defined on Γ.
One can define differential operators on Γ. Main operator
(Hamiltonian) H acts on each edge on a function u as
d2u(x)
+ q(x)u(x),
Hu = −
dx2
∞ (Γ). Schrödinger operator on
(Γ)
∩
L
real potential q ∈ L2
loc
the graph Γ. Domain D(H) consists of u s.t.


u ∈ H 2(e) for each edge e ∈ E,

 P
kuk2
2 (e) < ∞ (needed for infinite graphs only),
H

e∈E


u satisfies “appropriate” boundary conditions at each v ∈ V.
We will describe all vertex conditions that give rise to selfadjoint operators H. The simplest: Neumann conditions


u is continuous at each vertex, and thus on the whole Γ,


 P du
dxe (v) = 0 for each v ∈ V

e∈Ev


 (derivatives taken in outgoing directions from v).
Definition. A quantum graph Γ is a metric graph equipped
with a self-adjoint operator H.
More general operators are also considered in applications.
E.g., magnetic Schrödinger operators
2
d
+ A(x) + q(x)u(x),
−i
dx
operators of higher orders, pseudo-differential operators (see
later on in these lectures and also in Exner’s lectures), etc.
5. Why are quantum graphs interesting?
♣ Modeling waves in thin branching “graph-like ”structures:
narrow waveguides, quantum wires, photonic crystals, blood
vessels, lungs. Applications in nanotechnology, optics, chemistry, medicine.
♣ Modeling electron spectra of aromatic molecules
Naphthalene
and Carbon nano-structures
Graphene
Nanotube
♣ Averaging in dynamical systems.
♣ Toy models in quantum mechanics and other areas
6. Justification of “fattened graph” models
“Fattened graph” domain Ω around Γ. Laplace operator
−∆N, (or more general Schrödinger) with Neumann b.c.
in Ω. Does σ(−∆N,) converge when → 0 to the spectrum of a quantum graph Γ?
Yes, it does (Freidlin&Wentzel, Rubinstein&Schatzmann,
P.K.&Zeng, Saito, Exner&Post, Post).
When neighborhoods of vertices decay slowly, additional spectrum attached to the vertices arises, or the graph completely
decouples (Freidlin&Wentzel, P.K.&Zeng, Exner&Post).
Technique used:
Constructing operators of extension E : H 1(Γ) 7→ H 1(Ωε)
and averaging P : H 1(Ωε) 7→ H 1(Γ) such that they can only
slightly increase the quadratic forms and only slightly decrease L2-norms. Then the Rayleigh-Ritz representation of
eigenvalues is used:
(Hu, u)
.
inf
sup
λn =
dimW =n u∈W,u6=0 (u, u)
Similar question about the Dirichlet b.c..
Yes in smooth case (i.e., no vertices allowed). (Exner et
al.)
Graph case: the answer is in general a No.
Partial results show different spectral behavior below the lowest energy of transversal modes (around π 2/2) and above it.
Below: additional spectrum arises “attached to vertices”
(Exner et al.)
Above: a graph model might be possible (Post,
Molchanov&Vainberg).
Electromagnetic waves
Spectral problems for the operators:
Maxwell
1
~ = λU
~,∇ · U
~ =~
∇U
0.
∇×
ε(x)
Divergence type
−∇ ·
1
∇H = λH.
ε(x)
Helmholtz type
−∆E = λε(x)E.
ε(x) - periodic electric permittivity.
For the Helmholtz case, some modes converge to the ones
of the problem
−∆E = µδΓ(x)E + cE,
δΓ(x) – the delta-function supported on the graph Γ.(Figotin
& P.K., P.K.&Kunyansky)
When c = 0, this reduces to the spectrum of the (pseudodifferential) Dirichlet-to-Neumann operator on the graph Γ:
∂E
]Γ = µE.
[
∂ν
Similar spectral problem arises also for the “leaky wires” (see
Exner’s lectures).
Relations to quantum graphs with differential Hamiltonians
(sometimes of higher order) exist (P.K.&Kunaynsky).
7. Self-adjoint vertex conditions
♣ Local (vertex) conditions involve values at a single vertex
at a time.
Av F (v) + Bv F 0(v) = 0
with d × d matrices Av , Bv . d - degree of v.
Theorem 1. (Kostrykin&Schrader ’99) Let Γ - finite. H is
self-adjoint iff matrix (Av Bv ) has the maximal rank and
the matrix Av Bv∗ is self-adjoint.
(1)
Parametrization (A, B) is redundant.
♣ Non-local conditions - the same, matrices involve all
vertex values. Theorem still holds.
Conclusions:
(i) topology of a quantum graph is encoded in boundary
conditions.
(ii) Allowing arbitrary conditions, every quantum graph is a
rose.
8. Alternative descriptions of self-adjoint vertex
conditions
♣ (M. Harmer) Let U = Uv - unitary d × d.
i(Uv − I)F (v) + (Uv + I)F 0(v) = 0.
Alternatively, Pv - ortho-projector onto eigenspace of Uv corresponding to −1 (Pv could be zero), Qv = I − Pv . Then

P F (v) = 0 (the “Dirichlet” part of conditions),
v
Qv F 0(v) = Sv QF (v),
Uv −I is s.-a. in Q C|B| . Quadratic form:
where Sv = −i U
v
v +I
X Z
X
du 2
| | dx −
hSv U (v), U (v)i
h[u, u] =
e∈E e dx
v∈V
Domain: u ∈ H 1(e) ∀e ∈ E(G), P Sv F (v) = 0 ∀v ∈ V (G).
So called vertex and bond scattering matrices, as well as
trace formulas will not be covered in these lectures.
9. “Popular” types of vertex conditions
δ-type conditions:

 f (x) is continuous on Γ
P df
at
each
vertex
v
,

dxe (v) = αv f (v), αv ∈
e∈Ev
R .
(2)
Quadratic form defined on functions from H 1(Γ):
X Z
X
df 2
| | dx +
αv |f (v)|2.
e∈E e dx
v∈V
(3)
When αv = 0, we get the most common case of
Neumann (Kirchhoff) conditions:

 f (x) is continuous on Γ
P df
.
 at each vertex v ,
dxe (v) = 0
e∈Ev
(4)
Conditions of δ 0-type:

df

 Derivative dxe (v) is the same for all edges e s.t. v ∈ e
e
P
.
df
f
(v)
=
α
(v)

e
v

dx
e∈Ev
(5)
The quadratic form
X Z
X
X
1
df 2
| | dx +
|
fe(v)|2.
α
e∈E e dx
{v∈V | αv =
6 0} v {e∈Ev }
is defined on functions in
L
e
H 1(e) that have at each vertex v
where αv = 0 the sum of the vertex values along all entering
edges equal to zero.
Vertex Dirichlet conditions: at each vertex the boundary
values of the function are equal to zero. Operator decouples
into direct sum of Dirichlet Laplacians on the edges.
Quadratic form
X Z df
| |2dx
e∈E e dx
is defined on f ∈ H 1(Γ) s.t. f (v) = 0 at any vertex v.
Vertex Neumann conditions: derivatives fe0 (v) at all vertices are equal to zero. Operator decouples into the sum of
Neumann problems on each edge.
Quadratic form
X Z df
| |2dx
e∈E e dx
on
L
e∈E(Γ)
H 1(e).
10. Dirichlet-to-Neumann map
A graph G with a subset B ⊂ V (G) designated as a boundary:
Graph G
Blue vertices - the “boundary”

Hu = λu on G
“Dirichlet problem”
u| = 0
B
Σ = {λn}.
has discrete spectrum
Let λ ∈
/ Σ, φ : B 7→ C. Dirichlet-to-Neumann map Λ(λ) is:
∂u
|B ,
Λ(λ)φ :=
∂ν
where

Hu = λu on G
u| = φ.
B
For v ∈ B, ∂u
∂ν (v) is the sum of ingoing derivatives of u at v.
Λ(λ) is a meromorphic |B| × |B| - matrix function.
11. Scattering graphs
Changing the view of the graph,
the problem can be recast as a vector-valued on R+:

− d2 U = λU, t > 0
dt2
U 0(0) = Λ(λ)U (0)
I.e., spectral problems for scattering graphs reduce to star
graphs & eigenvalue-dependent vertex conditions
Related to the Schur complement in the matrix theory.
12. Some “graphic” peculiarities :-)
♣ Unlike for ODEs or second order elliptic PDEs, there is
no uniqueness of continuation. One can, for instance,
have compactly supported eigenfunctions (bound states)
for quantum graph operators:
This influences many aspects of quantum graph research.
♣ No monodromy.
Consider a graph like this:
Whatever graph
Easy to check: there are values of λ such that given initial
data left from the obstacle, solution of Hu = λu cannot be
extended to the right.
13. Spectra of quantum graphs
♣ Standard decomposition of the spectrum:
σ(H) = σpp(H) ∪ σac(H) ∪ σsc(H)
1
0.8
0.6
0.4
0.2
Pure point σpp (localized states)
-4
-2
0
0
2
4
x
1
0.5
x
-20
-10
0
0
10
20
-0.5
Absolutely continuous σac (extended states)
-1
Singular continuous σsc (?)
Discrete σdis - isolated eigenvalues of finite multiplicity
Essential σess = σ − σdisc
♣ A connection between quantum and discrete graph
spectra
Γ - finite. λ ∈ σ(H) ⇔ ∃u ∈ L2, Hu = λu. Assume Neumann
conditions: u is continuous, sum of outgoing derivatives at
each v is 0.
On edges, resolve Hu = λu, assuming vertex values of u
known. This gives sum of vertex derivatives of u at v ∈ V (Γ)
as combination (with coefficients depending on λ) of values
of u at v and at neighboring vertices.
We need to make the sum of the derivatives equal to zero.
This produces discrete linear equation T (λ){u|V (Γ)} = 0.
Thus, quantum graph spectral problem reduces to a discrete
one, at the price of λ entering nonlinearly.
Observation: this works when λ is not in the Dirichlet spectrum of H on any edge e. Incorporating “fake” vertices of
degree 2 along edges, one can avoid this difficulty.
2
d
H = − dx2 , all edge lengths are equal to 1, all
Example:
valences are d.
Solve along edge e = (v, w):
ue(x) =
sin
1
√
λle
√
√ ue(v) sin λ(le − x) + ue(we) sin λx . (6)
This works if λ 6= n2π 2 with an integer n 6= 0.
Simple calculation shows that T (λ)u = 0 becomes
√
!
√
X
sin l λ
u(w) = α √
+ dv cos λ u(v).
l λ
{w| e=(v,w)∈E }
v
I.e., λ 6= n2π 2 is in σ(H) iff
√
√
α sin√l λ + d cos λ is in σ(∆),
l λ
∆ being the discrete Laplace operator on Γ.
This reduction to discrete problems is harder to deduce for
infinite graphs, due to appearance of continuous spectrum.
It is established in several cases:
1. Graphs with equal lengths of edges (Pankrashkin).
2. Periodic graphs (P.K., must be folklore, uses Floquet
theory, see later in the lectures).
3. Graphs of sub-exponential growth (P.K., uses Shnol’s
Theorem and generalized eigenfunctions, described later in
the lectures).
4. Point spectrum (Cattaneo).
♣ Genericity of simple spectrum
There are known results by Albert and by Uhlenbeck claiming
that generically the spectrum of an elliptic operator on a
compact manifold is simple. Is this true for quantum graphs?
Theorem 2. (L. Friedlander)
d2 and Neumann
Let Γ be a finite quantum graph with H = − dx
2
conditions. Consider the set M of points {le}|e∈E(Γ) ∈ R|E(Γ)|
such that the spectrum of H is simple. If Γ is not topologically a circle, then M is residual.
The statement is incorrect for Γ a circle.
♣ Shnol’s Theorem
How can one detect that a particular λ belongs to σ(H)?
Theorem 3.(a) λ is an eigenvalue ⇔ there exists an eigenfunction φ ∈ L2(Γ) (suffices for compact graphs).
(b) λ ∈ σ(H) ⇔ there exists a sequence φn ∈ L2 of approximate eigenfunctions, i.e. kφnkL2(Γ) = 1 and
k(H − λ)φnk →n→∞ 0 (Weyl’s criterion).
(c) If for every > 0 there exists a generalized eigenfunction
φ (i.e., satisfying Hφ = λφ) such that kφkL2(Br ) < Cer ,
then λ ∈ σ(H). (Shnol’s Theorem)
♣ Generalized eigenfunctions
Shnol’s theorem: having a generalized eigenfunction of a subexponential growth implies being on spectrum. Is converse
true: does one have generalized eigenfunctions of a controllable growth? Yes (Gelfand-Kostyuchenko, Berezansky).
Let 0 < Φ ∈ L∞(Γ) ∩ L2(Γ).
Theorem 4. (Hislop&Post ’06) For almost every (w.r.t. spectral measure) λ ∈ σ(H) there exists a generalized eigenfunction ψ s.t. Φψ ∈ L2(Γ). The system of such functions is
complete.
In particular, if Φ decays slower than exponentially, i.e. min Φ ≥
Br
Ce−r for any positive , then the generalized eigenfunction
is of Shnol’s Theorem’s class.
♣ Spectral gaps
It is often desirable to have gaps in the spectrum of a graph
operator (photonic crystals, expander graphs, etc.)
Some known mechanisms of opening gaps:
• Periodic media. According to Floquet theory, spectra
of operators in periodic media tend (but do not have) to
have gaps. Hard to control.
• Resonant gap opening by “decorating” graphs (Pavlov,
Aizenman&Schenker, P.K.& Ong):
or
This mandatorily opens spectral gaps at locations that
depend on the decoration only.
This is how it works: equation of the type
∆u = λu
on the decorated graph after replacing the decoration by
its Dirichlet-to-Neumann map, reduces to
∆u + Λ(λ)u = λu.
The energy dependent “potential” Λ(λ) has poles at certain locations, which makes the equality impossible near
these poles, and thus no spectrum arises there.
UCLA antenna ground plane.
• Gap opening by using appropriate edge potentials (Pankrashkin).
♣ Spectra of periodic graphs.
Periodic graph Γ: free co-compact action of Zn:
Edge lengths and operators periodic w.r.t. the action.
Floquet-Bloch theory: let k ∈ Rn - quasimomentum. Restrict the operator to functions u such that
u(x + p) = eik·pu(x), x ∈ Γ, p ∈ Zn.
Get the (Bloch Hamiltonian) operator H(k).
Theorem 5. (Floquet-Bloch theorem):
i) H(k) has discrete spectrum λm(k) that tends to ∞.
ii) σ(H) = ∪ σ(H(k)).
k
E.g., dispersion curves for graphene and carbon nano-tubes
can be easily found explicitly (Korotyaev&Lobanov, P.K.&Post).
Theorem 6. (L. Thomas) For a periodic elliptic operator
(including periodic quantum graph Hamiltonians), σsc(H) = ∅
Theorem 7. (L. Thomas, Birman&Suslina, A. Sobolev, L. Friedlander, P.K.& Levendorskii, ...) For periodic elliptic operators of 2nd order, σpp(H) = ∅, i.e. σ(H) = σac(H).
This statement does not hold for quantum graphs:
Theorem 8. (P.K.) If λ ∈ σpp(H), then the (∞-dimensional)
eigenspace is generated by compactly supported functions.
Amenable case due to Veselic.
Embedded eigenvalues. Localized perturbation of a periodic quantum graph by changing topology, vertex conditions,
or potential in a compact domain can only add isolated eigenvalues either in gaps, or inside the spectral bands (embedded
eigenvalues). (Weyl’s theorem)
Theorem 9. (1D Rofe-Beketov, 2D, 3D P.K.&Vainberg)
There are no embedded eigenvalues due to localized perturbations of a periodic Schrödinger operator.
This is not true for periodic quantum graphs. Embedded
eigenvalues can be constructed.
Theorem 10. (P.K.&Vainberg)
Eigenfunctions corresponding to the embedded eigenvalues
of a locally perturbed quantum graph are compactly supported near the support of the perturbation.
♣ Slowing down wave packets
ψ(x, ω) - Bloch solution with quasimomentum k(ω) of Hψ =
ω 2ψ (x ∈ Γ, ω ∈ R). v = ψe−iωt solves non-stationary wave
equation vtt = −Hv and “propagates with the velocity” ω/k(ω).
Wave packet
u=
Z
|ω−ω0 |<
ψ(x, ω)α(ω − ω0)e−iωtdω.
Group velocity Vg = (k0(ω0))−1. It can be shown that
−i(k(ω0 )x−ω0 t)
e
u ∼ f (x)α(x/V
g − t)e
So, flat pieces of dispersion curves lead to low group velocities.
Possible/suggested ways to slow down packets:
• Near spectral edges. Does not work well.
• Near inflection points of the dispersion (Figotin&Vitebsky).
Uses layered anisotropic media.
• Necklace structures (using poles of the Dirichlet-to-Neumann
map). (Molchanov&Vainberg)
• CROWS = coupled resonator optical waveguides (uses
slow tunnelling between resonators).
(Yariv et al, P.K.&Kunyansky)
• SCISSORS = side-coupled integrated spaced sequence of
resonators(Heebner&Boyd)
14. Inverse problems: can one hear the shape of a
quantum graph?
One wanders whether the spectrum σ(H) contains sufficient
information for recovery of the topology of graph, vertex
conditions, and potentials along the edges. Without extra
assumptions on the graph, none of these is impossible (von
Below, Kurasov&Stenberg).
2
d and Neumann conditions.
Consider the case of H = − dx
2
It is clear that vertices of degree 2 need to be excluded (a
graph without such vertices is sometimes called clean).
Theorem 11. (Gutkin&Smilansky) If a clean compact graph
has no loops and multiple edges, and if the lengths of its
edges are rationally independent, then the topology of the
graph is uniquely determined by its spectrum.
Later extension to clean graphs with incommensurate lengths
of the edges (Kurasov&Nowaczyk).
The proofs are based on trace formulas due to Smilansky et
al that allow one to recover the lengths of periodic orbits in
Γ.
See also work by Pivovarchik and by Yurko.
15. Generalizations: “leaky graphs”, higher dimensional
“quantum buildings”, multi-structures.
There are various generalizations of quantum graphs:
♣ Pseudo-differential Hamiltonians, photonic crystals, and
leaky wires.
Study of high contrast photonic crystals leads to
−∆u = λδΓu + cu (Figotin & P.K.)
Parameter c - coupling of two dielectrics. When c = 0 (no
coupling), −∆u = λδΓu - spectral problem for D-to-N operator on Γ.
Same spectral problem describes “leaky wires”, while there
c is the spectral parameter and λ is a coupling constant.
♣ Higher order Hamiltonians.
The pseudo-differential problem was studied by P.K. & Kunyansky. Existence of good differential approximations was
shown for some geometries (the square of D-to-N operator
is “almost” Laplacian). This lead to energy dependent vertex
conditions.
In some cases, e.g. for Γ = Z2, the resulting differential
operator is of fourth order.
These results were heuristic and numerical, and there is no
rigorous treatment of this problem. An analytic approach
was suggested by Ola and Paivarinta.
♣ Quantum buildings and multistructures
Some popular photonic crystal structures look like thin surface (rather than graph) structures in 3D. Such is, for instance, the inverse opal structure (what is left after close
packing by spheres):
(The picture is from work of Ames group headed by Prof.
K. Soukoulis.)
Thus (and for some other reasons) it is worthwhile to attempt a generalizations of quantum graph theory to higher
dimension CW complexes (“quantum buildings”).
In various applications there is an interest towards equations
on “multistructures,” i.e. complexes combining cells of different dimensions. See, e.g. books by Mehmeti et al and by
V. A. Kozlov, V. G. Mazýa, and A. B. Movchan.
16. The outer space
We mention here a moduli space of metric graphs, which
might be useful in quantum graph studies. It has been
studied intensively as a tool for investigating automorphism
groups of free groups.
The fundamental group of a connected graph is a free group.
Indeed, contracting a maximal tree T in Γ into a point produces a rose Rn of n petals and π1(Rn) is isomorphic to the
free group Fn with n generators.
Consider the space of all connected graphs Γ whose fundamental group is Fn and whose total length is normalized to
be equal to 1.
We call a homotopy equivalence mapping g : Rn 7→ Γ a marking. A pair (g, Γ) is a marked metric graph.
Two such marked graphs (g1, Γ1), (g2, Γ2) are said to be
7
Γ2 such
equivalent, if there exists an isometry i : Γ1 →
that the diagram
is commutative.
The set Xn of these equivalence classes is the Outer space.
If C is the set of conjugacy classes in Fn, one can naturally
embed Xn into the infinite dimensional projective space RP C .
This is done by coordinate functions: let c ∈ C, w - a representative of c, and (g, Γ) - marked graph. One can construct
the number l(g,Γ)(c) = the length of the shortest loop in Γ
in the free homotopy class of g∗(w).
It has been shown that Xn consists of simplices, is contractible, and has dimension 3n − 4. Shrinking one edge to
a point moves a graph to the boundary of a simplex, where
several simplices meet.
There is a natural compactification of Xn - its closure in RP C .
The topology of Xn and of its boundary has been studied
intensively. This knowledge might prove to be useful for the
quantum graph research.
An incomplete bibliography on quantum
graphs, their applications, generalizations, and
related topics
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