Groups, Graphs

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Groups, Graphs Isospectrality

• Rami Band
• Ori Parzanchevski
• Gilad Ben-Shach
• Uzy Smilansky

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What is a graph ?

• The graphs spectrum is the sequence of basic
  frequencies at which the graph can vibrate.
• The spectrum depends on the shape of the graph.

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Can one hear the shape of a graph ?

• Can one hear the shape of a drum ?was first
  asked by Marc Kac (1966).
• Can one hear the shape of a graph?
• Can one deduce the shape from the spectrum ?
• Is it possible to have two different graphs with
  the same spectrum (isospectral graphs) ?

Marc Kac (1914-1984)
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Metric Graphs - Introduction
L23

• A graph G consists of a finite set of vertices
  Vvi and a finite set of undirected edges
  Eej.
• A metric graph has a finite length (Legt0)
  assigned to each edge.
• Let Ev be the set of all edges connected to a
  vertex v.
• The degree of v is
• A function on the graph is a vector of
  functions on the edges

L34
L35
L25
L12
L14
L45
L15
L13
L45
L34
L23
L46
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Quantum Graphs - Introduction

• A quantum graph is a metric graph equipped with
  an operator, such as the negative Laplacian
• For each vertex v, define
• We impose boundary conditions of the
  formWhere Av and Bv are complex dv x dv
  matrices.
• To ensure the self-adjointness of the Laplacian,
  we require that the matrix (AvBv) has rank dv,
  and that the matrix AvBv is self-adjoint
  (Kostrykin and Schrader, 1999).

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Quantum Graphs - Introduction

• A quantum graph is a metric graph equipped with
  an operator, such as the negative Laplacian
• For each vertex v, define
• We impose boundary conditions of the
  formWhere Av and Bv are complex dv x dv
  matrices.
• A common boundary condition is the Kirchhoff
  condition, namely all functions agree at each
  vertex, and the sum of the derivatives vanishes.
• This corresponds to the matrices
• For dv 1 vertices, there are two specialcases
  of boundary conditions, denoted
• Dirichlet Av (1), Bv(0) (so that
  )
• Neumann Av (0), Bv(1) (so that
  )

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Quantum Graphs - Introduction

• A quantum graph is a metric graph equipped with
  an operator, such as the negative Laplacian
• For each vertex v, define
• We impose boundary conditions of the
  formWhere Av and Bv are complex dv x dv
  matrices.
• A common boundary condition is the Kirchhoff
  condition, namely all functions agree at each
  vertex, and the sum of the derivatives vanishes.
• For dv 1 vertices, there are two special cases
  of boundary conditions, denoted
• Dirichlet Av (1), Bv(0) (so that
  )
• Neumann Av (0), Bv(1) (so that
  )

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The Spectrum of Quantum Graphs
The spectrum is With the set of corresponding
eigenfunctions Use the notation
v2
Examples of several functions of the graph

?1324.0
?89.0
?1637.2
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Can one hear the shape of a graph ?

• One can hear the shape of a simple graph if the
  lengths are incommensurate (Gutkin, Smilansky
  2001)
• Otherwise, we do have isospectral graphs
• Von Below (2001)
• Band, Shapira, Smilansky (2006)
• There are several methods for construction of
  isospectrality the main is due to Sunada
  (1985).
• We present a method based on representation
  theory arguments.

D
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Groups Graphs

• Example The Dihedral group
  the symmetry group of the squareD4
  e , a , a2 , a3 , rx , ry , ru , rv

How does the Dihedral group act on a square ?
a
rx
e
y
u
v
x

• A few subgroups of the Dihedral group H1 e
  , a2 , rx , ryH2 e , a2 , ru , rv H3
  e , a , a2 , a3

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Groups - Representations

• Representation Given a group G, a
  representation R is an assignment of a matrix
  R(g) to each group element g ? G, such that ?
  g1,g2 ? G R(g1)R(g2)R(g1g2).
• Example 1 - D4 has the following 1-dimensional
  rep. S1
• Example 2 - D4 has the following 2-dimensional
  rep. S2
• Restriction - is the following
  rep. of H1
• Induction - is the following rep. of
  G

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Groups Graphs

• Example The Dihedral group
  the symmetry group of the squareD4
  e , a , a2 , a3 , rx , ry , ru , rv

• The group can act on a graph and the group
  can act on a function which is defined on the
  graph and may give new functions
• We have So that, form a
  representation of the group.

• Knowing the matrix representation gives us
  information on the functions.

How does the group act on the function ?
F
a
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Groups Graphs

• Consider the following rep. R1 of the subgroup
  H1
• H1 e , a2, rx , ry
• R1
• This is a 1d rep. what do we know about the
  function F ?

Examine the graph G

• Consider the following rep. R2 of the subgroup
  H2
• H2 e , a2, ru , rv
• R2
• This is a 1d rep. what do we know about the
  function F ?

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Groups, Graphs Isospectrality

• Theorem Let G be a graphwhich obeys a
  symmetry group G.Let H1, H2 be two subgroupsof
  G with representations R1, R2 that obey then
  the graphs , are isospectral.
• In our example
• G
• H1 e , a2 , rx , ry R1
• H2 e , a2, ru , rv R2
• And it can be checked that

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G D4
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Extending the Isospectral pair
Extending our example G H1 e ,
a2, rx , ry R1 H2 e , a2, ru , rv
R2 H3 e , a, a2 , a3 R3
G D4
N
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Extending the Isospectral pair
Extending our example G H1 e ,
a2, rx , ry R1 H2 e , a2, ru , rv
R2 H3 e , a , a2 , a3 R3
G D4
N
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Groups, Graphs Isospectrality

• Theorem Let G be a graph which obeys a
  symmetry group G.Let H1, H2 be two subgroups of
  G with representations R1, R2 that obey
  then the graphs , are
  isospectral.
• ProofLemma There exists a quantum graph
  such that Using the Lemma and
  Frobenius reciprocity theorem givesHence
  , are isospectral.Applying the same
  for the rep. R2 and using finishes the proof.

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Groups, Graphs Isospectrality

• Theorem Let G be a graph which obeys a
  symmetry group G.Let H1, H2 be two subgroups of
  G with representations R1, R2 that obey
  then the graphs , are
  isospectral.
• ProofLemma There exists a quantum graph
  such thatInteresting issues in the proof of
  the lemma
• A group which does not act freely on the edges.
• Representations which are not 1-d.
• The dependence of in the choice of basis for
  the representation.

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Arsenal of isospectral examples
ry
rv
G is the Cayley graph of D4(with respect to the
generators a, rx) Take again GD4 and
the same subgroups H1 e , a2, rx , ry with
the rep. R1 H2 e , a2, ru , rv with the rep.
R2 H3 e , a , a2 , a3 with the rep. R3
a3
a2
e
L1
a
L2
L2
rx
ru
L1
The resulting quotient graphs are
L1
L1
L2
L1
L2
L2
L2
L2
L1
L1
L1
L2
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Arsenal of isospectral examples
G D6 e, a, a2, a3, a4, a5, rx, ry, rz, ru,
rv, rw with the subgroups H1 e, a2, a4,
rx, ry, rz with the rep. R1 H2 e, a2, a4,
ru, rv, rw with the rep. R2 H3 e, a, a2,
a3, a4, a5 with the rep. R3
2L3
The resulting quotient graphs are
L1
2L2
2L2
2L2
2L2
2L3
2L3
2L3
2L1
L1
2L2
2L1
2L1
2L1
L2
2L2
L2
2L3
L1
2L1
L3
2L3
L3
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Arsenal of isospectral examples
4
G S4 acts on the tetrahedron. with the
subgroups H1 S3 with the rep. R1 H2 S4 with
the rep. R2 S424 , S36
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1
3
2
The resulting quotient graphs are
D
D
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L/2
D
D
L/2
L/2
L/2
L/2
L/2
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Arsenal of isospectral examples
G is the a graph which obeys the Oh (octahedral
group with reflections). Oh48. Take G Oh
and the subgroups H1 O, the octahedral
group. H2 Td, the tetrahedral group with
reflections. H1 H2 24
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Arsenal of isospectral examples
A puzzle construct an isospectral pair out of
the following familiar graph
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Arsenal of isospectral examples
G S3 (D3) acts on G with no fixed points. To
construct the quotient graph, we take the same
rep. of G, but use two different bases for the
matrix representation.
L3
L2
L3
L1
L2
The resulting quotient graphs are
L3
L2
L2
L2
L3
L1
L3
L2
L2
L3
L2
L3
L1
L1
L2
L3
L2
L2
L3
L3
L1
L1
L1
L3
L2
L3
L2
L2
L3
L3
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Arsenal of isospectral examples
Isospectral drums

• One cannot hear the shape of a drum Gordon,
  Webb and Wolpert (1992)

G0444 (using Conways orbifold notation) acts
on the hyperbolic plane. Considering a
homomorphism of G0 onto GPSL(3,2)and taking two
subgroups H1, H2 such that
and R1, R2 are the sign representations, we
obtain the known isospectral drums of Gordon et
al.but with new boundary conditions
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Arsenal of isospectral examples
Isospectral drums
Spectral problems with mixed Dirichlet-Neumann
boundary conditions isospectrality and beyond
D. Jacobson, M. Levitin,
N. Nadirashvili, I.
Polterovich (2004) Isospectral domains with
mixed boundary conditions
M. Levitin, L. Parnovski, I.
Polterovich (2005)
This isospectral quartet can be obtained when
acting with the group D4xD4 on the following
torus and considering the subgroups H1X H1, H1X
H2, H2X H1, H2X H2 with the reps. R1X R1, R1X
R2, R2X R1, R2X R2 (using the notation
presented before for the main dihedral example).
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The relation to Sunadas construction
Let G be a group. Let H1, H2 be two subgroups of
G. Then the triple (G, H1, H2) satisfies Sunadas
condition ifwhere g is the cunjugacy class
of g in G.For such a triple (G, H1, H2) we get
that , are isospectral.Pesce (94)
proved Sunadas theorem using the observation
that Sunadas condition is equivalent to the
following The relation to the construction
method presented so far is via the
identification
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Further on

• What is the strength of this method ?
• Having two isospectral graphs how to construct
  the parent graph from which they were born ?
• Having such a parent graph, can it be shown
  that it obeys a symmetry group such that the
  conditions of the theorem are fulfilled ?
• What are the conditions which guarantee that the
  quotient graphs are not isometric ?
• A graph with a self-adjoint operator might be
  isospectral to a graph with a non self-adjoint
  one.
• What other properties of the functions can be
  used to resolve isospectrality ?

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Groups, Graphs Isospectrality

• Rami Band
• Ori Parzanchevski
• Gilad Ben-Shach
• Uzy Smilansky

Acknowlegments M. Sieber, I. Yaakov.