Groups, Graphs & Isospectrality Rami Band Ori Parzanchevski Gilad Ben-Shach Uzy Smilansky.

Groups, Graphs Groups, Graphs & Isospectrality& Isospectrality

Rami BandRami Band

Ori ParzanchevskiOri Parzanchevski

Gilad Ben-ShachGilad Ben-Shach

Uzy SmilanskyUzy Smilansky

What is a graph ?What is a graph ?

The graph’s The graph’s spectrumspectrum is the is the sequence of basic sequence of basic frequenciesfrequencies at which the at which the graph can vibrate.graph can vibrate.

The spectrum depends on The spectrum depends on the shape of the graph.the shape of the graph.

‘‘Can one hear the shape of a drum ?’Can one hear the shape of a drum ?’was first asked by Marc Kac (1966). was first asked by Marc Kac (1966).

‘‘Can one hear the shape of a graph?’Can one hear the shape of a graph?’

Can one deduce the shape from the spectrum ?Can one deduce the shape from the spectrum ? Is it possible to have two different graphs with the Is it possible to have two different graphs with the

same spectrum (same spectrum (isospectral graphsisospectral graphs) ? ) ?

‘‘Can one hear the shape of a Can one hear the shape of a graph ?’graph ?’

Marc Kac (1914-1984)

Metric Graphs - IntroductionMetric Graphs - Introduction

A A graphgraph ΓΓ consists of a finite set consists of a finite set of vertices of vertices V={vV={vii}} and and a finite set of undirected edges E={eE={ejj}.}.

A metric graph has a finite length (Le>0) assigned to each edge.

Let EEvv 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:

2

1

3 4

5

6

3

4

5

1

2

L13

L23

L34

L45

L46

L15

L45

L34

L23

L12L25

L35

L14

),,(Eee ffF

1

C],[: jj ee Lf 0

vv Ed :

Quantum Graphs - Quantum Graphs - IntroductionIntroduction

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 form:Where Av and Bv are complex dv x dv matrices.

To ensure the self-adjointness of the Laplacian, we require that the matrix (Av|Bv) has rank dv, and that the matrix AvBv

† is self-adjoint (Kostrykin and Schrader, 1999).

),...,(Eee ffF

1

0 'vvvv FBFA

,))(),...,(( teev vfvfFvd1

teev vfvfFdv

))(),...,(( '''

1





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 )

),...,(Eee ffF

1

0 'vvvv FBFA

00

101

1

101

0011

vA

111

000

00

vB

0ef v 0ef v


teev vfvfFdv

))(),...,(( '''

1





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 )

),...,(Eee ffF

1

0 'vvvv FBFA


teev vfvfFdv

))(),...,(( '''

1

0ef v 0ef v

The The spectrumspectrum is is

With the set of corresponding eigenfunctions:With the set of corresponding eigenfunctions:

Use the notation: Use the notation:

The Spectrum of Quantum The Spectrum of Quantum GraphsGraphs

Examples of several functions of the graph:

λλ88≈9.0≈9.0 λλ1313≈24≈24.0.0

λλ1616≈3≈37.27.2

2√2

D

D

N

N√3√3

√2

211 ; nn

nn FF n

FF|:)( F

One can hear the shape of a simple graph One can hear the shape of a simple graph if the lengths are incommensurate if the lengths are incommensurate (Gutkin, Smilansky 2001)(Gutkin, Smilansky 2001)

Otherwise, Otherwise, we do have isospectral graphs:we do have isospectral graphs: Von Below (2001)Von Below (2001) Band, Shapira, Smilansky (2006)Band, Shapira, Smilansky (2006)

There are several methods for There are several methods for construction of isospectralityconstruction of isospectrality – the main is due to Sunada (1985). – the main is due to Sunada (1985).

We present a method based on We present a method based on representation theory arguments.representation theory arguments.

a2b

a

2c

D N 2abb

cc

D

D

N

N

‘‘Can one hear the shape of a Can one hear the shape of a graph ?’graph ?’

rrxxee

Groups & GraphsGroups & Graphs Example: The Dihedral group – Example: The Dihedral group –

the symmetry group of the square the symmetry group of the squareDD44 = { e , a , a = { e , a , a22 , a , a33 , r , rxx , r , ryy , r , ruu , r , rvv } }

yy

xx

uuvv

aaHow does the Dihedral group act on a square ?How does the Dihedral group act on a square ?

A few subgroups of the Dihedral group: A few subgroups of the Dihedral group: HH11 = { e , a = { e , a2 2 , r, rxx , r , ryy}}HH22 = { e , a = { e , a2 2 , r, ruu , r , rvv } }HH33 = { e , a , a = { e , a , a22 , a , a33}}

RepresentationRepresentation – Given a group – Given a group GG, a representation , a representation RR is an assignment of a is an assignment of a matrixmatrix R(g)R(g) to each group element to each group element g g G G, such that: , such that: g g11,g,g2 2 G G R(gR(g11)R(g)R(g22)=R(g)=R(g11gg22))..

Example 1 - DExample 1 - D44 has the following 1-dimensional rep. S has the following 1-dimensional rep. S11::

Example 2 - DExample 2 - D44 has the following 2-dimensional rep. S has the following 2-dimensional rep. S22::

Restriction Restriction - is the following rep. of H- is the following rep. of H11::

InductionInduction - is the following rep. of G: - is the following rep. of G:

Groups - RepresentationsGroups - Representations

1e 1a 12 a 13 a 1xr 1yr 1ur 1vr

10

01e

01

10a

10

012a

01

103a

10

01xr

10

01yr

01

10ur

01

10vr

14

1SD

HResQ 1e 1a 12 a 13 a 1xr 1yr 1ur 1vr

QDH4

1Ind

10

01e

10

01a

10

012a

10

013a

10

01xr

10

01yr

10

01ur

10

01vr

F =F =F =F =

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

The group can act on a graph and …The group can act on a graph and … the group can act on a function which is defined on the graph the group can act on a function which is defined on the graph and may give new functions: and may give new functions:

We have We have

So that, form a representation of the group.So that, form a representation of the group.

Groups & GraphsGroups & Graphs

...,)()(, aFaFFF

How does the group act on the function ?How does the group act on the function ?

aa

Example: The Dihedral group – Example: The Dihedral group – the symmetry group of the square the symmetry group of the squareDD44 = { e , a , a = { e , a , a22 , a , a33 , r , rxx , r , ryy , r , ruu , r , rvv } }

)xF(a(aF)(x) ,x -1

)(

Consider the following rep. Consider the following rep. RR11 of the subgroup of the subgroup HH11::

HH11 = { e , a = { e , a22, r, rxx , r , ryy}}

RR11::

This is a 1d rep. – what do we know This is a 1d rep. – what do we know about the function F ?about the function F ?

Consider the following rep. Consider the following rep. RR22 of the subgroup of the subgroup HH22::

HH22 = { e , a = { e , a22, r, ruu , r , rvv}}

RR22::

This is a 1d rep. – what do we know This is a 1d rep. – what do we know about the function F ?about the function F ?

D

N

N

D

Groups & GraphsGroups & Graphs

1R

Γ

1e 12 a 1xr 1yr

FF xr FFyr

1e 12 a 1ur 1vr

FFur FF vr

Examine the graphExamine the graph ΓΓ::

D N

2RΓ

DDDD

N

N

N

N

N

N

D

D

Theorem – Let Theorem – Let ΓΓ be a graph be a graphwhich obeys a symmetry group which obeys a symmetry group GG..Let Let HH11, , HH22 be two subgroups be two subgroupsof G with representations of G with representations RR11, , RR22 that obey that obey then the graphs , are isospectral.then the graphs , are isospectral.

In our example:In our example:

ΓΓ = =

HH11 = {e , a = {e , a2 2 , r, rx x , r, ryy} R} R11::

HH22 = {e , a = {e , a22, r, ruu , r , rvv} R} R22::

And it can be checked thatAnd it can be checked that

Groups, Graphs & Groups, Graphs & IsospectralityIsospectrality

1RΓ

2RΓ

D

N

N

D

D N

1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1RΓ

2RΓ

21 21IndInd RR G

HGH

214

2

4

1RR D

HDH IndInd

G = DG = D44

Extending our example:Extending our example:

ΓΓ = =

HH11 = { e , a = { e , a22, r, rxx , r , ryy} R} R11::

HH22 = { e , a = { e , a22, r, ruu , r , rvv} R} R22::

HH33 = { e , a, a = { e , a, a22 , a , a33} R} R33::

Extending the Isospectral pairExtending the Isospectral pair

D

N

N

D

D N

1RΓ

1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

FF ia

××ii××ii××ii

3RΓ

3RΓ

2RΓ

3214

3

4

2

4

1RRR D

HDH

DH IndIndInd

G = DG = D44

3RΓ

3RΓ

Extending our example:Extending our example:

ΓΓ = =

HH11 = { e , a = { e , a22, r, rxx , r , ryy} R} R11::

HH22 = { e , a = { e , a22, r, ruu , r , rvv} R} R22::

HH33 = { e , a , a = { e , a , a22 , a , a33} R} R33::

Extending the Isospectral pairExtending the Isospectral pair

D

N

N

D

D N

1RΓ

1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

××ii ××ii××ii

2RΓ

1f 2fv

(v)f

(v)fF

2

1v

(v)'f

(v)'f'F

2

1v

0

0'FBFA vvvv

00

1Av

i

i1

00Bv

G = DG = D44

3214

3

4

2

4

1RRR D

HDH

DH IndIndInd

TheoremTheorem – Let – Let ΓΓ be a graph which obeys a symmetry group be a graph which obeys a symmetry group GG..Let Let HH11, , HH22 be two subgroups of G with representations be two subgroups of G with representations RR11, , RR22 that obey then the graphs , are isospectral.that obey then the graphs , are isospectral.

ProofProof::LemmaLemma: There exists a quantum graph such that:: There exists a quantum graph such that:

Using the Lemma and Frobenius reciprocity theorem gives:Using the Lemma and Frobenius reciprocity theorem gives:

Hence , are isospectral.Hence , are isospectral.Applying the same for the rep. Applying the same for the rep. RR22 and using and using finishes the proof.finishes the proof.


1RΓ

2RΓ

21 21IndInd RR G

HGH

1RΓ

)(,Hom)(1H 1

1

RC CR

)()(,IndHom

)(,Hom)(

G

H1

1

1

1

1

R

RC

GHC

CR

1RΓ

1RGH1

IndΓ

21 21IndInd RR G

HGH

TheoremTheorem – Let – Let ΓΓ be a graph which obeys a symmetry group be a graph which obeys a symmetry group GG..Let Let HH11, , HH22 be two subgroups of G with representations be two subgroups of G with representations RR11, , RR22 that obey then the graphs , are that obey then the graphs , are isospectral.isospectral.

ProofProof::LemmaLemma: There exists a quantum graph such that:: There exists a quantum graph such that:

Interesting issues in the proof of the lemma:Interesting issues in the proof of the lemma: A group which does not act freely on the edges.A group which does not act freely on the edges. Representations which are not 1-d.Representations which are not 1-d. The dependence of in the choice of basis for the The dependence of in the choice of basis for the

representation.representation.


1RΓ

2RΓ

21 21IndInd RR G

HGH

1RΓ

)(,Hom)(1H 1

1

RC CR

1RΓ

ΓΓ is the Cayley graph of is the Cayley graph of DD44

(with respect to the generators (with respect to the generators aa, , rrxx): ):

Take again Take again G=DG=D44 and the same subgroups: and the same subgroups:HH11 = { e , a = { e , a22, r, rxx , r , ryy} } with the rep.with the rep. R R11

HH22 = { e , a = { e , a22, r, ruu , r , rvv} } with the rep.with the rep. R R22

HH33 = { e , a , a = { e , a , a22 , a , a33} } with the rep.with the rep. R R33

Arsenal of isospectral Arsenal of isospectral examplesexamples

1RΓ

2RΓ

ea

a3 a

2

rx

ryrv

ruL1

L2

L2

L1

The resulting quotient graphs are:The resulting quotient graphs are:

3RΓ

L1

L1

L1

L1

L2

L2

L2

L2

L1

L1

L2

L2

G = DG = D66 = {e, a, a = {e, a, a22, a, a33, a, a44, a, a55, r, rxx, r, ryy, r, rzz, r, ruu, r, rvv, r, rww} } with the subgroups:with the subgroups:

HH11 = { e, a = { e, a22, a, a44, r, rxx, r, ryy, r, rz z } } with the rep.with the rep. R R11

HH22 = { e, a = { e, a22, a, a44, r, ruu, r, rvv, r, rw w } } with the rep.with the rep. R R22

HH33 = { e, a, a = { e, a, a22, a, a33, a, a44, a, a5 5 } } with the rep.with the rep. R R33


1RΓ

2RΓ


3RΓ

L2

L1 2L2

2L3

2L1L2

L3L3

2L2

2L3

2L1

2L3

2L3

2L2

2L2

2L1

L1

L1

2L2

2L2

2L3

2L3

2L1

2L1

G = SG = S44 acts on the tetrahedron.acts on the tetrahedron.

with the subgroups:with the subgroups:HH11 = S = S33 with the rep.with the rep. R R11

HH22 = S = S44 with the rep.with the rep. R R22

|S|S44|=24 , |S|=24 , |S33|=6|=6



L

3

4

1

2

1RΓ

D

L

L/2

L/2

2RΓ

D

N

L/2L/2

L/2

D

DN

D

L/2

(v)f

(v)fF

2

1v

(v)'f

(v)'f'F

2

1v

0

0'FBFA vvvv

00

11Av

21

00Bv

1f

2f

v

vvvv ABBA

ΓΓ is the a graph which obeys is the a graph which obeys the the OOh h (octahedral group (octahedral group with reflections).with reflections). |O |Ohh|=48. |=48.

Take Take G= OG= Ohh and the subgroups: and the subgroups:HH11 = O, = O, the octahedral group.the octahedral group.HH22 = T = Tdd, , the tetrahedral groupthe tetrahedral group with reflections.with reflections. |H|H11|= |H|= |H22|= 24|= 24



A puzzleA puzzle: construct an isospectral pair : construct an isospectral pair out of the following familiar graphout of the following familiar graph: :

G = SG = S33 (D (D33) ) acts on acts on ΓΓ with no fixed points. with no fixed points.

To construct the quotient graph, To construct the quotient graph, we take the same rep. of we take the same rep. of GG, , but use two different bases but use two different bases for the matrix representation. for the matrix representation.



L1

L2

L3

L3 L2

L1L3

L3

L3

L2L2

L2

L1L3

L3

L3

L2L2

L2

L1

L2

L2

L3

L3

L1

L2 L3

L1

L2 L3

L1 L2

L3

L3

L2


‘‘One One cannotcannot hear the shape of a drum’ hear the shape of a drum’ Gordon, Webb and Wolpert (1992)

Isospectral drumsIsospectral drums

D

N

D

N

GG00=*444=*444 (using Conway’s orbifold notation) acts on the hyperbolic plane.

Considering a homomorphism of GG00 onto G=PSL(3,2)G=PSL(3,2)and taking two subgroups HH11, HH22 such that: and RR11, RR22 are the sign representations, we obtain the known isospectral drums of Gordon et al.but with new boundary conditions:

4241 SHSH ;


‘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)

Isospectral drumsIsospectral drums

This isospectral quartet can be obtained when This isospectral quartet can be obtained when acting with the group Dacting with the group D44xDxD44 on the following on the following torus and considering the subgroups torus and considering the subgroups HH11X HX H11, H, H11X HX H22, H, H22X HX H11, H, H22X HX H22 with the reps. with the reps. RR11X RX R11, R, R11X RX R22, R, R22X RX R11, R, R22X RX R22 (using the notation presented before for the (using the notation presented before for the main dihedral example).main dihedral example).

The relation to Sunada’s The relation to Sunada’s constructionconstruction

21 HH 11 GH

GH IndInd

21

Let G be a group. Let HH11, , HH22 be two subgroups of G be two subgroups of G.

Then the triple (G, H1, H2) satisfies Sunada’s condition if:

where [g] is the cunjugacy class of g in G.For such a triple (G, H1, H2) we get that , are isospectral.

Pesce (94) proved Sunada’s theorem using the observation that Sunada’s condition is equivalent to the following:

The relation to the construction method presented so far is via the identification:

21 HgHgGg ][][

1H

21, iH i iH1

2H

Further on …Further on … What is the strength of this method ?What is the strength of this method ?

Having two isospectral graphs – how to construct the ‘parent’ graph Having two isospectral graphs – how to construct the ‘parent’ graph from which they were born ?from which they were born ?

Having such a ‘parent’ graph, can it be shown that it obeys a Having such a ‘parent’ graph, can it be shown that it obeys a symmetry group such that the conditions of the theorem are symmetry group such that the conditions of the theorem are fulfilled ?fulfilled ?

What are the conditions which guarantee that the quotient What are the conditions which guarantee that the quotient graphs are not isometric ?graphs are not isometric ?

A graph with a self-adjoint operator might be isospectral to a A graph with a self-adjoint operator might be isospectral to a graph with a non self-adjoint one.graph with a non self-adjoint one.

What other properties of the functions can be used to What other properties of the functions can be used to resolve isospectrality ?resolve isospectrality ?

Groups, Graphs Groups, Graphs & Isospectrality& Isospectrality

Rami BandRami Band

Ori ParzanchevskiOri Parzanchevski

Gilad Ben-ShachGilad Ben-Shach

Uzy SmilanskyUzy Smilansky

Acknowlegments: M. Sieber, I. Yaakov.Acknowlegments: M. Sieber, I. Yaakov.

Groups, Graphs & Isospectrality Rami Band Ori Parzanchevski Gilad Ben-Shach Uzy Smilansky.

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Transcript of Groups, Graphs & Isospectrality Rami Band Ori Parzanchevski Gilad Ben-Shach Uzy Smilansky.