Resolving the isospectrality of the dihedral graphsRam Band, Uzy Smilansky

• A graph G is made of V vertices and B bonds and a connectivity matrix Cij

• Cij≡ # of bonds connecting i and j.

• Bond notation is (i,j)• Valency of the vertex i:

• Boundary vertex is a vertex with vi=1.

• Interior vertex is a vertex with vi>1.2

3

4

5

1

1

3 4

5

6

2

Introduction to graphs

V

jiji Cv

1

• On the bond (i,j) use the coordinate xij:

• The wave function ψij

on each bond obeys

On the interior vertices (vi>1) we demand

• Continuity

• Current conservation

On the boundary vertices (vi=1) we demand

Dirichlet → or Neumann →

Introduction to quantum graphs

ijij

ij

kdx

d 22

2

ijjijiij Lxx ,0

0..)0( ijiji Ctsj

0:0

0;

ij

ij xCj ij

ij

dx

di

0i 00

ijx

ijijdx

d

Examples of several wave functions of kk88 kk1313 kk1616

Resolving isospectrality by counting nodal domains

• Conjecture (by U. Smilansky et. al): The nodal count sequence resolves isospectrality.

• Two types of nodal domains:

1) Metric domains – The connected domains where the wave function is of constant sign.

2) Discrete domains – A discrete domainconsists of a maximal set of connectedinterior vertices where the vertex wave function has the same sign.

• Discrete nodal count resolves isospectrality for some isospectral pairs. Numerical evidence was found [2].

• We look for rigorous proof of resolving isospectrality by counting nodal domains.

Search for new simpler isospectral graphs

a

2b2c

• Constructing isospectral pair using the Dihedral D4 symmetry

• The work was done Following [3] and the notes of Martin Sieber

• We obtain the Dihedral graphs

Theorem 1 – Resolution of the isospectrality by the discrete count

• Theorem 1 – Let GI and GII the graphs below. Denote with {in} the sequence of

discrete nodal count of the graph Gi.

Then {In} is different from {II

n} for half of the spectrum.

Following is a sketch of the proof of theorem 1:

GII

a2ba

2c

D N

GI

2abb

cc

D

D

N

N

The transplantation that takes an eigenfunction of GI with eigenvalue k and transforms it to eigenfunction of GII with eigenvalue k is:

• Cut the graph GI with its eigenfunction along the dashed line. Let , be the two functions defined on the subgraphs.

• Obtain two new functions , by: (appropriate reflections are be needed).

• Glue , together to obtain an eigenfunction on GII

minus

2

1

2

1

11

11~

~

1

• The action of the transplantation on the vertex wave function is

• The transplantation implies that the vector is obtained by rotating counterclockwise by (this is true since the eigenfunction is defined up to a multiplicative factor).

• The number of nodal domains is

• Therefore if the transplantation rotates the vector across the quadrant borders. In other words:

2

1

4

))(tan1(2

11))(1(

2

11

2

1 signsign

III

),1()0,1()tan( III The vector belongs to the colored domains

I=1

I=1

I=2

I=2

2

1

)()( xhxh • Calculation of h(x) the distribution function of yields tan

1

2

{In} is different from {II

n} for half of the spectrum.

• Theorem 2 – Let GI and GII the graphs below. Denote with {in} the sequence of

metric nodal count of the graph Gi.

Then { In} is different from { II

n} for half of the spectrum.

References

Metric count → 8

Wave functionWave function

++

++--

--

++

--Discrete count → 3

Vertex wave Vertex wave functionfunction

Isospectral graphs are graphs that have the same spectrum in spite of being different.

• Below is an example of such isospectral pair of graphs which is constructed out of isospectral domains [1]

• These graphs are not isometricAnd indeed the isospectrality of the dihedral graphs is

resolved by counting nodal domains

kk11

44

kk11

44

kk11

22

kk11

22

Discrete → 1

Metric → 12

Discrete → 2

Metric → 12

++++

++ --

++ ++

++

++

Discrete → 1

Metric → 14

Discrete → 1

Metric → 13

Theorem 2 – Resolution of the isospectrality by the metric count

1 2

1

1

212211~,~

• How can we distinguish between isospectral pairs ?- How can isospectrality be resolved ?

GII

D N

GI

D

D

N

N

N

N

2

1

22

D

D

1~

1

~D

N

N

11

22

2

~2~

D

D

plus

N

1~ 2

~1

~ 2

~

2

1~ 2~

1~ 2~

2

1~

~

2

1

2

1

[1] P. Buser, J. Conway, P. Doyle and K-D Semmler, Int. Math. Res. Notices 9 (1994), 391-400.

[2] T. Shapira and U. Smilansky, Proceedings of the NATO advanced research workshop, Tashkent, Uzbekistan, 2004, In Press.

[3] D. Jakobson, M. Levitin, N. Nadirashvili and I. Polterovich, J. Comp. and Appl. Math. 194 (2006), 141-155.and M. Levitin, L. Parnovski and I. Polterovich, j. Phys. A: Math. Gen., 39 (2006), 2073-2082.