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

What is a graph ? The graph’s spectrum is the sequence of basic frequencies at which the graph can vibrate. The graph’s spectrum is the sequence of basic frequencies at which the graph can vibrate. The spectrum depends on the shape of the graph. The spectrum depends on the shape of the graph.

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

Metric Graphs - Introduction A graph Γ consists of a finite set of vertices V={v i }and E={e j }. A graph Γ consists of a finite set of vertices V={v i } and a finite set of undirected edges E={e j }. A metric graph has a finite length (L e >0) assigned to each edge. E v Let E v 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: L 13 L 23 L 34 L 45 L 46 L 15 L 45 L 34 L 23 L 12 L 25 L 35 L 14

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 form: Where A v and B v are complex d v x d v matrices. To ensure the self-adjointness of the Laplacian, we require that the matrix (A v |B v ) has rank d v, and that the matrix A v B v † is self-adjoint (Kostrykin and Schrader, 1999).

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 form: Where A v and B v are complex d v x d v 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 d v =1 vertices, there are two special cases of boundary conditions, denoted Dirichlet: A v = (1), B v =(0) (so that ) Neumann: A v = (0), B v =(1) (so that )

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 form: Where A v and B v are complex d v x d v matrices. A common boundary condition is the Kirchhoff condition, namely all functions agree at each vertex, and the sum of the derivatives vanishes. For d v =1 vertices, there are two special cases of boundary conditions, denoted Dirichlet: A v = (1), B v =(0) (so that ) Neumann: A v = (0), B v =(1) (so that )

The spectrum is With the set of corresponding eigenfunctions: Use the notation: The Spectrum of Quantum Graphs Examples of several functions of the graph: λ 8 ≈9.0 λ 13 ≈24.0 λ 16 ≈ √2 D D N N √3 √2

One can hear the shape of a simple graph if the lengths are incommensurate (Gutkin, Smilansky 2001) One can hear the shape of a simple graph if the lengths are incommensurate (Gutkin, Smilansky 2001) Otherwise, we do have isospectral graphs: Otherwise, we do have isospectral graphs: Von Below (2001) Von Below (2001) Band, Shapira, Smilansky (2006) Band, Shapira, Smilansky (2006) There are several methods for construction of isospectrality – the main is due to Sunada (1985). There are several methods for construction of isospectrality – the main is due to Sunada (1985). We present a method based on representation theory arguments. We present a method based on representation theory arguments. a 2b a 2c DN 2a b b c c D D N N ‘Can one hear the shape of a graph ?’

rxrxrxrx e Groups & Graphs Example: The Dihedral group – the symmetry group of the square D 4 = { e, a, a 2, a 3, r x, r y, r u, r v } Example: The Dihedral group – the symmetry group of the square D 4 = { e, a, a 2, a 3, r x, r y, r u, r v } y x u v a How does the Dihedral group act on a square ? A few subgroups of the Dihedral group: H 1 = { e, a 2, r x, r y } H 2 = { e, a 2, r u, r v } H 3 = { e, a, a 2, a 3 } A few subgroups of the Dihedral group: H 1 = { e, a 2, r x, r y } H 2 = { e, a 2, r u, r v } H 3 = { e, a, a 2, a 3 }

Representation – Given a group G, a representation R is an assignment of a matrix R(g) to each group element g  G, such that:  g 1,g 2  G R(g 1 )R(g 2 )=R(g 1 g 2 ). Representation – Given a group G, a representation R is an assignment of a matrix R(g) to each group element g  G, such that:  g 1,g 2  G R(g 1 )R(g 2 )=R(g 1 g 2 ). Example 1 - D 4 has the following 1-dimensional rep. S 1 : Example 1 - D 4 has the following 1-dimensional rep. S 1 : Example 2 - D 4 has the following 2-dimensional rep. S 2 : Example 2 - D 4 has the following 2-dimensional rep. S 2 : Restriction - is the following rep. of H 1 : Restriction - is the following rep. of H 1 : Induction - is the following rep. of G: Induction - is the following rep. of G: Groups - Representations

F = Knowing the matrix representation gives us information on the functions. Knowing the matrix representation gives us information on the functions. 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: 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. We have So that, form a representation of the group. Groups & Graphs How does the group act on the function ? a Example: The Dihedral group – the symmetry group of the square D 4 = { e, a, a 2, a 3, r x, r y, r u, r v } Example: The Dihedral group – the symmetry group of the square D 4 = { e, a, a 2, a 3, r x, r y, r u, r v }

Consider the following rep. R 1 of the subgroup H 1 : Consider the following rep. R 1 of the subgroup H 1 : H 1 = { e, a 2, r x, r y } R 1 : This is a 1d rep. – what do we know about the function F ? Consider the following rep. R 2 of the subgroup H 2 : Consider the following rep. R 2 of the subgroup H 2 : H 2 = { e, a 2, r u, r v } R 2 : This is a 1d rep. – what do we know about the function F ? D N N D Groups & Graphs Examine the graph Γ: D N D D D D N N N N N N D D

Theorem – Let Γ be a graph which obeys a symmetry group G. Let H 1, H 2 be two subgroups of G with representations R 1, R 2 that obey then the graphs, are isospectral. Theorem – Let Γ be a graph which obeys a symmetry group G. Let H 1, H 2 be two subgroups of G with representations R 1, R 2 that obey then the graphs, are isospectral. In our example: In our example: Γ = H 1 = {e, a 2, r x, r y } R 1 : H 2 = {e, a 2, r u, r v } R 2 : And it can be checked that Groups, Graphs & Isospectrality D N N D D N G = D 4

Extending our example: Γ = H 1 = { e, a 2, r x, r y } R 1 : H 2 = { e, a 2, r u, r v } R 2 : H 3 = { e, a, a 2, a 3 } R 3 : Extending the Isospectral pair D N N D D N ×i×i×i×i ×i×i×i×i ×i×i×i×i G = D 4

Extending our example: Γ = H 1 = { e, a 2, r x, r y } R 1 : H 2 = { e, a 2, r u, r v } R 2 : H 3 = { e, a, a 2, a 3 } R 3 : Extending the Isospectral pair D N N D D N ×i×i×i×i ×i×i×i×i ×i×i×i×i G = D 4

Theorem – Let Γ be a graph which obeys a symmetry group G. Let H 1, H 2 be two subgroups of G with representations R 1, R 2 that obey then the graphs, are isospectral. Theorem – Let Γ be a graph which obeys a symmetry group G. Let H 1, H 2 be two subgroups of G with representations R 1, R 2 that obey then the graphs, are isospectral. Proof: Lemma : There exists a quantum graph such that: Using the Lemma and Frobenius reciprocity theorem gives: Hence, are isospectral. Applying the same for the rep. R 2 and using finishes the proof. Proof: Lemma : There exists a quantum graph such that: Using the Lemma and Frobenius reciprocity theorem gives: Hence, are isospectral. Applying the same for the rep. R 2 and using finishes the proof. Groups, Graphs & Isospectrality

Theorem – Let Γ be a graph which obeys a symmetry group G. Let H 1, H 2 be two subgroups of G with representations R 1, R 2 that obey then the graphs, are isospectral. Theorem – Let Γ be a graph which obeys a symmetry group G. Let H 1, H 2 be two subgroups of G with representations R 1, R 2 that obey then the graphs, are isospectral. Proof: Lemma : There exists a quantum graph such that: Interesting issues in the proof of the lemma: Proof: Lemma : There exists a quantum graph such that: 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 representation. The dependence of in the choice of basis for the representation. Groups, Graphs & Isospectrality

Γ is the Cayley graph of D 4 (with respect to the generators a, r x ): Take again G=D 4 and the same subgroups: H 1 = { e, a 2, r x, r y } with the rep. R 1 H 2 = { e, a 2, r u, r v } with the rep. R 2 H 3 = { e, a, a 2, a 3 } with the rep. R 3 Arsenal of isospectral examples e a a3a3 a2a2 rxrx ryry rvrv ruru L1L1 L2L2 L2L2 L1L1 The resulting quotient graphs are: L1L1 L1L1 L1L1 L1L1 L2L2 L2L2 L2L2 L2L2 L1L1 L1L1 L2L2 L2L2

G = D 6 = {e, a, a 2, a 3, a 4, a 5, r x, r y, r z, r u, r v, r w } with the subgroups: H 1 = { e, a 2, a 4, r x, r y, r z } with the rep. R 1 H 2 = { e, a 2, a 4, r u, r v, r w } with the rep. R 2 H 3 = { e, a, a 2, a 3, a 4, a 5 } with the rep. R 3 Arsenal of isospectral examples The resulting quotient graphs are: L2L2 L1L1 2L 2 2L 3 2L 1 L2L2 L3L3 L3L3 2L 2 2L 3 2L 1 2L 3 2L 2 2L 1 L1L1 L1L1 2L 2 2L 3 2L 1

G = S 4 acts on the tetrahedron. with the subgroups: H 1 = S 3 with the rep. R 1 H 2 = S 4 with the rep. R 2 |S 4 |=24, |S 3 |=6 Arsenal of isospectral examples The resulting quotient graphs are: L D L L/2 D N D D N D

Γ is the a graph which obeys the O h (octahedral group with reflections). |O h |=48. Take G= O h and the subgroups: H 1 = O, the octahedral group. H 2 = T d, the tetrahedral group with reflections. with reflections. |H 1 |= |H 2 |= 24 |H 1 |= |H 2 |= 24 Arsenal of isospectral examples

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

G = S 3 (D 3 ) acts on Γ 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. Arsenal of isospectral examples The resulting quotient graphs are: L1L1 L2L2 L3L3 L3L3 L2L2 L1L1 L3L3 L3L3 L3L3 L2L2 L2L2 L2L2 L1L1 L3L3 L3L3 L3L3 L2L2 L2L2 L2L2 L1L1 L2L2 L2L2 L3L3 L3L3 L1L1 L2L2 L3L3 L1L1 L2L2 L3L3 L1L1 L2L2 L3L3 L3L3 L2L2

Arsenal of isospectral examples ‘One cannot hear the shape of a drum’ ‘One cannot hear the shape of a drum’ Gordon, Webb and Wolpert (1992) Isospectral drums D N D N G 0 =*444 G 0 =*444 (using Conway’s orbifold notation) acts on the hyperbolic plane. G 0 G=PSL(3,2) H 1 H 2 R 1 R 2 Considering a homomorphism of G 0 onto G=PSL(3,2) and taking two subgroups H 1, H 2 such that: and R 1, R 2 are the sign representations, we obtain the known isospectral drums of Gordon et al. but with new boundary conditions:

Arsenal of isospectral examples ‘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 drums This isospectral quartet can be obtained when acting with the group D 4 xD 4 on the following torus and considering the subgroups H 1 X H 1, H 1 X H 2, H 2 X H 1, H 2 X H 2 with the reps. R 1 X R 1, R 1 X R 2, R 2 X R 1, R 2 X R 2 (using the notation presented before for the main dihedral example).

The relation to Sunada’s construction H 1, H 2 be two subgroups of G Let G be a group. Let H 1, H 2 be two subgroups of G. Then the triple (G, H 1, H 2 ) satisfies Sunada’s condition if: where [g] is the cunjugacy class of g in G. For such a triple (G, H 1, H 2 ) 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:

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 from which they were born ? 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 ? 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 ? 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. 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 ? What other properties of the functions can be used to resolve isospectrality ?

Groups, Graphs & Isospectrality Rami Band Ori Parzanchevski Gilad Ben-Shach Uzy Smilansky Acknowlegments: M. Sieber, I. Yaakov.