Groups Graphs Isospectrality Rami Band Ori Parzanchevski Gilad

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

‘Can one hear the shape of a graph ? ’ n ‘Can one hear the shape of a drum ? ’ was first asked by Marc Kac (1966). Marc Kac (1914 -1984) n ‘Can one hear the shape of a graph? ’ Can one deduce the shape from the spectrum ? n Is it possible to have two different graphs with the same spectrum (isospectral graphs) ? n

Metric Graphs - Introduction n n A graph Γ consists of a finite set of vertices V={vi} and a finite set of undirected edges E={ej}. A metric graph has a finite length (Le>0) assigned to each edge. 2 L 25 L 12 3 L 23 L 14 1 4 L 45 L 15 5 Let Ev be the set of all edges connected to a vertex v. The degree of v is 5 1 A function on the graph is a vector of functions on the edges: L 34 L 35 L 13 L 45 3 L 23 2 4 L 34 L 46 6

Quantum Graphs - Introduction n n 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 matrices. n To ensure the self-adjointness of the Laplacian, we require that the matrix (Av|Bv) has rank dv, and that the matrix Av. Bv† is self adjoint (Kostrykin and Schrader, 1999).

Quantum Graphs - Introduction n n 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 matrices. n A common boundary condition is the Kirchhoff condition, namely all functions agree at each vertex, and the sum of the derivatives vanishes. n n This corresponds to the matrices: For dv =1 vertices, there are two special cases of boundary conditions, denoted n n Dirichlet: Av= (1), Bv=(0) (so that Neumann: Av= (0), Bv=(1) (so that ) )

Quantum Graphs - Introduction n n 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 matrices. n A common boundary condition is the Kirchhoff condition, namely all functions agree at each vertex, and the sum of the derivatives vanishes. n For dv =1 vertices, there are two special cases of boundary conditions, denoted n n Dirichlet: Av= (1), Bv=(0) (so that Neumann: Av= (0), Bv=(1) (so that ) )

The Spectrum of Quantum Graphs The spectrum is With the set of corresponding eigenfunctions: Use the notation: D N √ 2 Examples of several functions of the graph: N λ 8≈9. 0 λ 13≈24. 0 √ 3 2 √ 3 D λ 16≈37. 2

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

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

Groups - Representations n 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). n Example 1 - D 4 has the following 1 -dimensional rep. S 1: n Example 2 - D 4 has the following 2 -dimensional rep. S 2: n Restriction - n Induction - is the following rep. of H 1: is the following rep. of G:

Groups & Graphs n n n Example: The Dihedral group – the symmetry group of the square D 4 = { e , a 2 , a 3 , 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, n form a representation of the group. Knowing the matrix representation us How does the group act on gives the function ? information on the functions. F= a

Groups & Graphs Consider the following rep. R 1 of the subgroup H 1: the graph Γ: Examine H 1 = { e , a 2, rx , ry} NN R 1: NN This is a 1 d rep. – what do we know D D about the function F ? D n D N N Consider the following rep. R 2 of the subgroup H 2: H 2 = { e , a 2, ru , rv} DD R 2: This is a 1 d rep. – what do we know about the function F ? n N NN D

Groups, Graphs & Isospectrality n 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: G = D 4 Γ = n H 1 = {e , a 2 , rx , ry} R 1: H 2 = {e , a 2, ru , rv} R 2: And it can be checked that N N D D D N

Extending the Isospectral pair Extending our example: G = D 4 Γ = N N H 1 = { e , a 2, rx , ry} R 1: D D H 2 = { e , a 2, ru , rv} R 2: D H 3 = { e , a, a 2 , a 3} R 3: i ×i × N

Extending the Isospectral pair Extending our example: G = D 4 Γ = N N H 1 = { e , a 2, rx , ry} R 1: H 2 = { e , a 2, ru , rv} R 2: D D N D H 3 = { e , a 2 , a 3} R 3: i ×i ×

Groups, Graphs & Isospectrality n n 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.

Groups, Graphs & Isospectrality n n 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: n n n 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.

Arsenal of isospectral examples rv Γ is the Cayley graph of D 4 (with respect to the generators a, rx): ry a 3 a 2 Take again G=D 4 and the same subgroups: H 1 = { e , a 2, rx , ry} with the rep. R 1 H 2 = { e , a 2, ru , rv} with the rep. R 2 H 3 = { e , a 2 , a 3} with the rep. R 3 e L 1 a L 2 rx ru L 1 The resulting quotient graphs are: L 1 L 2 L 2 L 1

Arsenal of isospectral examples G = D 6 = {e, a, a 2, a 3, a 4, a 5, rx, ry, rz, ru, rv, rw} with the subgroups: H 1 = { e, a 2, a 4, rx, ry, rz } with the rep. R 1 H 2 = { e, a 2, a 4, ru, rv, rw } with the rep. R 2 H 3 = { e, a, a 2, a 3, a 4, a 5 } with the rep. R 3 2 L 3 The resulting quotient graphs are: L 1 2 L 2 2 L 3 2 L 1 L 2 L 3 2 L 3 L 2 L 3 2 L 1 2 L 2 2 L 3 L 1 2 L 2 2 L 3 2 L 1

Arsenal of isospectral examples 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 4 L 1 3 |S 4|=24 , |S 3|=6 2 The resulting quotient graphs are: D D L L/2 L/2 N D L/2 D

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

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

Arsenal of isospectral examples 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. L 3 L 2 L 1 L 2 The resulting quotient graphs are: L 3 L 2 L 1 L 3 L 2 L 1 L 2 L 3 L 3 L 3 L 2 L 3 L 1 L 2 L 3

Arsenal of isospectral examples Isospectral drums ‘One cannot hear the shape of a drum’ Gordon, Webb and Wolpert (1992) G 0=*444 (using Conway’s orbifold notation) acts on the hyperbolic plane. 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: D D N N

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 D 4 x. D 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 Let G be a group. Let H 1, H 2 be two subgroups of G. 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 … n What is the strength of this method ? n n 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 theorem are fulfilled ? n What are the conditions which guarantee that the quotient graphs are not isometric ? n A graph with a self-adjoint operator might be isospectral to a graph with a non self-adjoint one. n 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.