Groups, Graphs & Isospectrality

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

2. What is a graph ? • 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.

3. ‘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)

4. 3 2 4 1 5 5 1 3 4 2 6 Metric Graphs - Introduction L23 • A graphΓ 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. • 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

5. 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 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).

6. 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 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 )

7. 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 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 )

8. D N √2 2 √3 √3 N D The Spectrum of Quantum Graphs The spectrum is With the set of corresponding eigenfunctions: Use the notation: √2 Examples of several functions of the graph: λ13≈24.0 λ8≈9.0 λ16≈37.2

9. ‘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. N D 2b b b D N 2a a a c c 2c N D

10. 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 , ry}H2 = { e , a2 , ru , rv }H3 = { e , a , a2 , a3}

11. Groups - Representations • Representation – Given a group G, a representation R is an assignment of a matrixR(g) to each group element g  G, such that:  g1,g2  GR(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:

12. F = 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

13. N N N N D D D D D D N N D N N D 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 graphΓ: • 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 ? N D

14. N N D D Groups, Graphs & Isospectrality • Theorem – Let Γ be a graphwhich obeys a symmetry group G.Let H1, H2 be two subgroupsof G with representations R1, R2that obey then the graphs , are isospectral. • In our example: Γ = H1 = {e , a2 , rx , ry} R1: H2 = {e , a2, ru , rv} R2: And it can be checked that N D G = D4

15. Extending the Isospectral pair Extending our example: Γ = H1 = { e , a2, rx , ry} R1: H2 = { e , a2, ru , rv} R2: H3 = { e , a, a2 , a3} R3: G = D4 N N D D N D ×i ×i ×i

16. Extending the Isospectral pair Extending our example: Γ = H1 = { e , a2, rx , ry} R1: H2 = { e , a2, ru , rv} R2: H3 = { e , a , a2 , a3} R3: G = D4 N N D D N D ×i ×i ×i

17. Groups, Graphs & Isospectrality • Theorem – Let Γ be a graph which obeys a symmetry group G.Let H1, H2 be two subgroups of G with representations R1, R2that 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. R2and using finishes the proof.

18. Groups, Graphs & Isospectrality • Theorem – Let Γ be a graph which obeys a symmetry group G.Let H1, H2 be two subgroups of G with representations R1, R2that obey then the graphs , are isospectral. • 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. • Representations which are not 1-d. • The dependence of in the choice of basis for the representation.

19. Arsenal of isospectral examples rv ry Γ is the Cayley graph of D4(with respect to the generators a, rx): Take again G=D4 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 L1 e a L2 L2 rx ru L1 The resulting quotient graphs are: L1 L1 L2 L1 L2 L2 L2 L2 L1 L1 L1 L2

20. 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

21. Arsenal of isospectral examples 4 G = S4acts on the tetrahedron. with the subgroups: H1 = S3with the rep. R1 H2 = S4with the rep. R2 |S4|=24 , |S3|=6 L 1 3 2 The resulting quotient graphs are: D D L L/2 D D L/2 L/2 L/2 L/2 L/2 N D N

22. 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: H1 = O, the octahedral group. H2 = Td, the tetrahedral group with reflections. |H1|= |H2|= 24

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

24. Arsenal of isospectral examples G = S3 (D3) acts on Γwith 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: L2 L3 L2 L2 L3 L1 L2 L3 L3 L2 L2 L3 L1 L1 L2 L2 L3 L2 L3 L3 L1 L1 L1 L3 L2 L3 L2 L2 L3 L3

25. Arsenal of isospectral examples Isospectral drums ‘One cannot hear the shape of a drum’Gordon, Webb and Wolpert (1992) G0=*444 (using Conway’s orbifold notation) acts on the hyperbolic plane. Considering a homomorphism of G0 onto G=PSL(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: D D N N

26. 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).

27. The relation to Sunada’s construction Let G be a group. Let H1, H2 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:

28. 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 ?

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