Resolving the isospectrality of the dihedral graphs Ram Band, Uzy Smilansky A graph G is made of V vertices and B bonds and a connectivity matrix C ij C ij ≡ # of bonds connecting i and j. Bond notation is (i,j) Valency of the vertex i: Boundary vertex is a vertex with v i =1. Interior vertex is a vertex with v i > Introduction to graphs On the bond (i,j) use the coordinate x ij : The wave function ψ ij on each bond obeys On the interior vertices (v i >1) we demand Continuity Current conservation On the boundary vertices (v i =1) we demand Dirichlet → or Neumann → Introduction to quantum graphs Examples of several wave functions of k8k8k8k8 k 13 k 16 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 domain consists of a maximal set of connected interior 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 2b 2c Constructing isospectral pair using the Dihedral D 4 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 G I and G II the graphs below. Denote with {  i n } the sequence of discrete nodal count of the graph G i. Then {  I n } is different from {  II n } for half of the spectrum. Following is a sketch of the proof of theorem 1: G II a 2b a 2c D N GIGI 2a b b cc D D N N The transplantation that takes an eigenfunction of G I with eigenvalue k and transforms it to eigenfunction of G II with eigenvalue k is: Cut the graph G I 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 G II minus 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: The vector belongs to the colored domains  I =1  I =2 Calculation of h(x) the distribution function of yields {  I n } is different from {  II n } for half of the spectrum. Theorem 2 – Let G I and G II the graphs below. Denote with { i n } the sequence of metric nodal count of the graph G i. Then { I n } is different from { II n } for half of the spectrum. References Metric count → 8 Wave function Discrete count → 3 Vertex wave function 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 isometric And indeed the isospectrality of the dihedral graphs is resolved by counting nodal domains k 14 k 12 Discrete → 1 Metric → 12 Discrete → 2 Metric → Discrete → 1 Metric → 14 Discrete → 1 Metric → 13 Theorem 2 – Resolution of the isospectrality by the metric count How can we distinguish between isospectral pairs ? - How can isospectrality be resolved ? G II D N GIGI D D N N N N D D D N N D D plus N [1] P. Buser, J. Conway, P. Doyle and K-D Semmler, Int. Math. Res. Notices 9 (1994), [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), and M. Levitin, L. Parnovski and I. Polterovich, j. Phys. A: Math. Gen., 39 (2006),