Resolving the isospectrality of the dihedral graphs Ram Band, Uzy Smilansky

1. k13 I=1 I=2 I=2 I=1 k16 k8 Wave function Metric count → 8 Vertex wave function + - + + - - Discrete count → 3 a 2c 2b GII GI N D b b N D 2a a a 2b c c 2c N D Theorem 1 – Resolution of the isospectrality by the discrete count Resolving the isospectrality of the dihedral graphsRam Band, Uzy Smilansky • 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 {IIn} for half of the spectrum. Following is a sketch of the proof of theorem 1: Resolving isospectrality by counting nodal domains Introduction to graphs 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 GI • 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. N 3 • Conjecture (by U. Smilansky et. al):The nodal count sequence resolves isospectrality. • Two types of nodal domains: • Metric domains – The connected domains where the wave function is of constant sign. • Discrete domains – A discrete domainconsists of a maximal set of connectedinterior vertices where the vertex wave function has the same sign. D 2 4 N D 1 5 5 1 N N 3 4 N N plus minus D D 2 6 Introduction to quantum graphs GII D D • 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 → N D N D • 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: • 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 • 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 Examples of several wave functions of The vector belongs to the colored domains • Calculation of h(x) the distribution function of yields 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 {In} is different from {IIn} for half of the spectrum. Theorem 2 – Resolution of the isospectrality by the metric count And indeed the isospectrality of the dihedral graphs is resolved by counting nodal domains • 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 { IIn} for half of the spectrum. k12 k14 + + + Discrete → 1 Metric → 12 + Discrete → 1 Metric → 14 References [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. k14 k12 + Discrete → 2 Metric → 12 Discrete → 1 Metric → 13 - + + • How can we distinguish between isospectral pairs ?- How can isospectrality be resolved ?