Can One Hear the Shape of a Graph?

Details Can One Hear the Shape of a Graph? Can One Hear the Shape of a Graph?

2001-05-08

arxiv.org/abs/nlin/0105020v1

Nonlinear Sciences

Chaotic Dynamics

9 pages, 1 figure

Scientific paper

10.1088/0305-4470/34/31/301

We show that the spectrum of the Schrodinger operator on a finite, metric graph determines uniquely the connectivity matrix and the bond lengths, provided that the lengths are non-commensurate and the connectivity is simple (no parallel bonds between vertices and no loops connecting a vertex to itself). That is, one can hear the shape of the graph! We also consider a related inversion problem: A compact graph can be converted into a scattering system by attaching to its vertices leads to infinity. We show that the scattering phase determines uniquely the compact part of the graph, under similar conditions as above.

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