Isospectral graphs and isospectral surfaces. (English) Zbl 0910.05042

Brooks, Robert

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 105-113 (1997).

Spectral theory of Riemannian surfaces and spectral theory of graphs are closely related. There has been mutual influence and interaction in both directions. Isospectral surfaces have been investigated in relation to the well-known question “Can one hear the shape of a drum?” or how does the geometric structure influence the spectrum. Isospectral graphs also received much attention in the past, originally due to possible applications to the reconstruction conjecture. The author relates a theorem of Sunada on covering spaces and isospectrality of quotients to known results in graph theory.
For the entire collection see [Zbl 0882.00016].

Reviewer: B.Mohar (Ljubljana)

MSC:

05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

Laplacian; isospectral surfaces; isospectral graphs; Riemannian surfaces; spectral theory of graphs; isospectrality