M I Belishev and N Wada 2009 Inverse Problems 25 105011 doi:10.1088/0266-5611/25/10/105011
On revealing graph cycles via boundary measurements
M I Belishev1 and N Wada2
Show affiliationsThis paper deals with boundary value inverse problems on a metric graph, the structure of the graph being assumed unknown. The question under consideration is how to detect from the dynamical and/or spectral inverse data whether the graph contains cycles (is not a tree). For any graph Ω, the dynamical as well as spectral boundary inverse data determine the so-called wave diameter 
defined on functionals supported in the graph. The known fact is that if Ω is a tree then dw ≥ 0 holds and, in this case, the inverse data determine Ω up to isometry. A graph Ω is said to be coordinate if the functions {distΩ(
, γ)}γ
∂Ω constitute a coordinate system on Ω. For such graphs, we propose a procedure, which reveals the presence/absence of cycles. The hypothesis is that Ω contains cycles if and only if dw takes negative values. We do not justify this hypothesis in the general case but reduce it to a certain special class of graphs (suns).
- PACS
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02.60.Lj Ordinary and partial differential equations; boundary value problems
- MSC
- Subjects
- Dates
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Issue 10 (October 2009)
Received 29 June 2009, in final form 11 August 2009
Published 1 October 2009
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On revealing graph cycles via boundary measurements
M I Belishev and N Wada 2009 Inverse Problems 25 105011
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Charge transfer in heterostructures of strongly correlated materials
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Dynamic fragmentation of ductile materials
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Magnetic memory in discrete media observed by coherent soft x-ray resonant scattering
G Beutier et al 2009 New J. Phys. 11 113026
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SIXA—Silicon X-Ray Array
O Vilhu et al 1998 Phys. Scr. 1998 27
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