Abstract
In this paper, we try to put the results of Smilansky and al. on Topological on a mathematical basis. A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exists compactly supported eigenfunctions. We give several estimate of the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case of trees is also discussed.
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