  In this paper, we try to put the results of Smilansky and al. on "Topological
resonances" 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 estimateof the
dimension of this semi-algebraic set, in particular in terms of the girth of
the graph. The case oftrees is also discussed.
