  Trace formulas play a central role in the study of spectral geometry and in
particular of quantum graphs. The basis of our work is the result by Kurasov
which links the Euler characteristic $\chi$ of metric graphs to the spectrum of
their standard Laplacian. These ideas were shown to be applicable even in an
experimental context where only a finite number of eigenvalues from a physical
realization of quantum graph can be measured.
  In the present work we analyse sufficient hypotheses which guarantee the
successful recovery of $\chi$. We also study how to improve the efficiency of
the method and in particular how to minimise the number of eigenvalues
required. Finally, we compare our findings with numerical
examples---surprisingly, just a few dozens of eigenvalues can be enough.
