Concrete method for recovering the Euler characteristic of quantum graphs

Abstract

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 χ 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 χ. 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.