Trace formula and topological properties of graphs
Celebrated trace formula first proven by Kottos-Smilansky connects the energy spectrum with the length spectrum of periodic orbits and is therefore an efficient tool allowing one to study the relations between spectral and topological properties of quantum graphs. It will be shown how this formula can be used to determine the number of cycles in a graph, i.e. its Euler characteristics, from the spectrum of the Schrödinger operator.