  We report on a hitherto unnoticed type of resonances occurring in scattering
from networks (quantum graphs) which are due to the complex connectivity of the
graph - its topology. We consider generic open graphs and show that any cycle
leads to narrow resonances which do not fit in any of the prominent paradigms
for narrow resonances (classical barriers, localization due to disorder,
chaotic scattering). We call these resonances `topological' to emphasize their
origin in the non-trivial connectivity. Topological resonances have a clear and
unique signature which is apparent in the statistics of the resonance
parameters (such as e.g., the width, the delay time or the wave-function
intensity in the graph). We discuss this phenomenon by providing analytical
arguments supported by numerical simulation, and identify the features of the
above distributions which depend on genuine topological quantities such as the
length of the shortest cycle (girth). These signatures cannot be explained
using any of the other paradigms for narrow resonances. Finally, we propose an
experimental setting where the topological resonances could be demonstrated,
and study the stability of the relevant distribution functions to moderate
dissipation.
