Non-Weyl Resonance Asymptotics for Quantum Graphs

We consider the resonances of a quantum graph \(\mathcal G\) that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of \(\mathcal G\) in a disc of a large radius. We call \(\mathcal G\) a \emph{Weyl graph} if the coefficient in front of this leading term coincides with the volume of the compact part of \(\mathcal G\). We give an explicit topological criterion for a graph to be Weyl. In the final section we analyze a particular example in some detail to explain how the transition from the Weyl to the non-Weyl case occurs.