On the effective size of a non-Weyl graph

We show how to find the coefficient of the leading term of the resonance asymptotics using the method of pseudo-orbit expansion for quantum graphs which do not obey Weyl asymptotics. For a non-Weyl graph we develop a method to reduce the number of edges of a corresponding directed graph. Through this method we prove bounds on the above coefficient depending on the structure of the graph, for graphs with the same lengths of internal edges. We explicitly find the positions of the resolvent resonances. (paper)