**Abstract:** In the talk we will focus mainly on the resonance condition and resonance
asymptotics on quantum graphs. We are interested in the number of resolvent
resonances enclosed in the circle of radius $R$ in the $k$-plane in the
limit $R\to \infty$. In some cases the leading term of the asymptotics is
smaller than expected by Weyl asymptotics. We will recall the main results
for these non-Weyl graphs. Using the method of pseudo-orbit expansion we
will construct the resonance condition and find the expression of the
effective size of the graph, which is proportional to the coefficient by the
leading term of the asymptotics. The main results are bounds on the
effective size.

The talk will be based on the publications

J. Lipovsky, Pseudo orbit expansion for the resonance condition on quantum
graphs and the resonance asymptotics, Acta Physica Polonica A 128 (2015),
no. 6, p. 968-973 [arXiv: 1507.06845]

J. Lipovsky, On the effective size of a non-Weyl graph [arXiv: 1507.04176]