  One of the most important characteristics of a quantum graph is the average
density of resonances, $\rho = \frac{\mathcal{L}}{\pi}$, where $\mathcal{L}$
denotes the length of the graph. This is a very robust measure. It does not
depend on the number of vertices in a graph and holds also for most of the
boundary conditions at the vertices. Graphs obeying this characteristic are
called Weyl graphs. Using microwave networks which simulate quantum graphs we
show that there exist graphs which do not adhere to this characteristic. Such
graphs will be called non-Weyl graphs. For standard coupling conditions we
demonstrate that the transition from a Weyl graph to a non-Weyl graph occurs if
we introduce a balanced vertex. A vertex of a graph is called balanced if the
numbers of infinite leads and internal edges meeting at a vertex are the same.
Our experimental results confirm the theoretical predictions of [E. B. Davies
and A. Pushnitski, Analysis and PDE 4, 729 (2011)] and are in excellent
agreement with the numerical calculations yielding the resonances of the
networks.
