  Quantum graphs and their experimental counterparts, microwave networks, are
ideally suited to study the spectral statistics of chaotic systems. The graph
spectrum is obtained from the zeros of a secular determinant derived from
energy and charge conservation. Depending on the boundary conditions at the
vertices, there are Neumann and Dirichlet graphs. The first ones are realized
in experiments, since the standard junctions connecting the bonds obey Neumann
boundary conditions due to current conservation. On average, the corresponding
Neumann and Dirichlet eigenvalues alternate as a function of the wave number,
with the consequence that the Neumann spectrum is described by random matrix
theory only locally, but adopts features of the interlacing Dirichlet spectrum
for long-range correlations. Another spectral interlacing is found for the
Green's function, which in contrast to the secular determinant is
experimentally accessible. This is illustrated by microwave studies and
numerics.
