  We present experimental and theoretical results for the fluctuation
properties in the incomplete spectra of quantum systems with symplectic
symmetry and a chaotic dynamics in the classical limit. To obtain theoretical
predictions, we extend the random-matrix theory (RMT) approach introduced in
[O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)] for incomplete
spectra of quantum systems with orthogonal symmetry. We validate these RMT
predictions by randomly extracting a fraction of levels from complete sequences
obtained numerically for quantum graphs and experimentally for microwave
networks with symplectic symmetry and then apply them to incomplete
experimental spectra to demonstrate their applicability. Independently of their
symmetry class quantum graphs exhibit nongeneric features which originate from
nonuniversal contributions. Part of the associated eigenfrequencies can be
identified in the level dynamics of parameter-dependent quantum graphs and
extracted, thereby yielding spectra with systematically missing
eigenfrequencies. We demonstrate that, even though the RMT approach relies on
the assumption that levels are missing at random, it is possible to determine
the fraction of missing levels and assign the appropriate symmetry class by
comparison of their fluctuation properties with the RMT predictions.
