Statistical properties of quantum graph spectra

A general analytical approach to the statistical description of quantum graph spectra is discussed. It is based on the exact expansions of quantum levels in series in periodic orbits. The exact and approximate expressions obtained for the probability distribution functions using the spectral hierarchy method [Y. Dabaghian and R. Blümel, Phys. Rev. E 68, 055201(R) (2003); JETP Lett. 77, 530 (2003); Phys. Rev. E 70, 046206 (2004)] are analyzed. In addition, the mechanism of appearance of the universal statistical properties of spectral fluctuations of quantum-chaotic systems is considered in terms of the semiclassical theory of periodic orbits.