Spectral properties of chaotic microwave networks and quantum graphs under an edge swap transformation

We discuss the spectral properties of microwave networks and quantum graphs with preserved and partially violated T invariance (time-reversal invariance) under an edge swap transformation. A chaotic tetrahedral quantum graph was simulated by the tetrahedral microwave network in which the swap transformation was realized by replacing one pair of edges {e_{i}} and {e_{j}} adjacent to different vertices {v_{i}} and {v_{j}}. We show that the spectra {ν_{n}}_{n=1}^{∞} and {ν[over ̃]_{n}}_{n=1}^{∞} before and after an edge swap operation are level-2 interlaced. The experimental distribution P(ΔN) of the spectral shift ΔN=N(ν)-N[over ̃](ν), where N(ν) and N[over ̃](ν) are the counting functions for the original and swapped networks, was confirmed in the numerical calculations. For chaotic systems with partially violated T invariance, the cases of level-2 interlacing (ΔN=±2) appear less frequently than for the systems with time reversal symmetry. Furthermore, the widths of the overlapping areas with ΔN=±2 are getting narrower than for the system with preserved T invariance. The smaller susceptibility to the swap operation for networks and graphs with partially violated T invariance is likely caused by the stronger level repulsion in their spectra than in the systems with preserved T invariance leading to smaller rigidity of their spectra.