Time-reversal symmetry has major consequences in the field of disordered or chaotic quantum systems, as it defines the statistical properties of energy levels in such systems. In particular, if the time-reversal operator squares to minus unity then one expects to observe GSE statistics (one of the Wigner-Dyson random matrix ensembles). Such time-reversal operators are commonly associated with particles of half-integer spin, however I will show how to create quantum graphs that possess such a time-reversal operator, yet only require single-component wavefunctions. I will also explain how we reached this result through the investigations of quantum graphs with discrete symmetries.
