Title: Quantum Graphs: $ \mathcal{PT}$-symmetry and reflection symmetry of the spectrum,
Abstract: Not necessarily self-adjoint quantum graphs -- differential operators on
metric graphs -- are considered. Assume in addition that the underlying metric
graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential
operator is $ \mathcal P \mathcal T$-symmetric, then its spectrum has
reflection symmetry with respect to the real line. Our goal is to understand
whether the opposite statement holds, namely whether the reflection symmetry of
the spectrum of a quantum graph implies that the underlying metric graph
possesses a non-trivial automorphism and the differential operator is $
\mathcal P \mathcal T$-symmetric. We give partial answer to this question by
considering equilateral star-graphs. The corresponding Laplace operator with
Robin vertex conditions possesses reflection-symmetric spectrum if and only if
the operator is $ \mathcal P \mathcal T$-symmetric with $ \mathcal P $ being an
automorphism of the metric graph.