On symmetric spectra of Hermitian adjacency matrices for non-bipartite mixed graphs

We study the equivalence between bipartiteness and symmetry of spectra of mixed graphs, for $\theta$-Hermitian adjacency matrices defined by an angle $\theta \in (0,\pi ]$. We show that this equivalence holds when, for example, an angle $\theta$ is an algebraic number, while it breaks down for any angle $\theta \in \mathbb{Q}\pi$. Furthermore, we construct a family of non-bipartite mixed graphs having the symmetric spectra for given $\theta \in \mathbb{Q}\pi$.