Switching equivalence of Hermitian adjacency matrices of mixed graphs

Let $0 \in \Gamma$ and $\Gamma \setminus \{0\}$ be a subgroup of the complex numbers of unit modulus. Define $\mathcal{H}_{n}(\Gamma)$ to be the set of all $n\times n$ Hermitian matrices with entries in $\Gamma$, whose diagonal entries are zero. The matrices $A,B\in \mathcal{H}_{n}(\Gamma)$ are said to be switching equivalent if there is a diagonal matrix $D$, in which the diagonal entries belong to $\Gamma \setminus \{0\}$, such that $D^{-1} A D=B$. We find a characterization, in terms of fundamental cycles of graphs, of switching equivalence of matrices in $\mathcal{H}_{n}(\Gamma)$. We give sufficient conditions to characterize the cospectral matrices in $\mathcal{H}_{n}(\Gamma)$. We find bounds on the number of switching equivalence classes of all mixed graphs with the same underlying graph. We also provide the size of all switching equivalence classes of mixed cycles, and give a formula that calculates the size of a switching equivalence class of a mixed plane graph. We also discuss an action of the automorphism group of a graph on switching equivalence classes of matrices in $\mathcal{H}_{n}(\Gamma)$.