  It is shown that an undirected graph $G$ is cospectral with the Hermitian
adjacency matrix of a mixed graph $D$ obtained from a subgraph $H$ of $G$ by
orienting some of its edges if and only if $H=G$ and $D$ is obtained from $G$
by a four-way switching operation; if $G$ is connected, this happens if and
only if $\lambda_1(G)=\lambda_1(D)$. All mixed graphs of rank 2 are determined
and this is used to classify which mixed graphs of rank 2 are cospectral with
respect to their Hermitian adjacency matrix. Several families of mixed graphs
are found that are determined by their Hermitian spectrum in the sense that
they are cospectral precisely to those mixed graphs that are switching
equivalent to them.
