  A mixed graph is obtained from a graph by orienting some of its edges. The
Hermitian adjacency matrix of a mixed graph with the vertex set $ \{v_{1},
\ldots , v_{n}\} $, is the matrix $ H=[h_{ij}]_{n \times n} $, where $
h_{ij}=-h_{ji}=i $ if there is a directed edge from $ v_{i} $ to $ v_{j} $, $
h_{ij}=1 $ if there exists an undirected edge between $v_i$ and $v_{j}$, and
$h_{ij}=0$ otherwise. The Hermitian spectrum of a mixed graph is defined to be
the spectrum of its Hermitian adjacency matrix. In this paper we study mixed
graphs which are determined by their Hermitian spectrum (DHS). First, we show
that each mixed cycle is switching equivalent to either a mixed cycle with no
directed edges ($C_{n}$), a mixed cycle with exactly one directed edge
($C_{n}^{1}$), or a mixed cycle with exactly two consecutive directed edges
with the same direction ($C_{n}^{2}$) and we determine the spectrum of these
three types of cycles. Next, we characterize all DHS mixed paths and mixed
cycles. We show that all mixed paths of even order, except $P_{8}$ and
$P_{14}$, are DHS. It is also shown that mixed paths of odd order, except
$P_{3}$, are not DHS. Also, all cospectral mates of $P_{8}$, $P_{14}$ and
$P_{4k+1}$ and two families of cospectral mates of $P_{4k+3}$, where $k\geq1$,
are introduced. Finally, we show that the mixed cycles $C_{2k}$ and
$C_{2k}^{2}$, where $k\geq3$, are not DHS, but the mixed cycles $C_{4}$,
$C_{4}^{2}$, $C_{2k+1}$, $C_{2k+1}^{2}$, $C_{2k+1}^{1}$ and $C_{2j}^{1}$ except
$C_{7}^{1}$, $C_{9}^{1}$, $C_{12}^{1}$ and $C_{15}^{1}$, are DHS, where
$k\geq1$ and $j\geq2$.
