  A mixed graph is called \emph{second kind hermitian integral}(or
\emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of
second kind are integers. A mixed graph is called \emph{Eisenstein integral} if
the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We
characterize the set $S$ for which a mixed circulant graph
$\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed
circulant graph is Eisenstein integral if and only if it is HS-integral.
Further, the eigenvalues and the HS-eigenvalues of some oriented circulant
graphs are expressed in terms of generalized M$\ddot{\text{o}}$bius function.

Combinatorics