Integral mixed circulant graph

A mixed graph is said to be $\mathit{\text{integral}}$ if all the eigenvalues of its Hermitian adjacency matrix are integer. The $\mathit{\text{mixed}}\mathit{\text{circulant}}\mathit{\text{graph}}$ $Circ({\mathbb{Z}}_n,\mathcal{C})$ is a mixed graph on the vertex set ${\mathbb{Z}}_n$ and edge set $\{(a,b):b-a\in \mathcal{C}\}$, where $0\not\in $\mathcal{C}$$\mathrm{.}If$$\mathcal{C}$$isclosedunderinverse,then$Circ($\mathbb{Z}$_n,$\mathcal{C}$)$ is called a $\mathit{\text{circulant}}\mathit{\text{graph}}$. We express the eigenvalues of $Circ({\mathbb{Z}}_n,\mathcal{C})$ in terms of primitive $n$-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of $Circ({\mathbb{Z}}_n,\mathcal{C})$. For $n\equiv 0\text{Mod}4$, we factorize the cyclotomic polynomial into two irreducible factors over $\mathbb{Q}(i)$. Using this factorization, we characterize integral mixed circulant graphs in terms of its symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties.