Integral mixed circulant graph

Abstract: A mixed graph is said to be \textit{integral} if all the eigenvalues of its Hermitian adjacency matrix are integer. The \textit{mixed circulant 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 $0\not\in \mathcal{C}$. If $\mathcal{C}$ is closed under inverse, then $Circ(\mathbb{Z}_n,\mathcal{C})$ is called a \textit{circulant 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 $n\equiv 0 \Mod 4$, 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.