A formula for eigenvalues of integral Cayley graphs over abelian groups

Let $Z$ be an abelian group, $x \in Z$, and $[x] = \{ y : \langle x \rangle = \langle y \rangle \}$. A graph is called integral if all its eigenvalues are integers. It is known that a Cayley graph is integral if and only if its connection set can be express as union of the sets $[x]$. In this paper, we determine an algebraic formula for eigenvalues of the integral Cayley graph when the connection set is $[x]$. This formula involves an analogue of M$\ddot{\text{o}}$bius function.