Circulant digraphs integral over number fields

A number field K is a finite extension of rational number field Q . A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K . In this paper we give the necessary and sufficient condition for circulant digraphs which are integral over a number field K . We also solve Conjecture 3.3 in Xu and Meng (2011) [11] and find that it is affirmative.