The unitary Cayley graph on n vertices, Xn, has vertex set Zn, and two vertices a and b are connected by an edge if and only if they differ by a multiplicative unit modulo n, i.e. gcd(ab, n) = 1. A k-regular graph X is Ramanujan if and only if (X) 2pk − 1 where (X) is the second largest absolute value of the eigenvalues of the adjacency matrix of X. We obtain a complete characterization of the cases in which the complements of unitary Cayley graph ¯Xn is a Ramanujan graph.
