A classification of Ramanujan unitary cayley graphs

Abstract

The unitary Cayley graph on n vertices, Xn, has vertex set Z/nZ, 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(a - b, n) = 1. A k-regular graph X is Ramanujan if and only i (X) ≤ 2√k - 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 unitary Cayley graph Xn is a Ramanujan graph.