A Classification of Ramanujan Complements of Unitary Cayley Graphs

Abstract

Abstract:The unitary Cayley graph on n vertices, Xn, has vertex set Zn, where 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 λ(X) ≤ 2 √ k − 1 where λ(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 graphXn is a Ramanujan graph.