A Classification of Ramanujan Unitary Cayley Graphs

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 if λ(X) 6 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 unitary Cayley graph Xn is a Ramanujan graph. 1 Unitary Cayley graphs Given a finite additive abelian group G and a symmetric subset S of G, we define the Cayley graph X(G, S) to be the graph whose vertex set is G, and in which two vertices v and w in G are connected by an edge if and only if v − w is in S. A Cayley graph of the form X(G, S) with G = Z nZ is called a circulant graph. The unitary Cayley graph on n vertices, Xn, is defined to be the undirected graph whose vertex set is Z nZ , and in which two vertices a and b are connected by an edge if and only if gcd(a− b, n) = 1. This can also be stated as Xn = X (