Integral Circulant Ramanujan Graphs of Prime Power Order

A connected ρ-regular graph G has largest eigenvalue ρ in modulus. G is called Ramanujan if it has at least 3 vertices and the second largest modulus of its eigenvalues is at most 2 √ ρ− 1. In 2010 Droll classified all Ramanujan unitary Cayley graphs. These graphs of type ICG(n, {1}) form a subset of the class of integral circulant graphs ICG(n,D), which can be characterised by their order n and a set D of positive divisors of n in such a way that they have vertex set Z/nZ and edge set {(a, b) : a, b ∈ Z/nZ, gcd(a− b, n) ∈ D}. We extend Droll’s result by drawing up a complete list of all graphs ICG(ps,D) having the Ramanujan property for each prime power ps and arbitrary divisor set D.