Integral circulant Ramanujan graphs of prime power order

Abstract

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.