Integral circulant Ramanujan graphs of prime power order (Q396862)

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