On the kernel of integral circulant graphs

Given a positive integer n, every integral circulant graph on n vertices is isomorphic to some graph ICG(n, D) having vertex set Z/nZ and edge set {(a, b): a, b is an element of Z/nZ, gcd(a-b, n) is an element of D} for a uniquely determined set D of positive divisors of n. By virtue of its adjacency matrix, one defines the spectrum of a graph G and, naturally, can ask to which degree the eigenvalues of G determine the graph itself. With respect to integral circulant graphs little is known about this question, which is related to a conjecture of So. In this note we examine the role of the eigenvalue 0 and clarify the interrelation between the dimension of the kernel of ICG(n, D) and the graph itself for all prime powers n = p(k) and for all positive integers n in case D is a multiplicative divisor set. (C) 2018 Elsevier Inc. All rights reserved.