Abstract
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∈Z/nZ,gcd(a−b,n)∈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=pk and for all positive integers n in case D is a multiplicative divisor set.
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