How many non-isospectral integral circulant graphs are there?

The answer to the question in the title is contained in the following conjecture by So [Discrete Math. 306 (2006), 153-158]: There are exactly 2(t(n)- 1) non-isospectral integral circulant graphs of order n, where t (n) is the number of divisors of n. In this paper we review some background about this conjecture, which is still open. Moreover, we affirm this conjecture for some special cases of n, namely, n = p(k), pqk, p(2)q with primes 2 <= p < q and integer k >= 1; and n = pqr with primes 2 <= p < q < r. Our approach is basically a case-by-case study, but a common technique used in the proofs of these different cases is the notion of a super sequence: a positive sequence in which each term is greater than the partial sum of all previous terms. An immediate consequence of this conjecture is a result of Klin and Kovacs [Electron. J. Combin. 19 (2012), #P35], which asserts that there are exactly 2(t(n)- 1) non-isomorphic integral circulant graphs of order n.