So's Conjecture for Integral Circulant Graphs of Order p^aq

RESUMO

So conjectured that, for a fixed positive integer n, the ordinary adjacency spectrum of an integral circulant graph of order n determines its divisor set. We prove this for graphs of order p to the power a times q, where p and q are primes with p less than q and a is at least one. To handle coincident eigenvalues arising from distinct greatest-common-divisor classes, we use a spectral counting measure. For connected graphs, an exact identity recovers the part of the divisor set consisting of one and q, when present, together with the counting measure for a graph of order p to the power a minus one times q. Strong induction and decomposition into connected components then recover the full divisor set, including the case p equals two and disconnected graphs.