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In this paper, we determine the bound of the valency of the odd circulantgraph which guarantees to be a Ramanujan graph for each fixed number ofvertices. In almost of the cases, the bound coincides with the trivial bound,which comes from the trivial estimate of the largest non-trivial eigenvalue ofthe circulant graph. As exceptional cases, the bound in fact exceeds thetrivial one by two. We then prove that such exceptionals occur only in thecases where the number of vertices has at most two prime factors and isrepresented by a quadratic polynomial in a finite family and, moreover, underthe conjecture of Hardy-Littlewood and Bateman-Horn, exist infinitely many.

Ramanujan circulant graphs and the conjecture of Hardy-Littlewood and Bateman-Horn