Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ramanujan circulant graphs and the conjecture of Hardy-Littlewood and
  Bateman-Horn   In this paper, we determine the bound of the valency of the odd circulant
graph which guarantees to be a Ramanujan graph for each fixed number of
vertices. In almost of the cases, the bound coincides with the trivial bound,
which comes from the trivial estimate of the largest non-trivial eigenvalue of
the circulant graph. As exceptional cases, the bound in fact exceeds the
trivial one by two. We then prove that such exceptionals occur only in the
cases where the number of vertices has at most two prime factors and is
represented by a quadratic polynomial in a finite family and, moreover, under
the conjecture of Hardy-Littlewood and Bateman-Horn, exist infinitely many.

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The categories of this paper are: hep-ph
3
The categories of this paper are: physics
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an