Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ramanujan Graphs and Digraphs   Ramanujan graphs have fascinating properties and history. In this paper we
explore a parallel notion of Ramanujan digraphs, collecting relevant results
from old and recent papers, and proving some new ones. Almost-normal Ramanujan
digraphs are shown to be of special interest, as they are extreme in the sense
of an Alon-Boppana theorem, and they have remarkable combinatorial features,
such as small diameter, Chernoff bound for sampling, optimal covering time and
sharp cutoff. Other topics explored are the connection to Cayley graphs and
digraphs, the spectral radius of universal covers, Alon's conjecture for random
digraphs, and explicit constructions of almost-normal Ramanujan digraphs.

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The categories of this paper are: chem-ph
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
18
The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph