Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Alon-Boppana theorem for powered graphs and generalized Ramanujan
  graphs   The r-th power of a graph modifies a graph by connecting every vertex pair
within distance r. This paper gives a generalization of the Alon-Boppana
Theorem for the r-th power of graphs, including irregular graphs. This leads to
a generalized notion of Ramanujan graphs, those for which the powered graph has
a spectral gap matching the derived Alon-Boppana bound. In particular, we show
that certain graphs that are not good expanders due to local irregularities,
such as Erdos-Renyi random graphs, become almost Ramanujan once powered. A
different generalization of Ramanujan graphs can also be obtained from the
nonbacktracking operator. We next argue that the powering operator gives a more
robust notion than the latter: Sparse Erdos-Renyi random graphs with an
adversary modifying a subgraph of log(n)^c$ vertices are still almost Ramanujan
in the powered sense, but not in the nonbacktracking sense. As an application,
this gives robust community testing for different block models.

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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
18
The categories of this paper are: econ
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The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn