Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for
  Laplacian Eigenvalues   Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any
undirected graph with $n$ vertices or more has at least $k$ nonpositive
eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of
$NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we
prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$
is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular
number. This implies new lower bounds for eigenvalues of Laplacian matrices:
the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest
degree, which generalizes some prior results.

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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con