Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gap sets for the spectra of regular graphs with minimum spectral gap   Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra
of large connected cubic and quartic graphs with minimum spectral gap. We focus
on two sequences of graphs, $\Delta_n$ and $\Gamma_n$ which are more
`symmetric' among these two families, respectively. We prove that
$(1,\sqrt{5}]$ is a gap interval for $\Delta_n$, and $[(-1+\sqrt{17})/2,3]$ is
a gap interval for $\Gamma_n$. We conjecture that these two are indeed maximal
gap intervals. As a by-product, we show that the eigenvalues of $\Delta_n$
lying in the interval $\left[-3,-\sqrt{5}\right]$ (in particular, its minimum
eigenvalue) converge to $(1-\sqrt{33})/2$ and the eigenvalues of $\Gamma_n$
lying in the interval $[-4,-(1+\sqrt{17})/2]$ (and in particular, its minimum
eigenvalue) converge to $1-\sqrt{13}$ as $n$ tends to infinity. The proofs of
the above results heavily depend on the following property which can be of
independent interest: with few exceptions, all the eigenvalues of connected
cubic and quartic graphs with minimum spectral gap are simple.

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The categories of this paper are: astro-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: cs
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: bayes-an