Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Preprojective roots and cycle graphs   We study $c$-preprojective roots for a Coxeter element $c$ of infinite
Coxeter group $W$. In particular, we consider the case when any positive root
is $c$-preprojective for some Coxeter element $c$. In this paper, by assuming
that the Coxeter graph of $W$ is cyclic, we establish that any positive root is
$c$-preprojective for some Coxeter element $c$ if and only if $W$ is an affine
Coxeter group.

25
The categories of this paper are: patt-sol
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph