Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regular graphs of large girth and arbitrary degree   For every integer d > 9, we construct infinite families {G_n}_n of
d+1-regular graphs which have a large girth > log_d |G_n|, and for d large
enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special
set of d+1 generators whose choice is related to the arithmetic of integral
quaternions. These graphs are inspired by the Ramanujan graphs of
Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime.
When d is not equal to the power of an odd prime, this improves the previous
construction of Imrich in 1984 where he obtained infinite families {I_n}_n of
d+1-regular graphs, realized as Cayley graphs on SL_2(q), and which are
displaying a girth > 0,48 log_d |I_n|. And when d is equal to a power of 2,
this improves a construction by Morgenstern in 1994 where certain families
{M_n}_n of 2^k+1-regular graphs were shown to have a girth > 2/3 log_d |M_n|.

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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga