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Small vertex-transitive and Cayley graphs of girth six and given degree: an algebraic approach

✍ Scribed by Eyal Loz; Martin Mačaj; Mirka Miller; Jana Šiagiová; Jozef Širáň; Jana Tomanová

Publisher: John Wiley and Sons
Year: 2010
Tongue: English
Weight: 184 KB
Volume: 68
Category: Article
ISSN: 0364-9024
DOI: 10.1002/jgt.20556

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✦ Synopsis

We examine the existing constructions of the smallest known vertex-transitive graphs of a given degree and girth 6. It turns out that most of these graphs can be described in terms of regular lifts of suitable quotient graphs. A further outcome of our analysis is a precise identification of which of these graphs are Cayley. We also investigate higher level of transitivity of the smallest known vertex-transitive graphs of a given degree and girth 6 and relate their constructions to near-difference sets. ᭧ 2010 Wiley

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## Abstract For any __d__⩾5 and __k__⩾3 we construct a family of Cayley graphs of degree __d__, diameter __k__, and order at least __k__((__d__−3)/3)^__k__^. By comparison with other available results in this area we show that our family gives the largest currently known Cayley graphs for a wide ra