2-Arc-transitive hexavalent Cayley graphs on nonabelian simple groups

Abstract

A plenty of contributions have been done on symmetric Cayley graphs on nonabelian simple groups, but the only known complete classification of such graphs with composite valency is of valency 4 (provided 2-arc-transitivity) by Fang et al. [Europ. J. Combin. 25 (2004), 1107–1116] and Du and Feng [Comm. Algebra 47 (2019), 4565–4574]. This naturally motivates this work for classifying 2-arc-transitive hexavalent Cayley graphs on nonabelian simple groups. It is proved that these graphs are either normal or $({\mathrm{A}}_n,2)$-arc-transitive Cayley graphs on ${\mathrm{A}}_{n-1}$ where n is among 11 specific numbers dividing $2^7·3^3·5^3.$ A specific example is also constructed.

Keywords:

• 2-arc-transitive graph
• Cayley graph
• nonabelian simple group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20B15
• 20B30
• 05C25

Acknowledgements

The authors thank the referee for some helpful comments.

Additional information

Funding

This paper was supported by NSFC (11901512, 11961076) and NSF of Yunnan (2019FD116).