Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T17:07:54.257Z Has data issue: false hasContentIssue false
  • English
  • Français

1-REGULAR CAYLEY GRAPHS OF VALENCY 7

Part of: Graph theory

Published online by Cambridge University Press:  08 March 2013

JING JIAN LI ,
GENG RONG ZHANG  and
BO LING
JING JIAN LI*
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650031, PR China College of Mathematics and Information Science, Guangxi University, Nanning 530004, PR China email zgrzaw@gxu.edu.cnbolinggxu@163.com
GENG RONG ZHANG
Affiliation:
College of Mathematics and Information Science, Guangxi University, Nanning 530004, PR China email zgrzaw@gxu.edu.cnbolinggxu@163.com
BO LING
Affiliation:
College of Mathematics and Information Science, Guangxi University, Nanning 530004, PR China email zgrzaw@gxu.edu.cnbolinggxu@163.com
*
✉lijjhx@163.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A graph $\mit{\Gamma} $ is called $1$-regular if $ \mathsf{Aut} \mit{\Gamma} $ acts regularly on its arcs. In this paper, a classification of $1$-regular Cayley graphs of valency $7$ is given; in particular, it is proved that there is only one core-free graph up to isomorphism.

Keywords

1-regularCayley graphcore-free

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 479 - 485
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Conway, J. H., Curtis, R. T., Noton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Clarendon Press, Oxford, 1985).Google Scholar
Conder, M. D. E. and Praeger, C. E., ‘Remarks on path-transitivity in finite graphs’, European J. Combin. 17 (1996), 371–378.CrossRefGoogle Scholar
Dixon, J. D. and Mortimer, B., Permutation Groups, 2nd edition (Springer-Verlag, New York, 1996).CrossRefGoogle Scholar
Frucht, R., ‘A one-regular graph of degree three’, Canad. J. Math. 4 (1952), 240–247.CrossRefGoogle Scholar
Li, C. H., ‘Finite s-arc transitive graphs of prime-power order’, Bull. Lond. Math. Soc. 33 (2001), 129–137.Google Scholar
Li, C. H., ‘Finite $s$-arc transitive Cayley graphs and flag-transitive projective planes’, Proc. Amer. Math. Soc. 133 (2005), 31–40.Google Scholar
Li, C. H., ‘Finite edge-transitive Cayley graphs and rotary Cayley maps’, Trans. Amer. Math. Soc. 359 (2006), 4605–4635.CrossRefGoogle Scholar
Li, J. J. and Lu, Z. P., ‘Cubic s-transitive Cayley graphs’, Discrete Math. 309 (2009), 6014–6025.CrossRefGoogle Scholar
Marus˘ic˘, D., ‘A family of one-regular graphs of valency 4’, European J. Combin. 18 (1997), 59–64.Google Scholar
Malnic˘, A., Marus˘ic˘, D. and Seifter, N., ‘Constructing infinite one-regular graphs’, European J. Combin. 20 (1999), 845–853.CrossRefGoogle Scholar
Praeger, C. E., ‘Finite normal edge-transitive Cayley graphs’, Bull. Aust. Math. Soc. 60 (1999), 207–220.Google Scholar
Xu, S. J., Fang, X. G., Wang, J. and Xu, M. Y., ‘5-arc transitive cubic Cayley graphs on finite simple groups’, European J. Combin. 28 (2007), 1023–1036.Google Scholar