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ON ARC-REGULAR FROBENIUS METACIRCULANTS

Part of: Graph theory Permutation groups

Published online by Cambridge University Press:  10 February 2016

JIANGMIN PAN ,
ZHAOHONG HUANG  and
SHIQIN PENG
JIANGMIN PAN*
Affiliation:
School of Mathematics and Statistics, Yunnan University of Finance and Economics, Kunming, PR China email [email protected]
ZHAOHONG HUANG
Affiliation:
School of Mathematics and Statistics, Yunnan University, Kunming, PR China
SHIQIN PENG
Affiliation:
School of Mathematics and Statistics, Yunnan University of Finance and Economics, Kunming, PR China
*
✉[email protected]
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Abstract

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A graph is called arc-regular if its full automorphism group acts regularly on its arc set. In this paper, we completely determine all the arc-regular Frobenius metacirculants of prime valency.

Keywords

arc-regular graphmetacirculantFrobenius groupCayley graph

MSC classification

Primary: 20B15: Primitive groups
Secondary: 20B30: Symmetric groups 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 20 - 29
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Alspach, B. and Parsons, T. D., ‘A construction for vertex-transitive graphs’, Canad. J. Math. 34 (1982), 307–318.Google Scholar
Biggs, N., Algebraic Graph Theory, 2nd edn (Cambridge University Press, New York, 1992).Google Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Oxford University Press, London–New York, 1985).Google Scholar
Dixon, J. and Mortimer, B., Permutation Groups (Springer, New York, 1996).Google Scholar
Dobson, E., ‘Isomorphism problem for metacirculant graphs of order a product of distinct primes’, Canad. J. Math. 50(6) (1998), 1176–1188.Google Scholar
Fang, X. G., Wang, J. and Xu, M. Y., ‘On 1-arc-regular graphs’, European J. Combin. 23 (2002), 785–791.Google Scholar
Feng, Y. Q. and Li, Y. T., ‘One-regular graphs of square-free order of prime valency’, European J. Combin. 32 (2011), 265–275.Google Scholar
Giudici, M., Li, C. H. and Praeger, C. E., ‘Analysing finite locally s-arc-transitive graphs’, Trans. Amer. Math. Soc. 356 (2004), 291–317.Google Scholar
Godsil, C. D., ‘On the full automorphism group of a graph’, Combinatorica 1 (1981), 243–256.Google Scholar
Huppert, B., Finite Groups (Springer, Berlin, 1967).Google Scholar
Hwak, J. H., Kwon, Y. S. and Oh, J. M., ‘Infinitely many finite one regular Cayley graphs on dihedral groups of any prescribed valency’, J. Combin. Theory Ser. B 98 (2008), 585–598.Google Scholar
Kovács, I., ‘Classifying arc-transitive circulants’, J. Algebraic Combin. 20 (2004), 353–358.Google Scholar
Li, C. H., Isomorphisms of Finite Cayley Graphs, PhD Thesis, The University of Western Australia, 1996.Google Scholar
Li, C. H., ‘Permutation groups with a cyclic regular subgroup and arc transitive circulants’, J. Algebraic Combin. 21 (2005), 131–136.Google Scholar
Li, C. H., Lu, Z. P. and Pan, J. M., ‘Finite vertex-primitive edge-transitive metacirculants’, J. Algebraic Combin. 40 (2014), 785–804.CrossRefGoogle Scholar
Li, C. H. and Pan, J. M., ‘Finite 2-arc-transitive abelian Cayley graphs’, European J. Combin. 29 (2008), 148–158.Google Scholar
Li, C. H., Pan, J. M., Song, S. J. and Wang, D. J., ‘A characterization of a family of edge-transitive metacirculants’, J. Combin. Theory Ser. B 107 (2014), 12–25.Google Scholar
Li, C. H. and Sim, H. S., ‘On half-transitive metacirculant graphs of prime-power order’, J. Combin. Theory Ser. B 81 (2001), 45–57.Google Scholar
Malnič, A., Marušič, D. and Seifter, N., ‘Constructing infinite one-regular graphs’, European J. Combin. 20 (1999), 845–853.Google Scholar
Marušič, D., ‘A family of one-regular graphs of valency 4’, European J. Combin. 18 (1997), 59–64.Google Scholar
Marušič, D. and Scapellato, R., ‘Classifying vertex-transitive graphs whose order is a product of two primes’, Combinatorica 14(2) (1994), 187–201.Google Scholar
Oh, J. M. and Hwang, K. W., ‘Construction of one-regular graphs of valency 4 and 6’, Discrete Math. 278 (2004), 195–208.Google Scholar
Pan, J. M., Huang, Z. H. and Liu, Z., ‘Arc-transitive regular cyclic covers of the complete graphs K p, p ’, J. Algebraic Combin. 42 (2015), 619–633.CrossRefGoogle Scholar
Song, S. J., Li, C. H. and Wang, D. J., ‘Classifying a family of edge-transitive metacirculant graphs’, J. Algebraic Combin. 35 (2012), 497–513.CrossRefGoogle Scholar
Wielandt, H., Finite Permutation Groups (Academic Press, London, 1964).Google Scholar