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LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

Part of: Graph theory

Published online by Cambridge University Press:  01 February 2009

CAI HENG LI ,
JIANGMIN PAN  and
LI MA
CAI HENG LI*
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia (email: li@maths.uwa.edu.au)
JIANGMIN PAN
Affiliation:
Department of Mathematics, Yunnan University, Kunming 650031, PR China (email: jmpan@ynu.edu.cn)
LI MA
Affiliation:
Department of Mathematics, Yunnan University, Kunming 650031, PR China
*
✉For correspondence; e-mail: li@maths.uwa.edu.au
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Abstract

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Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

Keywords

locally primitive graphsCayley graphscovers

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 111 - 122
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work forms a part of the PhD project of Jiangmin Pan. It was partially supported by a NNSF and an ARC Discovery Project Grant.

References

[1]Biggs, N., Algebraic Graph Theory, 2nd edn (Cambridge University Press, New York, 1992).Google Scholar
[2]Brouwer, A. E. and Wilbrink, H. A., ‘Ovoids and fans in the generalized quadrangle GQ(4, 2)’, Geom. Dedicata 36 (1990), 121–124.CrossRefGoogle Scholar
[3]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Oxford University Press, London, 1985).Google Scholar
[4]Dixon, J. D. and Mortimer, B., Permutation Groups (Springer, Berlin, 1996).CrossRefGoogle Scholar
[5]Guralnick, R. M., ‘Subgroups of prime power index in a simple group’, J. Algebra 225 (1983), 304–311.CrossRefGoogle Scholar
[7]Huppert, B., Finite Groups (Springer, Berlin, 1967).Google Scholar
[8]Ivanov, A. A. and Praeger, C. E., ‘On finite affine 2-arc transitive graphs’, European J. Combin. 14 (1993), 421–444.CrossRefGoogle Scholar
[9]Li, C. H., ‘Finite s-arc transitive graphs of prime-power order’, Bull. London Math. Soc. 33 (2001), 129–137.CrossRefGoogle Scholar
[10]Li, C. H., ‘The finite primitive permutation groups containing an abelian regular subgroup’, Proc. London Math. Soc. 87 (2003), 725–748.CrossRefGoogle Scholar
[11]Li, C. H., ‘Finite edge-transitive Cayley graphs and rotary Cayley maps’, Trans. Amer. Math. Soc. 358 (2006), 4605–4635.CrossRefGoogle Scholar
[12]Li, C. H., Praeger, C. E., Venkatesh, A. and Zhou, S., ‘Finite locally-primitive graphs’, Discrete Math. 246 (2002), 197–218.CrossRefGoogle Scholar
[13]Liebeck, M. W., Praeger, C. E. and Saxl, J., On regular subgroups of primitive permutation groups, Mem. Amer. Math. Soc. to appear.Google Scholar
[14]Praeger, C. E., ‘Primitive permutation groups with a doubly transitive subconstituent’, J. Austral. Math. Soc. Ser. A 45 (1988), 66–77.CrossRefGoogle Scholar
[15]Praeger, C. E., ‘On the O’Nan–Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs’, J. London. Math. Soc. 47 (1992), 227–239.Google Scholar
[16]Praeger, C. E., ‘Finite normal edge-transitive Cayley graphs’, Bull. Austral. Math. Soc. 60 (1999), 207–220.CrossRefGoogle Scholar
[17]Xu, M. Y., ‘Automorphism groups and isomorphisms of Cayley digraphs’, Discrete Math. 182 (1998), 309–319.CrossRefGoogle Scholar