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ON THE ORDER OF ARC-STABILIZERS IN ARC-TRANSITIVE GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  02 October 2009

GABRIEL VERRET
GABRIEL VERRET*
Affiliation:
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia (email: gabriel.verret@fmf.uni-lj.si)
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Abstract

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Let p be a prime. We say that a transitive action of a group L on a set Ω is p-sub-regular if there exist x,y∈Ω such that 〈Lx,Ly〉=L and LYx≅ℤp, where Y =yLx is the orbit of y under Lx. Our main result is that if Γ is a G-arc-transitive graph and the permutation group induced by the action of Gv on Γ(v) is p-sub-regular, then the order of a G-arc-stabilizer is equal to ps−1 where s≤7, s≠6, and moreover, if p=2, then s≤5. This generalizes a classical result of Tutte on cubic arc-transitive graphs as well as some more recent results. We also give a characterization of p-sub-regular actions in terms of arc-regular actions on digraphs and discuss some interesting examples of small degree.

Keywords

arc-transitive grapharc-stabilizer

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 498 - 505
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Conder, M. D. E. and Dobcsányi, P., ‘Trivalent symmetric graphs on up to 768 vertices’, J. Combin. Math. Combin. Comput. 40 (2002), 41–63.Google Scholar
[2]Conder, M. D. E., Li, C. H. and Praeger, C. E., ‘On the Weiss conjecture for finite locally-primitive graphs’, Proc. Edinb. Math. Soc. (2) 43 (2000), 129–138.CrossRefGoogle Scholar
[3]Conder, M. D. E., Malnič, A., Marušič, D. and Potočnik, P., ‘A census of semisymmetric cubic graphs on up to 768 vertices’, J. Algebraic Combin. 23 (2006), 255–294.CrossRefGoogle Scholar
[4]Gardiner, A., ‘Arc transitivity in graphs’, Q. J. Math. 24 (1973), 399–407.Google Scholar
[5]Gardiner, A., ‘Doubly primitive vertex stabilisers in graphs’, Math. Z. 135 (1974), 257–266.CrossRefGoogle Scholar
[6]Glauberman, G., ‘Normalizers of p-subgroups in finite groups’, Pacific J. Math. 29 (1969), 137–144.CrossRefGoogle Scholar
[7]Iranmanesh, M. A., ‘On finite G-locally primitive graphs and the Weiss conjecture’, Bull. Aust. Math. Soc. 70 (2004), 353–356.CrossRefGoogle Scholar
[8]Marušič, D. and Potočnik, P., ‘Bridging semisymmetric and half-arc-transitive actions on graphs’, European J. Combin. 23 (2002), 719–732.Google Scholar
[9]Potočnik, P., ‘A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index (4,2)’, European J. Combin. 30 (2009), 1323–1336.Google Scholar
[10]Sami, A. Q., ‘Locally dihedral amalgams of odd type’, J. Algebra 298 (2006), 630–644.Google Scholar
[11]Sims, C. C., ‘Graphs and finite permutation groups’, Math. Z. 95 (1967), 76–86.Google Scholar
[12]Trofimov, V. I., ‘Stabilizers of the vertices of graphs with projective suborbits’, Soviet Math. Dokl. 42 (1991), 825–828.Google Scholar
[13]Trofimov, V. I., ‘Graphs with projective suborbits’, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 890–916 (in Russian).Google Scholar
[14]Trofimov, V. I., ‘Vertex stabilizers of locally projective groups of automorphisms of graphs. A summary’, in: Groups, Combinatorics and Geometry, Durham (World Scientific, River Edge, NJ, 2001), pp. 313–326.Google Scholar
[15]Trofimov, V. I. and Weiss, R., ‘Graphs with a locally linear group of automorphisms’, Math. Proc. Cambridge Philos. Soc. 118 (1995), 191–206.CrossRefGoogle Scholar
[16]Tutte, W. T., ‘A family of cubical graphs’, Proc. Cambridge Philos. Soc. 43 (1947), 459–474.CrossRefGoogle Scholar
[17]Tutte, W. T., ‘On the symmetry of cubic graphs’, Canad. J. Math. 11 (1959), 621–624.Google Scholar
[18]Weiss, R., ‘s-transitive graphs’, Colloq. Math. Soc. János Bolyai 25 (1978), 827–847.Google Scholar
[19]Wielandt, H., Finite Permutation Groups (Academic Press, New York, 1964).Google Scholar