Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T12:04:01.532Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Weiss conjecture for finite locally primitive graphs

Published online by Cambridge University Press:  20 January 2009

Marston D. Conder ,
Cai Heng Li  and
Cheryl E. Praeger
Marston D. Conder
Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand
Cai Heng Li
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia
Cheryl E. Praeger
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6907, Australia
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 Γ is said to be locally primitive if, for each vertex α, the stabilizer in Aut Γ of α induces a primitive permutation group on the set of vertices adjacent to α. In 1978, Richard Weiss conjectured that for a finite vertex-transitive locally primitive graph Γ, the number of automorphisms fixing a given vertex is bounded above by some function of the valency of Γ. In this paper we prove that the conjecture is true for finite non-bipartite graphsprovided that it is true in the case in which Aut Γ contains a locally primitive subgroup that is almost simple.

Keywords

vertex-transitive graphslocally primitive graphsquasiprimitive permutation groupsalmost simple groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 129 - 138
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Cameron, P. J., Praeger, C. E., Saxl, J. and Seitz, G. M., On the Sims conjecture and distance transitive graphs, Bull. Lond. Math. Soc. 15 (1983), 499–506.CrossRefGoogle Scholar
2.Dixon, J. D. and Mortimer, B., Permutation groups (Springer, New York and Berlin, 1996).CrossRefGoogle Scholar
3.Gardiner, T., Arc transitivity in graphs, Q. J. Math. Oxford (2) 24 (1973), 399–407.CrossRefGoogle Scholar
4.Gardiner, T., Doubly primitive vertex stabilizers in graphs, Math. Z. 135 (1974), 157–166.CrossRefGoogle Scholar
5.Gardiner, T., Arc transitively in graphs, II, Q. J. Math. Oxford (2) 25 (1974), 163–167.CrossRefGoogle Scholar
6.Gardiner, T., Arc transitively in graphs, III, Q. J. Math. Oxford (2) 27 (1976), 313–323.CrossRefGoogle Scholar
7.Godsil, C. D., On the full automorphism group of a graph, Combinatorica 1 (1981), 243–256.CrossRefGoogle Scholar
8.Praeger, C. E., Imprimitive symmetric graphs, Ars Combin. 19A (1985), 149–163.Google Scholar
9.Praeger, C. E., An O'Nan-Scott Theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. Lond. Math. Soc. (2) 47 (1993), 227–239.CrossRefGoogle Scholar
10.Praeger, C. E., On a reduction theorem for finite, bipartite, 2-arc transitive graphs, Australas. J. Combin. 7 (1993), 21–36.Google Scholar
11.Praeger, C. E., Finite quasiprimitive graphs, in Surveys in combinatorics, London Mathematical Society Lecture Note Series, vol. 24 (1997), pp. 65–85.Google Scholar
12.Sabidussi, G., Vertex-transitive graphs, Monatsh. Math. 68 (1964), 426–438.CrossRefGoogle Scholar
13.Sims, C. C., Graphs and finite permutation groups, Math. Z. 95 (1967), 76–86.CrossRefGoogle Scholar
14.Suzuki, M., Group theory, I (Springer, New York, 1982).CrossRefGoogle Scholar
15.Thompson, J. G., Bounds for the orders of maximal subgroups, J. Algebra 14 (1970), 135–138.CrossRefGoogle Scholar
16.Trofimov, V. I., Stabilizers of the vertices of graphs with projective suborbits, Soviet Math. Dokl. 42 (1991), 825–828.Google Scholar
17.Trofimov, V. I., Graphs with projective suborbits (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 890–916.Google Scholar
18.Trofimov, V. I., Graphs with projective suborbits, cases of small characteristics, I (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 58 (1994), 124–171.Google Scholar
19.Trofimov, V. I., Graphs with projective suborbits, cases of small characteristics, II (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 58 (1994), 137–156.Google Scholar
20.Trofimov, V. I. and Weiss, R., Graphs with a locally linear group of automorphisms, Math. Proc. Camb. Phil. Soc. 118 (1995), 191–206.CrossRefGoogle Scholar
21.Weiss, R., s-transitive graphs, Colloq. Math. Soc. János Bolyai 25 (1978), 827–847.Google Scholar
22.Weiss, R., Groups with a (B, N)-pair and locally transitive graphs, Nagoya Math. J. 74 (1979), 1–21.CrossRefGoogle Scholar
23.Weiss, R., An application of p-factorization methods to symmetric graphs, Math. Proc. Camb. Phil. Soc. 85 (1979), 43–48.CrossRefGoogle Scholar
24.Weiss, R., Permutation groups with projective unitary subconstituents, Proc. Am. Math. Soc. 78 (1980), 157–161.CrossRefGoogle Scholar