Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T01:20:03.328Z Has data issue: false hasContentIssue false
  • English
  • Français

Graphs with a locally linear group of automorphisms

Published online by Cambridge University Press:  24 October 2008

V. I. Trofimov  and
R. M. Weiss
V. I. Trofimov
Affiliation:
Institute of Mathematics and Mechanics, Russian Academy of Sciences, Ural Branch 620066 Ekaterinburg, Russia
R. M. Weiss
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, USA
Get access Rights & Permissions [Opens in a new window]

Extract

Let Γ be an undirected graph, V(Γ) the vertex set of Γ and G a subgroup of aut(Γ). For each vertex x ↦ V(Γ), let Γx denote the set of vertices adjacent to x in Γ and the permutation group induced on Γx. by the stabilizer Gx. For each i ≥ 1, will denote the pointwise stabilizer in Gx of the set of vertices at distance at most i from x in Γ. Let

for each i ≥ 1 and any set of vertices x, y, …, z of Γ. An s-path (or s-arc) is an (s + 1)-tuple (x0, x1, … xs) of vertices such that xi ↦ Γxi–1 for 1 ≤ i ≤ s and xi ╪ xi–2 for 2 ≤ i ≤ s.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 2 , September 1995 , pp. 191 - 206
Copyright
Copyright © Cambridge Philosophical Society 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Delgado, A. and Stellmacher, B.. Weak (B, N)-pairs of rank 2; in Group and graphs: new results and methods (Birkhäuser, 1985).Google Scholar
[2]Gardiner, A.. Arc transitivity in graphs. Quart. J. Math. Oxford 24 (1973), 399–407.CrossRefGoogle Scholar
[3]Gardiner, A., Arc transitivity in graphs II. Quart. J. Math. Oxford 25 (1974), 163–167.CrossRefGoogle Scholar
[4]Goldschmidt, D. M.. Automorphisms of trivalent graphs. Annals Math. 111 (1980), 377–407.CrossRefGoogle Scholar
[5]Serre, J.-P.. Trees (Springer-Verlag, 1980).CrossRefGoogle Scholar
[6]Thompson, J. G.. Bounds for the order of maximal subgroups. J. Algebra 14 (1970), 135–138.CrossRefGoogle Scholar
[7]Tits, J.. Non-existence de certains polygones généralisés, I-II. Inventiones Math. 36 (1976), 275–284 and 51 (1979), 267–269.CrossRefGoogle Scholar
[8]Trofimov, V. I.. Graphs with projective suborbits. Math. USSR Izvestiya 39 (1992), 869–893.CrossRefGoogle Scholar
[9]Trofimov, V. I.. Stabilizers of the vertices of graphs with projective suborbits. Soviet Math. Dokl. 42 (1991), 825–828.Google Scholar
[10]Tutte, W. T.. A family of cubical graphs. Proc. Cambridge Phil. Soc. 43 (1947), 459–474.CrossRefGoogle Scholar
[11]Weiss, R.. The nonexistence of certain Moufang polygons. Inventiones Math. 51 (1979), 261–266.CrossRefGoogle Scholar
[12]Weiss, R.. Groups with a (B, N)-pair and locally transitive graphs. Nagoya Math. J. 74 (1979), 1–21.CrossRefGoogle Scholar
[13]Weiss, R.. An application of p–factorization methods to symmetric graphs. Math. Proc. Cambridge Phil. Soc. 85 (1979), 43–48.CrossRefGoogle Scholar
[14]Weiss, R.. The nonexistence of 8-transitive graphs. Combinatorica 1 (1981), 309–311.CrossRefGoogle Scholar
[15]Weiss, R.. On a theorem of Goldschmidt. Annals Math. 126 (1987), 429–438.CrossRefGoogle Scholar
[16]Weiss, R.. Graphs which are locally Grassmann. Math. Ann. 297 (1993), 325–334.CrossRefGoogle Scholar
[17]Wielandt, H.. Subnormal subgroups and permutation groups. Ohio State University Lecture Notes, Columbus, 1971.Google Scholar