Abstract
A graph is s-transitive if its automorphism group acts transitively on the set of paths of length s. This is a notion due to William Tutte who showed in 1947 that a finite trivalent graph can never be 6-transitive. We examine connections between the theory of s-transitive graphs and the classification of Moufang polygons, a class of graphs exhibiting “local” s-transitivity for large values of s. Moufang polygons are examples of buildings. Both of these notions were introduced by Jacques Tits in the study of algebraic groups. We give an overview of Tits’ classification results in the theory of spherical buildings (which include the classification of Moufang polygons as a special case) and describe, in particular, the classification of finite buildings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bruhat, F., Tits, J.: Groupes réductifs sur un corps local, I, Données radicielles valuées. Publ. Math. I.H.E.S. 41, 5–252 (1972)
Delgado, A., Stellmacher, B.: Weak (B, N)-pairs of rank 2. In: Groups and Graphs: New Results and Methods. Birkhäuser, Basel (1985)
De Medts, T., Segev, Y.: A course on Moufang sets. Innov. Incid. Geom. 9, 79–122 (2009)
Gardiner, A.: Arc transitivity in graphs. Q. J. Math. Oxford 24, 399–407 (1973)
Ronan, M.: Lectures on Buildings. Academic, Boston (1989)
Sims, C.C.: Graphs and finite permutation groups. Math. Z. 95, 76–86 (1967)
Tits, J.: Sur la trialité et certains groupes qui s’en déduisent. Publ. Math. I.H.E.S. 2, 14–60 (1959)
Tits, J.: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol. 386. Springer, Heidelberg (1974)
Tits, J.: Non-existence de certains polygones généralisés, I-II. Invent. Math. 36, 275–284 (1976) and 51, 267–269 (1979)
Tits, J.: A local approach to buildings. In: Davis, C., Grünbaum, B., Sherk, F.A. (eds.) The Geometric Vein: The Coxeter Festschrift, pp. 519–547. Springer, Heidelberg (1981)
Tits, J.: Immeubles de type affine. In: Rosati, L.A. (ed.) Buildings and the Geometry of Diagrams. Lecture Notes in Mathematics, vol. 1181, pp. 159–190. Springer, Heidelberg (1986)
Tits, J., Weiss, R.M.: Moufang Polygons. Springer, Heidelberg (2002)
Tutte, W.T.: A family of cubical graphs. Proc. Camb. Philos. Soc. 43, 621–624 (1947)
Tutte, W.T.: Connectivity in Graphs. University of Toronto Press, Toronto (1966)
Weiss, R.M.: Elations of graphs. Acta Math. Acad. Sci. Hung. 34, 101–103 (1979)
Weiss, R.M.: Groups with a (B, N)-pair and locally transitive graphs. Nagoya J. Math. 74, 1–21 (1979)
Weiss, R.M.: The nonexistence of certain Moufang polygons. Invent. Math. 51, 261–266 (1979)
Weiss, R.M.: The nonexistence of 8-transitive graphs. Combinatorica 1, 309–311 (1981)
Weiss, R.M.: The Structure of Spherical Buildings. Princeton University Press, Princeton (2003)
Weiss, R.M.: The Structure of Affine Buildings. Princeton University Press, Princeton (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Weiss, R.M. (2014). Buildings and s-Transitive Graphs. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_18
Download citation
DOI: https://doi.org/10.1007/978-1-4939-0781-6_18
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-0780-9
Online ISBN: 978-1-4939-0781-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)