Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-27T21:30:29.858Z Has data issue: false hasContentIssue false
  • English
  • Français

Distance-transitive graphs of valency five

Published online by Cambridge University Press:  20 January 2009

A. Gardiner  and
Cheryl E. Praeger
A. Gardiner
Affiliation:
Department of Mathematics, university of Birmingham, Birmingham B15 2TT, U.K.
Cheryl E. Praeger
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia
Rights & Permissions [Opens in a new window]

Extract

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.

If u and v are vertices of the (finite, connected) graph Γ, let d(u, v) denote the length of the shortest path joining u to v in Γ. The graph Γ is said to be distance-transitive if whenever d(u, v) = d(u′, v′), there exists an automorphism g of Γ such that ug = u′ and if vg = v′. Distance-transitive graphs of valency 3 and 4 were originally classified [2, 11, 12, 13] by using a computer to generate all “feasible intersection arrays” (cf. [1, Chapter 20]). In both cases a classification has since been given by hand [4, 5]. Wecontinue this latter tradition and prove the following theorem—which was recently proved independently by Ivanov et al. using a computer [10].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 1 , February 1987 , pp. 73 - 81
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

REFERENCES

1.Biggs, N. L., Algebraic Graph Theory (Cambridge University Press, 1974).CrossRefGoogle Scholar
2.Biggs, N. L. and Smith, D. H., On trivalent graphs, Bull. London Math. Soc. 3 (1971), 155–158.CrossRefGoogle Scholar
3.Gardiner, A., Classifying distance-transitive graphs, in Combinatorial Mathematics IX, Eds. Billington, E. J., Oates-Williams, S. and Street, A. P. (Springer Lecture Notes in Mathematics 952, 1982), 67–88.Google Scholar
4.Gardiner, A., On trivalent graphs, J. London Math. Soc. (2) 10 (1975), 507–512.CrossRefGoogle Scholar
5.Gardiner, A., An elementary classification of distance-transitive graphs of valency four, Ars Combin., to appear.Google Scholar
6.Gardiner, A., Antipodal covering graphs, J. Combinatorial Theory B 16 (1974), 255–273.CrossRefGoogle Scholar
7.Gardiner, A., Arc transitivity in graphs III, Quart. J. Math. Oxford (2) 27 (1976), 313–323.Google Scholar
8.Hoffman, A. J. and Singleton, R. R., On Moore graphs of diameters 2 and 3, IBM J. Res. Develop. 4 (1960), 497–504.CrossRefGoogle Scholar
9.Huppert, B., Endliche Gruppen (Springer, 1967).CrossRefGoogle Scholar
10.Ivanov, A. A., Ivanov, A. V. and FaradŽev, I. A., Distance-transitive graphs of valency 5, 6 and 7, Zh. Vychisl. Mat. i Mat. Fiz. 24 (1984), 1704–1718 (in Russian).Google Scholar
11.Smith, D. H., On tetravalent graphs, J. London Math. Soc. (2) 6 (1973), 659–662.Google Scholar
12.SMITH, D. H., Distance-transitive graphs of valency four, J. London Math. Soc. (2) 8 (1974), 377–384.CrossRefGoogle Scholar
13.Smith, D. H., On bipartite tetravalent graphs, Discrete Math. 10 (1974), 167–172.CrossRefGoogle Scholar
14.Weiss, R., Distance-transitive graphs and generalised polygons, Arch. Math., to appear.Google Scholar
15.Wielandt, H., Permutation Groups (Academic Press, 1964).Google Scholar