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Graphs With Stability Index One

Published online by Cambridge University Press:  09 April 2009

D. A. Holton  and
J. A. Sims
D. A. Holton
Affiliation:
Department of Mathematics, University of MelbourneVictoria 3052, Australia.
J. A. Sims
Affiliation:
Linacre College, Oxford University Oxford, England.
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Abstract

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We consider the effect on the stability properties of a graph G, of the presence in the automorphism group of G of automorphisms (uv)h, where u and v are vertices of G, and h is a permutation of vertices of G excluding u and v. We find sufficient conditions for an arbitrary graph and a cartesian product to have stability index one, and conjecture in the latter case that they are necessary. Finally we exhibit explicitly a large class of graphs which have stability index one.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 2 , September 1976 , pp. 212 - 220
Copyright
Copyright © Australian Mathematical Society 1976

References

Grant, D. D. (1974), ‘The stability index of graphs’, in Combinatorial Mathematics, Lecture Notes in Mathematics No. 403, Springer-Verlag, Berlin, 1974, 29–52.CrossRefGoogle Scholar
Grant, D. D. (1975), ‘Stability and operations on graphs’, Combinatorial Mathematics III, Lecture Notes in Mathematics No. 452, Springer-Verlag, Berlin, 1975), 116–135.CrossRefGoogle Scholar
Harary, F. (1969), Graph Theory (Addison-Wesley, Reading, Mass., 1969).CrossRefGoogle Scholar
Holton, D. A. (1973), ‘A report on stable graphs’, J. Austral. Math. Soc. 15, 163–171.CrossRefGoogle Scholar
Holton, D. A. (1973a), ‘Two applications of semi-stable graphs’, Discrete Math. 4, 151–158.CrossRefGoogle Scholar
Holton, D. A. and Grant, D. D. (1975), ‘Regular graphs and stability’, J. Austral. Math. Soc. 20 (Series A), 377–384.CrossRefGoogle Scholar
Turner, J. (1967), ‘Point-symmetric graphs with a prime number of points’, J. Combinatorial Theory 3, 136–145.CrossRefGoogle Scholar
Wielandt, H. (1964), Finite Permutation Groups (Academic Press, New York, 1964).Google Scholar