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On the Minimum Order of Graphs with Given Group

Published online by Cambridge University Press:  20 November 2018

László Babai
László Babai*
Affiliation:
Eötvös L. University, BudapestHungary, Université de Montréal, MontréalQuébeck
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For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.

G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 4 , 01 December 1974 , pp. 467 - 470
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Sabidussi, G., On the minimum order of graphs with given automorphism group, Monatsh. Math. 63, (1959) pp. 124-127.Google Scholar
2. Harary, F. and Palmer, E., The smallest graph whose group is cyclic, Czechoslovak Math. J. 16,(91)(1966), pp. 70-71.Google Scholar
3. Sabidussi, G., Review 2563, Math. Rev. 33, No. 3, March 1967. 1500Google Scholar