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A report on stable graphs

Published online by Cambridge University Press:  09 April 2009

D. A. Holton
D. A. Holton
Affiliation:
Department of Mathematics University of Melbourne Parkville, Vic 3052, Australia
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It is the aim of this paper to introduce a new concept relating various subgroups of the automorphism group of a graph to corresponding subgraphs. Throughout g will denote a (Michigan) graph on a vertex set V(¦V¦ =n) and Γ(g)=G will be the automorphism group of G considered as a permutation group on V.En, Cn, Dn and Sn are the identity, cyclic, dihedral, and symmetric groups acting on a set of size n, while Sp(q) is the permutation group of pq objects which is isomorphic to Sp but is q-fold in the sense that the objects are permuted q at a time [6]. H ≦ G means that H is a subgroup of G. Other group concepts can be found in Wielandt [7]. The graphs G1 ∪ G2, G1 + G2, G1 × G2, and G1[G2] along with their corresponding groups are as defined in, for example, Harary [4]. Finally we use Kn for the complete graph on n vertices.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 2 , March 1973 , pp. 163 - 171
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Frucht, R., ‘Die Gruppe des Petersenchen Graphen und der Kantensysteme der regulären Polyeder’, Comm. Math. Helv. 9 (1937), 217–223.CrossRefGoogle Scholar
[2]Frucht, R., ‘On the groups of repeated graphs’, Bull. Amer. Math. Soc. 55 (1949), 418–420.CrossRefGoogle Scholar
[3]Harary, F., ‘On the group of the composition of two graphs’, Duke Math. Journal 26 (1959), 29–34.CrossRefGoogle Scholar
[4]Harary, F., Graph Theory (Addison-Wesley (1969).CrossRefGoogle Scholar
[5]Kagno, I., ‘Desargues’ and Pappus ‘graphs and their groups’, Amer. J. Math. 69 (1947), 859–862.CrossRefGoogle Scholar
[6]Uhlenbeck, G. and Ford, L., Theory of Linear Equations in Studies in Statistical Mechanics, Vol. 1 Ed. Boer, and Uhlenbeck, G., (North Holland Publishing Co. (1962)).Google Scholar
[7]Wielandt, H., Finite Permutation Groups (Academic Press (1964).Google Scholar