Biprimitive Graphs of Smallest Order
Shao-Fei Du
and Dragan Marušič
Department of Mathematics, Capital Normal University, Beijing, 100037, Peoples Republic of China DRAGAN MARU \check SI \check C\dagger
DOI: 10.1023/A:1018625926088
Abstract
A regular and edge-transitive graph which is not vertex-transitive is said to be semisymmetric. Every semisymmetric graph is necessarily bipartite, with the two parts having equal size and the automorphism group acting transitively on each of these parts. A semisymmetric graph is called biprimitive if its automorphism group acts primitively on each part. In this paper biprimitive graphs of smallest order are determined.
Pages: 151–156
Keywords: primitive group; semisymmetric graph; biprimitive graph
Full Text: PDF
References
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2. I.Z. Bouwer, “On edge but not vertex transitive regular graphs,” J. Combin. Theory Ser. B 12 (1972), 32-40.
3. P.J. Cameron, “Finite permutation groups and finite simple groups,” Bull. London Math. Soc. 13 (1981), 1-22.
4. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
5. J.D. Dixon and B. Mortimer, “The primitive permutation groups of degree less than 1000,” Math. Proc. Camb. Soc. 103 (1988), 213-238.
6. S.F. Du, “Construction of semisymmetric graphs,” Graph Theory Notes of New York XXIX, 1995, 48-49.
7. S.F. Du and M.Y. Xu, “A Classification of Semisymmetric Graphs of Order 2 pq (I),” submitted.
8. J. Folkman, “Regular line-symmetric graphs,” J. Combin. Theory Ser. B 3 (1967), 215-232.
9. B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967.
10. M.E. Iofinova and A.A. Ivanov, “Biprimitive cubic graphs” (Russian), Investigation in Algebric Theory of Combinatorial Objects, Proceedings of the seminar, Institute for System Studies, Moscow, 1985, pp. 124-134.
11. I.V. Ivanov, “On edge but not vertex transitive regular graphs,” Comb. Annals of Discrete Mathematices 34 (1987), 273-286.
12. M.H. Klin, “On edge but not vertex transitive regular graphs,” Colloquia Mathematica Societatis Janos Bolyai,
25. Algebric Methods in Graph Theory, Szeged (Hungary), 1978 Budapest, 1981, pp. 399-403.
13. W.A. Manning, “On the order of primitive groups III,” Tran. Amer. Math. Soc. 19 (1918), 127-142.
14. V.K. Titov, “On symmetry in the graphs” (Russian), Voprocy Kibernetiki (15), Proceedings of the II All Union Seminar on Combinatorial Mathematices, part 2, Nauka, Moscow, 1975, pp. 76-109.
15. H. Wielandt, Permutation Groups, Academic Press, New York, 1966.