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CUBIC EDGE-TRANSITIVE GRAPHS OF ORDER 8p2

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  01 April 2008

MEHDI ALAEIYAN  and
MOHSEN GHASEMI
MEHDI ALAEIYAN*
Affiliation:
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran (email: alaeiyan@iust.ac.ir)
MOHSEN GHASEMI
Affiliation:
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844, Iran (email: m.ghasemi@mail.urmia.ac.ir)
*
✉For correspondence; e-mail: alaeiyan@iust.ac.ir
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Abstract

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A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime. It was shown by Folkman [J. Folkman, ‘Regular line-symmetric graphs’, J. Combin. Theory3 (1967), 215–232] that a regular edge-transitive graph of order 2p or 2p2 is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that every cubic edge-transitive graph of order 8p2 is vertex-transitive.

Keywords

semisymmetric graphssymmetric graphsregular coverings

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 315 - 323
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Bouwer, I. Z., ‘An edge but not vertex transitive cubic graph’, Bull. Can. Math. Soc. 11 (1968), 533–535.CrossRefGoogle Scholar
[2]Bouwer, I. Z., ‘On edge but not vertex transitive regular graphs’, J. Combin. Theory, B 12 (1972), 32–40.CrossRefGoogle Scholar
[3]Conder, M., Malnic, A., Marusic, D. and Potocnik, P., ‘A census of semisymmetric cubic graphs on up to 768 vertices’, J. Algebraic Combin. 23 (2006), 255–294.CrossRefGoogle Scholar
[4]Du, S. F., Kwak, J. H. and Xu, M. Y., ‘Lifting of automorphisms on the elementary abelian regular covering’, Linear Algebra Appl. 373 (2003), 101–119.CrossRefGoogle Scholar
[5]Du, S. F. and Xu, M. Y., ‘A classification of semisymmetric graphs of order 2pq’, Comm. Algebra 28(6) (2000), 2685–2715.CrossRefGoogle Scholar
[6]Feng, Y. Q., Kwak, J. H. and Wang, K., ‘Classifying cubic symmetric graphs of order 8p or 8p 2’, European J. Combin. 26 (2005), 1033–1052.CrossRefGoogle Scholar
[7]Feng, Y. Q. and Wang, K., ‘s-Regular cyclic coverings of the three-dimensional hypercube Q 3’, European J. Combin. 24 (2003), 719–731.CrossRefGoogle Scholar
[8]Folkman, J., ‘Regular line-symmetric graphs’, J. Combin. Theory 3 (1967), 215–232.CrossRefGoogle Scholar
[9]Gorenstein, D., Finite simple groups (Plenum Press, New York, 1982).CrossRefGoogle Scholar
[10]Gross, J. L. and Tucker, T. W., ‘Generating all graph coverings by permutation voltage assignments’, Discrete Math. 18 (1977), 273–283.CrossRefGoogle Scholar
[11]Iofinova, M. E. and Ivanov, A. A., Biprimitive cubic graphs, an investigation in algebraic theory of combinatorial objects (Institute for System Studies, Moscow, 1985), pp. 124–134 (in Russian).Google Scholar
[12]Ivanov, A. V., ‘On edge but not vertex transitive regular graphs’, Comb. Annals Discrete Math. 34 (1987), 273–286.Google Scholar
[13]Klin, M. L., ‘On edge but not vertex transitive regular graphs’, in: Algebric methods in graph theory, Colloq-Math. Soc. Janos Bolyai, 25 (North-Holland, Amsterdam, 1981), pp. 399–403.Google Scholar
[14]Lu, Z., Wang, C. Q. and Xu, M. Y., ‘On semisymmetric cubic graphs of order 6p 2’, Sci. China Ser. A Math. 47 (2004), 11–17.Google Scholar
[15]Malnic, A., ‘Group actions, covering and lifts of automorphisms’, Discrete Math. 182 (1998), 203–218.CrossRefGoogle Scholar
[16]Malnic, A., Marusic, D. and Potocnik, P., ‘On cubic graphs admitting an edge-transitive solvable group’, J. Algebraic Combin. 20 (2004), 99–113.CrossRefGoogle Scholar
[17]Malnic, A., Marusic, D. and Wang, C. Q., ‘Cubic edge-transitive graphs of order 2p 3’, Discrete Math. 274 (2004), 187–198.CrossRefGoogle Scholar
[18]Skoviera, M., ‘A construction to the theory of voltage groups’, Discrete Math. 61 (1986), 281–292.CrossRefGoogle Scholar
[19]Titov, V. K., ‘On symmetry in graphs’, Proc. 2nd All Union Seminar on Combinatorial Mathematics, Vopracy Kibernetiki, 9150, part 2 (Nauka, Moscow, 1975), pp. 76–109 (in Russian).Google Scholar