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Cubic symmetric graphs of order twice an odd prime-power

Part of: Graph theory Permutation groups

Published online by Cambridge University Press:  09 April 2009

Yan-Quan Feng  and
Jin Ho Kwak
Yan-Quan Feng
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R., China, e-mail: yqfeng@center.njtu.edu.cn
Jin Ho Kwak
Affiliation:
Combinatorial and Computational, Mathematics Center, Pohang University of Science and Technology, Pohang, 790-784, Korea, e-mail: jinkwak@postech.ac.kr
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Abstract

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An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.

Keywords

Symmetric graphss-regular graphss-arc-transitive graphs.

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
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 153 - 164
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Alspach, B., Marušič, D. and Nowitz, L., ‘Constructing graphs which are ½-transitive’, J. Aust. Math. Soc. 56 (1994), 391–402.CrossRefGoogle Scholar
[2]Biggs, N., Algebraic graph theory, 2nd edition (Cambridge University Press, Cambridge, 1993).Google Scholar
[3]Cheng, Y. and Oxley, J., ‘On weakly symmetric graphs of order twice a prime’, J. Combin. Theory Ser. B 42 (1987), 196–211.CrossRefGoogle Scholar
[4]Conder, M. D. E. and Dobcsányi, P., ‘Trivalent symmetric graphs on up to 768 vertices’, J. Combin. Math. Combin. Comput. 40 (2002), 41–63.Google Scholar
[5]Conder, M. D. E. and Praeger, C. E., ‘Remarks on path-transitivity in finite graphs’, European J. Combin. 17 (1996), 371–378.CrossRefGoogle Scholar
[6]Djoković, D. Ž. and Miller, G. L., ‘Regular groups of automorphisms of cubic graphs’, J. Combin. Theory Ser. B 29 (1980), 195–230.CrossRefGoogle Scholar
[7]Fang, X. G., Wang, J. and Xu, M. Y., ‘On 1-arc-regular graphs’, European J. Combin. 23 (2002), 785–791.CrossRefGoogle Scholar
[8]Feng, Y.-Q. and Kwak, J. H., ‘Cubic symmetric graphs of order a small number times a prime or a prime square’, submitted.Google Scholar
[9]Feng, Y.-Q. and Kwak, J. H., ‘Constructing an infinite family of cubic 1-regular graphs’, European J. Combin. 23 (2002), 559–565.CrossRefGoogle Scholar
[10]Feng, Y.-Q. and Kwak, J. H., ‘An infinite family of cubic one-regular graphs with unsolvable automorphism groups’, Discrete Math. 269 (2003), 281–286.CrossRefGoogle Scholar
[11]Feng, Y.-Q. and Kwak, J. H., ‘One-regular cubic graphs of order a small number times a prime or a prime square’, J. Aust. Math. Soc. 76 (2004), 345–356.CrossRefGoogle Scholar
[12]Feng, Y.-Q. and Kwak, J. H., ‘s-Regular cubic graphs as coverings of the complete bipartite graph K33’, J. Graph Theory 45 (2004), 101–112.CrossRefGoogle Scholar
[13]Feng, Y.-Q., Kwak, J. H. and Wang, K. S., ‘Classifying cubic symmetric graphs of order 8p or 8p 2’, European J. Combin. 26 (2005), 1033–1052.CrossRefGoogle Scholar
[14]Feng, Y.-Q. and Wang, K. S., ‘s-Regular cyclic coverings of the three-dimensional hypercube Q 3’, European J. Combin. 24 (2003), 719–731.CrossRefGoogle Scholar
[15]Frucht, R., ‘A one-regular graph of degree three’, Canad. J. Math. 4 (1952), 240–247.CrossRefGoogle Scholar
[16]Godsil, C. D., ‘On the full automorphism group of a graph’, Combinatorica 1 (1981), 243–256.CrossRefGoogle Scholar
[17]Gorenstein, D., Finite simple groups (Plenum Press, New York, 1982).CrossRefGoogle Scholar
[18]Lorimer, P., ‘Vertex-transitive graphs: symmetric graphs of prime valency’, J. Graph Theory 8 (1984), 55–68.CrossRefGoogle Scholar
[19]Miller, R. C., ‘The trivalent symmetric graphs of girth at most six’, J. Combin. Theory Ser B 10 (1971), 163–182.CrossRefGoogle Scholar
[20]Robinson, D. J., A course in the theory of groups (Springer, New York, 1982).CrossRefGoogle Scholar
[21]Marušič, D. and Nedela, R., ‘Maps and half-transitive graphs of valency 4’, European J. Combin. 19 (1998), 345–354.CrossRefGoogle Scholar
[22]Marušič, D. and Pisanski, T., ‘Symmetries of hexagonal graphs on the torus’, Croat. Chemica Acta 73 (2000), 69–81.Google Scholar
[23]Marušič, D. and Xu, M. Y., ‘A ½-transitive graph of valency 4 with a nonsolvable group of automorphisms’, J. Graph Theory 25 (1997), 133–138.3.0.CO;2-N>CrossRefGoogle Scholar
[24]Suzuki, M., Group theory I (Springer, New York, 1982).CrossRefGoogle Scholar
[25]Tutte, W. T., ‘A family of cubical graphs’, Proc. Cambridge Philos. Soc. 43 (1947), 459–474.CrossRefGoogle Scholar
[26]Tutte, W. T., ‘On the symmetry of cubic graphs’, Canad. J. Math. 11 (1959), 621–624.CrossRefGoogle Scholar
[27]Xu, M. Y., ‘Automorphism groups and isomorphisms of Cayley digraphs’, Discrete Math. 182 (1998), 309–319.CrossRefGoogle Scholar