A census of semisymmetric cubic graphs on up to 768 vertices
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A list is given of all semisymmetric (edge- but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3,3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K 1,3 as a normal quotient is investigated in detail.
KeywordsSemisymmetric graphs Edge-transitive graphs Amalgams
- 6.M.D.E. Conder, P. Dobcsányi, B. Mc Kay and G. Royle, The Extended Foster Census, http://www.cs.uwa.edu.au/gordon/remote/foster.
- 12.P. Dobcsányi, Home Page, http://www.scitec.auckland.ac.nz/peter.
- 14.S.F. Du and M.Y. Xu, “A classification of semisymmetric graphs of order 2pq (I),“ Comm. Algebra 28 (2000), 2685–2715. J. Combin. Theory, Series B 29 (1980), 195–230.Google Scholar
- 17.M. Giudici, C.H. Li and C.E. Praeger, “Characterising finite locally s-arc transitive graphs with a star normal quotient,“ preprint.Google Scholar
- 18.C. Godsil, “On the full automorphism group of Cayley graphs,“ Combinatorica 1 (1981), 143–156.Google Scholar
- 19.C. Godsil and G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer-Verlag, New York, 2001.Google Scholar
- 24.M.E. Iofinova and A.A. Ivanov, Biprimitive cubic graphs, Investigations in Algebraic Theory of Combinatorial Objects (Proceedings of the seminar, Institute for System Studies, Moscow, 1985) Kluwer Academic Publishers, London, 1994, pp 459–472.Google Scholar
- 26.M.H. Klin, “On edge but not vertex transitive graphs,“ Coll. Math. Soc. J. Bolyai, (25. Algebraic Methods in Graph Theory, Szeged, 1978), Budapest, 1981, 399–403.Google Scholar