Census of cubic edge-transitive graphs

by Marston Conder and Primož Potočnik

The graphs

This site contains the list of all cubic edge-transitive graphs on at most 10000 vertices. Such a graph can be either vertex-transitive (and thus arc-transitive), or not (in this case it is said to be semisymmetric).

The graphs are available for download in the sparse6 format. This zip file contains files of the form "CAT_n.s6" and "CSS_n.s6" for each admissible order n, containing the sparse6 codes of the cubic arc-transitive and the cubic semisymmetric graphs of order n, respectively; one graph per line. The sparse6 formats were generated by Nauty, version 2_8_8.

We also provide a magma package containing the graphs from these two extended censuses:

  • Cubic arc-transitive graphs up to 10000 vertices (128 MB).
  • Cubic semisymmetric graphs up to 10000 vertices (38 MB).
  • Instructions, how to use the packages, are available here.

    Tables

    Some graph invariants are provided in the following two tables:

  • Table of cubic arc-transitive graphs
  • Table of cubic semisymmetric graphs
  • History

    Cubic arc-transitive graphs form a widely studied class of graphs, and first attempts of compiling a comprehensive list goes back to Ronald Foster. Even though Foster started collecting these graph back in the 1930's a printed copy of his collection was first published in 1988. In 2002, Marston Conder and Peter Dobcsanyi presented a computer generated census of cubic arc-transitive graphs on up to 768 vertices (see M. Conder and P. Dobcsanyi, Trivalent symmetric graphs on up to 768 vertices, J. Combinatorial Mathematics & Combinatorial Computing 40 (2002), 41-63). Conder later extended this list, first to 2048 vertices, and in 2011, up to 10000 vertices. A summary of the census is available on Marston's homepage.

    Similarly, the first census of cubic semisymmetric graphs appeared in a paper by Conder, Malnič, Marušič, and Potočnik, A census of cubic semisymmetric graphs on up to 768 vertices, J. Alg. Combin. 23 (2006), 255-294, and was later extended on up to 10000 vertices by Conder and Potočnik. A summary of the extended census is available here.

    Web design by Gregor Potočnik