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:
Instructions, how to use the packages, are available here.
All the graphs, except the two well known exceptions, the Petersen graph, denoted here as CAT(10,1), and the Coxeter graph, denoted CAT(28,1), are Hamiltonian and thus posses LCF codes. You can download the LCF codes of cubic arc-transitive graphs here and the LCF codes of cubic semisymmetric graphs here. Each line of the file represents one graph in the census and starts with the character "!", "?" or "#", depending on whether we can guarantee that the LCF notation exhibits the highest possible rotational symmetry of the corresponding Hamiltonian cycle ("!") or not ("?"). If the graph posseses no Hamiltonian cycle, the line starts with "#". Computation of the LCF codes is still in progress and more information will be included shortly. (We thank Gregor Potočnik for his help in this part of the project.)
Some graph invariants are provided in the following two tables:
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