Cubic vertex-transitive graphs on up to 1280 vertices.

The Census of cubic vertex-transitive graphs contains the list of all cubic vertex-transitive graphs on at most 1280 vertices, together with a number of graph-theoretic properties (listed below). 

The list of graphs was originally compiled by Pablo Spiga, Gabriel Verret, and Primož Potočnik. The authors described the theoretical results and computations that were needed to compile the list in the paper Cubic vertex-transitive graphs on up to 1280 vertices. The original dataset files are available on Potočnik's website.

The graph-theoretic properties were either computed or verified with the SageMath system, with a few exceptions. Some of the properties were not supported by SageMath at the time of computation, and were contributed: is_cayley, odd_girth, is_partial_cube (based on some previous code). The methods is_moebius_ladder, is_prism, and is_spx (together with SPX constructions) were computed in DiscreteZOO.

A searchable version of this dataset is available on an instance of the MathDataHub platform hosted at mdh.graphsym.net/collection/CVT.

You can download this collection either as an SQLite database or as CSV files. Contents:

• master_db_zenodo.sqlite3: the SQLite database,
• create_database.sql: the SQL script that creates empty database schema used in the SQLite database (not necessary for opening the database),
• main_Graph.txt: the main table containing graphs as a CSV file,
• main_Graph.sample.txt: a sample of the main table,
• main_CLTime.txt: the table containing canonical labelling computation times as a CSV file.
• main_CLTime.sample.txt: a sample of the canonical labeling table

Structure and contents

Each graph is given a census-specific identifier CVT[n,i] (the i-th graph of order n in the census). Some graphs have a list of more easily recognized names (such as the Petersen graph). The dataset contains graphs in two formats: the sparse6 format and a format readable by the computer algebra system Magma. For compatibility with SQLite, boolean values are represented with ones and zeroes (true and false, respectively).

Data columns with types:

• canonical_label (string): the graph in sparse6 format, canonically labelled with nauty (call with no additional arguments), version: nauty-27r3,
• cvt_index (string): the census-specific identifier CVT[n,i] (the i-th graph of order n in the census),
• data (string): the graph in sparse6 format,
• raw_magma_code (string): the list of vertex neighbourhoods, readable by the computer algebra system Magma,
• name (string): a list of names of the graph,
• number_of_vertices (integer): number of vertices in the graph,
• clique_number (integer): the number of vertices in the largest clique subgraph,
• diameter (integer): the greatest distance between any pair of points,
• girth (integer): the length of the shortest cycle in the graph,
• is_arc_transitive (boolean): for every two ordered pairs of adjacent vertices, does there exist an automorphism, mapping one to the other,
• is_bipartite (boolean): can the vertices of the graph be partitioned into two sets, such that every edge connects a vertex in one set to a vertex in the other set,
• is_cayley (boolean): can the graph be constructed as a Cayley graph of some group for some generating set,
• is_distance_regular (boolean): for any two vertices v and w, does the number of vertices at distance j from v and at distance k from w depend only upon j, k, and i = d(v, w),
• is_distance_transitive (boolean): for any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, is there an automorphism of the graph that carries v to x and w to y,
• is_edge_transitive (boolean): for every two edges, does there exist an automorphism, mapping one to the other,
• is_hamiltonian (boolean): does the graph have a cycle that visits each vertex exactly once,
• is_partial_cube (boolean): is the graph isometric to a subgraph of a hypercube,
• is_split (boolean): can the vertices of the graph be partitioned into a clique and an independent set,
• is_strongly_regular (boolean): do there exists \(\lambda\) and \(\mu\) such that every two adjacent vertices have \(\lambda\) common neighbours and every two non-adjacent vertices have \(\mu\) common neighbours,
• odd_girth (integer): length of the shortest odd cycle,
• triangles_count (integer): the number of cycles of length 3 in the graph,
• is_moebius_ladder (boolean): is the graph a Möbius ladder,
• is_prism (boolean): is the graph a skeleton of a prism,
• is_spx (boolean): does the graph belong to the Split Praeger-Xu family of graphs,
• vertex_stabilizer (list of integer pairs): the order of the vertex stabilizer of the graph's automorphism group, given as a prime factorization; a list of pairs \((p_i, e_i)\) of primes and exponents such that \(\prod_i p_i^{e_i}\) is the order of the vertex stabilizer.

This work was partially supported by ARRS research project no. J1-1691.