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Cubic vertex-transitive graphs on up to 1280 vertices.

<p>The <strong>Census of cubic vertex-transitive graphs</strong>&nbsp;contains the list of all cubic vertex-transitive graphs on at most 1280 vertices, together with a number of graph-theoretic properties (listed below).&nbsp;</p> <p>The&nbsp;list of graphs was&nbsp;originally compiled by Pablo Spiga, Gabriel Verret, and Primož Potočnik. The authors described&nbsp;the theoretical results and computations that were needed to compile the list in the paper&nbsp;<a href="https://doi.org/10.1016/j.jsc.2012.09.002">Cubic vertex-transitive graphs on up to 1280 vertices</a>.&nbsp;The original dataset files are available on <a href="https://www.fmf.uni-lj.si/~potocnik/work.htm">Potočnik&#39;s website</a>.</p> <p>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,&nbsp;odd_girth,&nbsp;is_partial_cube (based on some previous code). The methods&nbsp;<a href="https://github.com/DiscreteZOO/DiscreteZOO-sage/blob/df8c8368a4912bd5396001139a13a9f3b9b2863f/discretezoo/entities/cvt/cvtgraph.py#L177-L249">is_moebius_ladder, is_prism, and is_spx</a>&nbsp;(together with <a href="https://github.com/DiscreteZOO/DiscreteZOO-sage/blob/df8c8368a4912bd5396001139a13a9f3b9b2863f/discretezoo/entities/spx/spxgraph.py#L182-L300">SPX constructions</a>) were computed in DiscreteZOO.</p> <p>A searchable version of this dataset is available on an instance of the <a href="https://data.mathhub.info/">MathDataHub</a> platform hosted at <a href="http://mdh.graphsym.net/collection/CVT">mdh.graphsym.net/collection/CVT</a>.</p> <p>You can download this collection either as an&nbsp;SQLite database or as CSV files. Contents:</p> <ul> <li>master_db_zenodo.sqlite3: the&nbsp;SQLite database,</li> <li>create_database.sql: the&nbsp;SQL script that creates empty database schema used in the SQLite database (not necessary for opening the database),</li> <li>main_Graph.txt: the main table containing graphs as a CSV file,</li> <li>main_Graph.sample.txt: a sample of the&nbsp;main table,</li> <li>main_CLTime.txt: the table containing canonical labelling computation times as a CSV file.</li> <li>main_CLTime.sample.txt:&nbsp;a sample of the canonical labeling&nbsp;table</li> </ul> <p><strong>Structure and contents</strong></p> <p>Each graph is given a census-specific identifier&nbsp;<em>CVT[n,i]</em>&nbsp;(the <em>i</em>-th graph of order <em>n</em> in the census).&nbsp;Some graphs have a list of more easily recognized names&nbsp;(such as the Petersen graph).&nbsp;The dataset contains graphs in two formats: the&nbsp;<a href="https://users.cecs.anu.edu.au/~bdm/data/formats.html">sparse6 format</a>&nbsp;and&nbsp;a format readable by the&nbsp;computer algebra system Magma.&nbsp;For compatibility with SQLite, boolean values are represented with ones and zeroes&nbsp;(true and false, respectively).</p> <p>Data columns with types:</p> <ul> <li>canonical_label (string): the graph in sparse6 format, canonically labelled with nauty (call with no additional arguments), version: nauty-27r3,</li> <li>cvt_index (string): the census-specific identifier&nbsp;<em>CVT[n,i]</em>&nbsp;(the <em>i</em>-th graph of order <em>n</em> in the census),</li> <li>data (string): the graph in sparse6 format,</li> <li>raw_magma_code (string): the list of vertex neighbourhoods, readable by the computer algebra system&nbsp;Magma,</li> <li>name (string): a&nbsp;list of names of the graph,</li> <li>number_of_vertices (integer): number of vertices&nbsp;in the graph,</li> <li>clique_number (integer): the number of vertices in the largest clique subgraph,</li> <li>diameter (integer): the greatest distance between any pair of points,</li> <li>girth (integer): the length of the shortest cycle in the graph,</li> <li>is_arc_transitive (boolean): for every&nbsp;two ordered pairs of adjacent vertices, does there exist an automorphism, mapping one to the other,</li> <li>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,</li> <li>is_cayley (boolean): can the graph be constructed as a Cayley graph of some group for some generating set,</li> <li>is_distance_regular (boolean): for any two vertices <em>v</em>&nbsp;and <em>w</em>, does the number of vertices at distance <em>j</em>&nbsp;from <em>v</em>&nbsp;and at distance <em>k</em>&nbsp;from <em>w</em>&nbsp;depend&nbsp;only upon <em>j</em>, <em>k</em>, and <em>i = d(v, w)</em>,</li> <li>is_distance_transitive (boolean): for any two vertices <em>v</em>&nbsp;and <em>w</em> at any distance <em>i</em>, and any other two vertices <em>x</em>&nbsp;and <em>y</em>&nbsp;at the same distance, is there&nbsp;an automorphism of the graph that carries <em>v</em>&nbsp;to <em>x</em>&nbsp;and <em>w</em>&nbsp;to <em>y</em>,</li> <li>is_edge_transitive (boolean):&nbsp;for every two edges, does there exist an automorphism, mapping one to the other,</li> <li>is_hamiltonian (boolean): does the graph have a cycle that visits each vertex exactly once,</li> <li>is_partial_cube (boolean): is&nbsp;the graph isometric to a subgraph of a hypercube,</li> <li>is_split (boolean): can the vertices of the graph be partitioned into a clique and an independent set,</li> <li>is_strongly_regular (boolean): do there exists <span class="math-tex">\(\lambda\)</span>&nbsp;and <span class="math-tex">\(\mu\)</span>&nbsp;such that every two adjacent vertices have <span class="math-tex">\(\lambda\)</span> common neighbours and every&nbsp;two non-adjacent vertices have <span class="math-tex">\(\mu\)</span> common neighbours,</li> <li>odd_girth (integer): length of the shortest odd cycle,</li> <li>triangles_count (integer): the number of cycles of length <em>3</em>&nbsp;in the graph,</li> <li>is_moebius_ladder (boolean): is&nbsp;the graph a M&ouml;bius ladder,</li> <li>is_prism (boolean): is the graph a skeleton of a prism,</li> <li>is_spx (boolean): does the graph belong to the Split Praeger-Xu family of graphs,</li> <li>vertex_stabilizer (list of integer pairs): the order of the vertex stabilizer of the graph&#39;s automorphism group, given as a prime factorization; a list of pairs <span class="math-tex">\((p_i, e_i)\)</span>&nbsp;of primes and exponents such that <span class="math-tex">\(\prod_i p_i^{e_i}\)</span>&nbsp;is the order of the vertex stabilizer.</li> </ul> <p>This work was partially supported by ARRS research project no. J1-1691.</p>