Cubic vertex-transitive graphs on up to 1280 vertices.

DOI10.5281/zenodo.6576526Zenodo6576526MaRDI QIDQ6710596

Dataset published at Zenodo repository.

Author name not available (Why is that?)

Publication date: 27 May 2022

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Description

The Census of cubic vertex-transitive graphscontains the list of all cubic vertex-transitive graphs on at most 1280 vertices, together with a number of graph-theoretic properties (listed below). Thelist of graphs wasoriginally compiled by Pablo Spiga, Gabriel Verret, and Primož Potočnik. The authors describedthe theoretical results and computations that were needed to compile the list in the paperCubic vertex-transitive graphs on up to 1280 vertices.The original dataset files are available on Potočniks 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 methodsis_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 anSQLite database or as CSV files. Contents: master_db_zenodo.sqlite3: theSQLite database, create_database.sql: theSQL 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 themain table, main_CLTime.txt: the table containing canonical labelling computation times as a CSV file. main_CLTime.sample.txt:a sample of the canonical labelingtable Structure and contents Each graph is given a census-specific identifierCVT[the i-th graph of order n in the census n,i].Some graphs have a list of more easily recognized names(such as the Petersen graph).The dataset contains graphs in two formats: thesparse6 formatanda format readable by thecomputer 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 identifierCVT[the i-th graph of order n in the census n,i], data (string): the graph in sparse6 format, raw_magma_code (string): the list of vertex neighbourhoods, readable by the computer algebra systemMagma, name (string): alist of names of the graph, number_of_vertices (integer): number of verticesin 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 everytwo 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 vand w, does the number of vertices at distance jfrom vand at distance kfrom wdependonly upon j, k, and i = d(v, w), is_distance_transitive (boolean): for any two vertices vand w at any distance i, and any other two vertices xand yat the same distance, is therean automorphism of the graph that carries vto xand wto 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): isthe 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 everytwo 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 3in the graph, is_moebius_ladder (boolean): isthe graph a Mbius 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 graphs 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.

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