Primož Potočnik's home page

First name: Primož Family name: Potočnik (If you wonder, how to type-set my name in LaTex, here’s a solution: Primo\v{z} Poto\v{c}nik) Email: primoz.potocnik@fmf.uni-lj.si Affiliations: Faculty of Mathemathics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia and Institute of Mathematics,Physics, and Mechanics, Jadranska 19, Ljubljana, Slovenia

Education: PhD in Mathematics, University of Ljubljana, year 2000. Post doctoral training: University of Ottawa, August 2003 – January 2004; University of Auckland, August 2004 – August 2005; Academic degree: Full Professor, University of Ljubljana Research interests: permutation groups, graph theory, symmetries of combinatorial objects (such as graphs, maps, codes, finite geometries),…

A photo of me, taken in Auckland, New Zealand

RESEARCH:

• My CV with a list of publications.

• Several lists of graphs of a prescribed symmetry type and valence, and some other combinatorial and algebraic structures:
  
  
• Census of rotary maps. Based on the census of (2,*)-groups (see below), a census of all rotary maps was constructed. Recall that a rotary map is either chiral (in which case its group of automorphisms acts regularly on the oriented edges of the graph but contains no orientation reversing automorphism), or regular (in which case the automorphism group acts regularly on the flags of the map), the latter either orientable or non-orientable. The census is complete up to 1500 edges in the case of non-orientable maps, and up to 3000 edges for orientable maps (both chiral and regular). The previously known census of Marston Conder contained the maps up to 1000 edges. A census of the underlying skeleton graphs of these maps was also computed. For more information and instruction how to use the census, please read the README file. The following files are available for download (note that some of these files are rather large – largest is 34MB long):
• zipped magma files containing the census of chiral maps;
• zipped magma files containing the census of regular maps (both orientable and non-orientable);
• zipped magma files containing the census of skeletons (up to 1500 edges) or rotary maps;
• zipped (and non-zipped) csv file containing some precomputed data about the chiral maps in the census;
• zipped (and non-zipped) csv file containing some precomputed data about the regular maps in the census;
• zipped (and non-zipped) csv file containing some precomputed data about the skeleton graphs.
• Census of cubic vertex-transitive graphs. A census (compiled by Pablo Spiga, Gabriel Verret and myself) of cubic vertex-transitive graphs on at most 1280 vertices is available at this link or here (zipped file with a slightly impoved magma code for uploading the graphs into computer memory).
• Census of (2,*)-groups. A (2,*)-generated group is a group that can be generated by an involution and one extra element. A magma file containing all (2,*)-groups of order at most 6000 is available for download here. This mgm file requires also this txt file (zipped). Moreover, the mgm file contains a function that will generate all the corresponding generating pairs of the (2,*)-groups (modulo the action of the automorphism group of the group). This function will need the following, rather lengthy file (about 585Mb zipped). To read more about this, click here. This work makes part of a larger project (by Pablo Spiga, Gabriel Verret and myself) of constructing complete lists of 4-valent half-arc-transitive graphs and cubic vertex-transitive graphs (see below).
• Census of 2-valent arc-transitive digraphs. Pablo Spiga, Gabriel Verret and myself have compiled a complete list of all connected arc-transitive digraphs on at most 1.000 vertices. As a byproduct, we've computed all connected 4-valent graphs with at most 1.000 vertices that admit a half-arc-transitive action of a group of automorphisms. In particular, all tetravalent half-arc-transitive graphs are there. The graphs are available in magma format (note that the graphs might fail to load into some older versions of magma, or magma for Windows). Click here to get more information about the organisation of the census. To download a complete zipped file with the lists of graphs and accompanying data, plese click here (note that this file has more than 100Mb). The csv files containing the list of 2-valent arc-transitive digraphs, the list of 4-valent arc-transitive G-half-arc-transitive graphs and a list of 4-valent half-arc-transitive graphs, together with some graph-theoretical invariants is also available. More about this can be found in the paper: P. Potočnik, P. Spiga, G. Verret, A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two, Ars Mathematica Contemporanea, vol 8 (2015) (Please cite this paper if you use the data.)
• Census of arc-transitive tetravalent graphs. Based on some theoretical work of Pablo Spiga, Gabriel Verret and myself, we were able to construct a complete list of all tetravalent arc-transitive graphs on at most 640 vertices. The magma code which generates the sequence of these graphs can be found here. If you use this list for research purposes, please cite the following papers: P. Potočnik, P. Spiga, G. Verret, Bounding the order of the vertex-stabiliser in 3-valent vertex transitive and 4-valent arc-transitive graphs, J. Combin. Theory Ser B 111 (2015), 148-180, and P. Potočnik, P. Spiga, G. Verret, Cubic vertex-transitive graphs on up to 1280 vertices, J. Symbolic. Comp. 50 (2013) 465-477. We would also like to thank Marston Conder who provided the list of all regular maps with at most 640 edges, which were needed to compile this census.
• Census of 2-arc-transitive tetravalent graphs. Here can be found a list of 2-arc-transitive tetravalent graphs on up to 2000 vertices (in magma code). The list is complete on up to 727 vertices, but misses some 7-arc-transitive graphs that admit no s-arc-transitive group for s less than 7 on more than 727 vertices, and also some 4-arc-transitive graphs that admit no s-arc-transitive group for s less than 4 on more than 1157 vertices. If you use this list for research purposes, please cite the paper P. Potočnik, A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index (4, 2). Eur. j. comb. 2009, vol. 30, no. 5, 1323—1336.
• Census of edge-transitive tetravalent graphs. I am currently involved in an ambitious project, suggested by Steve Wilson of Northern Arizona University, whose aim is to compile a complete list (census) of all edge-transitive tetravalent graphs on at most 512 vertices. The current (uncomplete) version of the census can be found here. There are several segments, though, in which the current version is complete. For example, it contains all 2-arc-transitive graphs up to that order, as well as all arc-transitive and half-arc-transitive graphs (due to the joint work with G. Verret and P. Spiga mentioned above). The census is, however, incomplete for the case of semisymmetric graphs. Some of the semisymmetric graphs from the table are available in magma code here (after unzipping the downloaded file, copy the *.mgm and *.txt file to the magma work directory and load the *.mgm file into magma – a double sequence TetraSS will be created with TetraSS[n,k] being the k-th graph on n vertices). For more details on this census, see the paper P. Potočnik, S. Wilson, Recipes for edge-transitive tetravalent graphs, Art of Discrete and Applied Mathematics.
• Census of pentavalent arc-transitive graphs. I have started constructing some examples of pentavalent arc-transitive graphs. The current list (in magma code) contains all those pentavalent arc-transitive graphs on up to 500 vertices which admit an arc-transitive group G with the vertex stabiliser G_v acting faithfully on the neighbourhood of v and being solvable (i.e. G_v is one of C_5, D_5, or Aff(1,5)). Beware, this is work in progress—codenames of the graphs might change in the future.
• Some other edge-transitive graphs. Here's a magma code generating the bipartite biregular graphs of valence {3,4} that are locally G-arc-transitive for some group of automorphisms G with the edge stabiliser G_uv of an edge uv acting faithfully on the union of the neighbourhoods of u and v. This list contains graphs up to 1050 and is complete up to 350 vertices, but might miss some graphs in the range of 350 and 1050 vertices. More about the list can be found in the paper: P. Potočnik, Locally arc-transitive graphs of valence {3,4} with trivial edge kernel, to appear in J. Alg. Combin.