Coherent Configurations with at most 13 vertices

This is a classification of coherent configurations with at most 13 vertices.

First, Megumi Shiratsuchi classified coherent configurations at most 11 vertices in her master thesis in 1997. It was not complete. One of the reasons of the incompleteness is that classification of association schemes was not complete then.

In 2009, Asumi Nagatomo and Junichi Shigezumi made a correction of her result, and classified coherent configurations with 12 and 13 vertices. Most of them are published in the master thesis of Asumi Nagatomo. Most of the classification was done by hand, and we also use `Mathematica' partially.

The following table is the total numbers of coherent configurations:

 1 2 3 4 5 6 7 8 9 10 11 12 13
 1 2 4 10 15 38 57 143 227 491 766 1821 2771

Every coherent configuration with at most 13 vertices is Schurian. (i.e. each configuration corresponds to some permutation group.)

The following datas include lists of relation matirices of coherent configurations with at most 13 vertices, which are not association schemes and have no fiver of order 1:

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Additional remark

The above table and this database of coherent configuration are not incomplete.
Prof. Matan Ziv-Av told us some example of coherent configuration which are not included in our list.
However, we could not check and correct our list yet.

Also, there is another project by Sven Reichard:
http://conferences2.imfm.si/conferenceDisplay.py/getPic?amp;confId=12&picId=20
 

Data of Coherent Configurations

検索検索
123456

123456

References

"Association schemes"
[1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, CA, 1984.
[2] A. Hanaki, Representations of finite association schemes, European J. Combin., 30(2009), 1477-1496.

"Classification of association schemes"
[1] E. Nomiyama, Classification of association schemes with at most ten vertices, Kyushu J. Math., 49(1995), 163-195.
[2] M. Hirasaka, The classification of association schemes with 11 or 12 vertices, Kyushu J. Math., 51(1997), 413-428.
[3] Y. Sakita, Classification of association schemes with 14 vertices, M.S. thesis, 1994.
[4] M. Hirasaka and Y. Suga, The classification of association schemes with 13 or 15 points, proceedings of Kyoto Univ. Res. Inst. Math. Sci., 962(1996), 71-80. (Proceeding of `Algebraic combinatorics' (Japanese), Kyoto, 1995.)
[5] K. See and S. Y. Song, Association schemes of small order, J. Statist. Plann. Inference, 73(1998), 225-271. (Proceeding of `R. C. Bose Memorial Conference', FortCollins, CO, 1995.)
[6] A. Hanaki, I. Miyamoto, Classification of association schemes with 16 and 17 vertices, Kyushu J. Math., 52(1998), 383-395.
[7] A. Hanaki, I. Miyamoto, Classification of association schemes with 18 and 19 vertices, Korean J. Comput. Appl. Math., 5(1998), 543-551.
[8] A. Hanaki, I. Miyamoto, Classification of association schemes of small order, Discrete Math. 264(2003), 75-80. (Proceeding of `The 2000 Com^2 MaC Conference on Association Schemes, Codes and Designs', Pohang, 2000.)
[9] A. Hanaki and I. Miyamoto, Classification of association schemes with small vertices, published on WWW, since 1999:

"Coherent configurations"
[1] D. G. Higman, Coherent configurations, I. Ordinary representation theory, Geometriae Dedicata, 4(1975), 1-32; II. Weights, Geometriae Dedicata, 5(1976), 413-424.
[2] D. G. Higman, Coherent Algebras, Linear Algebra Appl., 93(1987), 209-239.

"Classification of coherent configurations"
[1] M. Shiratsuchi, The coherent configurations with at most 11 points and their constructions, M.S. thesis, Kyushu Univ., 1997.
[2] A. Nagatomo, Classification of coherent configuration with at most 13 points (Japanese), M.S. thesis, Kyushu Univ., 2009.