## Non-symmetric association schemes with
3 classes.

We consider association schemes with three relations (or graphs) R_{1}, R_{2} and R_{3}.

We assume that R_{1}and R_{2} are non-symmetric. R_{1}^{T}= R_{2} and that R_{3} is symmetric. Then R_{3} is a strongly regular graph.

Papers:

[J1] Leif K Jørgensen, Algorithmic Approach to Non-symmetric
3-class association schemes, in Algorithmic algebraic combinatorics and Gröbner bases, eds.: M. Klin et al., Springer 2009, 251-268.

[J2] Leif K. Jørgensen, Schur rings and non-symmetric association
schemes on 64 vertices, Discrete Mathematics (2010),
doi:10.1016/j.disc.2010.03.002

Two new
association schemes/Bush-type Hadamard matrices of order 36 constructed in [J1].

New associations schemes with a regular group of order 64 constructed in [J2].

Table: Feasible parameter sets for primitive (all relations are connected graphs)

non-symmetric association schemes with three classes and less than 100

vertices.

R_{3 }parameters | p^{1}_{12} | p^{3}_{12} | scheme exists | S-ring exists | reference |

(36,21,12,12) | 0 | 2 | yes, 1 | no | Iwasaki |

(64,35,18,20) | 4 | 2 | yes | yes | Enomoto and Mena |

(64,27,10,12) | 4 | 6 | yes | yes | [J2] |

(64,21,8,6) | 7 | 6 | ? | no | |

(81,30,9,12) | 9 | 5 | ? | no | |

(85,20,3,5) | 13 | 8 | ? | no | |

(85,14,3,2) | 13 | 20 | ? | no | |

(96,57,36,30) | 3 | 4 | ? | ? | |

(96,19,2,4) | 16 | 10 | ? | ? | |

In the most interesting case of imprimitive non-symmetric association schemes with classes

the relation R_{3} is isomorphic to m disjoint copies of the complete graph K_{r} and a vertex in one K_{r} has exactly r/2 out-neighbours in R_{1} in any other K_{r}.

In this m and r are even and r-1 divides m-1. If r=2 then m is a multiple of 4.

Table: Feasible imprimitive non-symmetric association schemes with r>2, rm<100.

r | m | p^{1}_{12} | p^{3}_{12} | scheme exists | S-ring exists | reference |

4 | 4 | 2 | 2 | yes, 2 | yes, 1 scheme | |

6 | 6 | 6 | 6 | yes | no | [J1] |

4 | 10 | 8 | 6 | ? | no | |

4 | 16 | 14 | 10 | yes | yes, 40 schemes | [J2] |

8 | 8 | 12 | 12 | yes | yes, 46 schemes | [J2] |

4 | 22 | 20 | 14 | ? | no | |

6 | 16 | 21 | 18 | ? | ? | |