Non-symmetric association schemes with 3 classes.

We consider association schemes with three relations (or graphs) R1, R2 and R3.
We assume that R1and  R2 are non-symmetric. R1T= R2 and that R3 is symmetric. Then R3 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.
R3 parametersp112p312scheme existsS-ring existsreference
(36,21,12,12)02yes, 1noIwasaki
(64,35,18,20)42yesyesEnomoto and Mena
(64,27,10,12)46yesyes[J2]
(64,21,8,6)76?no
(81,30,9,12)95?no
(85,20,3,5)138?no
(85,14,3,2)1320?no
(96,57,36,30)34??
(96,19,2,4)1610??


In the most interesting case of imprimitive non-symmetric association schemes with classes
the relation R3 is isomorphic to m disjoint copies of the complete graph Kr  and a vertex in one Kr has exactly r/2 out-neighbours in R1 in any other Kr.
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.
rmp112p312scheme existsS-ring existsreference
4422yes, 2yes, 1 scheme
6666yesno[J1]
41086?no
4161410yesyes, 40 schemes[J2]
881212yesyes, 46 schemes[J2]
4222014?no
6162118??