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 parameters p112 p312 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 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.
 r m p112 p312 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 ? ?