Normally Regular Digraphs, Association Schemes and Related combinatorial Structures
Leif Kjær Jørgensen, Gareth Aneurin Jones, Mikhail H. Klin, Sung‐Yell Song · 2014
This paper reports the characteristics of and mutual relationships between various combinatorial structures that give rise to certain imprimitive nonsymmetric three-class association schemes. Nontrivial relation graphs of an imprimitive symmetric 2-class association scheme are m ◦ Kr, (the union of m copies of the complete graph on r vertices) and its complement m ◦Kr, (the complete m-partite strongly regular graph) for some positive integers m and r. The set of nontrivial relation graphs of nonsymmetric three-class fission scheme of such a 2-class association scheme contains a complementary pair of oriented graphs of either m◦Kr or m ◦Kr depending on m and r. For suitable parameters m and r, these graphs arise from various combinatorial objects, such as, doubly regular tournaments, normally regular digraphs, skew Hadamard matrices, Cayley graphs of dicyclic groups and certain group rings. The construction and the characteristics of these objects are investigated combinatorially and algebraically, and their mutual relationships are discussed.