Séminaire Lotharingien de Combinatoire, B71c (2014), 39 pp.

Leif Kjær Jørgensen, Gareth A. Jones, Mikhail H. Klin and Sung Y. Song

Normally Regular Digraphs, Association Schemes and Related Combinatorial Structures

Abstract. This paper reports on the characteristics of and mutual relationships between various combinatorial structures that give rise to certain imprimitive non-symmetric three-class association schemes. The relation graphs of an imprimitive symmetric 2-class association scheme are isomorphic to m o Kr (the union of m copies of the complete graph on r vertices) and its complement (the complete m-partite strongly regular graph) for some positive integers m and r. The set of relation graphs of a non-symmetric three-class fission scheme of such a 2-class association scheme contains a pair of opposite orientations of either m o Kr or its complement, 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.

Received: March 21, 2014. Accepted: October 13, 2014. Final Version: December 9, 2014.

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