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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 *K*_{r}
(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 *K*_{r} 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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