Directed strongly regular graphs obtained from coherent algebras

The notion of a directed strongly regular graph was introduced by A. Duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case. We investigate this generalization with the aid of coherent algebras in the sense of D.G. Higman. We show that the coherent algebra of a mixed directed strongly regular graph is a non-commutative algebra of rank at least 6. With this in mind, we examine the group algebras of dihedral groups, the flag algebras of a Steiner 2-designs, in search of directed strongly regular graphs. As a result, a few new infinite series of directed strongly regular graphs are constructed. In particular, this provides a positive answer to a question of Duval on the existence of a graph with certain parameter set having 20 vertices. One more open case This paper is a revised and shortened version of the preprint [29]. This preprint stimulated a new wave of interest in the investigations of d.s.r.g.’s, in particular results by L. Jorgensen, S. A. Hobart and T.J. Shaw, A. Duval and D. Iourinski. We refer to [11,21,25,30] where a part of recent investigations is discussed with more details. We also mention that the most of the current status of the theory of d.s.r.g.’s can be found in Andries Brouwer’s home page http://www.cwi.nl/∼aeb/math/dsrg/. ∗ Corresponding author. E-mail address: [email protected] (M. Klin). 1 Partially supported by DAAD allowance for study visit to Germany, by the Israeli Ministry of Absorption and by the Department of Mathematical Sciences, University of Delaware. 2 Supported by the research grant no. 3869 from the Israeli Ministry of Science. 0024-3795/$ see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2003.06.020 84 M. Klin et al. / Linear Algebra and its Applications 377 (2004) 83–109 with 14 vertices listed in Duval’s paper is ruled out, while new interpretations in terms of coherent algebras are given for many of Duval’s results. © 2003 Elsevier Inc. All rights reserved.