Small vertex-transitive directed strongly regular graphs

We consider directed strongly regular graphs de2ned in 1988 by Duval. All such graphs with n vertices, n6 20, having a vertex-transitive automorphism group, are determined with the aid of a computer. As a consequence, we prove the existence of directed strongly regular graphs for three feasible parameter sets listed by Duval. For one parameter set a computer-free proof of the nonexistence is presented. This, together with a recent result by JHrgensen, gives a complete answer on Duval’s question about the existence of directed strongly regular graphs with n6 20. The paper includes catalogues of all generated graphs and certain theoretical generalizations based on some known and new graphs. c © 2002 Elsevier Science B.V. All rights reserved.