Small vertex-transitive directed strongly regular graphs

We consider directed strongly regular graphs defined in 1988 by Duval. All such graphs with n vertices, n ≤ 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 Jørgensen, gives a complete answer on Duval's question about the existence of directed strongly regular graphs with n ≤ 20. The paper includes catalogues of all generated graphs and certain theoretical generalizations based on some known and new graphs.