Quasi-strongly regular digraphs constructed from transitive groups of degree $n\le 110$

Abstract

In this paper we present a method for constructing directed regular graphs from a transitive permutation group. This method is a generalization of a construction method for transitive 1-designs from a finite group, described in \cite{dean1}. Using this construction, we prove the existence of directed strongly regular graphs with parameters $(72,25,15,6,10)$, $(96,11,4,3,1)$, $(96,22,16,8,4)$, $(96,26,11,7,7)$, $(96,29,23,10,8)$, $(96,42,32,16,20)$, $(96,45,35,22,20)$ and $(165,60,36,23,21)$. Finally, we classify quasi-strongly regular digraphs arising from transitive permutation groups of degree at most $30$ and from primitive permutation groups of degrees from $31$ to $110$ and rank at most $30$.