Directed strongly regular graphs from groups, loops and quasigroups

Abstract

We introduce four infinite families of directed strongly regular graphs of orders $2n^2$ and $3n^2$. The constructions are described in terms of groups, quasigroups, loops and their Latin squares. Two preliminary Cayley digraph constructions over wreath products are extended to arbitrary quasigroups and loops, yielding directed strongly regular graphs with parameters $(2n^2,3n-2,2n-1,n-1,3),(2n^2,4n-2,2n+2,n+2,6),(3n^2,4n-2,2n,n,4),(3n^2,6n-2,2n+6,n+6,10)$.

10 pages