Partial sum quadruples and bi-Abelian digraphs

As a generalization of undirected strongly regular graphs, a digraph X without loops, of valency k and order v is said to be a (v,k,μ,λ,t)-directed strongly regular graph whenever for any vertex u of X there are t undirected edges having u as an endvertex and for every two different vertices u and w of X the number of paths of length 2 starting at u and ending at w is λ or μ depending only on whether uw is an arc of X or not. An m-Cayley digraph of a group H is a digraph admitting a semiregular group of automorphisms having m orbits, all of equal length, isomorphic to H. In this paper, the structure of directed strongly regular 2-Cayley graphs of cyclic groups is investigated. In particular, the arithmetic conditions on parameters v, k, μ, λ, and t are given. Also, several infinite families of directed strongly regular graphs which are also 2-Cayley digraphs of abelian groups are constructed.