Strongly Regular Graphs (SRGs) can be defined as $k$-regular graphs with exactly two distinct eigenvalues different from $k.$
Define Strongly Regular Digraphs (SRDs) as $k$-regular digraphs with exactly two distinct eigenvalues different from $k.$ Can we characterize SRDs?
For instance DSRG (Directed Strongly Regular Graphs) as defined by Duval are SRDs. Is the converse true?