  The concept of directed strongly regular graphs (DSRG) was introduced by
Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of
Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several
construction methods for directed strongly regular graphs. In this paper, We
construct several new classes of directed strongly regular graphs which are
obtained by using Kronecker matrix product, Semidirect product and Cayley coset
graph. At the same time, using group representation, for two special cases, we
give some other sufficient and necessary conditions of Cayley graphs to be
DSRG. At last, we finish this paper with a discussion of some propositions of
in(out)-neighbours and automorphism group in directed strongly regular graphs.
