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

The Constructions of directed strongly regular graph by algebraic method