The Constructions Of Some Classes Of Weakly Distance-regular Digraphs

Let G be a finite group and S a subset of G not containing the identity element.Let V be a finite set of vertices and AT a set of arcs.We define the Cayley digraph? = Cay(G,S)of G with respect to S by V? = G and A? = {(x,sx)| x ?G,s?S}.Let(?)(x,y)be the distance f'rom x to y in ? and(?)(x,y)=((?)(x,y),(?)(y,x))the two-way distance of x and y.For simplicity,let h denote the two-way distance of two vertices.A strongly connected digraph ? is said to be weakly distance-regular if Pi,jh(x,y)= |{z?V?|(?)(x,z)=i,(?)(z,y)=j}|depends only on i,j,h and does not depend on the choices of x and y with(?){x,y=h.In this paper,we give a new construction of weakly distance-regular digraphs using the theory of Cayley digraph and obtain some examples of weakly distance-regular di-graphs.Moreover,we determine the conditions that some digraphs are weakly distance-regular digraphs using the theory of the direct product and lexicographic product of Cayley digraphs.