Weak Distance Is To The Structure Of The Graph,

In this paper , we discuss the constructions of lexicographic product and directed product from weakly distance-regular digraphs to undirected graphs, and of lexicographic product from a digraph to a digraph. We give the equivalent conditions that they are weakly distance-regular digraphs. We deserve the main results as followings:※ Let Γ be a strongly connected digraph of girth g ≠ 2 and C^*_r be an undirected circuit of length r. Then Γ' = Γ[C^*_r] is a weakly distance-regular digraph if and only if Γ is a weakly distance-regular digraph and one of the following holds. 1. r≤2[g/2]-1,2. 2[g/2] - 1< r ≤< 2g + 1, Γ_m,m(x) = φ, (m ≤[(r-1)/2]) for any x ∈ VΓ. ※. Let Γ be a strongly connected digraph of girth g ≠ 2 and K_s be a completed graph with s vertexes. Then Γ[K_s] is a weakly distance-regular digraph if and only if Γ is a weakly distance-regular digraph.※ Let C_r = Cay(Z_r, 1) and K_t be a coclique of size t. Then C_r[K_t] is a t-valent weakly distance-regular digraph.※ Let Γ be a weakly distance-regular digraph of valent k = t. Then T (?) C_r[K_t] if every arc of Γ is contained in a minimal circuit and . ※ Let Γ is a strongly connected digraph of girth g ≠ 2 and C^*_r be an undirected graph of length r. Then Γ'=Γ × C^*_r is a weakly distance-regular digraph if and only if Γ is a weakly distance-regular digraph and the following hold . 1.Γ_m,m = φ for m≤ [(r-1)/2], 2. If (i, j) ∈ (?)(Γ), then (i + s, j+ s)(?){Γ), where 0 ≤ s ≤ [(r-1)/2].