Connectivity of lexicographic product and direct product of graphs

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs G1 and G2 are equal to κ1v2 and min{λ1v 2 2 , δ2 +δ1v2}, respectively, where δi, κi, λi and vi denote the minimum degree, the connectivity, the edge-connectivity and the number of vertices of Gi, respectively. We also obtain that the edge-connectivity of the direct product of K2 and a graph H is equal to min{2λ, 2β,minδj=λ{j + 2βj}}, where β is the minimum size of a subset F ⊂ E(H) such that H − F is bipartite and βj = min{β(C)}, where C takes over all components of H − B for all edge-cuts B of size j > λ = λ(H).