Connectivity of Kronecker products with complete multipartite graphs

We study connectivity of a Kronecker product of a general graph GG with a complete multipartite graph Kp1,p2,…,prKp1,p2,…,pr, where the parameters pk,k=1,…,rpk,k=1,…,r satisfy certain conditions. Precisely, we prove that κ(G×Kp1,p2,…,pr)=min{∑i=1rpiκ(G),∑i=1r−1piδ(G)}, where the sequence p1,p2,…,prp1,p2,…,pr satisfies (1) r≥3r≥3, (2) p1≤p2≤⋯≤prp1≤p2≤⋯≤pr, (3) ∑i=1r−2pi≥pr−1 and (4) ∑i=1r−1pi≥pr.