Super-connectivity of Kronecker products of split graphs, powers of cycles, powers of paths and complete graphs

The Kronecker product of two connected graphs G1,G2, denoted by G1 × G2, is the graph with vertex set V (G1 ×G2) = V (G1)×V (G2) and edge set E(G1 ×G2) = {(u1, v1)(u2, v2) : u1u2 ∈ E(G1), v1v2 ∈ E(G2)}. The kth power Gk of G is the graph with vertex set V (G) such that two distinct vertices are adjacent in Gk if and only if their distance apart in G is at most k. A connected graph G is called super-κ if every minimal vertex cut of G is the set of neighbors of some vertex in G. In this note, we consider the super-connectivity of the Kronecker products of several kinds of graphs and complete graphs. We show that D = G × Km is super-κ for m ≥ 3 and G satisfying one of the following conditions: (1) G is a non-complete split graph with |C | ≥ 5; (2) G is a power graph of a path Pk n such that n ≥ 2k; (3) G is a power graph of a cycle C r n such that n ≥ m and n ≥ 2r + 1. © 2012 Elsevier Ltd. All rights reserved.