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

The Kronecker product of two connected graphs G 1 , G 2 , denoted by G 1 × G 2 , is the graph with vertex set V ( G 1 × G 2 ) = V ( G 1 ) × V ( G 2 ) and edge set E ( G 1 × G 2 ) = { ( u 1 , v 1 ) ( u 2 , v 2 ) : u 1 u 2 ∈ E ( G 1 ) , v 1 v 2 ∈ E ( G 2 ) } . The k th power G k of G is the graph with vertex set V ( G ) such that two distinct vertices are adjacent in G k 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 × K m 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 P n k such that n ≥ 2 k ; (3) G is a power graph of a cycle C n r such that n ≥ m and n ≥ 2 r + 1 .