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,G2G1,G2, denoted by G1×G2G1×G2, is the graph with vertex set V(G1×G2)=V(G1)×V(G2)V(G1×G2)=V(G1)×V(G2) and edge set E(G1×G2)={(u1,v1)(u2,v2):u1u2∈E(G1),v1v2∈E(G2)}E(G1×G2)={(u1,v1)(u2,v2):u1u2∈E(G1),v1v2∈E(G2)}. The kkth power GkGk of GG is the graph with vertex set V(G)V(G) such that two distinct vertices are adjacent in GkGk if and only if their distance apart in GG is at most kk. A connected graph GG is called super-κκ if every minimal vertex cut of GG is the set of neighbors of some vertex in GG. 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×KmD=G×Km is super-κκ for m≥3m≥3 and GG satisfying one of the following conditions: (1) GG is a non-complete split graph with |C|≥5|C|≥5; (2) GG is a power graph of a path Pnk such that n≥2kn≥2k; (3) GG is a power graph of a cycle Cnr such that n≥mn≥m and n≥2r+1n≥2r+1.