A vertex-cut $S$ is called a {\it super vertex-cut} if $G-S$ is

disconnected and it contains no isolated vertices. The {\it

super-connectivity}, $\kappa^{'}$,  is the minimum cardinality over

all super vertex-cuts. This article provides bounds for the super

connectivity of the direct product of an arbitrary

graph and the complete graph $K_n$.  Among other results, we show that if $G$ is a non-complete graph with  girth$(G)=3$

and $\kappa^{'}(G)=\infty$, then  $\kappa^{'}(G\times K_n) \leq

{\rm min}\{mn-6, m(n-1)+5, 5n+m-8\}$, where $|V(G)|=m$.

Super connectivity of graphs is a crucial issue in graph theory, and this paper provides bounds for the super connectivity of an arbitrary graph and a complete graph.

On the super connectivity of direct product of graphs

Key Takeaway: Super connectivity of graphs is a crucial issue in graph theory, and this paper provides bounds for the super connectivity of an arbitrary graph and a complete graph.