Vertex cut of a graph and connectivity of its neighbourhood complex

We show that if a graph G satisfies certain conditions, then the connectivity of neighbourhood complex N(G) is strictly less than vertex G. As an application, we give relation between and for stiff chordal graphs, weakly triangulated graphs satisfying properties. Next, consider with v such any k-subset S neighbours v, there exists vS≠v subset vS. prove as above, N(G−{v}) (k−1)-connected implies (k−1)-connected. use this to that:(i) complexes queen king are simply connected (ii) non-complete graph, n+1 only Conn(N(G))=n.