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 $\mathcal{N}(G)$ is strictly less than the vertex connectivity of $G$. As an application, we give a relation between the connectivity of the neighbourhood complex and the vertex connectivity for stiff chordal graphs, and for weakly triangulated graphs satisfying certain properties. Further, we prove that for a graph $G$ if there exists a vertex $v$ satisfying the property that for any $k$-subset $S$ of neighbours of $v$, there exists a vertex $v_S \neq v$ such that $S$ is subset of neighbours of $v_S$, then $\mathcal{N}(G-\{v\})$ is $(k-1)$-connected implies that $\mathcal{N}(G)$ is $(k-1)$-connected. As a consequence of this, we show that:(i) neighbourhood complexes of queen and king graphs are simply connected and (ii) if $G$ is a $(n+1)$-connected chordal graph which is not folded onto a clique of size $n+2$, then $\mathcal{N}(G)$ is $n$-connected.