Vertex connectivity of chordal graphs

Let $G$ be a finite graph and $\kappa (G)$ the vertex connectivity of $G$. A chordal graph $G$ is called chordal${}^{\ast }$ if no vertex of $G$ is adjacent to all other vertices of $G$. Using the syzygy theory in commutative algebra, it is proved that every chordal${}^{\ast }$ graph $G$ on $n$ vertices satisfies $\kappa (G)\le (n-1)-\lceil 2\sqrt{n}-2\rceil$. Furthermore, given an integer $0\le \kappa \le (n-1)-\lceil 2\sqrt{n}-2\rceil$, a chordal${}^{\ast }$ graph $G$ on $n$ vertices satisfying $\kappa (G)=\kappa$ is constructed.