Binomial edge ideals and rational normal scrolls

‎Let $X=\left(‎ ‎\begin{array}{llll}‎ ‎ x_1 & \ldots & x_{n-1}& x_n\\‎ ‎ x_2& \ldots & x_n & x_{n+1}‎ ‎\end{array}\right)$ be the Hankel matrix of size $2\times n$ and let $G$ be a closed graph on the vertex set $[n].$ We study the binomial ideal $I_G\subset K[x_1,\ldots,x_{n+1}]$ which is generated by all the $2$-minors of $X$ which correspond to the edges of $G.$ We show that $I_G$ is Cohen-Macaulay‎. ‎We find the minimal primes of $I_G$ and show that $I_G$ is a set theoretical complete intersection‎. ‎Moreover‎, ‎a sharp upper bound for the regularity of $I_G$ is given‎.‎