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 $2times n$ and let $g$ be a closed graph on the vertex set $[n].$ we study the binomial ideal $i_gsubset k[x_1,ldots,x_{n+1}]$ 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.‎