Regularity of binomial edge ideals of Cohen-Macaulay bipartite graphs

Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$Inthisarticle,weprovethatif$G$ is a fan graph of a complete graph, then $reg(S\mathrm{/}{J}_G)\le c(G)$, where $c(G)$ denote the number of maximal cliques in $G$. Further, we show that if $G$ is a $k$-pure fan graph, then $reg(S\mathrm{/}{J}_G)=k+1$. We then compute a precise expression for the regularity of Cohen-Macaulay bipartite graphs.