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