  Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding
binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1,\ldots, y_n].$ We prove
that the Castelnuovo-Mumford regularity of $J_G$ is bounded above by $c(G)+1$
when $G$ is a quasi-block graph or semi-block graph. We give another proof of
Saeedi Madani-Kiani regularity upper bound conjecture for chordal graphs. We
obtain the regularity of binomial edge ideals of Jahangir graphs. Later, we
establish a sufficient condition for Hibi-Matsuda conjecture to be true.
