Binomial edge ideals and bounds for their regularity

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S=K[x_1,\dots ,x_n,y_1,\dots ,y_n]\mathrm{.}$ 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.