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Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the correspondingbinomial edge ideal in $S = K[x_1, \ldots, x_n, y_1,\ldots, y_n].$ We provethat 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 ofSaeedi Madani-Kiani regularity upper bound conjecture for chordal graphs. Weobtain the regularity of binomial edge ideals of Jahangir graphs. Later, weestablish a sufficient condition for Hibi-Matsuda conjecture to be true.

Binomial edge ideals and bounds for their regularity