LAUSR

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.