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.

中文翻译：

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二项式边缘理想和规则性的界限

令G为n个顶点上的简单图，\（J_G \）表示\（S = K [x_1，\ ldots，x_n，y_1，\ ldots，y_n]。\）中的相应二项式边理想。）我们证明了Castelnuovo –当G为准方块图或半方块图时，\（J_G \）的芒福德正则性由\（c（G）+1 \）限定。我们给出弦图的Saeedi Madani–Kiani正则性上限猜想的另一种证明。我们获得了Jahangir图的二项式边理想的正则性。后来，我们为Hibi-Matsuda猜想成立提供了充分的条件。