Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the second Betti number and obtain a minimal presentation of trees and unicyclic graphs. We also classify all graphs whose binomial edge ideals are almost complete intersection and we prove that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.

中文翻译：

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几乎完全相交二项式边理想及其里斯代数

令 $G$ 是 $n$ 个顶点上的简单图，$J_G$ 表示多项式环 $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n] 中对应的二项边理想。$ 在这个在文章中，我们计算第二个 Betti 数并获得树和单环图的最小表示。我们还对二项式边理想几乎完全交集的所有图进行分类，并证明其二项式边理想的里斯代数是 Cohen-Macaulay。我们还获得了这些二项式边缘理想的里斯代数的定义理想的明确描述。