Let G be a connected graph on the vertex set [n]. Then depth(S/JG) ≤ n + 1. In this paper, we prove that if G is a unicyclic graph, then the depth of S/JG is bounded below by n. Also, we characterize G with depth(S/JG) = n and depth(S/JG) = n + 1. We then compute one of the distinguished extremal Betti numbers of S/JG. If G is obtained by attaching whiskers at some vertices of the cycle of length k, then we show that k − 1 ≤reg(S/JG) ≤ k + 1. Furthermore, we characterize G with reg(S/JG) = k − 1, reg(S/JG) = k and reg(S/JG) = k + 1. In each of these cases, we classify the uniqueness of the extremal Betti number of these graphs.

中文翻译：

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单环图的二项式边理想

让G是顶点集上的连通图[n]. 然后深度(小号/ĴG) ≤ n + 1. 在本文中，我们证明如果G是一个单环图，那么深度小号/ĴG下界为n. 此外，我们表征G和深度(小号/ĴG) = n和深度(小号/ĴG) = n + 1. 然后我们计算一个显着的极值 Betti 数小号/ĴG. 如果G通过在长度循环的某些顶点处附加晶须获得ķ，然后我们证明ķ - 1 ≤注册(小号/ĴG) ≤ ķ + 1. 此外，我们表征G和注册(小号/ĴG) = ķ - 1,注册(小号/ĴG) = ķ和注册(小号/ĴG) = ķ + 1. 在每种情况下，我们对这些图的极值 Betti 数的唯一性进行分类。