A graded ideal I in \(\mathbb {K}[x_1,\ldots ,x_n]\), where \(\mathbb {K}\) is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree \(\mathrm {pd}(I)-2\), while it is not linear at the homological degree \(\mathrm {pd}(I)-1\), where \(\mathrm {pd}(I)\) denotes the projective dimension of I. In this paper, we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of \(\mathbb {K}\). We also compute the nonlinear Betti numbers of these ideals. Finally, we show that for the edge ideal I of a graph G with almost maximal finite index, the ideal \(I^s\) has a linear resolution for \(s\ge 2\) if and only if the complementary graph \(\bar{G}\) does not contain induced cycles of length 4.

渐变理想我在\（\ mathbb {K} [X_1，\ ldots，x_n] \） ，其中\（\ mathbb {K} \）是一个字段，被认为具有几乎最大有限索引如果其最小分辨率免在齐次度\（\ mathrm {pd}（I）-2 \）处是线性的，而在齐次度\（\ mathrm {pd}（I）-1 \）处不是线性的，其中\（\ mathrm {pd}（I）\）表示I的投影维。在本文中，我们对边缘理想具有此属性的图进行分类。这尤其表明，对于边缘理想而言，具有几乎最大的有限索引的属性不取决于\（\ mathbb {K} \）的特征。我们还计算了这些理想的非线性贝蒂数。最后，我们表明，对于边缘理想我的曲线图的ģ几乎最大有限指数，理想\（I，2S \）具有用于线性分辨率\（S \ GE 2 \）当且仅当所述补图\ （\ bar {G} \）不包含长度为4的诱导循环。