Abstract:For any graph $G$, assume that $J(G)$ is the cover ideal of $G$. Let $J(G)^{(k)}$ denote the $k$th symbolic power of $J(G)$. We characterize all graphs $G$ with the property that $J(G)^{(k)}$ has a linear resolution for some (equivalently, for all) integer $k\geq 2$. Moreover, it is shown that for any graph $G$, the sequence $\big ({\mathrm {reg}}(J(G)^{(k)})\big )_{k=1}^{\infty }$ is nondecreasing. Furthermore, we compute the largest degree of minimal generators of $J(G)^{(k)}$ when $G$ is either an unmixed of a claw-free graph.

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中文翻译：

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图的覆盖理想符号幂的最小自由分辨率

摘要：对于任何图$G$，假设$J(G)$ 是$G$ 的覆盖理想。让 $J(G)^{(k)}$ 表示 $J(G)$ 的第 $k$ 个符号力量。我们用 $J(G)^{(k)}$ 对某些（等效地，对于所有）整数 $k\geq 2$ 具有线性分辨率的属性来表征所有图 $G$。此外，还表明对于任何图 $G$，序列 $\big ({\mathrm {reg}}(J(G)^{(k)})\big )_{k=1}^{\ infty }$ 是非递减的。此外，当 $G$ 是无爪图的未混合时，我们计算 $J(G)^{(k)}$ 的最小生成器的最大程度。

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