Depth of edge ideals and vertex connectivity of finite graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909016643338240 |
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| author | Hibi, Takayuki Fakhari, Seyed Amin Seyed |
| author_facet | Hibi, Takayuki Fakhari, Seyed Amin Seyed |
| contents | Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $κ(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq κ(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $κ(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $κ(G)$ is given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_04444 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Depth of edge ideals and vertex connectivity of finite graphs Hibi, Takayuki Fakhari, Seyed Amin Seyed Commutative Algebra Combinatorics Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $κ(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq κ(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $κ(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $κ(G)$ is given. |
| title | Depth of edge ideals and vertex connectivity of finite graphs |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2605.04444 |