The edge rings of compact graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911869855334400 |
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| author | Wang, Zexin Lu, Dancheng |
| author_facet | Wang, Zexin Lu, Dancheng |
| contents | We define a simple graph as compact if it lacks even cycles and satisfies the odd-cycle condition. Our focus is on classifying all compact graphs and examining the characteristics of their edge rings. Let $G$ be a compact graph and $\mathbb{K}[G]$ be its edge ring. Specifically, we demonstrate that the Cohen-Macaulay type and the projective dimension of $\mathbb{K}[G]$ are both equal to the number of induced cycles of $G$ minus one, and that the regularity of $\mathbb{K}[G]$ is equal to the matching number of $G_0$. Here, $G_0$ is obtained from $G$ by removing the vertices of degree one successively, resulting in a graph where every vertex has a degree greater than 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07587 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The edge rings of compact graphs Wang, Zexin Lu, Dancheng Commutative Algebra Primary 05E40, 13A02, Secondary 06D50 We define a simple graph as compact if it lacks even cycles and satisfies the odd-cycle condition. Our focus is on classifying all compact graphs and examining the characteristics of their edge rings. Let $G$ be a compact graph and $\mathbb{K}[G]$ be its edge ring. Specifically, we demonstrate that the Cohen-Macaulay type and the projective dimension of $\mathbb{K}[G]$ are both equal to the number of induced cycles of $G$ minus one, and that the regularity of $\mathbb{K}[G]$ is equal to the matching number of $G_0$. Here, $G_0$ is obtained from $G$ by removing the vertices of degree one successively, resulting in a graph where every vertex has a degree greater than 1. |
| title | The edge rings of compact graphs |
| topic | Commutative Algebra Primary 05E40, 13A02, Secondary 06D50 |
| url | https://arxiv.org/abs/2309.07587 |