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arXiv · 2309.07587

The edge rings of compact graphs

Zexin Wang·Dancheng Lu
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Abstract

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.

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Zexin Wang, Dancheng Lu. 2024-05-07. The edge rings of compact graphs. https://arxiv.org/abs/2309.07587

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