Graded Betti numbers of a hyperedge ideal associated to join of graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912589229850624 |
|---|---|
| author | Patra, Sourav Kanti Roy, Amit |
| author_facet | Patra, Sourav Kanti Roy, Amit |
| contents | Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $r$ be a positive integer. We consider the hypergraph $\mathrm{Con}_r(G)$ whose vertices are the vertices of $G$ and the (hyper)edges are all $A\subseteq V(G)$ such that $|A|=r+1$ and the induced subgraph $G[A]$ is connected. The (hyper)edge ideal $I_r(G)$ of $\mathrm{Con}_r(G)$ is also the Stanley-Reisner ideal of a generalisation of the independence complex of $G$, called the $r$-independence complex $\mathrm{Ind}_r(G)$. In this article we make extensive use of the Mayer-Vietoris sequence to find the graded Betti numbers of $I_r(G_1*G_2)$ in terms of the graded Betti numbers of $I_r(G_1)$ and $I_r(G_2)$, where $G_1*G_2$ is the join of $G_1$ and $G_2$. Moreover, we find formulas for all the graded Betti numbers of $I_r(G)$, when $G$ is a complete graph, complete multipartite graph, cycle graph and the wheel graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07308 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Graded Betti numbers of a hyperedge ideal associated to join of graphs Patra, Sourav Kanti Roy, Amit Commutative Algebra Combinatorics 13F55, 05E45 Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $r$ be a positive integer. We consider the hypergraph $\mathrm{Con}_r(G)$ whose vertices are the vertices of $G$ and the (hyper)edges are all $A\subseteq V(G)$ such that $|A|=r+1$ and the induced subgraph $G[A]$ is connected. The (hyper)edge ideal $I_r(G)$ of $\mathrm{Con}_r(G)$ is also the Stanley-Reisner ideal of a generalisation of the independence complex of $G$, called the $r$-independence complex $\mathrm{Ind}_r(G)$. In this article we make extensive use of the Mayer-Vietoris sequence to find the graded Betti numbers of $I_r(G_1*G_2)$ in terms of the graded Betti numbers of $I_r(G_1)$ and $I_r(G_2)$, where $G_1*G_2$ is the join of $G_1$ and $G_2$. Moreover, we find formulas for all the graded Betti numbers of $I_r(G)$, when $G$ is a complete graph, complete multipartite graph, cycle graph and the wheel graph. |
| title | Graded Betti numbers of a hyperedge ideal associated to join of graphs |
| topic | Commutative Algebra Combinatorics 13F55, 05E45 |
| url | https://arxiv.org/abs/2311.07308 |