Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866907821005602816 |
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| author | Faridi, Sara Hewalage, Iresha Madduwe |
| author_facet | Faridi, Sara Hewalage, Iresha Madduwe |
| contents | In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as $(\depth R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron--Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this paper, for the edge ideal $I$ of a Cameron--Walker graph on $n$ vertices, we find how many points in $\mathbb{N}^2$ appear as $(\depth(R/I),\dim(R/I))$, and how many points in $\mathbb{N}^4$ appear as $(\depth(R/I),\reg(R/I),\dim(R/I),\degh(R/I)).$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_02557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs Faridi, Sara Hewalage, Iresha Madduwe Commutative Algebra Combinatorics In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as $(\depth R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron--Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this paper, for the edge ideal $I$ of a Cameron--Walker graph on $n$ vertices, we find how many points in $\mathbb{N}^2$ appear as $(\depth(R/I),\dim(R/I))$, and how many points in $\mathbb{N}^4$ appear as $(\depth(R/I),\reg(R/I),\dim(R/I),\degh(R/I)).$ |
| title | Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2403.02557 |