The vertex covers, Betti numbers and projective dimensions of perfect binary trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914053746589696 |
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| author | Hang, Nguyen Thu Dung, Tran Duc Van Kien, Do |
| author_facet | Hang, Nguyen Thu Dung, Tran Duc Van Kien, Do |
| contents | Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a consequence, we compute the depth and projective dimension of $S/I$ and show that the total Betti number of $S/I$ at the highest homological degree always equals one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19683 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The vertex covers, Betti numbers and projective dimensions of perfect binary trees Hang, Nguyen Thu Dung, Tran Duc Van Kien, Do Commutative Algebra 13A15, 13C15, 05C90, 13D45 Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a consequence, we compute the depth and projective dimension of $S/I$ and show that the total Betti number of $S/I$ at the highest homological degree always equals one. |
| title | The vertex covers, Betti numbers and projective dimensions of perfect binary trees |
| topic | Commutative Algebra 13A15, 13C15, 05C90, 13D45 |
| url | https://arxiv.org/abs/2509.19683 |