Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908589368541184 |
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| author | Haq, Ahtsham Ul Rashid, Muhammad Usman Ishaq, Muhammad |
| author_facet | Haq, Ahtsham Ul Rashid, Muhammad Usman Ishaq, Muhammad |
| contents | This work establishes combinatorial bounds on the Castelnuovo-Mumford regularity of edge ideals for trees and their multi-whiskered variants. For a tree \( T \), we give bounds for the Castelnuovo-Mumford regularity of \( I(T) \) in terms of the order, diameter, and number of pendant vertices. Furthermore, we present an upper bound for multi-whiskered trees \( T_{\mathbf{a}} \), demonstrating that the Castelnuovo-Mumford regularity of \( I(T_{\mathbf{a}}) \) is bounded by the same invariants of the underlying tree \( T \). A principal consequence of this work is the derivation of corresponding inequalities for two key combinatorial invariants of \( T \), namely the induced matching number \( \operatorname{im}(T) \) and the independence number \( α(T) \). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_11265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees Haq, Ahtsham Ul Rashid, Muhammad Usman Ishaq, Muhammad Commutative Algebra Combinatorics This work establishes combinatorial bounds on the Castelnuovo-Mumford regularity of edge ideals for trees and their multi-whiskered variants. For a tree \( T \), we give bounds for the Castelnuovo-Mumford regularity of \( I(T) \) in terms of the order, diameter, and number of pendant vertices. Furthermore, we present an upper bound for multi-whiskered trees \( T_{\mathbf{a}} \), demonstrating that the Castelnuovo-Mumford regularity of \( I(T_{\mathbf{a}}) \) is bounded by the same invariants of the underlying tree \( T \). A principal consequence of this work is the derivation of corresponding inequalities for two key combinatorial invariants of \( T \), namely the induced matching number \( \operatorname{im}(T) \) and the independence number \( α(T) \). |
| title | Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2510.11265 |