Lattice points arising from regularity and $\mathrm{v}$-number of Graphs: Whisker and Cameron-Walker
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arXiv
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2026
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| author | Biswas, Prativa Mandal, Mousumi Saha, Kamalesh |
| author_facet | Biswas, Prativa Mandal, Mousumi Saha, Kamalesh |
| contents | Let $G$ be a simple graph on $n$ vertices and $I(G)\subseteq R$ be its edge ideal. In this paper, we initiate the study of determining lattice points in $\mathbb{N}^2$ that appear as a pair $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$, where $G$ ranges over all connected graphs on $n$ vertices, and we denote this set by $\mathcal{RV}(n)$. Here `$\mathrm{reg}$' denotes the (Castelnuovo-Mumford) regularity and `$\mathrm{v}$' denotes the $\mathrm{v}$-number. We establish general bounds for $\mathcal{RV}(n)$ by identifying two sets $A(n)$ and $B(n)$ satisfying $A(n)\subseteq \mathcal{RV}(n)\subseteq B(n)$. Furthermore, we explicitly determine the subsets of $\mathcal{RV}(n)$ consisting of all possible pairs $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$ arising from whisker graphs and Cameron-Walker graphs on $n$ vertices. Finally, we propose a conjecture on the subset of $\mathcal{RV}(n)$ arising from connected chordal graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_04876 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lattice points arising from regularity and $\mathrm{v}$-number of Graphs: Whisker and Cameron-Walker Biswas, Prativa Mandal, Mousumi Saha, Kamalesh Commutative Algebra Combinatorics 05C69, 05C25, 05E40, 13D02, 13F55 Let $G$ be a simple graph on $n$ vertices and $I(G)\subseteq R$ be its edge ideal. In this paper, we initiate the study of determining lattice points in $\mathbb{N}^2$ that appear as a pair $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$, where $G$ ranges over all connected graphs on $n$ vertices, and we denote this set by $\mathcal{RV}(n)$. Here `$\mathrm{reg}$' denotes the (Castelnuovo-Mumford) regularity and `$\mathrm{v}$' denotes the $\mathrm{v}$-number. We establish general bounds for $\mathcal{RV}(n)$ by identifying two sets $A(n)$ and $B(n)$ satisfying $A(n)\subseteq \mathcal{RV}(n)\subseteq B(n)$. Furthermore, we explicitly determine the subsets of $\mathcal{RV}(n)$ consisting of all possible pairs $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$ arising from whisker graphs and Cameron-Walker graphs on $n$ vertices. Finally, we propose a conjecture on the subset of $\mathcal{RV}(n)$ arising from connected chordal graphs. |
| title | Lattice points arising from regularity and $\mathrm{v}$-number of Graphs: Whisker and Cameron-Walker |
| topic | Commutative Algebra Combinatorics 05C69, 05C25, 05E40, 13D02, 13F55 |
| url | https://arxiv.org/abs/2603.04876 |