On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909210004946944 |
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| author | Dey, Deblina Jayanthan, A. V. Saha, Kamalesh |
| author_facet | Dey, Deblina Jayanthan, A. V. Saha, Kamalesh |
| contents | Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain the $\mathrm{v}$-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for $\mathrm{v}(J_G)$ using the number of vertices of the graph. We characterize all connected graphs $G$ with $\mathrm{v}(J_G) = 2$. We show that for a given pair $(k,m), k\leq m$, there exists a graph $G$ with an associated monomial edge ideal $I$ having $\mathrm{v}$-number equal to $k$ and regularity $m$. If $2k \leq m$, then there exists a binomial edge ideal with $\mathrm{v}$-number $k$ and regularity $m$. Finally, we compute $\mathrm{v}$-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the $\mathrm{v}$-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15354 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs Dey, Deblina Jayanthan, A. V. Saha, Kamalesh Commutative Algebra 05E40, 13F20(Primary), 05C25, 05C69(Secondary) Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain the $\mathrm{v}$-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for $\mathrm{v}(J_G)$ using the number of vertices of the graph. We characterize all connected graphs $G$ with $\mathrm{v}(J_G) = 2$. We show that for a given pair $(k,m), k\leq m$, there exists a graph $G$ with an associated monomial edge ideal $I$ having $\mathrm{v}$-number equal to $k$ and regularity $m$. If $2k \leq m$, then there exists a binomial edge ideal with $\mathrm{v}$-number $k$ and regularity $m$. Finally, we compute $\mathrm{v}$-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the $\mathrm{v}$-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals. |
| title | On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs |
| topic | Commutative Algebra 05E40, 13F20(Primary), 05C25, 05C69(Secondary) |
| url | https://arxiv.org/abs/2405.15354 |