Induced Matchings and the v-Number of Graded Ideals
We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | article |
| Language: | EN |
| Published: |
MDPI AG
2021
|
| Subjects: | |
| Online Access: | https://doaj.org/article/659ec7b0da7e42ca9d4fefed4cd2bc1e |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We give a formula for the v-number of a graded ideal that can be used to compute this number. Then, we show that for the edge ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a graph <i>G</i>, the induced matching number of <i>G</i> is an upper bound for the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <i>G</i> is very well-covered, or <i>G</i> has a simplicial partition, or <i>G</i> is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a lower bound for the regularity of the edge ring of <i>G</i>. We classify when the induced matching number of <i>G</i> is an upper bound for the v-number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> when <i>G</i> is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mn>2</mn></msub></semantics></math></inline-formula>-graphs. |
|---|