Induced Matchings and the v-Number of Graded Ideals

Gonzalo Grisalde, Enrique Reyes and Rafael H. Villarreal
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Gonzalo Grisalde: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, Mexico City 07000, Mexico
Enrique Reyes: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, Mexico City 07000, Mexico
Rafael H. Villarreal: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, Mexico City 07000, Mexico

Mathematics, 2021, vol. 9, issue 22, 1-16

Abstract: 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 I ( G ) of a graph G , the induced matching number of G is an upper bound for the v-number of I ( G ) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contains neither four, nor five cycles. In all these cases, the v-number of I ( G ) is a lower bound for the regularity of the edge ring of G . We classify when the induced matching number of G is an upper bound for the v-number of I ( G ) when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of W 2 -graphs.

Keywords: graded ideals; v-number; induced matchings; edge ideals; regularity; very well-covered graphs; W 2 -graphs; simplicial vertices (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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