Induced Matchings and the v-Number of Graded Ideals - Gonzalo Grisalde; Enrique Reyes; Rafael H. Villarreal (Mathematics, 2021)
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 <semantics>I(G)</semantics> of a graph G, the induced matching number of G is an upper bound for the v-number of <semantics>I(G)</semantics> 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 <semantics>I(G)</semantics> 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 <semantics>I(G)</semantics> when G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of <semantics>W2</semantics>-graphs.