  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 contain neither $4$- nor $5$-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 upper bound holds when $G$ is a cycle, and
classify when all vertices of a graph are shedding vertices to gain insight on
$W_2$-graphs.
